Sailing yacht performance in calm water and in waves

SAILING YACHT PERFORMANCE I N

CALM WATER AND I N WAVES

J . G e r r i t s m a , J.A. Keuning

and A. v e r s l u i s

Report No. 938-P

J a n u a r y 1993

Delft University of Technology Ship Hydromechanics Laboratory IVIekelweg 2

2628 CD Delft The Netherlands Phone 015 - 78 68 82

SMLING YACHT PERFOEMANCE I N CflIiM WZVTER AND I N WAVES b y J . G e r r i t s m a , J.A. K e u n i n g and A. V e r s l u i s S h i p H y d r o m e c h a n i c s L a b o r a t o r y D e l f t U n i v e r s i t y o f T e c h n o l o g y Mekelweg 2, 2628 CD D e l f t , The N e t h e r l a n d s Summary The D e l f t S y s t e m a t i c Y a c h t H u l l S e r i e s has been e x t e n d e d t o a t o t a l o f 3 9 h u l l f o r m v a r i a t i o n s , c o v e r i n g a w i d e range o f l e n g t h - d i s p l a c e m e n t r a t i o s a n d o t h e r f o r m p a r a m e t e r s . The t o t a l s e t o f model-e x p model-e r i m model-e n t r model-e s u l t s , i n c l u d i n g u p r i g h t and h e e l e d r e s i s t a n c e as w e l l as s i d e -f o r c e a n d s t a b i l i t y , has been a n a l y s e d and p o l y n o m i a l e x p r e s s i o n s t o a p p r o x i -mate t h e s e q u a n t i t i e s a r e p r e s e n t e d . I n v i e w o f t h e c u r r e n t i n t e r e s t i n t h e p e r f o r m a n c e o f s a i l i n g y a c h t s i n waves, t h e added r e s i s t a n c e i n i r r e g u l a r waves o f 8 w i d e l y d i f f e r e n t h u l l f o r m v a r i a -t i o n s has been c a l c u l a -t e d . A n a l y s i s o f t h e r e s u l t s shows t h a t t h e added r e s i s -t a n c e i n waves s -t r o n g l y depends on -t h e p r o d u c t o f d i s p l a c e m e n t - l e n g t h r a t i o and t h e g y r a d i u s o f t h e p i t c h i n g m o t i o n . C o n t e n t s 1 . I n t r o d u c t i o n 2. V e l o c i t y p r e d i c t i o n i n c a l m w a t e r 2 . 1 . M a i n d i m e n s i o n s a n d f o r m c o e f f i -c i e n t s 2 . 2 . D e t e r m i n a t i o n o f t h e h y d r o d y n a -mic r e s i s t a n c e 2 . 2 . 1 . U p r i g h t r e s i s t a n c e 2 . 2 . 2 . I n d u c e d r e s i s t a n c e 2 . 2 . 3 . R e s i s t a n c e due t o h e e l 2 . 3 . S i d e f o r c e as a f u n c t i o n o f h e e l and l e e w a y 2 . 4 . S t a b i l i t y 3. P r e d i c t i o n o f added r e s i s t a n c e i n waves 4. R e f e r e n c e s N o m e n c l a t u r e A„ - w a t e r l i n e a r e a hy- - maximum c r o s s - s e c t i o n a r e a AR - a s p e c t r a t i o % L " w a t e r l i n e b r e a d t h ^MAX ' maximum beam

Cp - f r i c t i o n a l r e s i s t a n c e c o e f f i c i e n t Cjj - h e e l e d r e s i s t a n c e c o e f f i c i e n t Cjyi maximum c r o s s s e c t i o n c o e f f i -c i e n t Cp - p r i s m a t i c c o e f f i c i e n t CQ-[ - i n d u c e d r e s i s t a n c e c o e f f i c i e n t C L - l i f t c o e f f i c i e n t Ffj - s i d e f o r c e Fn - Froude number GM - m e t a c e n t r i c h e i g h t g - a c c e l e r a t i o n due t o g r a v i t y ^ 1 / 3 ' s i g n i f i c a n t wave h e i g h t k y y - p i t c h g y r a d i u s LCB - l o n g i t u d i n a l c e n t e r o f b u o y a n c y i n % Lp^L o f L;^L/2 ^WL " w a t e r l i n e l e n g t h MN - r e s i d u a r y s t a b i l i t y q - s t a g n a t i o n p r e s s u r e - ^pV^ R^ - t o t a l r e s i s t a n c e w i t h h e e l and l e e w a y Rrp - t o t a l r e s i s t a n c e i n u p r i g h t p o s i t i o n Rp - f r i c t i o n a l r e s i s t a n c e Rj^ - r e s i d u a r y r e s i s t a n c e Rj^ - i n d u c e d r e s i s t a n c e Rjj - r e s i s t a n c e due t o h e e l '^AW " added r e s i s t a n c e i n waves S]^ - w e t t e d a r e a k e e l S,-, - w e t t e d a r e a canoe body SJ- - w e t t e d a r e a r u d d e r S^ - s p e c t r a l d e n s i t y T l - wave p e r i o d = 2 TT mg/m^^ Tg - p e r i o d o f e n c o u n t e r Tg - e f f e c t i v e d r a u g h t T - t o t a l d r a u g h t T,-, - d r a u g h t o f canoe body V - speed - wave a m p l i t u d e (p - h e e l a n g l e X - wave l e n g t h p - d e n s i t y o f w a t e r u - c i r c u l a r f r e q u e n c y - volume o f d i s p l a c e m e n t A - w e i g h t o f d i s p l a c e m e n t /3 - l e e w a y a n g l e liyj - wave d i r e c t i o n f - k i n e m a t i c v i s c o s i t y 1 . I n t r o d u c t i o n On t h e t e n t h Chesapeake S a i l i n g Y a c h t Symposium t h e r e s u l t s o f an e x t e n s i o n o f t h e D e l f t S y s t e m a t i c Y a c h t h u l l s e r i e s h a s been p r e s e n t e d .

T h i s S e r i e s I I c o n s i s t s o f 6 medium- t o l i g h t d i s p l a c e m e n t - l e n g t h h u l l f o r m s i n v i e w o f t h e t r e n d t o w a r d s l i g h t e r s a i l -i n g y a c h t s w h -i c h have a g r e a t e r speed p o t e n t i a l , i n p a r t i c u l a r i n ' r e a c h i n g c o n d i t i o n s .

The S e r i e s has been e x t e n d e d a n d comp l e t e d w i t h model t e s t s o f an a d d i t i o -n a l S e r i e s o f 11 h u l l f o r m v a r i a t i o -n s : S e r i e s I I I .

The t o t a l S e r i e s now c o n s i s t s o f t h i r t y n i n e models.

S e r i e s I I I has been added t o i n c r e a s e t h e r e l i a b i l i t y o f t h e u p r i g h t r e s i s -t a n c e p r e d i c -t i o n f o r l i g h -t d i s p l a c e m e n -t y a c h t s , i n p a r t i c u l a r i n t h e h i g h speed r a n g e w i t h Fn > 0.45 [ 1 ] . The need f o r t h e e x t e n s i o n o f t h e S e r i e s w i t h t h e l i g h t - d i s p l a c e m e n t h u l l f o r m s can be i l l u s t r a t e d by a c o m p a r i s o n o f t h e p r e d i c t e d u p r i g h t r e s i s t a n c e f o r model 25 a c c o r d i n g t o t h e e x p r e s s i o n as u s e d b y t h e IMS and by t h e new D e l f t f o r m u l a t i o n v e r s u s t h e e x p e r i m e n t a l r e s u l t , see F i g u r e 1 [ 2 ] . 300 250 CI 0 0 I 25 -1-c a l -1-c u l a t e d { D E T i F T - -1-c a e f f i -1-c i e n t s c a l c u l a t e d XKS o meaauxed !

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0 0.1 0.2 0.3 O . i O.S 0.6 Fn

F i g . 1 : Comparison o f IMS and D e l f t ap-p r o x i m a t i o n s o f t h e r e s i d u a r y r e s i s t a n c e w i t h e x p e r i m e n t s [ 2 ] . A n o t h e r a s p e c t o f l i g h t d i s p l a c e m e n t h u l l f o r m s i s t h e s a i l c a r r y i n g c a p a c i t y . L i g h t s a i l i n g y a c h t s w i t h a l a r g e B^i^/T,^ r a t i o l o o s e 20 t o 30 % s t a b i l i t y a t h i g h speed compared w i t h t h e r e s u l t o f a h y d r o s t a t i c s t a b i l i t y c a l c u l a t i o n . T h i s seams t o be caused by more e x t r e m e d i s -t o r -t i o n o f -t h e f r e e s u r f a c e when h e e l e d a t h i g h speeds. N e g l e c t i n g t h e s e e f f e c t s c o u l d l e a d t o e r r o n e o u s v e l o c i t y p r e d i c -t i o n s .

