Analysis of the resistance increases in waves of a fast cargo ship

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LABORATORIUM VOOR

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TECHNISCHE HOGESCHOOL DELFT

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ANALYSIS OF THE RESISTANCE INCREASE I N WAVES OF A Fi^ST CARGO SHIP

b y P r o f . i r . J . G e r r i t s m a

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a n d Vl, B e u l i e l r n a n S e p t e m b e r 1971

"ANALYSIS OF THE RESISTANCE INCREASE I N WAVES OF A FAST CARGO S H I P " , "by P r o f . i r . J . G e r r i t s m a a n d W. B e u l i e l m a n . 1 . I n t r o d u c t i o n . One o f t h e f i r s t a t t e m p t s 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 o f a s h i p i n w a v e s was c a r r i e d o u t b y H a v e l o c k [ l ] He d e t e r m i n e d t h e mean v a l u e o f t h e l o n g i t u -d i n a l c o m p o n e n t o f t h e p r e s s u r e f o r c e s i n t e g r a t e -d o v e r t h e w e t t e -d p a r t o f t h e o s c i l l a t i n g s h i p ' s h u l l . I n h i s t r e a t m e n t o f t h e p r o b l e m t h e w a t e r p r e s -s u r e wa-s t a k e n a-s t h e u n d i -s t u r b e d p r e -s -s u r e o f t h e i n c i d e n t w a v e , w h i c h i m p l i e -s t h e u s e o f t h e w e l l - k n o w n F r o u d e - K r y l o f f h y p o t h e s i s . C l e a r l y t h i s was d o n e t o a v o i d t h e d i f f i c u l t p r o b l e m o f t h e e v a l u a t i o n o f t h e c o m p l i c a t e d d i f f r a c t e d w a v e s 5 w h i c h o r i g i n a t e f r o m t h e o s c i l l a t i n g s h i p i n t h e i n c i d e n t w a v e s . T h e r e f o r e H a v e l o c k c o n s i d e r e d h i s s o l u t i o n as a f i r s t a p p r o x i m a t i o n o n l y . An i n t e r e s t i n g d i s c u s s i o n o n t h e - u s e o f t h e u n d i s t u r b e d w a v e p r e s s u r e a n d t h e e f f e c t u p o n t h e c a l c u l a t e d a d d e d r e s i s t a n c e i n w a v e s w a s g i v e n b y F i r s o f f [ 2 ] . He s t a t e s t h a t t h e F r o u d e - K r y l o f f h y p o t h e s i s i s n o t a p p l i c a b l e i n t h i s c a s e . H a v e l o c k ' s e x p r e s s i o n f o r t h e a d d e d r e s i s t a n c e i n w a v e s r e a d s as f o l l o w s : %W -

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+ M^Qa w h e r e : F a n d M a r e t h e a m p l i t u d e s o f t h e e x c i t a t i o n f o r c e a n d moment i n h e a v e a n d p i t c h , z a n d 9 a r e t h e c o r r e s p o n d i n g m o t i o n a m p l i t u d e s 3 / 3 . w i t h p h a s e l a g s a n d e^^. An a l t e r n a t i v e m e t h o d t o f i n d t h e e x p r e s s i o n ( I ) i s t o e q u a l i z e t h e w o r k d o n e b y t h e e x c i t i n g f o r c e a n d moment t o t h e w o r k d o n e b y t h e f o r c e w h i c h i s n e c e s -s a r y t o t o w t h e -s h i p t h r o u g h t h e g i v e n wave f i e l d [3^ . A c c o r d i n g t o e q u a t i o n ( I ) t h e a d d e d r e s i s t a n c e i s z e r o w h e n t h e m o t i o n o f t h e s h i p i s v a n i s h e d . T h e e x p e r i m e n t shows t h a t a n a d d e d r e s i s t a n c e f o r c e c a n b e p r e s e n t i n s u c h a c a s e . F u r t h e r m o r e t h e H a v e l o c k v a l u e s do n o t a g r e e w i t h e x p e r i m e n t a l r e s u l t s o v e r a l a r g e r a n g e o f w a v e l e n g t h r a t i o ' s . I n o r d e r t o g e t a m o r e s a t i s f a c t o r y a g r e e m e n t t h e c o n c e p t o f t h e r e l a t i v e v e r t i c a l m o t i o n o f t h e s h i p w i t h r e s p e c t t o t h e w a t e r h a s t o b e u s e d , a s shown i n t h e f o l l o w i n g c h a p t e r s . T h e m e t h o d i s a l s o a p p l i c a b l e f o r t h e d e t e r m i n a t i o n o f t h e d r i f t f o r c e o n t h e m o t i o n l e s s s h i p w h i c h c o r r e s p o n d s t o t h e s i t u a t i o n o f r e l a t i v e l y s m a l l w a v e l e n g t h s . To c h e c k t h e c a l c u l a t e d v a l u e s a n a c c u r a t e e x p e r i m e n t h a s b e e n c a r r i e d o u t w i t h a 3 m e t e r m o d e l o f a f a s t c a r g o s h i p "S.A. v a n d e r S t e l " . 1

-E x i s t i n g e x p e r i m e n t a l r e s u l t s , as f o u n d i n l i t e r a t u r e do n o t c o v e r a s u f f i c i e n t l y l a r g e s p e e d - w a v e l e n g t h a n d w a v e h e i g h t r a t i o f o r d e t a i l e d c o m p a r i s o n p u r p o s e s ; a l s o t h e r e s u l t s w i t h s t a n d a r d s h i p f o r m s , s u c h as t h e S e r i e s S i x t y , d i f f e r c o n s i d e r a b l y f r o m t a n k t o t a n k . I n a d d i t i o n t h e r e i s some d i s c u s s i o n w i t h r e g a r d t o t h e a s s u m p t i o n t h a t a d d e d r e s i s t a n c e i n w a v e s v a r i e s as t h e s q u a r e d wave h e i g h t , p a r t i c u l a r l y i n t h e c a s e o f s l e n d e r s h i p f o r m s [k^ . An a c c u r a t e e x p e r i m e n t w i t h a m o d e l o f a p a s s e n g e r -c a r g o s h i p , h a v i n g a b l o -c k -c o e f f i -c i e n t = O.65 c o n f i r m e d t h e s q u a r e " l a w t o a l a r g e e x t e n t . B a s e d o n t h i s p r i n c i p l e a p r e d i c t i o n o f t h e mean a d d e d r e s i s t a n c e i n a s p e c i f i e d i r r e g u l a r s e a was i n g o o d a g r e e m e n t w i t h c o r r e s -p o n d i n g m o d e l t e s t r e s u l t s [5 . I t s h o u l d b e e m p h a s i z e d t h a t i n t h i s c a s e e x t r e m e l y l o w a n d h i g h w a v e h e i g h t s w e r e e x c l u d e d . The p r e s e n t a n a l y s i s i n c l u d e s a n e x t e n s i v e s e r i e s o f a d d e d r e s i s t a n c e a n d m o t i o n m e a s u r e m e n t s i n a l a r g e r a n g e o f w a v e l e n g t h s , f o r w a r d s p e e d s a n d t h r e e o r f o u r w a v e h e i g h t s . I n one p a r t i c u l a r c a s e t h e i n f l u e n c e o f s u r g e was a l s o i n v e s t i g a t e d . To a l a r g e e x t e n t t h e r e s u l t s c o n f i r m a g a i n t h e l i n e a r " r e l a t i o n b e t w e e n a d d e d r e s i s t a n c e a n d w a v e h e i g h t s q u a r e d , a t c o n s t a n t s p e e d a n d w a v e l e n g t h , a t l e a s t as a v e r y g o o d a p p r o x i m a t i o n w h i c h c a n b e u s e d f o r p r a c t i c a l p u r p o s e s . 2

-2 . M e a s u r e m e n t o f t h e a d d e d r e s i s t a n c e a n d m o t i o n s i n w a v e s . I n T a b l e 1 t h e m a i n p a r t i c u l a r s o f t h e s h i p a n d t h e c o n s i d e r e d m o d e l a r e g i v e n . T h e m o d e l was made o f g l a s s f i b r e r e i n f o r c e d p o l y e s t e r o n a s c a l e o f 1:50. T a b l e 1 . M a i n p a r t i c u l a r s o f M.V. "S.A. v a n d e r S t e l " a n d m o d e l . SHIP _{M O D E L} L PP 152 5 m 3.050 m ^ W L 7 m 3.09^+ m B _{22. 8 m 0.1+56 m} T _{9. 1 m 0.183 m} V ₁₇₉₃₁ m3 _{0. I l i 3 l | m-^} 3