T h e r e f o r e t h e s y s t e m a t i c S e r i e s r e s u l t s a l s o have been used t o r e a n a l y s e t h e f o r w a r d speed e f f e c t s on s t a b i l i t y f o r a l l c o n s i d e r e d h u l l f o r m v a r i a t i o n s . The u p r i g h t r e s i s t a n c e , t h e h e e l e d r e s i s t a n c e , t h e s i d e f o r c e and t h e s t a b i -l i t y c o u -l d be e x p r e s s e d i n s i m p -l e h u -l -l f o r m p a r a m e t e r s : LWL/^C^''^ % L / T C - T C / T , L „ L / B „ L , ^w/^c^'^' LCB, Cp a t c o n s t a n t V / / g L ^ . I n g e n e r a l , v e l o c i t y p r e d i c t i o n s o f s a i l i n g y a c h t s c o n c e r n t h e c a l m w a t e r p e r f o r m a n c e . However t h e i n f l u e n c e o f seawaves on t h e r e s i s t a n c e can be v e r y s u b s t a n t i a l , i n p a r t i c u l a r when r e s o n a n c e c o n d i t i o n s f o r t h e p i t c h i n g m o t i o n a r e e n c o u n t e r e d . A l s o i n t h i s r e s p e c t , l i g h t - d i s p l a c e m e n t y a c h t s d i f f e r f r o m medium- a n d heavy-d i s p l a c e m e n t y a c h t s because t h e n a t u r a l p e r i o d s o f p i t c h and heave w i l l be s m a l l e r . C o n s e q u e n t l y , r e s o n a n c e c o n d i -t i o n s w i l l o c c u r i n a d i f f e r e n -t p a r -t o f t h e wave s p e c t r u m r e s u l t i n g , i n many cases, i n l o w e r added r e s i s t a n c e f o r t h e l i g h t d i s p l a c e m e n t y a c h t , i n c o m p a r i s o n w i t h a l a r g e r d i s p l a c e m e n t y a c h t w i t h c o m p a r a b l e m a i n d i m e n s i o n s . The t o t a l r e s i s t a n c e , i n c l u d i n g t h i s added r e s i s t a n c e , c o u l d be t a k e n i n t o a c c o u n t i n a p e r f o r m a n c e c a l c u l a t i o n , when t h e r e l a t i v e m e r i t s o f y a c h t s i n a seaway have t o be compared, f o r i n s t a n c e i n t h e case o f r a c e h a n d i c a p p i n g . I n t h i s p a p e r t h e p o s s i b i l i t i e s t o i n -c l u d e s u -c h an added r e s i s t a n -c e i n waves c a l c u l a t i o n a r e d i s c u s s e d , a l s o w i t h r e g a r d t o t h e u s e o f t h e s y s t e m a t i c h u l l f o r m v a r i a t i o n s o f t h e D e l f t S e r i e s i n t h i s r e s p e c t . 2. V e l o c i t y p r e d i c t i o n i n c a l m w a t e r I n 1977 t h e r e s u l t s o f model e x p e r i m e n t s w i t h 9 s y s t e m a t i c v a r i a t i o n s o f s a i l i n g y a c h t h u l l f o r m s were p u b l i s h e d [ 5 ] . The measurements i n c l u d e d t h e d e t e r m i n a t i o n o f t h e u p r i g h t r e s i s t a n c e , t h e h e e l e d and i n d u c e d r e s i s t a n c e , t h e s i d e f o r c e and t h e s t a b i l i t y . An e x t e n s i o n o f t h i s r e s e a r c h w i t h a n o t h e r s e r i e s o f 12 h u l l f o r m s was p r e s e n t e d i n 1981 [ 6 ] . A l l o f t h e 22 h u l l f o r m v a r i a t i o n s were based on t h e s a i l i n g y a c h t S t a n d f a s t 43 d e s i g n e d by Frans Maas. ( S e r i e s I ) . I n v i e w o f t h e t r e n d t o w a r d s l i g h t d i s -p l a c e m e n t s a f u r t h e r e x t e n s i o n o f t h e s e r i e s w i t h 6 ifiodels ( S e r i e s I I ) was c o m p l e t e d p r o v i d i n g t h e same k i n d o f i n -f o r m a t i o n as -f o r S e r i e s I and p u b l i s h e d i n 1988 [ 7 ] and 1991 [ 8 ] . These h u l l f o r m v a r i a t i o n s were based on a v a n de S t a d t & P a r t n e r s d e s i g n e d p a r e n t f o r m . F i n a l l y a t h i r d s e r i e s ( S e r i e s I I I ) o f e l e v e n models has been t e s t e d , b u t o n l y i n t h e u p r i g h t c o n d i t i o n , w i t h o u t l e e -way.

The speed r a n g e f o r S e r i e s I i s l i m i t e d t o Fn = 0.45, b u t f o r t h e S e r i e s I I a n d I I I speeds c o r r e s p o n d i n g t o Fn = 0.75

have been i n c l u d e d . W i t h t h e p a r e n t mo-d e l o f S e r i e s I t h r e e m o mo-d i f i c a t i o n s o f t h e k e e l span have been t e s t e d .

S e r i e s I ( n r s . 1 - 22) i s 1.60 m e t e r s ; f o r t h e S e r i e s I I a n d I I I ( n r s . 23 - 28 and n r s . 29 - 39) t h e w a t e r l i n e l e n g t h i s 2.0 m e t e r s . 2 . 1 . M a i n d i m e n s i o n s a n d f o r m c o e f f i -c i e n t s