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2 i l 3 1 m2 _0.972^1 m2 2782993 m^ _{• 0.1+1+53} m^ S O.56I1 0.580 0.580 L C B L C F V s e r v i c e 1.68 ^ a f t L / 2 PP i| . 3 5 % a f t L / 2 PP 19.5 k n o t s 1.68 ^ a f t L / 2 PP I+.35 % a f t L / 2 PP 1 .1+21 m/sec k / L y y PP 0.219 0.219 The b o d y p l a n i s g i v e n i n f i g u r e 1 a n d t h e t e s t c o n d i t i o n s a r e s u m m a r i z e d i n T a b l e 2 . T a b l e 2 Wave c o n d i t i o n s A / L ?w/L 0.6 _{V 5 0} 0.8 _{V 5 0 V ^ o V 3 0} 1.0 _{V 5 0 V ^ o 1/30} 1.2 V15O 1/50 Vho 1/30 1 . ^ _{V 5 0 1/30} 1.6 _{V 5 0} 1.95 V 5 0 F o u r s h i p s p e e d s w e r e r e g a r d e d : F n = 0.15 - 0.20 - 0.25 - 0 . 3 0 . F o r F n = 0.15 o n l y o n e wave h e i g h t = L/ 5 0 h a s b e e n c o n s i d e r e d . 3

-The r e g u l a r waves w e r e m e a s u r e d a t a d i s t a n c e o f h m e t e r s i n f r o n t o f t h e m o d e l b y means o f a t w o - w i r e c o n d u c t a n c e w a v e p r o b e . A d d i t i o n a l m e a s u r e m e n t s w e r e c a r r i e d o u t w i t h a s o n i c w a v e p r o b e , w h i c h s h o w e d a v e r y s a t i s f a c t o r y a g r e e m e n t w i t h t h e r e s u l t s o f t h e f o r m e r one. The t e s t a r r a n g e m e n t i s g i v e n i n F i g u r e 2. The m a j o r i t y o f t h e r u n s i n w a v e s w e r e made w i t h t h e m o d e l r e s t r a i n e d i n s u r g e , b u t f r e e t o h e a v e a n d p i t c h , as

shown i n F i g u r e 2a a n d due c a r e was t a k e n t o e n s u r e a minimura o f f r i c t i o n i n t h e m e a s u r i n g p a r t o f t h e h e a v e g u i d e m e c h a n i s m . The mean r e s i s t a n c e i n w a v e s was m e a s u r e d b y means o f s t r a i n g a u g e d y n a m o m e t e r s o f w h i c h t h e o u t p u t was i n t e g r a t e d o v e r a f u l l n u m b e r o f w a v e p e r i o d s . A c o m p a r i s o n o f t h e r e s u l t s o f t h i s r e s i s t a n c e m e a s u r e m e n t m e t h o d w i t h r e s u l t s o f t h e m o r e c o n v e n t i o n a l s y s t e m , u s i n g a d e a d w e i g h t as shown i n F i g u r e 2b, s h o w e d o n l y v e r y m i n o r d i f f e r e n c e s . The t o w i n g a r r a n g e m e n t , g i v e n i n t h i s F i g u r e was a l s o u s e d t o i n v e s t i g a t e t h e i n f l u e n c e o f s u r g e o n t h e h e a v i n g a n d p i t c h i n g m o t i o n s a n d on t h e a d d e d r e s i s t a n c e . The o s c i l l a t o r y m o t i o n s o f t h e m o d e l w e r e m e a s u r e d b y t w o l o w - f r i c t i o n p o t e n t i o m e t e r s . I t h a s t o b e e m p h a s i z e d t h a t m o d e r a t e w a v e h e i g h t s w e r e c h o s e n i n v i e w o f t h e a p p l i c a b i l i t y o f t h e a d d e d w a v e r e s i s t a n c e o p e r a t o r s and f o r c o m p a r i s o n p u r p o s e s o f t h e r e s u l t s w i t h c a l c u l a t i o n s . I n F i g u r e 3 t h e e x p e r i m e n t a l a m p l i t u d e a n d p h a s e c h a r a c t e r i s t i c s o f t h e s h i p m o d e l a r e c o m p a r e d w i t h t h e c o r r e s p o n d i n g c a l c u l a t e d v a l u e s . The m e t h o d o f t h e c a l c u l a t i o n i s d e r i v e d f r o m [t] • I n v i e w - o f t h e b u l b o u s bow a c l o s e f i t p r o c e d u r e t o d e s c r i b e t h e c r o s s - s e c t i o n s was u s e d . T h e r e i s a s l i g h t i n d i c a t i o n o f n o n l i n e a r i t y i n t h e m e a s u r e d m o t i o n a m p l i -t u d e s , a n d a -t e n d e n c y -t o w a r d s b e -t -t e r a g r e e m e n -t w i -t h -t h e -t h e o r e -t i c a l v a l u e s i n t h e c a s e o f s m a l l w a v e h e i g h t s i s o b s e r v e d . F o r F n - 0.25 t h e i n f l u e n c e o f s u r g e o n t h e h e a v i n g a n d p i t c h i n g m o t i o n s i s v e r y s m a l l , as shown i n F i g u r e 3. T h e r e i s h a r d l y a n y d i f f e r e n c e i n t h e m o t i o n c h a r a c t e r i s t i c s , w h e t h e r t h e m o d e l i s f r e e t o s u r g e o r n o t . The a d d e d r e s i s t a n c e i n r e g u l a r w a v e s i s s h o w n i n F i g u r e h f o r t h e m o d e l w h i c h i s f r e e t o p i t c h a n d h e a v e , b u t r e s t r a i n e d i n s u r g e . I n a d d i t i o n t h e a d d e d r e s i s t a n c e i s g i v e n f o r t h e c a s e w h e r e t h e m o d e l i s r e s t r a i n e d i n h e a v e , p i t c h a n d s u r g e m o t i o n s . The r e s u l t s c o v e r t h e v a r i o u s w a v e h e i g h t c o n d i t i o n s as s-ummarized i n T a b l e 2. I n v i e w o f t h e s l i g h t n o n - l i n e a r i t y o f t h e v e r t i c a l m o t i o n s , t h e r e i s s u r p r i s i n g l y l i t t l e d e v i a t i o n f r o m t h e s q u a r e w a v e h e i g h t l a w . T h i s a l s o h o l d s f o r t h e a d d e d r e s i s t a n c e o f t h e m o t i o n l e s s m o d e l .

h

-F i g u r e 5 shows t h a t t h e i n f l u e n c e o f s u r g e o n t h e a d d e d r e s i s t a n c e i n how- w a v e s i s n e g l i g i b l e . T h i s may n o t b e t r u e f o r f o l l o w i n g w a v e s . The v a l i d i t y o f t h e s q u a r e wave h e i g h t l a w f o r t h i s p a r t i c u l a r m o d e l w i t h a l o w b l o c k c o e f f i c i e n t i s a l s o c l e a r l y shown i n t h e F i g u r e 6, w h e r e a d d e d r e s i s t a n c e due t o waves f o r t h e p i t c h i n g a n d h e a v i n g m o d e l a n d f o r t h e m o t i o n l e s s m o d e l i s p l o t t e d o n a b a s e o f wave h e i g h t s q u a r e d . The s t a n d a r d d e v i a t i o n f r o m t h e assumed q u a d r a t i c l a w i s i n t h e o r d e r o f 3l%, w h i c h i n c l u d e s t h e m e a s u r e m e n t e r r o r s b o t h i n w a v e h e i g h t a n d i n t h e a d d e d r e s i s t a n c e . 5