The main d i m e n s i o n s o f t h e models 1¬ 39, e x t r a p o l a t e d t o a w a t e r l i n e l e n g t h •'^WL ° m e t e r s a r e g i v e n i n T a b l e 2, whereas i n T a b l e 3 t h e f o r m c o e f f i c i e n t s and t h e l o n g i t u d i n a l p o s i t i o n o f t h e c e n t e r o f buoyancy a r e summarized. I n T a b l e 1 t h e r a n g e s o f some r a t i o ' s o f m a i n d i m e n s i o n s a n d f o r m c o e f f i c i e n t s a r e g i v e n . T a ü b l e 1 T a b l e 3 R a n g e s o f h u l l f o r m p a r a m e t e r s 2 . ,76 - 5 , ,00 2 , ,46 -- 19 , , 32 4 . ,34 • • 8 , ,50 0 . , 0 -- - 6 .0% 0 .52 -- 0 . 60 T a b l e 2 Main d i m e n s i o n s m o d e l T _{' c} S _{C A x} n o . m m m m 1 1 0 , 04 3 . 6 7 3 . 1 7 0 . 79 4 2 . I S 9 . 1 0 2 5 .4 1 . 6 2 2 1 8 2 1 0 .04 3 . 2 1 2 .76 0 . 9 0 7 2 .28 9 . 1 8 2 3 . 9 1 . 6 2 19 1 3 1 0 ,06 •1 . 25 3 .64 0 . 6 0 1 2 . 0 5 9 .16 2 7 . 6 1 . 6 3 2 5 2 4 1 0 . 0 6 3 . 3 2 2 . 0 5 0 . 7 2 2 2 . 0 9 7 . 5 5 2 3 . 0 1 . 3 4 1 9 0 5 1 0 . 0 5 4 .24 3 . 6 4 0 . 9 2 0 2 .29 1 2 . 1 0 2 9 . 1 2 . 1 5 2 5 3 6 1 0 ,00 3 . 6 6 3 . 1 7 1 . 0 6 4 2 .43 12 . 24 2 7 .5 2 . 1 6 2 1 9 7 1 0 . 0 6 3 . 6 8 3 . 1 7 0 . 6 4 0 2 . 0 1 7 . 3 5 2 4 .1 1 . 3 1 2 1 8 S 1 0 . 15 3 . 54 3 . 0 5 0 . 7 9 4 2 . 1 6 9 . 1 8 2 5 .4 1 . 5 7 2 2 1 9 1 0 .07 3 . 01 3 . 2 8 0 . 7 9 4 2 .16 9 . 1 0 2 5 .0 1 . SO 2 1 5 1 0 1 0 .00 3 . 6 0 3 . 1 7 0 . 7 9 4 2 . 1 6 9 . 19 2 5 . 6 1 . 6 2 2 2 0 1 1 1 0 0 0 3 . 6 8 3 . 1 7 0 . 7 9 4 2 . 1 6 9 . 1 9 2 5 . 3 1 . 6 2 2 1 6 1 2 1 0 0 0 3 . 3 0 2 . 0 5 0 . 7 2 4 2 .09 7 . 5 2 2 3 . 0 1 . 3 3 1 9 8 1 3 1 0 0 0 3 . 3 0 2 . 8 5 0 . 7 2 4 2 .09 7 . 5 2 2 2 . 8 1 . 3 3 1 9 4 1 4 1 0 0 0 3 .30 2 . 0 5 0 . 7 7 2 2 .14 7 5 2 2 2 . 4 1 . 4 2 1 8 7 I S 1 0 0 0 3 . 6 7 3 . 1 6 0 . 8 5 8 2 . 2 3 9 2 9 2 4 . 9 1 . 7 S 2 0 8 I S 1 0 0 0 3 68 3 1 7 1 . 1 2 8 2 . 6 5 12 2 3 2 7 .3 2 3 2 2 0 9 1 7 1 0 0 0 3 68 3 1 7 0 . 7 4 7 2 . 1 2 9 1 7 2G .3 1 . 5 3 2 3 0 1 8 1 0 0 0 3 6 0 3 1 7 0 . 7 4 7 2 . 12 9 1 7 2 6 .0 1 . 5 3 2 2 6 1 9 1 0 0 0 3 . 6 8 3 1 7 0 8 4 5 2 . 2 1 9 1 7 2 4 . 8 1 . 7 3 2 1 0 2 0 1 0 0 0 3 68 3 1 7 0 8 4 S 2 . 2 1 9 1 7 2 4 .6 1 7 3 2 0 6 2 1 1 0 0 0 3 3 0 2 8 5 0 6 8 4 2 . 0 5 7 5 4 2 3 .6 1 2 5 2 0 . 5 2 2 1 0 0 0 4 2 4 3 6 6 0 8 6 5 2 .23 12 2 S 3 0 .2 2 0 5 2 6 3 2 3 1 0 0 0 3 2 0 2 0 6 0 7 0 4 1 . 0 0 7 9 7 2 3 .3 1 4 6 1 9 3 2 4 1 0 0 0 3 3 0 2 8 6 0 2 6 1 1 . 3 6 3 0 0 19 9 0 5 5 1 9 0 2 5 • 1 0 00 2 8 0 2 5 0 0 4 6 4 1 .56 4 6 2 1 9 0 0 . 8 4 I S 7 2 6 1 0 00 2 9 0 2 5 0 0 1 9 4 1 .29 1 9 7 1 7 3 0 3 6 16 7 2 7 1 0 0 0 2 5 0 2 2 2 0 9 0 4 2 . 0 0 7 9 2 2 1 7 1 4 4 14 9 2 8 1 0 0 0 2 5 5 2 2 2 0 3 2 9 1 . 4 2 2 92 1 6 2 0 5 4 1 4 S 2 9 1 0 0 0 2 9 3 2 5 0 0 2 3 0 1 . 3 3 2 3 7 1 7 5 0 4 3 I S 5 3 0 1 0 0 0 2 9 3 2 5 0 0 3 5 0 1 4 5 3 C4 1 8 3 0 6 6 1 6 . 7 3 1 1 0 0 0 2 9G 2 5 0 0 1 6 0 1 . 2 5 1 63 1 7 1 0 3 0 1 4 5 3 2 1 0 00 2 9 5 2 5 0 0 2 3 0 1 . 3 3 2 3 7 1 7 8 0 4 3 1 5 5 3 3 1 0 00 2 9 4 2 5 0 0 2 3 0 1 . 3 3 2 3 7 1 7 2 0 4 4 I S . 3 3 4 1 0 0 0 2 9 5 2 5 0 0 2 4 0 1 3 4 2 3 7 1 7 0 0 4 6 I S . 2 3 5 1 0 0 0 2 9 5 2 5 0 0 2 2 0 1 3 2 2 3 7 1 8 0 0 4 1 1 7 3 3 6 1 0 0 0 3 0 4 2 5 0 0 2 5 0 1 34 2 3 7 1 7 2 0 4 3 1 6 . 3 3 7 1 0 0 0 3 2 2 2 5 0 0 2 7 0 1 3G 2 3 7 1 7 0 0 4 3 16 3 3 8 1 0 0 0 3 9 2 3 3 3 0 1 7 0 1 2 7 2 3 7 2 2 6 0 4 3 2 2 2 3 9 1 0 0 0 2 3 5 2 0 0 0 2 9 0 1 3 0 2 3 7 1 4 7 0 4 3 13 . 4 Form p a r a m e t e r s M o d e l no. BWL/TC ^ 3 1 3 1 7 2 7 3 3 9 9 0 5 6 8 4 7 8 - 2 . 3 2 3 6 4 3 1 2 3 0 4 0 3 6 9 4 7 8 - 2 . 3 3 7 6 2 3 5 5 3 3 0 S 6 5 4 7 8 - 2 . 3 4 3 5 3 3 0 1 3 9 5 0 5 6 4 5 1 0 - 2 . 3 S 2 7 6 2 3 6 3 9 6 0 5 7 4 4 3 6 - 2 . 4 6 3 1 3 2 7 3 2 9 8 0 5 6 8 4 3 4 - 2 . 4 7 3 1 7 2 7 2 4 9 3 0 5 6 2 5 1 4 - 2 . 3 S 3 3 2 2 8 2 3 0 4 0 5 8 5 4 7 8 - 2 . 4 9 3 0 7 2 6 2 4 1 3 0 5 4 6 4 7 9 -2.1 1 0 3 I S 2 7 2 3 9 9 0 3 6 5 4 7 7 0 . 0 1 1 3 1 5 2 7 2 3 9 9 0 5 6 5 4 7 7 - S . 0 1 2 3 5 1 3 0 3 3 9 4 0 S 6 S 5 1 0 0 . 0 1 3 3 5 1 3 0 3 3 9 4 0 5 6 5 5 1 0 - 5 . 0 1 4 3 5 1 3 0 3 3 6 9 0 3 3 0 5 1 1 - 2 . 3 I S 3 1 6 2 7 2 3 6 8 0 5 3 0 4 7 6 - 2 . 3 1 6 3 1 5 2 7 2 2 8 1 0 5 3 0 4 3 4 - 2 . 3 1 7 3 1 3 2 7 2 4 2 4 0 6 0 0 4 7 8 0 . 0 1 8 3 1 3 2 7 2 4 2 4 0 6 0 0 4 7 8 - 5 . 0 1 9 3 I S 2 7 2 3 7 5 0 3 3 0 4 7 0 0 . 0 2 0 3 1 5 2 7 2 3 7 5 0 S 3 0 4 7 8 - 5 . 0 2 1 3 5 1 3 0 3 4 1 7 0 6 0 0 3 1 0 - 2 . 3 22 2 7 3 2 3 6 4 2 3 0 6 0 0 4 3 4 - 2 . 3 2 3 3 S O 3 U 4 0 6 0 5 4 8 5 0 0 - 1 . 9 2 4 3 S O 3 0 3 1 0 9 6 0 5 4 8 6 9 3 - 2 . 1 2 5 4 0 0 3 5 7 3 3 9 0 5 4 8 6 0 1 - 1 . 9 2 6 4 0 0 3 4 3 1 2 8 9 0 5 4 3 7 9 7 - 2 . 1 2 7 4 S O 4 0 0 2 4 6 • 0 3 4 8 5 0 2 - 1 . 9 2 8 4 S O 3 9 2 6 7 5 0 5 4 6 6 9 9 - 1 . 9 2 9 4 0 0 3 4 1 1 0 8 7 0 3 4 9 7 5 0 - 4 . 4 3 0 4 0 0 3 4 1 7 0 7 0 3 4 9 6 5 0 - 4 . 4 3 1 4 0 0 3 3 8 I S 8 2 0 3 4 9 8 5 0 - 4 . 4 3 2 4 0 0 3 3 9 1 0 8 6 0 5 5 1 7 S O - 2 . 1 3 3 4 0 0 3 4 0 1 0 8 7 0 5 4 5 7 S O - 6 . 6 3 4 4 0 0 3 3 9 1 0 3 7 0 5 2 0 7 SO - 4 . 4 3 S 4 0 0 3 3 9 1 1 4 7 0 3 7 9 7 S O - 4 . 4 3 6 4 0 0 3 2 9 1 0 1 6 0 S 5 0 7 5 0 - 4 . 3 3 7 4 0 0 3 1 1 9 4 5 0 5 5 1 7 5 0 - 4 . 3 3 8 3 0 0 2 5 5 1 9 3 2 0 5 4 9 7 5 0 - 4 . 4 3 9 5 0 0 4 2 5 6 9 6 0 5 4 9 7 S O - 4 . 4 PARENT MODEL ( n r s . 1 - 2 2 ) PARENT MODEL ( n r s . 23 - 39) The p a r e n t body p l a n s f o r models 1 - 22

and 29 - 39 a r e d e p i c t e d i n F i g u r e 2. The w a t e r l i n e l e n g t h o f a l l models o f

F i g . 2: P a r e n t models f o r t h e D e l f t Sys-t e m a Sys-t i c Y a c h Sys-t H u l l S e r i e s .

A l l models were t e s t e d w i t h t h e same k e e l and r u d d e r a n d c o n s e q u e n t l y w i t h t h e u n i f o r m e x t r a p o l a t i o n t o % L = ^'^ m e t e r s t h e r e i s a d i f f e r e n c e . i n k e e l span f o r t h e S e r i e s I on t h e one hand and t h e S e r i e s I I and I I I on t h e o t h e r hand.

For S e r i e s I t h e k e e l span i s 1.37 me-t e r s a n d f o r S e r i e s I I and I I I me-t h i s i s 1.10 m e t e r s f o r t h e c o r r e s p o n d i n g wa-t e r l i n e l e n g wa-t h L^^^ = 10 m e wa-t e r s . The k e e l a n d r u d d e r l o c a t i o n i s g i v e n i n F i g u r e 3. For t h e a d d i t i o n a l k e e l span v a r i a t i o n s o f model 1 t h e f o l l o w i n g cases have been c o n s i d e r e d f o r t h e models l a , l b a n d I c r e s p e c t i v e l y : 1.25,1.45 a n d 0.69 m e t e r s . 2.2. D e t e r m i n a t i o n o f t h e h y d r o d y n a m i c r e s i s t a n c e The t o t a l h y d r o d y n a m i c r e s i s t a n c e o f a s a i l i n g y a c h t i n c a l m w a t e r may be s p l i t up i n t h r e e components: = R T + R i + R H (1) Where: Rj. u p r i g h t r e s i s t a n c e (no l e e -way) - i n d u c e d r e s i s t a n c e due t o t h e g e n e r a t i o n o f s i d e f o r c e Rjj - r e s i s t a n c e due t o h e e l (no s i d e f o r c e ) MODELS 1 - 22 O l i n . O O R D . 5 O R D . I O ORD 6h 0 . 2 2 j n