-3. C a l c u l a t i o n o f t h e a d d e d r e s i s t a n c e i n w a v e s . J o o s e n c a l c u l a t e d t h e added, r e s i s t a n c e o f a s h i p i n s h o r t w a v e s h y e x p a n d i n g M a r u o ' s e x p r e s s i o n i n t o a n a s y m p t o t i c s e r i e s w i t h r e s p e c t t o t h e s l e n d e r n e s s p a r a m e t e r 6 . T a k i n g i n t o a c c o u n t o n l y t h e f i r s t o r d e r t e r m s , h e f o u n d a r e a s o n a b l e a g r e e -m e n t w i t h t h e e x p e r i -m e n t , a l t h o u g h t h i s s i -m p l i f i e d t r e a t -m e n t r e s u l t s i n a s p e e d i n d e p e n d e n t a d d e d r e s i s t a n c e . I n f a c t u s e i s made o f t h e s t r i p t h e o r y t o d e t e r m i n e t h e r e s i s t a n c e f o r c e . O f p a r t i c u l a r i n t e r e s t i s t h e e x p r e s s i o n f o r t h e a d d e d r e s i s t a n c e g i v e n i n [è : w h i c h i s e q u i v a l e n t t o H a v e l o c k ' s e q u a t i o n ( l ) . E q u a t i o n ( 2 ) shows t h a t t h e a d d e d r e s i s t a n c e c a n b e r e g a r d e d as a r e s u l t f r o m t h e r a d i a t e d d a m p i n g w a v e s . A l t h o u g h n o t c o n s i s t e n t w i t h t h e m a t h e m a t i c a l t h e o r y t h e f r e q u e n c y o f e n c o u n t e r i s u s e d b y J o o s e n w h e n a s h i p w i t h f o r w a r d s p e e d i s c o n s i d e r e d . As i n t h e c a s e o f e q u a t i o n ( l ) t h i s e x p r e s s i o n does n o t t a k e i n t o a c c o u n t t h e r e l a t i v e v e r t i c a l m o t i o n o f t h e s h i p , w i t h r e s p e c t t o t h e w a t e r . T h e r e f o r e - t h e f o l l o w i n g p r o c e d u r e i s a d o p t e d f o r t h e c a l c u l a t i o n o f t h e r a d i a t e d e n e r g y P o f t h e o s c i l l a t i n g s h i p d u r i n g o n e p e r i o d o f e n c o u n t e r . We c o n s i d e r l o n g i t u d i n a l r e g u l a r bow w a v e s . Te L P = b ' . V^^ d x ^ d t ( 3 ) o o w h e r e : b ' = N' - V t h e s e c t i o n a l d a m p i n g c o e f f i c i e n t f o r s p e e d dxi3 a n d = z - X136 + VO - C , t h e v e r t i c a l r e l a t i v e w a t e r v e l o c i t y w h e r e : = c (1 -yw -T k^b , . . . y-^ e ^^l)) t h e e f f e c t i v e v e r t i c a l w a v e d i s p l a c e m e n t f o r a c r o s s - s e c t i o n . F o r t h i s c o n c e p t r e f e r e n c e i s made t o 7 . As V i s a h a r m o n i c . f u n c t i o n w i t h a m p l i t u d e V„ a n d a f r e q u e n c y e q u a l t o t h e f r e q u e n c y o f e n c o u n t e r oig, we f i n d : P = ^ L b ' V ^ a ^ d x ^ • ih) 6

-F o l l o w i n g t h e r e a s o n i n g g i v e n i n [3] t h e w o r k b e i n g d o n e b y t h e t o w i n g f o r c e Ry^y i s a l s o g i v e n b y : P = R A W ( V + c ) T e = R^w-A (5) w h e r e : c i s t h e w a v e c e l e r i t y a n d A i s t h e w a v e l e n g t h . F r o m (h) a n d (5) i t f o l l o w s t h a t : L R •AW - 2 ^ (6) F r o m (6) i t i s c l e a r , t h a t t h e a d d e d r e s i s t a n c e i n w a v e s v a r i e s as t h e s q u a r e d w a v e h e i g h t , b e c a u s e V ^ ^ i s p r o p o r t i o n a l t o t h e w a v e h e i g h t . We c a n d i s t i n g u i s h t w o e x t r e m e c a s e s : a. A s h i p w i t h o u t o s c i l l a t o r y m o t i o n s i n w a v e s . T h i s o c c u r s w h e n t h e s h i p s a i l s i n r e l a t i v e l y s h o r t w a v e s a n d p r a c t i c a l l y no s h i p m o t i o n e x i s t s . The r e s i s t a n c e i n c r e a s e i n t h i s c a s e i s c a u s e d b y w h a t i s commonly c a l l e d : d i f f r a c t i o n e f f e c t s . I n t h i s c a s e : AW -

2ü),

b'c dxi. ( T ) We may w r i t e : = w h e r e : 1 Tr B e c a u s e w -T We f i n d : _{R •AW} •u,_{b ' e dxx} -2kT , (8) b . A l s o i n t h e c a s e o f a n o s c i l l a t i n g s h i p i n c a l m w a t e r t h e r a d i a t e d e n e r g y c o r r e s p o n d i n g t o t h e m o t i o n o f t h e s h i p i s t h e c a u s e o f r e s i s t a n c e i n c r e a s e , H e r e t h e r e l a t i v e v e r t i c a l s p e e d o f a c r o s s - s e c t i o n x ^ w i t h r e g a r d t o t h e w a t e r i s g i v e n b y : V 2 = z - x^e + V 6 ( 9 : The r e s i s t a n c e i n c r e a s e f o l l o w s f r o m t h e e q u a t i o n (6) w h e n t h e a p p r o p r i a t e v a l u e o f V i s t a k e n . z 7

F r o m t h e f a c t t h a t t h e s p e e d a p p e a r s as a q u a d r a t i c f o r m i n t h e i n t e -g r a n d i n t h e e x p r e s s i o n f o r R A Ws i t f o l l o w s t h a t t h e r e s i s t a n c e i n c r e a s e i n w a v e s i s n o t m e r e l y t h e sum o f t h e r e s i s t a n c e i n c r e a s e o f a s h i p o s c i l l a t i n g i n c a l m w a t e r a n d t h e r e s i s t a n c e i n c r e a s e o f a m o t i o n l e s s s h i p i n w a v e s . The e q u a t i o n (6) i n d i c a t e s t h a t f o r t h e m o t i o n l e s s s h i p i n w a v e s a f i n i t e r e s i s t a n c e i s f o u n d w h i c h a p p r o a c h e s z e r o o n l y a t i n c r e a s i n g f r e q u e n c i e s o f e n c o u n t e r . The r e s i s t a n c e i n c r e a s e d u e t o w a v e s f o r t h e c o n s i d e r e d s h i p was c a l c u l a t e d a c c o r d i n g t o t h e e q u a t i o n s (6) a n d ( 8 ) . The r e s u l t s a r e shown i n t h e F i g u r e k. T h 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 s shows a v e r y s a t i s f a c t o r y a g r e e m e n t f o r t h e c a s e o f a p i t c h i n g a n d h e a v i n g s h i p . F o r t h e s h o r t w a v e s ('*'/L < 0.8) t h e e x p e r i m e n t a l v a l u e s a r e s o m e w h a t ' h i g h e r f o r s p e e d s h i g h e r t h a n F n = O. I 5 . A p o s s i b l e e x p l a n a t i o n c o u l d b e t h e i n -f l u e n c e o -f v i s c o u s e -f -f e c t s , w h i c h a r e n o t i n c l u d e d i n t h e c a l c u l a t i o n . B e c a u s e o f t h e i n c r e a s i n g f r e q u e n c y o f e n c o u n t e r t h e v e r t i c a l w a t e r v e l o c i t i e s i n c r e a s e a n d c o n s e q u e n t l y t h e v i s c o u s e f f e c t s c o u l d b e m o r e i m p o r t a n t . To show t h e i m p r o v e m e n t A^ i t h r e g a r d t o H a v e l o c k ' s f o r m u l a , t h e v a l u e s o f R_^y c o r r e s p o n d i n g t o e q u a t i o n ( I ) a r e g i v e n i n F i g u r e s h. The r e s i s t a n c e i n c r e a s e o f t h e m o t i o n l e s s s h i p i s s m a l l b u t s h o u l d n o t b e n e g l e c t e d . T h e r e i s a f a i r a g r e e m e n t w i t h t h e e x p e r i m e n t a t t h e l o w e r s p e e d s , b u t a t t h e h i g h e s t s p e e d , t h e d i f f r a c t i o n r e s i s t a n c e i s u n d e r e s t i m a t e d . H e r e a l s o t h e i n f l u e n c e o f v i s c o u s e f f e c t s may b e t h e m a i n r e a s o n f o r t h i s d e v i a t i o n , s p e c i a l l y f o r h i g h s p e e d s a n d f r e q u e n c i e s o f e n c o u n t e r . W i t h r e g a r d t o t h e t o t a l a d d e d r e s i s t a n c e i n w a v e s t h e d i f f e r e n c e s a r e o f m i n o r i m p o r t a n c e . 8

-h. C o n c l u s i o n s . F r o m t h e a n a l y s i s o f t h e e x p e r i m e n t s a n d t h e c a l c u l a t i o n s o f t h e a d d e d r e s i s t a n c e t h e f o l l o w i n g c o n c l u s i o n s may h e d r a w n . a. F o r t h e c o n s i d e r e d s h i p f o r m t h e a d d e d r e s i s t a n c e i n w a v e s v a r i e s l i n e a r l y as t h e s q u a r e d w a v e h e i g h t a t c o n s t a n t w a v e l e n g t h a n d c o n s t a n t f o r w a r d s p e e d . T h i s i s a l s o v a l i d f o r t h e r e s i s t a n c e i n c r e a s e o f a m o t i o n l e s s s h i p i n t h e same w a v e c o n d i t i o n s : l o n g i t u d i n a l bow w a v e s . b . The a d d e d r e s i s t a n c e i n w a v e s c a n b e c a l c u l a t e d b y d e t e r m i n i n g t h e r a d i a t e d e n e r g y o f t h e d a m p i n g w a v e s . To t h i s e n d t h e s t r i p t h e o r y i s u s e d , t a k i n g i n t o a c c o u n t t h e r e l a t i v e m o t i o n o f t h e w a t e r w i t h r e s p e c t t o t h e s h i p a n d a c l o s e f i t p r o c e d u r e t o d i s c r i b e a c c u r a t e l y t h e f o r m o f t h e c r o s s - s e c t i o n s , c. The i n f l u e n c e o f s u r g e o n b o t h t h e m o t i o n s a n d t h e a d d e d r e s i s t a n c e i n w a v e s may b e n e g l e c t e d . 9

-5. A c k n o w l e d g e m e n t .