L

/ -11 ACA 0 0 1 2 - l l A C A 6 3 , A 0 1 5 MODELS 23 - 39 om J J

1

L

_{- I I A C A 6 3 , A 0 1 5} Geometry o f k e e l a n d r u d d e r v o l u m e m3 w e t t e d a r e a r o o t c h o r d m K e e l Rudder 0.00262 0.00023 0.1539 0.0550 0 . 414 0 .124 t i p c h o r d m span m t sv;eep back a n g l e degrees Keel Rudder 0.262 0 . 096 0.219 0.266 45 5.4 2 . 2 . 1 . U p r i g h t r e s i s t a n c e The u p r i g h t r e s i s t a n c e i s s p l i t up i n f r i c t i o n a l r e s i s t a n c e Rp a n d r e s i d u a r y r e s i s t a n c e Rj^. The f r i c t i o n a l r e s i s t a n c e i s c a l c u l a t e d u s i n g t h e 1957 ITTC e x t r a p o l a t o r :

Cp =

0 . 075 VL ( l o g Rji - 2 ) : (2) where t h e R e y n o l d s number f o r t h e h u l l i s b a s e d o n L = 0.7 L ^ j ^ . F o r Iceel and r u d d e r t h e mean c h o r d l e n g t h s have been u s e d . I t has been c o n s i d e r e d t o u s e t h e so c a l l e d Prohaslca f o r m f a c t o r s i n t h e ext r a p o l a ext i o n p r o c e d u r e , b u ext ext h e d i f f e r ence i n t h e f i n a l r e s u l t i s n o t s i g n i f i -c a n t . For t h e a n a l y s i s o f t h e model e x p e r i m e n t r e s u l t s t h e k i n e m a t i c v i s c o s i t y v, c o r -r e s p o n d i n g t o t h e measu-red t a n k w a t e -r t e m p e r a t u r e , has been u s e d i n a l l c a s e s . For r e s i s t a n c e p r e d i c t i o n p u r p o s e s : f o r f r e s h w a t e r a n d s e a w a t e r r e s p e c t i v e -l y a t 15 d e g r e e s C e -l s i u s may be u s e d . The w e t t e d s u r f a c e o f t h e canoe body, w i t h o u t } t e e l a n d r u d d e r c a n be a p p r o x i -mated b y : S„= [1.97-10.171 — ] * [ - ^ ^ l ^ / ^ * [ V C * L W L ] ^ (3) w i t h : 'WL * Tc * Cp F i g u r e 3 : P o s i t i o n o f k e e l and r u d d e r . L W L * The f r i c t i o n a l r e s i s t a n c e f o l l o w s f r o m : Rp = ^pV2(Sc Cpc + S)^ Cp]^ + SJ. Cpj-) (4) where t h e i n d i c e s c, k a n d r r e f e r t o r e s p e c t i v e l y t h e canoe body, t h e Iceel and t h e r u d d e r .

U s i n g a l e a s t s q u a r e s method t h e r e s i d u -a r y r e s i s t -a n c e o f -a l l t e s t e d m o d e l s i s e x p r e s s e d i n a p o l y n o m i a l e x p r e s s i o n ,

u s i n g h u l l f o r m p a r a m e t e r s as v a r i a b l e s . For t h e speed r a n g e Fn = 0.125 (0.025) 0.450 t h e p a r a m e t e r s Cp, '^^y,/'^ c ' ^ , L C B and B^L/T,-, have been used:

— * 10' = ao+a, C „ + a 2 ( L C B ) + a 3 ( B „ L / T c ) + +a,(L„L/^c''')+a5 Cp2+a, Cp* ( L „ L / ^ C ' ' ' ) + +3^ ( L C B ) 2+a8 ( L „ L / V C ' / ' ) 2+a9 ( L „ L / V C ^ ' ^ ) ' (5) For t h e speed r a n g e Fn = 0.4 75 (0.025) 0.750 t h e p o l y n o m i a l f i t i s as f o l l o w s : ^ * 10^ = C O + C , ( L W L / % L ) - ^ C z l A ^ / V c ^ / ^ ) ^ + C 3 ( L C B ) + C ^ ( L „ L / B „ L ) 2 + + C 5 ( L „ L / B W L ) * ( A W / V C ' ' ' ) ' (6) The c o e f f i c i e n t s a and c a r e g i v e n i n t h e T a b l e s 4 and 5. T a b l e 4 R e s i d u a r y r e s i s t a n c e p o l y n o m i a l c o e f f i -c i e n t s T a b l e 5 R e s i d u a r y r e s i s t a n c e p o l y n o m i a l c o e f f i -c i e n t s Fn CO cl C2 C3 c4 c5 .475 ISO .1004 -31 .50257 -7. 451141 2 195042 2 689623 0 006400 .500 243 .9994 -44 .52551 -11 .15456 2 179046 3 857403 0 009676 .535 282 .9073 -51 .51953 -12 .97310 2 274505 4 343662 0 011066 .550 313 .4109 -56 .58257 -14 . 41970 2 326117 4 690432 0 012147 .575 337 .0038 -59 .19029 -16 .06975 2 419156 4 766793 0 014147 .600 356 .4572 -62 .85395 -16 . 05112 2 437056 5 078760 0 014980 .625 324 .7357 -51 .31252 -15 .34595 2 334146 3 055360 0 013695 .650 301 .1268 -39 .79631 -15 .02299 2 059657 2 545676 0 013508 .675 292 . 0571 -31 .85303 -15 .58540 1 847926 1 569917 0 014014 .700 284 .4641 -25 14558 -16 .15423 1 703981 0 817912 0 014575 .725 256 .6367 -19 .31922 -13 .00450 2 152024 0 340305 0 011343 .750 304 .1003 -30 11512 -15 .85429 2 863173 1 524379 0 014031 2.2.2. I n d u c e d r e s i s t a n c e The i n d u c e d r e s i s t a n c e c o e f f i c i e n t f o r a l i f t i n g s u r f a c e w i t h an e f f e c t i v e a s p e c t r a t i o ARg i s g i v e n b y : Fn aO a 5 a l a6 a 2 a 7 a 3 a 8 a 4 a 9 0 . 1 2 5 - 6 . 7 3 5 6 5 4 - 3 8 . 8 6 0 B 1 + 3 8 . 3 6 0 3 1 + 0 . 9 5 6 5 9 1 - 0 . 0 0 0 1 9 3 - 0 . 0 0 2 1 7 1 + 0 . 0 5 5 2 3 4 + 0 . 2 7 2 8 9 5 - 1 . 9 9 7 2 4 2 - 0 . 0 1 7 5 1 6 0 . 1 5 0 - 0 . 3 B 2 8 7 0 - 3 9 . 5 5 0 3 2 + 3 8 . 1 7 2 9 0 + 1 . 2 1 9 5 6 3 + 0 . 0 0 7 2 4 3 + 0 . 0 0 0 0 5 2 + 0 . 0 2 6 6 4 4 + 0 . 6 2 4 5 6 8 - 5 . 2 9 5 3 3 2 - 0 . 0 4 7 8 4 2 0 . 1 7 5 - 1 . 5 0 3 5 2 6 - 3 1 . 9 1 3 7 0 + 2 4 . 4 0 0 0 3 + 2 . 2 1 6 0 9 8 + 0 . 0 1 2 2 0 0 + 0 , 0 0 0 0 7 4 + 0 . 0 6 7 2 2 1 + 0 . 2 4 4 3 4 5 - 2 , 4 4 8 5 8 2 - 0 . 0 1 5 8 8 7 0 . 2 0 0 + 1 1 . 2 9 2 1 8 - 1 1 . 4 1 8 1 9 - 1 4 . 5 1 9 4 7 + 5 . 6 5 4 0 6 5 + 0 . 0 4 7 1 0 2 + 0 . 0 0 7 0 2 1 + 0 . 0 8 5 1 7 6 - 0 . 0 9 4 9 3 4 - 2 . 6 7 3 0 1 6 + 0 . 0 0 6 3 2 5 0 . 2 2 5 + 2 2 . 1 7 8 6 7 + 7 . 1 6 7 0 4 9 - 4 9 . 1 6 7 8 4 + 8 . 6 0 0 2 7 2 + 0 , 0 8 5 9 9 8 + 0 . 0 1 2 9 8 1 + 0 , 1 5 0 7 2 5 - 0 . 3 2 7 0 8 5 - 2 . 0 7 0 6 8 4 + 0 . 0 1 8 2 7 1 0 . 2 5 0 + 2 5 . 9 0 8 6 7 + 2 4 . 1 2 1 3 7 - 7 4 . 7 5 6 6 8 + 1 0 . 4 8 5 1 6 + 0 , 1 5 3 5 2 1 + 0 . 0 2 5 3 4 8 + 0 , 1 0 8 5 6 8 - 0 , 8 5 4 9 4 0 - 0 . 8 8 9 4 6 7 + 0 . 0 4 8 4 4 9 0 . 2 7 S + 4 0 . 9 7 5 5 9 + 5 3 . 0 1 5 7 0 - 1 1 4 . 2 6 5 5 + 1 3 . 0 2 1 7 7 + 0 . 2 0 7 2 2 6 + 0 . 0 3 5 9 3 4 + 0 , 2 5 0 8 2 7 - 0 . 7 1 5 4 5 7 - 3 . 0 7 2 6 6 2 + 0 . 0 3 9 8 7 4 0 . 3 0 0 + 4 5 . 8 3 7 5 9 + 1 3 2 . 2 5 6 8 - 1 0 4 . 7 6 4 6 + 1 0 . 0 6 0 5 4 + 0 . 3 5 7 0 3 1 + 0 . 0 6 6 8 0 9 + 0 . 3 3 8 3 4 3 - 1 , 7 1 9 2 1 5 + 3 . 8 7 1 6 5 8 + 0 . 0 9 5 9 7 7 0 . 3 2 S + 8 9 . 2 0 3 8 2 + 3 3 1 . 1 1 9 7 - 3 9 3 . 0 1 2 7 + 8 . 5 9 8 1 3 6 + 0 . 6 1 7 4 6 6 + 0 . 1 0 4 0 7 3 + 0 , 4 6 0 4 7 2 - 2 , 0 1 5 2 0 3 + 1 1 . 5 4 3 2 7 + 0 . 1 5 5 9 6 0 0 . 3 5 0 + 2 1 2 . 6 7 8 0 + 6 6 7 . 6 4 4 5 - 8 0 1 . 7 9 0 8 + 1 2 . 3 9 8 1 5 + 1 . 0 0 7 3 0 7 + 0 . 1 6 6 4 7 3 + 0 . 5 3 9 9 3 8 - 3 , 0 2 6 1 3 1 + 1 0 . 8 0 2 7 3 + 0 . 1 6 5 0 5 5 0 . 3 7 5 + 3 3 6 . 2 3 5 4 + 8 3 1 . 1 4 4 5 - 1 0 8 5 . 1 3 4 + 2 6 . 1 8 3 2 1 + 1 , 6 4 4 1 9 1 + 0 . 2 3 8 7 9 5 + 0 , 5 3 2 7 0 2 - 2 . 4 5 0 4 7 0 - 1 . 2 2 4 1 7 3 + 0 . 1 3 9 1 5 4 0 . 4 0 0 + 5 6 6 , 5 4 7 6 + 1 1 5 4 . 0 9 1 - 1 6 0 9 . 6 3 2 + 5 1 . 4 6 1 7 5 + 2 . 0 1 6 0 9 0 + 0 . 2 8 8 0 4 6 + 0 . 2 6 5 7 2 2 - 0 . 1 7 8 3 5 4 - 2 9 . 2 4 4 1 2 + 0 . 0 1 8 4 4 6 0 . 4 2 5 + 7 4 3 . 4 1 0 7 + 9 3 7 , 4 0 1 4 - 1 7 0 8 . 2 6 3 + 1 1 5 . 6 0 0 6 + 2 . 4 3 5 8 0 9 + 0 . 3 6 5 0 7 1 + 0 . 0 1 3 5 5 3 + 1 . 8 3 8 9 6 7 - 8 1 . 1 6 1 8 9 - 0 . 0 6 2 0 2 3 0 . 4 5 0 + 1 2 0 0 . 6 2 0 + 1 4 8 9 , 2 6 9 - 2 7 5 1 . 7 1 5 + 1 9 6 . 3 4 0 6 + 3 . 2 0 8 5 7 7 + 0 . 5 2 8 2 2 5 + 0 . 2 5 4 9 2 0 + 1 , 3 7 9 1 0 2 - 1 3 2 . 0 4 2 4 + 0 . 0 1 3 5 7 7 c I t s h o u l d be n o t e d t h a t A,-, i s t h e w e i g h t o f d i s p l a c e m e n t o f t h e canoe body, w i t h o u t k e e l a n d r u d d e r , i s t h e c o r r e s -p o n d i n g volume o f d i s -p l a c e m e n t .

The w a t e r p l a n e a r e a A„ may be a p p r o x i -mated w i t h s u f f i c i e n t a c c u r a c y b y : D i _AR (8) E S i m i l a r l y , f o r t h e h u l l , k e e l a n d r u d d e r c o m b i n a t i o n , t h e i n d u c e d r e s i s t a n c e r e -s u l t i n g f r o m t h e g e n e r a t e d -s i d e f o r c e Fjj can be w r i t t e n a s : AR _{E qSc} (9) where ARg i s t h e e f f e c t i v e a s p e c t r a t i o o f t h e h u l l , k e e l a n d r u d d e r combina-t i o n , and g = MpVz. U s i n g combina-t h e r e s u l combina-t s o f t h e r e s i s t a n c e measurements w i t h h e e l a n g l e a n d l e e w a y , t h e i n d u c e d r e s i s t a n c e c o u l d be e x p r e s s e d b y : R i = (Co ^ v ^ ) qSc (10)

where Cg and Cj depend on t h e g e o m e t r y o f t h e h u l l , k e e l a n d r u d d e r combina-t i o n .

The e x p r e s s i o n ( 1 0 ) w o r k s w e l l f o r S e r i e s I ( n r s . 1 - 22) b u t f o r t h e S e r i e s I I and I I I ( n r s . 23 - 39 an a d d i t i o n a l t e r m w i t h t h e Froude number Fn was n e c e s s a r y t o cope w i t h a s i g n i f i c a n t f r e e s u r f a c e i n f l u e n c e on t h e , i n d u c e d r e s i s t a n c e . Thus: (Co + Cj + Cj Fn) qSc (11) For S e r i e s I a f a i r a g r e e m e n t b e t w e e n (10) a n d (11) e x i s t s f o r Fn = 0.325. W i t h (9) a n d (10) we f i n d : -•WL 'WL 1.313Cp + 0.0371 (L„L/VC^^^)-1 0.0857Cn * ( L W L / V p ^ ^ ^ ) (7) ARr n (C(, -I- C 2 cp2 (12)

We now d e f i n e an e f f e c t i v e d r a u g h t Tg w i t h : ARr T p Z = and: , t h a n : n ( C o + Cp <p2 ) ^ Tg^ q (13) : i 4 ) t h e i n f l u e n c e o f deck i m m e r s i o n . By a n a l o g y w i t h t h e IMS f o r m u l a t i o n t h e f o l l o w i n g e x p r e s s i o n i s used f o r v e l o c i -t y p r e d i c -t i o n s : R rp ~ ",po [ 1 + 0.0004(,p R,. 'P i n d e g r e e s . 3 0) 2 ] (19) T h i s r e s u l t s i n a r e s i s t a n c e i n c r e a s e o f 1% and 4% f o r r e s p e c t i v e l y .p = 35 de-g r e e s and (p = 40 d e g r e e s . W i t h t h e measured v a l u e s f o r models 1 - 2 8 and model l a , model l b and model I c t h e e f f e c t i v e d r a u g h t s Tg have been d e t e r m i n e d f o r h e e l a n g l e s 0, 10, 20 and 30 d e g r e e s . The r e l a t i v e e f f e c t i v e d r a u g h t Tg/T ap-p e a r s t o be s t r o n g l y d e ap-p e n d e n t on T^/T, % L / T C and <p . A s a t i s f a c t o r y f i t t o t h e e x p e r i m e n t a l d a t a i s g i v e n b y : T T = A, ( — ^ ) + Aj (- ' T ' T T + A,(- (15) w i t h : A, = 4.080 + 0.0370 ip - 4.9830 ip^ Aj = -4.179 - 0.8090 q, + 9.9670 (p^ Aj = 0.055 - 0.0339 <p - 0.0522 q,^ <P i n r a d i a n s . 2.2.3. R e s i s t a n c e due t o h e e l 2.3. S i d e f o r c e as a f u n c t i o n o f h e e l and l e e w a y For t h e models 1 - 22 ( S e r i e s I ) a n d model I c ( h a l f )ceel span) t h e r e l a t i o n between l e e w a y a n d s i d e f o r c e i s a p p r o x -i m a t e d b y : FTT C O S w P = — ( B o + B, ¥ , 2 ) p and <p i n r a d i a n s (20) Due t o l a r g e r B ^ L / T ^ an a d d i t i o n a l t e r m d e p e n d i n g o f t h e h e e l a n g l e a n d t h e Froude number i s n e c e s s a r y f o r t h e mo-d e l s 23 - 28 ( S e r i e s I I ) t o s a t i s f y t h e e x p e r i m e n t a l e v i d e n c e w h i c h i n d i c a t e s f r e e s u r f a c e e f f e c t s . Thus: FTT C O S <P P = ( B 0 + B 2 <p^)+Bj (p2 Fn (21)

For each o f t h e models 1 28 t h e r e s i s t a n c e due t o h e e l , Rjj, has been d e t e r -mined . I t was f o u n d t h a t a r e a s o n a b l e a p p r o x i -m a t i o n o f Rfj i s g i v e n b y : = C H F n 2 qSc (p i n r a d i a n s . (16) I f t h e c o m b i n a t i o n o f h u l l , k e e l a n d r u d d e r i s c o n s i d e r e d as a s i d e f o r c e ( l i f t ) g e n e r a t i n g e l e m e n t , t h e " l i f t " s l o p e w i l l be g i v e n b y t h e f i r s t t w o terms o f ( 2 1 ) : FJJ cos (p P qSc Bo + B^<p^ (22)

The CJJ was e x p r e s s e d i n t h e k e e l and h u l l p a r a m e t e r s T^/T a n d B^i^/T^. T B CH * 10^ = 6 . 7 4 7 ( — ^ + 2 . 5 1 7 ( — ^ + T T^ B I- 3.710(- WL ) * { — ) (17) The r e s i s t a n c e due t o h e e l and s i d e f o r c e , t h e h e e l e d r e s i s t a n c e , i s g i v e n by: + ( C H F n 2 <p)qSc (18) n TgZ q w i t h Tg and C H as shown i n (15) a n d (17) For ip > 30 d e g r e e s an e x t r a r e s i s t a n c e i n c r e a s e c a n be i n c l u d e d t o a l l o w f o r The s l o p e depends on t h e e f f e c t i v e a s -p e c t r a t i o o f t h e u n d e r w a t e r -p a r t o f t h e h u l l , k e e l and r u d d e r , v/hich i n t h i s case i s r e l a t e d t o s i d e f o r c e g e n e r a -t i o n . I t was f o u n d t h a t t h e " l i f t " s l o p e c a n be e x p r e s s e d w i t h s u f f i c i e n t a c c u r a c y by: Tc/T a n d T^/S^: •H cos P q w i t h : = b i ( — ) + b 2 ( — ) ^ + b 3 ( ^ ) + T p T 2 b4 (-^) * ( — ) (23)