The a u t h o r s a r e i n d e b t e d t o V e r o l m e U n i t e d S h i p y a r d s who k i n d l y p r o v i d e d t h e d r a w i n g s a n d o t h e r d a t a o f M.V. "S.A. v a n d e r S t e l " .

3

-R e f e r e n c e s .

T.H. H a v e l o c k :

"The R e s i s t a n c e o f a S h i p among Waves". P r o c . Roy. Soc. A, V o l . l 6 l , 1 9 3 7 , P- 2 9 9 -G.A. F i r s o f f : D i s c u s s i o n o n V . I . P e r s h i n a n d A . I . V o s n e z s e n s k y S t u d y o f S h i p S p e e d D e c r e a s e i n I r r e g u l a r S e a . P r o c e e d i n g s S y m p o s i u m o n t h e B e h a v i o u r o f S h i p s i n a Seaway W a g e n i n g e n , 1957« 3 C h a p t e r 5, R e s i s t a n c e i n Waves. V o l u m e 8 , 6 o t h A n n i v e r s a r y S e r i e s The S o c i e t y o f N a v a l A r c h i t e c t s o f J a p a n "1903. " i n c r e a s e o f S h i p R e s i s t a n c e i n Waves" R e p o r t N A- 6 T - 2 , I 9 6 7 C o l l e g e o f E n g i n e e r i n g , U n i v e r s i t y o f C a l i f o r n i a [ 5 ] J . G e r r i t s m . a , J . J . v a n d e n B o s c h , W. B e u k e l m a n : " P r o p u l s i o n i n R e g u l a r a n d I r r e g u l a r Waves'.' I n t e r n a t i o n a l S h i p b u i l d i n g P r o g r e s s V o l . 8 , n o . 8 2 , 1 9 6 1 . 6 ] W.P.A. J o o s e n : "Added R e s i s t a n c e o f S h i p s i n Waves'.' S i x t h S y m p o s i u m N a v a l H y d r o d y n a m i c s W a s h i n g t o n I 9 6 6 7, J . G e r r i t s m a , W. B e i f k e l m a n : " A n a l y s i s o f a M o d i f i e d S t r i p T h e o r y f o r t h e C a l c u l a t i o n o f S h i p M o t i o n s a n d Wave B e n d i n g Moments'.' N e t h e r l a n d s S h i p R e s e a r c h C e n t e r , R e p o r t n o . 96 S, J u n e I 9 6 7 . O.J. S i b u l : 1 1

-L i s t o f S y m b o l s See a l s o A^j. A r e a o f w a t e r p l a n e . B B r e a d t h o f s h i p o r m o d e l . b ' S e c t i o n a l d a m p i n g c o e f f i c i e n t f o r s p e e d . C B B l o c k c o e f f i c i e n t . Cp L o n g i t u d i n a l p r i s m a t i c c o e f f i c i e n t . c Wave c e l e r i t y . Fg^ Wave f o r c e a m p l i t u d e . ( A m p l i t u d e o f e x c i t a t i o n f o r c e i n h e a v e ) F n F r o u d e nxmhex. g A c c e l e r a t i o n o f g r a v i t y . 1 ^ L o n g i t u d i n a l moment o f i n e r t i a o f w a t e r p l a n e w i t h r e s p e c t t o t h e y-|^ a x i s . 'k=^^/X Wave n u m b e r . k y y L o n g i t u d i n a l r a d i u s o f i n e r t i a o f t h e s h i p . L p p , L 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 . Mg_ Wave moment a m p l i t u d e . ( A m p l i t u d e o f e x c i t a t i o n moment i n p i t c h ) , m' S e c t i o n a l a d d e d m a s s . W' S e c t i o n a l d a m p i n g c o e f f i c i e n t f o r z e r o s p e e d . D a m p i n g c o e f f i c i e n t f o r h e a v e . 1^0 D a m p i n g c o e f f i c i e n t f o r p i t c h . A d d e d r e s i s t a n c e i n w a v e s . T . D r a u g h t o f s h i p . P e r i o d o f e n c o u n t e r , t T i m e . V F o r w a r d s p e e d o f s h i p . V e r t i c a l r e l a t i v e w a t e r v e l o c i t y . V^a A m p l i t u d e o f v e r t i c a l r e l a t i v e w a t e r v e l o c i t y . } R i g h t h a n d c o o r d i n a t e s y s t e m f i x e d t o t h e s h i p ^ b ' ^13' ^b H a l f w i d t h o f d e s i g n e d w a t e r l i n e . z H e a v e d i s p l a c e m e n t . Zg_ H e a v e a m p l i t u d e . E P h a s e a n g l e s . C I n s t a n t a n e o u s w a v e e l e v a t i o n , t; Wave a m p l i t u d e . C-^ Wave h e i g h t ( d o u b l e a m p l i t u d e ) , e P i t c h a n g l e . 1 2

-P i t c h a m p l i t u d e . Wave l e n g t h . D e n s i t y o f w a t e r . V o l u m e o f d i s p l a c e m e n t . C i r c u l a r f r e q u e n c y . C i r c u l a r f r e q u e n c y o f e n c o u n t

Resistance

dynamometer

Weight

Resistance

dynamometer

re 2 A r r a n g e m e n t f or r e s i s t a n c e t e s t s

s t i L l w a t e r and in w a v e s .

F n = . 1 5 O O \ 1.04 Ti u C A L C U L A T I O N O 2 r „ = L / 5 0 E X P E R I M E N T W I T H O U T F n = . 2 0 l . o J - O § 9 \ ? -S=-°-^-\ n C A L C U L A T I O N ^ ^ « ^ * " E X P E R I M E N T ° = \ ' 1° [ W I T H O U T \ ^ • = L / 30 -1 ^ " ' " ^ ^ \ V L A F n = . 3 0 V * ^> C A L C U L A T I O N

• ^^°=|-{"°„1

E X P E R I M E N T O = L / 5 0 . : : t; ^ 0 i

Figure 3 Experimental and calculated frequency cliaracteristics

for lieave and pitch

EXPERIMENT WITHOUT SURGE CALCULATION-DELFT

2Ca = L / 5 0 T gXpgPl^^f^.^ VVITH SURGE = L / 4 0 J

Fn

=.25

0.5

igure 5

1.5

Influence of surge on the added resistance

in waves and c a l c u l a t e d values

V L = I - 2 X / L = 1 - < .

Resistance

d y n a m o m e t e r

m

a.

Weight

Resistance

d y n a m o m e t e r

•

Figure 2 A r r a n g e m e n t f o r r e s i s t a n c e t e s t s in

stiLL w a t e r and in w a v e s .

1.5

Fn= .15

1.0-6 J N

1.04

n CD i f ) U J L U Ql LlJ ( J (D

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CALCULATION O

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WITHOUT

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n A T T — ^

i g u r e 2 Frequency c h a r a c t e r i s t i c s at Fn='15.

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F n = . 2 0

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= L/ 50

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—-Figure 5 Frequency c h a r a c t e r i s t i c s at Fn=-30.

T E C H N I S C H E H O G E S C H O O L DELFT

AFDELING DER MARITIEME TECHNIEK

L A B O R A T O R I U M V O O R S C H E E P S H Y D R O M E C H A N I C A A N A L Y S I S O F T H E R E S I S T A N C E I N C R E A S E I N W A V E S O F A F A S T C A R G O S H I P b y P r o f . i r . J . G e r r i t s m a a n d V J . B e u k e l m a n R e p o r t n o . 3 3 4 - P s e p t . 1 9 7 1

Delft University of Technology

Ship Hydromechanics Laboratory Mekelweg 2

2628 CD D E L F T The Netherlands Phone 0 1 5 - 7 8 6 8 8 2

285

ANALYSIS OF T H E RESISTANCE INCREASE IN WAVES OF

A FAST CARGO SHIP*)

by

Prof.ir. J. Gerritsma and W. Beukelman.

Siiinmai-y

A new method to calculate the added resistance o f a ship in longitudinal waves is discussed. For the particular case o f a fast cargo-ship the calculated values are compared with experimental results, and a satisfactory agreement is shown.

I n addition the experiments with the considered shipform confirm that added resistance varies as the squared wave height f o r constant speed and wave length.