<p = 0° <p = 10° (p = 20° <p = 30° b2 b3 b4 + 2.025 + 9.551 + 0.631 - 6.575 + 1.989 + 6.729 + 0.494 - 4.745 + 1.980 + 0.633 + 0.194 - 0.792 + 1.762 • - 4.957 - 0.087 + 2 . 766

The c o e f f i c i e n t Bj i n (21) has been de-t e r m i n e d w i de-t h de-t h e e x p e r i m e n de-t a l r e s u l de-t s o f models 23 - 28 ( S e r i e s I I ) : B, = 0 . 0 0 9 2 ( — ^ * ( ) ( 2 4 ) Tc Tc The c o n t r i b u t i o n o f t h e Bj i s r e l a t i v e l y s m a l l , e x c e p t i n t h e case o f v e r y l a r g e

^ W L / T C and T/Tc, s u c h as models 2 4 and 2 6 . Then t h e r e i s a c e r t a i n h e e l a n g l e a t w h i c h no s i d e f o r c e i s g e n e r a t e d , w h i c h f o l l o w s f r o m : /3 = BJ Fn. 2.4. S t a b i l i t y The d a t a r e d u c t i o n o f t h e e x p e r i m e n t a l s t a b i l i t y d a t a has b e e n c a r r i e d o u t as f o l l o w s (see F i g u r e 4 ) : GN s i n (p = GM s i n + MN s i n <p (25) where GM i s t h e c a l c u l a t e d h y d r o s t a t i c v a l u e a t V = 0.

The r e s i d u a r y s t a b i l i t y l e v e r can be ex-p r e s s e d i n : <p, Fn and Bj^^/Tc: MN s i n (p , , = (Dj * .p * Fri + Dj <p2 ) (26) L W L w i t h : Dp = -0.0406 + 0.0109 ( ^ ^ - 0 . 0 0 1 0 5 { - ^ ^ ^ 2 Tc Tc D , = + 0.0636 - 0.0196 ( — ^ T -^C <P i n r a d i a n s F i n a l l y t h e d i s t a n c e o f t h e c e n t e r o f l a t e r a l r e s i s t a n c e t o t h e w a t e r l i n e i s g i v e n by: * T (27) w i t h : Ta D , = 0.414 - 0.165 ( — ^ T A p p a r e n t l y f o r Tc/T — » • 0 D^ a p p r o a c h e s t h e v a l u e f o r an e l l i p t i c d i s t r i b u t i o n o f t h e s i d e f o r c e f r o m t h e t i p o f t h e k e e l t o t h e w a t e r l i n e . GN s i n <p = (GM + MN) s i n <p F i g . 4: D e f i n i t i o n o f r e s i d u a l s t a b i l i t y l e v e r MN s i n tp . To show t h e goodness o f f i t o f t h e v a r i -ous p o l y n o m i a l s as g i v e n f o r r e s i s t a n c e , s i d e f o r c e , and s t a b i l i t y , some r e s u l t s a r e g i v e n i n t h e F i g u r e s 5 - 8. I n F i g u r e 5 t h e measured and p r e d i c t e d u p - r i g h t r e s i s t a n c e f o r t h e models 16 and 37 (a heavy- and l i g h t - d i s p l a c e m e n t h u l l ) a r e compared. The t y p i c a l d i f f e r -ence i n c h a r a c t e r o f t h e r e s i s t a n c e c u r v e f o r speeds e x c l u d i n g Fn = 0.45 i s c l e a r l y shown. I n F i g u r e 6 t h e h e e l e d r e s i s t a n c e , p r e -d i c t e -d w i t h e q u a t i o n (18) i s compare-d w i t h t h e e x p e r i m e n t a l r e s u l t s f o r models 16 and 28, and i n F i g u r e 7 t h e g e n e r a t e d s i d e f o r c e as a f u n c t i o n o f l e e w a y and h e e l a n g l e p r e d i c t e d a c c o r d i n g t o equa-t i o n (21) i s compared w i equa-t h equa-t h e measure-ments . F i n a l l y a s i m i l a r c o m p a r i s o n has been made f o r t h e s t a b i l i t y l e v e r a t 10, 20 and 3 0 d e g r e e s as a f u n c t i o n o f t h e Froude number u s i n g e q u a t i o n s (25) and

( 2 6 ) .

The examples i n c l u d e some r a t h e r e x t r e m e h u l l f o r m s , b u t t h e p r e d i c t i o n i n a l l c o n s i d e r e d cases i s s a t i s f a c t o r y . The i m p o r t a n c e o f t h e l e n g t h - d i s p l a c e m e n t r a t i o L(^L/VC^'^ and t h e beam t o d r a u g h t r a t i o B J ^ L / T C i s c l e a r l y shown i n t h e F i g u r e s 5 - 8. I n p a r t i c u l a r t h e a t t e n t i o n i s drawn t o t h e l o s s o f s t a b i l i t y a t f o r w a r d speed f o r t h e w i d e beam models 3 1 and 33 as d e p i c t e d i n F i g u r e 8. 3. P r e d i c t i o n o f added r e s i s t a n c e i n waves To e s t i m a t e t h e added r e s i s t a n c e i n waves t h e r a d i a t e d damping e n e r g y o f t h e v e r t i c a l m o t i o n s (heave and p i t c h ) i s r e l a t e d t o t h e w o r k done by t h e e x t r a r e s i s t a n c e R/^^w as d e s c r i b e d i n [ 7 ] :

200 100 50 T 1 1 r — c a J . c i a a t l o n f l T r c n l c u l a t i o n o • m n n o i i r e n i G U t a J L O . 1 2 5 . 2 5 0 . 3 7 5 . 5 0 0 . 6 2 5 . 7 5 0 F n >— F i g . 5: Measured and p r e d i c t e d u p r i g h t r e s i s t a n c e . _ • - T - 0 » , 1 F a 1 - a . 3 0 , F n - 0.40 _ 4 - r - 1 0 ° , Fn - 0.30 _{- 2 _ f l l} a - r - 3 0 ° , Fn - 0 . 3 5 -• » - 3a<^, Fn - a. i Q - 0-530

-

-• ^^^^^"^ O 1 caJ-culaticnoa O • A D measurements — • " f - 0^, 1 Fn -1 0 . 3 0 , F n - 0 . 4 0 K l C l ! I i _ M _ A - r " 10", E n - 0.3G . B H I / T C - t . 7 S O - r - Fn - 0.3G _{^if^l^- - t . 3 3} •* - f -3 0 ° , Fn. - 0.36 Cj, - 0.546 O / • I - 1 1 . 0 2 . 0 ( F n / q S c l ' ' — * 1 0 ' F i g . 6: Measured and p r e d i c t e d h e e l e d r e s i s t a n c e . mrau irai.F_rmir,i W c * / ' Cp - O-JC» C3j.cnllatioaï3 H u r e j u e n t a d e g r e e s F i g . 7: Measured and p r e d i c t e d s i d e -f o r c e . S'S 1. 5 MODEL 3 1 1 - 1 1 1 _ G H 1 5 . 0 2 2 . 7 8 / / ^ — GH . s i n * -, F n . l 5 '—GÏI . s i n * , F n . l 5 '—GÏI . s i n * / 1 1 - . F n . 3 0 — ^ F n . 4 5 1 1 1 200 30" 4 0 ° SO" f c c n a 1 - 5 1 MODEL 33 1 1 1 B / T c = 10 07 1 92

-GH s i n 4 ( GH s i n f

-

= ^ ~ F i i 7 l 5

-' \ ^ F n . 3 0 / 1 1 F n . 4 5 1 1 0" 1 0 ° 20" 30" 40" F i g . 8: S t a b i l i t y l e v e r GN s i n tp as a f u n c t i o n o f Fn. The measured v a l u e s f o r Fn = .15, .30 a n d .45 c o i n c i d e w i t h t h e drawn l i n e s .

a c c o c d l n g f nt-mn I a O 1 . 0 2 . 0 3 . 0 4 . 0 u ( r a d / s ) accorrÜJig f o c m a l a M O D E L 2 5 • s t r i p t h e o i r y c a J . c u l a t d L o a / V ƒ X.

-/ 1 y / 1 1 1 O 1 . 0 2 . 0 3 - 0 4 . 0 u ( r a d / . . 5 )

As shown i n [ 7 ] and [ 9 ] c a l c u l a t e d added r e s i s t a n c e i n g e n e r a l a g r e e s q u i t e w e l l w i t h t h e r e s u l t s o f model r e s i s t a n c e ex-p e r i m e n t s i n r e g u l a r waves. I t i s shown t h a t t h e added r e s i s t a n c e i s p r o p o r t i o -n a l t o t h e wave h e i g h t s q u a r e d f o r co-n- con-s t a n t w a v e - l e n g t h a n d f o r w a r d con-speed. The added r e s i s t a n c e on a base o f wave-l e n g t h o r wave p e r i o d f o r a c e r t a i n y a c h t speed ( p r e f e r a b l y i n d i m e n s i o n l e s s q u a n t i t i e s ) i s t h e added wave r e s i s t a n c e r e s p o n s e o p e r a t o r . I n c o m b i n a t i o n w i t h a g i v e n wave spec-t r u m S f , w h i c h c o u l d be spec-t h e r e s u l spec-t o f a wave buoy measurement, o r a f o r m u l a t i o n based on o b s e r v e d wave p e r i o d and s i g -n i f i c a -n t wave h e i g h t , t h e mea-n added r e s i s t a n c e i n t h e c o n s i d e r e d wave spec-t r u m can be o b spec-t a i n e d b y s u p e r p o s i spec-t i o n : RAW RAW = 2 ~ * ((Je)da)e (29) o f a I n g e n e r a l t h e added r e s i s t a n c e o p e r a t o r ^Aw/fa^ o r a c o r r e s p o n d i n g d i m e n s i o n l e s s p r e s e n t a t i o n , depends on t h e h u l l geome-t r y , geome-t h e l o n g i geome-t u d i n a l p i geome-t c h g y r a d i u s , t h e wave p e r i o d o r f r e q u e n c y and t h e wave d i r e c t i o n For e i g h t models o f t h e D e l f t S y s t e m a t i c Y a c h t H u l l S e r i e s , n r s : 1 , 5, 6, 22, 25, 26, 30 and 3 1 , w h i c h c o n s t i t u t e a v e r y l a r g e r a n g e o f h u l l f o r m v a r i a t i o n s , t h e added r e s i s t a n c e r e s p o n s e o p e r a t o r s have been c a l c u l a t e d f o r wave d i r e c t i o n s / i ^ , , = 100, 115, 125 and 135 d e g r e e s , f o r w a r d speeds c o r r e s p o n d i n g t o Fn = 0.15(0.10) 0.45 and 0.60 and p i t c h g y r a d i u s k y v / ^ W L = 0.25, 0.27, and 0 . 3 1 . The c a l c u l a t i o n s c o n c e r n t h e u p r i g h t c o n d i t i o n s . Based on t h e model t e s t s i n [7] t h i s i s a r e a s o n a b l e e s t i m a t e a l s o f o r c o n d i t i o n s when h e e l e d . F i g . 9: Added r e s i s t a n c e o p e r a t o r s f o r models 1 and 25. 1 rljTATT, rTpi RAW = 7 J„ I , b'v^^ d x ^ d t (28) A U 0 where: A t b' •^e ^b wave l e n g t h t i m e c r o s s s e c t i o n a l damping c o e f -f i c i e n t , c o r r e c t e d -f o r t h e f o r w a r d speed r e l a t i v e v e r t i c a l v e l o c i t y o f t h e c o n s i d e r e d c r o s s - s e c t i o n w i t h r e s p e c t t o t h e w a t e r , p e r i o d o f wave e n c o u n t e r l e n g t h o r d i n a t e o f t h e h u l l . The v e r t i c a l r e l a t i v e m o t i o n depends on t h e v e r t i c a l m o t i o n s heave a n d p i t c h and t h e v e r t i c a l component o f t h e i n c i -d e n t wave v e l o c i t y . The c a l c u l a t i o n o f can be c a r r i e d o u t b y a s i m p l e s t r i p t h e o r y , i g n o r i n g 3 - d i m e n s i o n a l e f f e c t s .

The added r e s i s t a n c e o p e r a t o r s have been u s e d t o e s t i m a t e t h e mean added r e s i s -t a n c e i n a B r e -t s c h n e i d e r s p e c -t r u m d e f i n e d b y t h e s i g n i f i c a n t wave h e i g h t H,^3 and t h e mean wave p e r i o d T,,:

Aw'5 e x p( - B ü ) " ' ' ) (30)

w i t h : A = 173 Hi/j/T^'' and B = 6 9 1 / 5 / U s i n g e q u a t i o n (29) t h e mean added r e s i s t a n c e R;^^ has b e e n c a l c u l a t e d f o r Hi/3 = 1 m e t e r , T, = 2 ( 0 . 5 ) 6 seconds and t h e same wave d i r e c t i o n s as f o r t h e r e g u l a r wave cas?, a s s u m i n g u n i d i r e c -t i o n a l waves. I n p a r t i c u l a r t h e g y r a d i u s r a n g e i s v e r y w i d e , and p r e s u m a b l y exceeds t h e p r a c t i -c a l p o s s i b i l i t i e s . F o r s i g n i f i -c a n t wave h e i g h t s d i f f e r i n g f r o m H. 1/3 = 1 m e t e r t h e added r e s i s t a n c e has t o be m u l t i -p l i e d by t h e s q u a r e o f t h e c o n s i d e r e d wave h e i g h t . A s y s t e m a t i c a n a l y s i s o f t h e r e s u l t s o f t h i s c a l c u l a t i o n showed t h a t f o r con-s t a n t wave d i r e c t i o n , wave h e i g h t , wave p e r i o d and f o r w a r d speed t h e added

r e s i s t a n c e depends f o r t h e g r e a t e r p a r t on:

See F i g u r e 1 0 f o r = 4 seconds, Hy^ = 1 . 5 0 m e t e r s , Fn = 0 . 3 5 and % L = 1 ° m e t e r s , as an example. 1200 1000 600 400 WAVE DIRECTION ^ = 1 0 0 d G g r . "f- = 1 1 5 degr. ^ = 1 2 5 degr. • • • - 135 dcqr. sPBCTsnz-i: r , - s e c , K, ^ - i . s o M. FTT - . 3 5 , Lyj^ - 10 H . WAVE DIRECTION ^ = 1 0 0 d G g r . "f- = 1 1 5 degr. ^ = 1 2 5 degr. • • • - 135 dcqr. WAVE DIRECTION ^ = 1 0 0 d G g r . "f- = 1 1 5 degr. ^ = 1 2 5 degr. • • • - 135 dcqr. / m / m

/

- / é / » •y /

/r

/'

/ . . . • -• - • _{' : •} •

/

i • •1- :• 4 *t-, 0 1 2 3 4 S 6 7 8 F i g . 10: Added r e s i s t a n c e . The d a t a may be u s e d t o a n a l y s e t h e i n -f l u e n c e o -f t h e p i t c h g y r a d i u s on t h e added r e s i s t a n c e . A l s o , as an example, t h i s i s shown i n F i g u r e 11a where R^^^ i s p l o t t e d on a base o f T, f o r model no. 1 w i t h = 1 0 m e t e r s and g y r a d i i k v v / L w r = 0.23, 0.27 and 0.31. yy VMU A s i m i l a r p i c t u r e i s g i v e n i n F i g u r e l i b f o r model no 26, t h e p a r e n t model o f t h e l i g h t - d i s p l a c e m e n t m o d e l s . The t o t a l r e s u l t o f t h i s added r e s i s -t a n c e c a l c u l a -t i o n i n d i m e n s i o n l e s s f o r m can be summarized by:

RAW * 1 0 ^ / ' 9 % L H I / 3 ^ 1 0 2 V„ l/ 3 k -•WL --WL (31) where a and b a r e g i v e n i n T a b l e 7. The goodness o f f i t i s shown i n F i g u r e 12a, b f o r T, = 2.476 and 4 seconds and Fn = 0.3. T h i s s e t o f d a t a have b e e n u s e d i n t h e D e l f t V e l o c i t y P r e d i c t i o n Program t o e s t i m a t e t h e y a c h t speed i n a g i v e n wave c o n d i t i o n . The e x t r a i n p u t i s no more t h a n t h e s i g n i f i _ c a n t wave h e i g h t H^^j, t h e wave p e r i o d T, and t h e p i t c h g y r a -d i u s o f t h e c o n s i -d e r e -d y a c h t . t h e 457 MODEL 1 F n - 0 . 3 5 ( i ^ • 1 3 5 d e g x e e g = 10 m e t e r a ~ 1 \ kyy/Lm. - 0 . 3 1 k y y / L ^ - 0 . 2 7 .

Figure 11a: Added r e s i s t a n c e versus mean wave p e r i o d Model Nö.1. HODEL 26 — F n - 0 . 3 5 /iv = 1 3 5 d e g r e e s L ^ t . I D m e t e r n

T

T

fcyy/Lv,L . 0 . 3 1 I C y y / l W . - 0 . 2 7 • t / y / I w , " 0 . 2 3 T |

Figure l i b : Added, r e s i s t a n c e versus mean wave p e r i o d Model No.26.

I t s h o u l d be n o t e d t h a t t h e a n a l y s i s i s r e s t r i c t e d t o wave d i r e c t i o n s f o r w a r d o f t h e beam. For waves a f t o f t h e beam t h e c a l c u l a t i o n o f t h e e x t r a r e s i s t a n c e i n a g i v e n wave s p e c t r u m a c c o r d i n g t o t h e s t r i p t h e o r y i s n o t r e l i a b l e . However, i n g e n e r a l , t h e added r e s i s t a n c e i n t h i s r e g i o n i s r e l a t i v e l y s m a l l . A l s o , e f -f e c t s o -f s u r -f i n g a r e n o t i n c l u d e d .