1 Introduction

One of the first attempts to calculate the added resis-tance of a ship in waves was carried out by Havelock

[!]. He determined the mean value of the longitudinal component of the pressure forces integrated over the wetted part of the oscillating ship's hull. I n his treat-ment of the problem the water pressure was taken as the undisturbed pressure of the incident wave, vvhich implies the use of the well-known Froude-Kryloff hypothesis. Clearly this was done to avoid the difficult problem ofthe evaluation ofthe complicated diffracted waves, which originate from the oscillating ship in the incident waves. Therefore Havelock considered his solution as a first approximation only. A n interesting discussion on the use ofthe undisturbed wave pressure and the effect upon the calculated added resistance in waves was given bij Firsoff [2]. He states that the Froude-Kryloff hypothesis is not applicable in this case. Havelock's expression for the added resistance in waves reads as follows:

(1)

where: and M„ are the amplitudes of the excitation force and moment in heave and pitch, z„ and 0„ are the corresponding motion amplitudes with phase lags

£.f and Bo!,,.

A n alternative method to find the expression ( I ) is to equalize the work done by the exciting force and moment to the work done by the force which is neces-sary to tow the ship through the given wave field [3].

According to equation (1) the added resistance is

zero when the motion of the ship is vanished. The experiment shows that an added resistance force can be present in such a case.

• i U c p o r l N o . ShiiibuililiiiR l . a b D r a l u r y . D e l f l f n i v e r s i l y c,r T t c h -niilony,

• l l l f p o r l N n . IO!i S N e t h e r l a n d s Ship H e s e a r e h O e n l r e T N O . U e U t ,

Furthermore the Havelock values do not agree with experimental results over a large range of wave length ratio's.

In order to get a more satisfactory agreement the concept of the relative vertical motion of the ship with respect to the water has to be used, as shown in the following chapters. The method is also applicable for the determination of the drift force on the motionless ship which corresponds to the situation of relatively small wave lengths.

T o check the calculated values an accurate experi-ment has been carried out with a 3 meter model of a fast cargo ship "S. A . van der Stel".

Existing experimental results, as found in literature do not cover a sufficiently large speed-wave length and wave height ratio for detailed comparison purposes; also the results with standard ship forms, such as the Series Sixty, differ considerably from tank to tank. In addition there is some discussion with regard to the assumption that added resistance in waves varies as the squared wave height, particularly in the case of slender ship forms [4]. A n accurate experiment with a model of a passenger-cargo ship, having a block-coeRicient = 0.65 confirmed the square law to a large extent. Based on this principle a prediction of the mean added resistance in a specified irregular sea was in good agreement with corresponding model test results [5]. It should be emphasized that in this case extremely low and high wave heights were excluded. The present analysis includes an extensive series of added resistance and motion measurements in a large range of wave lengths, forward speeds and three or four wave heights. I n one particular case the influence of surge was also investigated.

T o a large extent the results confirm again the linear relation between added resistance and wave height squared, at constant speed and wave length, at least as a very good approximation which can be used for practical purposes,

286

2 Measurement of the added resistance and motions in waves

In Table I the main particulars of the ship and the considered model are given. The model was made of glass fibre reinforced polyester on a scale of 1:50.

Table I . Main particulars o f M . V . "S.A. van der Stel" and model ship model i-pp ( m ) 152.5 3.050 i l i x ( m ) 154.7 3.094 B ( m ) 22.8 0.456 T ( m ) 9.1 0.183 V (m=) 17931 0.1434 An- (m-) 2431 0.9724 Il ( m ' ) 2782993 0.4453 Cb 0.564 0.564 Cp 0.580 0,580 LCB 1.68% aft Z,pp/2 1.68% aft Lp^n LCF 4.35% aft Lppll 4.35% aft Lppll ^service 19.5 knots 1.421 m/sec

0.219 0.219

The bodyplan is given in figure I and the test conditions are summarized in Table I I .

Fig. 1. Body plan.

Table I I . Wave conditions

UL 0.6 0.8 1.0 1.2 1.4 1.6 1.95 1/150 1/50 1/50 1/40 1/30 1/50 1/40 1/30 1/50 1/40 1/30 1/50 1/40 1/30 1/50 1/50

Four ship speeds were regarded:

F„ = 0 . 1 5 - 0 . 2 0 - 0 . 2 5 - 0 . 3 0

F o r ƒ"„ = 0.15 only one wave height C„ = L / 5 0 has been considered.

The regular waves were measured at a distance of 4 meters in front ofthe model by means o f a two-wire conductance wave probe. Additional measurements were carried out with a sonic wave probe, which show-ed a very satisfactory agreement with the results of the former one. Resistance dynamometer a. W e i g h t . | - ^ Resistance d y n o m o m e t e r

Fig. 2. Arrangement f o r resistance tests i n still water and i n waves.

The test arrangement is given in figure 2. The major-ity of the runs in waves were made with the model restrained in surge, but free to heave and pitch, as shown in figure 2a and due care was taken to ensure a minimum of friction in the measuring part of the heave guide mechanism. The mean resistance in waves was measured by means of strain gauge dynamometers o f w h i c h the output was integrated over a full number of wave periods. A comparison of the results of this resistance measurement method with results of the more conventional system, using a dead weight as shown in figure 2b, showed only very minor diflferences. The towing arrangement, given in figure 2 was also used to investigate the influence of surge on the heaving and pitching motions and on the added resistance.

The oscillatory motions of the model were measure^, by two low-friction potentiometers. It has to be emphasized that moderate wave heights were chosen in view ofthe applicability of the added wave resistance operators and for comparison purposes of the results with calculations.

In the figures 3 the experimental amplitude and phase characteristics ofthe shipmodel are compared with the corresponding calculated values. The method of the calculation is derived from [7]. I n view of the bulbous

i i

287

\.S4 F n = .15 o \ ^ u —E o "0 C A L C U L A T I O N • o O O = L / 5 G E X P E R I M E N T W I T H O U T -300H VUT 1.0 CD -300H F n = . 2 0 / ® \ o - ° i * \ O 1

• \

* \ / C A L C U L A T I O N " E X P E R I M E N T L / 5 0 I = L / 4 0 f = L / 3 0 J 9 W I T H O U T S U R G E \A7x Fig. 3a. Experimental and calculated frequency characteristics f o r heave and pitch.

bow a close fit procedure to describe the cross-sections was used.

There is a slight indication of non-linearity in the measured motion amplitudes, and a tendency towards better agreement with the theoretical values in the case of small wave heights is observed. For F„ = 0.25 the influence of surge on the heaving and pitching motions is very small, as shown in figure 3b. There is hardly any difference in the motion characteristics, whether the model is free to surge or not.

The added resistance in regular waves is shown in figure 4 for the model which is free to pitch and heave, but restrained in surge. In addition the added resistance is given for the case where the model is restrained in heave, pitch and surge motions. The results cover the various wave height conditions as summarized in Table I I .

I n view of the slight non-linearity of the vertical motions, there is surprisingly little deviation from the square wave height law. This also holds for the added resistance of the motionless model.

Figure 5 shows that the influence of surge on the added resistance in bow waves is negligible. This may not be true for following waves.

The validity of the square wave height law for this particular model with a low blockcoefficient is also clearly shown in figure 6, where added resistance due to waves for the pitching and heaving model and for the motionless model is plotted on a base of wave height squared. The standard deviation from the assumed quadratic law is in the order of 3+%, which includes the measurement errors both in wave height and in the added resistance.

288

/LA

Fig. 3b. Experimental and calculated frequency characteristics for heave and pitch.

3 Calculation of the added resistance in waves Joosen calculated the added resistance of a ship in short waves by expanding Maruo's expression into an asymptotic series with respect to the slenderness parameter [6].

Taking into account only the first order terms, he found a reasonable agreement with the experiment, although this simplified treatment results in a speed independent added resistance. I n fact use is made of the strip theory to determine the resistance force. Of particular interest is the expression for the added resistance given in [6]:

Ra.- = ^{N,z'^+N,0'„) (2)

which is equivalent to Havelock's equation (1).

Equation (2) shows that the added resistance can be regarded as a result from the radiated damping waves. Although not consistent with the mathematical theory the frequency of encounter is used by Joosen when a ship forward speed is considered. A s in the case of equation (1) this expression does not take into account the relative vertical motion of the ship, with respect to the water. Therefore the following procedure is adopted for the calculation of the radiated energy P of the oscillating ship during one period of encounter. We consider longitudinal regular bow waves.

P = j ' j f c ' - K / d x , d r (3)

0 0

where:

290

Fig. 5. Influence of surge on the added resistance i n waves and calculated values.

tlie sectional damping coefficient for speed and

V, = z - x , Ó + V 9 - t \

the vertical relative water velocity where:

is the effective vertical wave displacement for a cross-section. F o r this concept reference is made to [7]. A s V. is a harmonic function with amplitude Kj„ and a frequency equal to the frequency of encounter co,, we find:

P = - j 5 ' 7 / , d x , (4)

Following the reasoning given in [3] the work being done by the towing force R^,y is also given by:

P = R^AV + c)Z = R^,y-X (5)

where: c is the wave celerity and X is the wave length. F r o m (4) and (5) it follows that:

We can distinguish two extreme cases:

a. A ship without oscillatory motions in waves. T h i s occurs when the ship sails in relatively short waves and practically no ship motion exists. The resistance increase in this case is caused by what is commonly called: diffraction effects.

In this case: R ^ y - r ^ ' i b T d x , (7) We may write: where: Because we find: RAw = ~Cl'ib'e-''''dx, (8)

F r o m equation (6) it is clear, that the added resistance b. Also in the case of an oscillating ship in calm water in waves varies as the squared wave height, because the radiated energy corresponding to the motion

292

the relative vertical speed of a cross-section with regard to the water is given by;

V, = z-.x,Ó+VO (9)

The resistance increase follows from the equation (6) when the appropriate value of is taken.

F r o m the fact that the speed appears as a qua-dratic form in the integrand in the expression for i?^„', it follows that the resistance increase in waves is not merely the sum of the resistance increase of a ship oscillating in calm water and the resistance increase of a motionless ship in waves.

The equation (6) indicates that for the motionless ship in waves a finite resistance is found which approaches zero only at increasing frequencies of encounter.

I

The resistance increase due to waves for the considered ship was calculated according to the equations (6) and (8).

The results are shown in figure 4. The comparison with the experiments shows a very satisfactory agree-ment for the case of a pitching and heaving ship.

For the short waves (XjL<O.S) the experimental values are somewhat higher for speeds higher than F„ = 0.\5. A possible explanation could be the in-fiuence of viscous etTects, which are not included in the calculation. Because of the increasing frequency of encounter the vertical water velocities increase and consequently the viscous eff"ects could be more im-portant. T o show the improvement with regard to Havelock's formula, the values of i?^,,. corresponding to equation ( 1 ) are given in figure 4.

The resistance increase of the motionless ship is small but should not be neglected. There is a fair agreement with the experiment at the lower speeds, but at the highest speed, the difTraction resistance is underestimated. Here also the influence of viscous effects may be the main reason for this deviation, specially for high speeds and frequencies of encounter. With regard to the total added resistance in waves the differences are of minor importance.

4 Conclusions

F r o m the analysis of the experiments and the calcula-tions of the added resistance the following conclusions may be drawn.

a. F o r the ship form considered the added resistance in waves varies linearly as the squared wave height at constant wave length and constant forward speed.

This is also valid for the resistance increase of a motionless ship in the same wave conditions: longitudinal bow waves.

b. The added resistance in waves can be calculated by determining the radiated energy of the damping waves. T o this end the strip theory is used, taking into account the relative motion of the water with respect to the ship and a close fit procedure to describe accurately the form of the cross-sections. c. The influence of surge on both the motions and the

added resistance in waves may be neglected.

5 Acknowledgement

The authors are indebted to Verolme United Ship-yards who kindly provided the drawings and other data of M . V . " S . A . van der Stel".

References

1. HA V E L O C K, T . H., The Resistance o f a Ship among Waves. Proc. Roy. Soc. A , V o l . 161, 1937, p . 299.

2. FI R S O F F, G. A . , Discussion on V . I . PERSHIN and A . I .

VOSNEZSENSKY, Study o f Ship Speed Decrease i n Irregular

Sea. Proceedings Symposium on the Behaviour o f Ships i n a Seaway, Wageningen, 1957.

3. Chapter 5, Resistance in Waves. Volume 8, 60th Anniversary Series. The Society o f Naval Architects o f Japan, 1963. 4. SiBUL, O. J., Increase o f Ship Resistance i n Waves. Report

NA-67-2, 1967. College o f Engineering, University o f California.

5. GERRFTSMA, J . , J . J . V A N D E N BOSCH and W . BE U K E L M A N , Propulsion i n Regular and Irregular Waves. International Shipbuilding Progress. V o l . 8, n o . 82, 1961.

6. JOOSEN, W. P. A . , Added Resistance o f Ships in Waves. Sixth Symposium. Naval Hydrodynamics, Washington, 1966. 7. GERRFFSMA, J . and W . BE U K E L M A N , Analysis o f a M o d i f i e d

Strip Theory f o r the Calculation o f Ship Motions and Wave Bending Moments. Netherlands Ship Research Center, Report no. 96 S, June 1967.

293

L I S T O F S Y M B O L S

Area of waterplane B Breadth of ship or model

b' Sectional damping coefRcient for speed Cg Block coefficient

Cp Longitudinal prismatic coefficient c Wave celerity

Wave force amplitude. (Amplitude of excitation force in heave) F„ Froude number

g Acceleration of gravity

Il Longitudinal moment of inertia of waterplane with respect to the ƒ(, axis

k = InjX Wave number

Longitudinal radius of inertia of the ship Lpp, L Length between perpendiculars

M„ Wave moment amplitude. (Amplitude of excitation moment in pitch)

m' Sectional added mass

N' Sectional damping coefRcient for zero speed Damping coefficient for heave

Damping coefRcient for pitch R^w Added resistance in waves T Draught of ship

Period of encounter / Time

V Forward speed of ship Vertical relative water velocity

Fj^ Amplitude of vertical relative water velocity Right hand coordinate system fixed to the ship y„ Half width of designed waterline

z Heave displacement z„ Heave amplitude e Phase angles

C Instantaneous wave elevation (a Wave amplitude

Wave height (double amplitude) 0 Pitch angle 9a Pitch amplitude 2 Wave length Q Density of water F Volume of displacement CO Circular frequency

TECHNISCHE H O G E S C H O O L DELFT

AFDELING DER MARITIEME TECHNIEK

L A B O R A T O R I U M V O O R S C H E E P S H Y D R O M E C H A N I C A A N A L Y S I S O F T H E R E S I S T A N C E I N C R E A S E I N W A V E S O F A F A S T C A R G O S H I P b y P r o f . i r . J . G e r r i t s m a a n d W . B e u k e l m a n R e p o r t n r . 3 3 4 - P S e p t e m b e r 1 9 7 2 1 3 t h I T T C - V o l . 2 .

Delft University of Technology

Ship Hydromechanics Laboratory Mekelweg 2

2628 CD D E L F T The Netherlands Phone 0 1 5 - 7 8 6 8 8 2

T H I R T E E N T H

I N T E R N A T I O N A L T O W I N G T A N K

C O N F E R E N C E

B E R L I N / H A M B U R G , SEPTEMBER 1972

PROCEEDINGS

Edited by

5. SCHUSTER and M . S C H M I E C H E N

VOLUME 2

The proceedings have been printed w i t h the financial

contribution of the

B U N D E S M I N I S T E R F Ü R V E R K E H R ,

A B T E I L U N G SEEVERKEHR, H A M B U R G

9 0 2 S E A K E E P I N G R E P O R T O F S E A K E E P I N G C O M M I T T E E A P P E N D I X 5 A N A L Y S I S O F T H E R E S I S T A N C E I N C R E A S E I N W A V E S O F A F A S T C A R G O S H I P b y J . G e r r i t s m a a n d W . B e u k e l m a n 1 . I n t r o d u c t i o n . One o f t h e f i r s t a t t e m p t s 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 o f a s h i p i n w a v e s w a s c a r r i e d o u t b y H a v e l o c k [ l . . He d e t e r m i n e d t h e m e a n v a l u e o f t h e l o n g i t u -d i n a l c o m p o n e n t o f t h e p r e s s u r e f o r c e s i n t e g r a t e -d o v e r t h e w e t t e -d p a r t o f t h e o s c i l l a t i n g s h i p ' s h u l l . I n h i s t r e a t m e n t o f t h e p r o b l e m t h e w a t e r p r e s -s u r e w a -s t a k e n a -s t h e u n d i -s t u r b e d p r e -s -s u r e o f t h e i n c i d e n t w a v e , w h i c h i m p l i e -s t h e u s e o f t h e w e l l - k n o w n F r o u d e - K r y l o f f h y p o t h e s i s . C l e a r l y t h i s w a s d o n e t o a v o i d t h e d i f f i c u l t p r o b l e m o f t h e e v a l u a t i o n o f t h e c o m p l i c a t e d d i f f r a c t e d w a v e s , w h i c h o r i g i n a t e f r o m t h e o s c i l l a t i n g s h i p i n t h e i n c i d e n t w a v e s . T h e r e f o r e H a v e l o c k c o n s i d e r e d h i s s o l u t i o n a s a f i r s t a p p r o x i m a t i o n o n l y . A n i n t e r e s t i n g d i s c u s s i o n -on t h e u s e o f t h e u n d i s t u r b e d w a v e p r e s s u r e a n d t h e e f f e c t u p o n t h e c a l c u l a t e d a d d e d r e s i s t a n c e i n w a v e s w a s g i v e n b y F i r s o f f [ 2 j . He s t a t e s t h a t t h e F r o u d e - K r y l o f f h y p o t h e s i s i s n o t a p p l i c a b l e i n t h i s c a s e . H a v e l o c k ' s e x p r e s s i o n f o r t h e a d d e d r e s i s t a n c e i n w a v e s r e a d s a s f o l l o w s : w h e r e : F a n d M a r e t h e a m p l i t u d e s o f t h e e x c i t a t i o n f o r c e a n d moment i n a a h e a v e a n d p i t c h , z a n d 6 a r e t h e c o r r e s p o n d i n g m o t i o n a m p l i t u d e s w i t h p h a s e l a g s e a n d e^^. A n a l t e r n a t i v e m e t h o d t o f i n d t h e e x p r e s s i o n ( l ) i s t o e q u a l i z e t h e w o r k d o n e b y t h e e x c i t i n g f o r c e a n d moment t o t h e w o r k d o n e b y t h e f o r c e w h i c h i s n e c e s -s a r y t o t o w t h e -s h i p t h r o u g h t h e g i v e n w a v e f i e l d -s ] . A c c o r d i n g t o e q u a t i o n ( l ) t h e a d d e d r e s i s t a n c e i s z e r o w h e n t h e m o t i o n o f t h e s h i p i s v a n i s h e d . T h e e x p e r i m e n t s h o w s t h a t a n a d d e d r e s i s t a n c e f o r c e c a n b e p r e s e n t i n s u c h a c a s e . F u r t h e r m o r e t h e H a v e l o c k v a l u e s d o n o t a g r f i e w i t h e x p f j r i r n c n t a l c.r.ul^.n ov--r a l a r g e r a n g e o f w a v e l e n g t h r a t i o' s . I n o r d e r t o g e t a m o r e s a t i s f a c t o r y a g r e e m e n t t h e c o n c e p t o f t h e r e l a t i v e v e r t i c a l m o t i o n o f t h e s h i p w i t h r e s p e c t t o t h e w a t e r h a s t o b e u s e d , a s s h o w n i n t h e f o l l o w i n g c h a p t e r s . T h e m e t h o d i s a l s o a p p l i c a b l e f o r t h e

A P P E N D I X 5 9 0 3 d e t e r m i o a t i o n o f t h e d r i f t f o r c e o n t h e m o t i o n l e s s s h i p w h i c h c o r r e s p o n d s t o t h e s i t u a t i o n o f r e l a t i v e l y s m a l l w a v e l e n g t h s . T o c h e c k t h e c a l c u l a t e d v a l u e s a n a c c u r a t e e x p e r i m e n t h a s b e e n c a r r i e d o u t w i t h a 3 m e t e r m o d e l o f a f a s t c a r g o s h i p " S . A . v a n d e r S t e l " . E x i s t i n g e x p e r i m e n t a l r e s u l t s , a s f o u n d i n l i t e r a t u r e do n o t c o v e r a s u f f i c i e n t l y l a r g e s p e e d - w a v e l e n g t h a n d w a v e h e i g h t r a t i o f o r d e t a i l e d c o m p a r i s o n p u r p o s e s ; a l s o t h e r e s u l t s w i t h s t a n d a r d s h i p f o r m s , s u c h a s t h e S e r i e s S i x t y , d i f f e r c o n s i d e r a b l y f r o m t a n k t o t a n k . I n a d d i t i o n t h e r e i s s o m e d i s c u s s i o n w i t h r e g a r d t o t h e a s s i m i p t i o n t h a t a d d e d r e s i s t a n c e i n w a v e s v a r i e s a s t h e s q u a r e d w a v e h e i g h t , p a r t i c u l a r l y i n t h e c a s e o f s l e n d e r s h i p f o r m s A n a c c u r a t e e x p e r i m e n t w i t h a m o d e l o f a p a s s e n g e r -c a r g o s h i p , h a v i n g a b l o -c k -c o e f f i -c i e n t = O.65 -c o n f i r m e d t h e s q u a r e l a w t o a l a r g e e x t e n t . B a s e d o n t h i s p r i n c i p l e a p r e d i c t i o n o f t h e m e a n a d d e d r e s i s t a n c e i n a s p e c i f i e d i r r e g u l a r s e a w a s i n g o o d a g r e e m e n t w i t h c o r r e s -p o n d i n g m o d e l t e s t r e s u l t s [5 . I t s h o u l d b e e m -p h a s i z e d t h a t i n t h i s c a s e e x t r e m e l y l o w a n d h i g h w a v e h e i g h t s w e r e e x c l u d e d . T h e p r e s e n t a n a l y s i s i n c l u d e s a n e x t e n s i v e s e r i e s o f a d d e d r e s i s t a n c e a n d m o t i o n m e a s u r e m e n t s i n a l a r g e r a n g e o f w a v e l e n g t h s , f o r w a r d s p e e d s a n d t h r e e o r f o u r w a v e h e i g h t s . I n o n e p a r t i c u l a r c a s e t h e i n f l u e n c e o f s u r g e w a s a l s o i n v e s t i g a t e d . T o a l a r g e e x t e n t t h e r e s u l t s c o n f i r m a g a i n t h e l i n e a r : r e l a t i o n b e t w e e n a d d e d r e s i s t a n c e a n d w a v e h e i g h t s q u a r e d , a t c o n s t a n t s p e e d a n d w a v e l e n g t h , a t l e a s t a s a v e r y g o o d a p p r o x i m a t i o n w h i c h c a n b e u s e d f o r p r a c t i c a l p u r p o s e s . 2. M e a s u r e m e n t o f t h e a d d e d r e s i s t a n c e a n d m o t i o n s i n w a v e s . I n T a b l e 1 t h e m a i n p a r t i c u l a r s o f t h e s h i p a n d t h e c o n s i d e r e d m o d e l a r e g i v e n . T h e m o d e l w a s m a d e o f g l a s s f i b r e r e i n f o r c e d p o l y e s t e r o n a s c a l e o f 1:50. T h e b o d y p l a n i s g i v e n i n F i g u r e 1 a n d t h e t e s t c o n d i t i o n s a r e s u m m a ; r i z e d i n T a b l e 2. F o u r s h i p s p e e d s w e r e r e g a r d e d : F n =

0.15

-

0.20

-

0.25

-

0 . 3 0 .

F o r F n = 0.15 o n l y o n e w a v e h e i g h t = h a s b e e n c o n s i d e r e d . T h e r e g u l a r w a v e s w e r e m e a s u r e d a t a d i s t a n c e o f h m e t e r s i n f r o n t o f t h e m o d e l b y m e a n s o f a t w o - w i r e c o n d u c t a n c e

wave

p r o b e ' . A d d i t i o n a l m e a s u r e m e n t s

9 0 4 S E A K E E P I N G T a b l e 1. M a i n p a r t i c i i l a r s o f M . V . " S . A . v a n d e r S t e l " a n d m o d e l . -S H I P MODEL L P P 152.5 m 3.050 m L P P 15^.7 m 3.09^ m B 22.8 m 0.il56 m T 9-1 m 0.183 m V 17931 m^ 0 . l i i 3 H m^ ^ v 2i;31 m2 0.972i+ m2 2782993 m^ 0.1|li53 m^ O.56U 0.56ii 0.580 0.580 L C B L C F V s e r v i c e 1.68 ^ a f t L / 2 P P i i . 3 5 % a f t L / 2 P P 19-5 k n o t s 1.68 ^ a f t L / 2 P P i|.35 % a f t L / 2 PP 1.^21 m / s e c k / L y y P P 0.219 0.219 T a b l e 2 , W a v e c o n d i t i o n s a / l 0 . 6 V 5 0 0 . 8 V 5 0 V ^ o V 3 0 1.0 V 5 0 VHo V 3 0 1.2 V i 5 0 V 5 0 Vi+0 V 3 0 1.I1 V 5 0 V H o V 3 0 1.6 V5O 1.95 V 5 0 w e r e c a r r i e d o u t w i t h a s o n i c w a v e p r o b e , w h i c h s h o w e d a v e r y s a t i s f a c t o r y a g r e e m e n t w i t h t h e r e s u l t s o f t h e f o r m e r o n e . T h e t e s t a r r a n g e m e n t i s g i v e n i n F i g u r e 2. T h e m a j o r i t y o f t h e r u n s i n w a v e s w e r e made w i t h t h e m o d e l r e s t r a i n e d i n s u r g e , - b u t f r e e t o h e a v e a n d p i t c h , a s s h o w n i n F i g u r e 2a a n d d u e c a r e w a s t a k e n t o e n s u r e a m i n i m u m o f f r i c t i o n i n t h e m e a s u r i n g p a r t o f t h e h e a v e g u i d e m e c h a n i s m . T h e m e a n r e s i s t a n c e i n w a v e s w a s m e a s u r e d b y m e a n s o f s t r a i n g a u g e d y n a m o m e t e r s o f w h i c h t h e o u t p u t w a s i n t e g r a t e d o v e r a f u l l n u m b e r o f w a v e p e r i o d s . A c o m p a r i s o n o f t h e r e s u l t s o f t h i s r e s i s t a n c e m e a s u r e m e n t m e t h o d w i t h r e s u l t s o f t h e m o r e c o n v e n t i o n a l

A P P E N D I X 5 9 0 5 s y s t e m , u s i n g a d e a d w e i g h t a s s h o w n i n F i g u r e 2b, s h o w e d o n l y v e r y m i n o r d i f f e r e n c e s . T h e t o w i n g a r r a n g e m e n t , g i v e n i n t h i s F i g u r e w a s a l s o u s e d t o i n v e s t i g a t e t h e i n f l u e n c e o f s u r g e o n t h e h e a v i n g a n d p i t c h i n g m o t i o n s a n d o n t h e a d d e d r e s i s t a n c e . T h e o s c i l l a t o r y m o t i o n s o f t h e m o d e l w e r e m e a s u r e d b y t w o l o w - f r i c t i o n p o t e n t i o m e t e r s . I t h a s t o b e e m p h a s i z e d t h a t m o d e r a t e w a v e h e i g h t s w e r e c h o s e n i n v i e w o f t h e a p p l i c a b i l i t y o f t h e a d d e d w a v e r e s i s t a n c e o p e r a t o r s a n d f o r c o m p a r i s o n p u r p o s e s o f t h e r e s u l t s w i t h c a l c u l a t i o n s . I n F i g u r e 3 t h e e x p e r i m e n t a l a m p l i t u d e a n d p h a s e c h a r a c t e r i s t i c s o f t h e s h i p m o d e l a r e c o m p a r e d w i t h t h e c o r r e s p o n d i n g c a l c u l a t e d v a l u e s . T h e m e t h o d o f t h e c a l c u l a t i o n i s d e r i v e d f r o m [ T . I n v i e w o f t h e b u l b o u s bow a c l o s e f i t p r o c e d u r e t o d e s c r i b e t h e c r o s s - s e c t i o n s w a s u s e d . T h e r e i s a s l i g h t i n d i c a t i o n o f n o n l i n e a r i t y i n t h e m e a s u r e d m o t i o n a m p l i -t u d e s , a n d a -t e n d e n c y -t o w a r d s b e -t -t e r a g r e e m e n -t w i -t h -t h e -t h e o r e -t i c a l v a l u e s i n t h e c a s e o f s m a l l w a v e h e i g h t s i s o b s e r v e d . F o r F n = 0.25 t h e i n f l u e n c e o f s u r g e o n t h e h e a v i n g a n d p i t c h i n g m o t i o n s i s v e r y s m a l l , a s s h o w n i n F i g u r e 3. T h e r e i s h a r d l y a n y d i f f e r e n c e i n t h e m o t i o n c h a r a c t e r i s t i c s , w h e t h e r t h e m o d e l i s • f r e e t o s u r g e o r n o t . T h e a d d e d r e s i s t a n c e i n r e g u l a r w a v e s i s s h o w n i n F i g u r e k f o r t h e m o d e l w h i c h i s f r e e t o p i t c h a n d h e a v e , b u t r e s t r a i n e d i n s u r g e . I n a d d i t i o n t h e a d d e d r e s i s t a n c e i s g i v e n f o r t h e c a s e w h e r e t h e m o d e l i s r e s t r a i n e d i n h e a v e , p i t c h a n d s u r g e m o t i o n s . T h e r e s u l t s - c o v e r t h e v a r i o u s w a v e h e i g h t c o n d i t i o n s a s s u m m a r i z e d i n T a b l e 2. I n v i e w o f t h e s l i g h t n o n - l i n e a r i t y o f t h e v e r t i c a l m o t i o n s , t h e r e i s s u r p r i s i n g l y l i t t l e d e v i a t i o n f r o m t h e s q u a r e w a v e h e i g h t l a w . T h i s a l s o h o l d s f o r t h e a d d e d r e s i s t a n c e o f t h e m o t i o n l e s s m o d e l . F i g u r e 5 s h o w s t h a t t h e i n f l u e n c e o f s u r g e o n t h e a d d e d r e s i s t a n c e i n bow- w a v e i s n e g l i g i b l e . T h i s m a y n o t b e t r u e f o r f o l l o w i n g w a v e s . T h e v a l i d i t y o f t h e s q u a r e w a v e h e i g h t l a w f o r t h i s p a r t i c u l a r m o d e l w i t h a l o w b l o c k c o e f f i c i e n t i s a l s o c l e a r l y s h o w n i n t h e F i g u r e 6, w h e r e a d d e d r e s i s t a n c e d u e t o w a v e s f o r t h e p i t c h i n g a n d h e a v i n g m o d e l a n d f o r t h e m o t i o n l e s s m o d e l i s p l o t t e d o n a b a s e o f w a v e h e i g h t s q u a r e d . T h e s t a n d a r d d e v i a t i o n f r o m t h e a s s u m e d q u a d r a t i c l a w i s i n t h e o r d e r o f 3l%, w h i c h i n c l u d e s t h e m e a s u r e m e n t e r r o r s b o t h i n w a v e h e i g h t a n d i n t h e a d d e d r e s i s t a n c e .

9 0 6 S E A K E E P I N G ' a l c u l a t i o n o f t h e a d d e d r e s i s t a n c e i n w a v e s . J o o s e n c a l c u l a t e d t h e a d d e d r e s i s t a n c e o f a s h i p i n s h o r t w a v e s "by e x p a n d i n g ;-:aruo's e x p r e s s i o n i n t o a n a s y m p t o t i c s e r i e s w i t h r e s p e c t t o t h e s l e n d e r n e s s p a r ar.'eter 6 . T a k i n g i n t o a c c o u n t o n l y t h e f i r s t o r d e r t e r m s , h e f o u n d a r e a s o n a b l e a g r e e -m e n t w i t h t h e e x p e r i m e n t , a l t h o u g h t h i s s i m p l i f i e d t r e a t m e n t r e s u l t s i n a s p e e d i n d e p e n d e n t a d d e d r e s i s t a n c e . I n f a c t u s e i s made o f t h e s t r i p t h e o r y x o d e t e r m i n e t h e r e s i s t a n c e f o r c e . O f p a r t i c u l a r i n t e r e s t i s t h e e x p r e s s i o n f o r t h e a d d e d r e s i s t a n c e g i v e n i n 6 j :

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( ^ z ^ ' - ^ N g e / ) ( 2 ) w h i c h i s e q u i v a l e n t t o H a v e l o c k ' s e q u a t i o n E q u a t i o n ( 2 ) shows t h a t t h e a d d e d r e s i s t a n c e c a n b e r e g a r d e d as a r e s u l t f r o. T . t h e r a d i a t e d d a m p i n g w a v e s . . A l t h o u g h n o t c o n s i s t e n t w i t h t h e m a t h e m a t i c a l t h e o r y t h e f r e q u e n c y o f e n c o u n t e r i s u s e d b y J o o s e n w h e n a s h i p w i t h f o r w a r d s p e e d i s c o n s i d e r e d . As i n t h e c a s e o f e q u a t i o n ( 1 ) t h i s e x p r e s s i o n d o e s n o t t a k e i n t o a c c o u n t zhe r e l a t i v e v e r t i c a l m o t i o n o f t h e s h i p , w i t h r e s p e c t t o t h e w a t e r . T h e r e f o r e t h e f o l l o v / i n g p r o c e d u r e i s a d o p t e d f o r t h e c a l c u l a t i o n o f t h e r a d i a t e d e n e r g y P o f t h e o s c i l l a t i n g s h i p d u r i n g o n e p e r i o d o f e n c o u n t e r . V7e c o n s i d e r l o n g i t u d i n a l r e g u l a r bow w a v e s . P = Te L f 2 b ' . d x ^ d t ( 3 ) w n e r e : b ' = N' - V t h e s e c t i o n a l d a m p i n g c o e f f i c i e n t f o r s p e e d a n d V = z - x-j^O + VG - ^ , t h e v e r t i c a l r e l a t i v e w a t e r v e l o c i t y w h e r e : = ? ( 1 - ~ ~ yw ^ ^ b . . y-^ e ^^\)) I S t h e e f f e c t i v e v e r t i c a l w a v e - T d i s p l a c e m e n t f o r a c r o s s - s e c t i o n . F o r t - h i s c o n c e p t r e f e r e n c e i s made t o 7 . As V^ i s a h a r m o n i c f u n c t i o n w i t h a m p l i t u d e V^g^ a n d a f r e q u e n c y e q u a l t o t h e f r e q u e n c y o f e n c o u n t e r cug, we f i n d : L P = ^ b' V z a ^ dx^ (U)