T a b l e 7 Added r e s i s t a n c e c o e f f i c i e n t s a and b (see e q u a t i o n ( 3 1 ) ) Fn a b .15 1 .981 135 0 .649 0 .440 .15 1 .981 125 0 .352 0 .593 .15 1 .981 115 0 .127 0 .841 .15 1 .981 100 0 .0011 2 .455 .15 2 .476 135 0 .209 0 .904 .15 2 .476 125 0 .114 1 .027 .15 2 .476 115 0 . 040 1 .247 .15 2 .476 100 0 .00032 2 .863 .15 2 .971 135 0 . 089 1 .124 .15 2 .971 125 0 .048 1 .229 .15 2 .971 115 0 .017 1 .437 .15 2 .971 100 0 .00014 3 .005 .15 3 .467 135 0 .045 1 .232 .15 3 .467 125 0 . 024 1 .329 .15 3 .467 115 0 .008 1 .532. .15 3 .467 100 0 . 00004 3 .391 .15 3 .962 135 0 . 026 1 .293 .15 3 .9 62 125 0 .014 1 .385 .15 3 .962 115 0 .005 1 .590 .15 3 .962 100 0 .00003 3 .259 .15 4 .457 135 0 .016 1 .329 .15 4 .457 125 0 .008 1 .417 .15 4 .457 115 0 .003 1 .601 .15 4 .457 100 0 .00004 2 .863 .15 4 .952 13 5 0 . 010 1 .345 .15 4 .952 125 0 .005 1 .428 .15 4 .952 115 0 .0019 1 .617 .15 4 .952 100 0 .00004 2 .618 .15 5 .447 13 5 0 .007 1 .364 .15 5 .447 125 0 .004 1 .445 .15 5 .447 115 0 .0012 1 .650 .15 5 .447 100 0 .00004 2 .414 .15 5 .943 135 0 .005 1 .371 .15 5 .943 125 0 .003 1 .441 .15 5 .943 115 0 .0009 1 .643 .15 5 .943 100 0 .00005 ' 2 .220 Fn _{a b} .25 1 . 981 135 0. .669 0. ,416 .25 1 . 981 125 0 . .380 0, .583 .25 1. 981 115 0 . ,148 0 . 832 .25 1 . 981 100 0 . 0021 2 . .195 .25 2 . 476 13 5 0 . 219 0 . 9 63 .25 2 . 476 125 0 . 124 1. 078 .25 2 . 476 115 0 . 048 1. 275 .25 2 . 476 100 0 . 0007 2 . 565 .25 2 . 971 135 0 . 095 1. 221 .25 2 . 971 125 0 . 053 1 . 308 .25 2 . 971 115 0 . 020 1 . 478 .25 2 . 971 100 0 . 00025 2. 789 .25 3 . 467 135 0 . 048 1 . 349 .25 3 .467 125 0 . 027 1 .424 .25 3 .467 115 0 .010 1 .580 .25 3 .467 100 0 .00011 2 .949 .25 3 .962 135 0 . 028 1 .418 .25 3 .962 125 0 . 015 1 .483 .25 3 .962 115 0 . 006 1 .636 .25 3 .962 100 0 .00006 3 .053 .25 4 .457 135 0 . 017 1 .456 .25 4 .457 125 0 .009 1 .519 .25 4 .457 115 0 . 004 1 .663 .25 4 .457 100 0 .00005 2 .905 .25 4 .952 135 0 . O i l 1 .482 .25 4 .952 125 0 .006 1 .540 .25 4 .952 115 0 . 0023 1 .686 .25 4 .952 100 0 . 00005 2 .675 .25 5 .447 135 0 .007 1 .501 .25 5 .447 125 0 .004 1 .548 .25 5 .447 115 0 .0015 1 .707 .25 5 .447 100 0 .00005 2 .397 .25 5 .943 13 5 0 .005 1 .516 .25 5 .943 125 0 . 003 1 .564 .25 5 .943 115 0 . 0011 1 .692 .25 5 .943 100 0 .0010 1 .000 Fn _Mw a : D .35 .981 135 0 . 843 0 .241 .35 1 .981 125 0 .487 0 .435 .35 1 .981 115 0 . 195 0 .710 .35 1 .981 100 0 . 004 1 .929 .35 2 .476 13 5 0 .283 0 .856 .35 2 .476 125 0 .162 0 .984 .35 2 . .476 115 0 .065 1 .182 .35 2 . .476 100 0 .0012 2 .283 .35 2 ,971 135 0 .014 1 .146 .35 2 , .971 125 0 . 071 1 .235 .35 2 . .971 115 0 . 028 1 .396 .35 2 , ,971 100 0 .0005 2 .445 .35 3 , .467 13 5 0 .064 1 .289 .35 3 . ,467 125 0 .036 1 .362 .35 3 , .467 115 0 . 014 1 .504 .35 3 . ,467 100 0 .00027 2 .519 .35 3 . .962 135 0 . 037 1 .367 .35 3 . .9 62 125 0 . 021 1 .43 0 .35 3 , ,9 62 115 0 . 008 1 .558 .35 3 . ,962 100 0 .00016 2 .548 .35 4 . ,457 135 0 . 023 1 .411 .35 4 . ,457 125 0 . 013 1 .465 .35 4. ,457 115 0 . 005 1 .586 .35 4 . ,457 10 0 0 .00008 2 . 691 .35 4 , ,952 135 0 . 015 1 .439 .35 4 . .952 125 0 . 008 1 .493 .35 4 . ,952 115 0 .003 1 .599 .35 4 . ,952 100 0 . 00005 2 .557 .35 5 . .447 135 0 . 010 1 .448 .35 5 . .447 125 0 , . 006 1 .513 .35 5 . 447 115 0 , . 0022 1 . 637 .35 5 . 447 100 0 ,00007 2 .293 .35 5 . 943 135 0 , ,007 1 .462 .35 5 . 943 125 0 , , 004 1 .513

.35 5 .943 115 0 . . 0015 1. .631 .35 5 .943 100 0 .00006 2. .228 Fn _(HT a b .45 1 .981 135 1 .117 0 .004 .45 1 .981 125 0 . 647 0 .232 .45 1 .981 115 0 .262 0 .543 .45 1 .981 100 0 .006 1 .657 .45 2 .476 13 5 0 .376 0 .693 .45 2 .476 125 0 .217 0 . 840 .45 2 .476 115 0 . 088 1 .050 .45 2 .476 100 0 .0022 2 .005 .45 2 .971 135 0 .166 1 .016 .45 2 .971 125 0 .096 1 .115 .45 2 .971 115 0 .039 1 .272 .45 2 .971 100 0 .0009 2 .173 .45 3 .467 13 5 0 .086 1 .172 .45 3 .467 125 0 .050 1 .247 .45 3 .467 115 0 .020 1 .382 .45 3 .467 100 0 .0005 2 .232 .45 3 .962 135 0 . 050 1 .254 .45 3 .962 125 0 . 029 1 .317 .45 3 .962 115 0 . 012 1 .436 .45 3 .962 100 0 .00029 2 .247 .45 4 .457 135 0 .031 1 .302 .45 4 .457 125 0 .018 1 .356 .45 4 .457 115 0 .007 1 .476 .45 4 .457 100 0 .00020 2 .196 .45 4 .952 135 0 .020 1 .327 .45 4 .952 125 0 .011 1 .382 .45 4 .952 115 0 .0046 . 1 .494 .45 4 .952 100 0 .00008 2 .49 8 .45 5 .447 13 5 0 . 014 1 .347 .45 5 .447 125 0 .008 1 .399 .45 5 .447 115 0 .003 1 .508 .45 5 .447 100 0 .00008 2 .318 .45 5 .943 13 5 0 .010 1 .358 .45 5 .943 125 0 .006 1 .405 . 45 5 .943 115 0 .0022 1 .514 .45 5 .943 100 0 .00011 1 .896 Fn _Mw a b .60 1. 981 135 1 . 677 -0 . 386 . 60 1. 981 125 0 . 975 -0 . 109 .60 1 . 981 115 0 . 394 0 . 263 .60 1 . 981 100 0 . 013 1 . 294 .60 2 . 476 135 0. 552 0 . 421 .60 2 . 476 125 0 . 315 0 . 603 .60 2 . 476 115 0 . 129 0 . 842 . 60 2. 476 100 0 . 004 1 . 641 .60 2 . 971 13 5 0 . 235 0 . 811 .60 2 . 971 125 0 . 137 0 . 925 . 60 2 . 971 115 0 . 057 1 . 087 .60 2 . 971 100 0 . 0020 1 . 790 .60 3 . 467 135 0 . 122 0 . 995 . 60 3 . 467 125 0 . 071 1 . 076 .60 3 . 467 115 0 . 030 1 . 201 60 3 .467 100 0 , .0010 1. ,871 60 3 .962 135 0 . 071 1. .087 60 3 .962 125 0 . .041 1. ,148 60 3 .962 115 0 .0017 1, .258 60 3 .962 100 0 .00056 1. .929 60 4 .457 135 0 .044 1, .134 60 4 .457 125 0 . 026 1. ,189 60 4 .457 115 0 .0107 1, .286 60 4 . 457 100 0 .00034 1. . 8 9 1 60 4 .952 135 0 . 029 1. .163 60 4 .952 125 0 .017 1 .210 60 4 .952 115 0 .007 1 .308 60 4 .952 100 0 .00027 1 . 837 60 5 .447 135 0 . 020 1 .176 60 5 .447 125 0 .012 1 .222 60 5 .447 115 0 .005 1, .323 60 5 .447 100 0 .00013 2. .046 60 5 .943 135 0 . 014 1, ,187 60 5 .943 125 0 .008 1. ,238 60 5 .943 115 0 .0034 1. ,315 60 5 .943 100 0 .00011 1. .921 O I

F i g . 12a: Mean added wave r e s i s t a n c e f o r T / g / % L = 2.475 and Fn = 0.3 5

When measured wave s p e c t r a a r e a v a i l a b l e a d i f f e r e n t a p p r o a c h t o t h e e s t i m a t i o n o f t h e added r e s i s t a n c e i n waves i s r e q u i r e d . I n t h i s case t h e added r e s i s t a n c e r e -sponse o p e r a t o r o f t h e c o n s i d e r e d y a c h t has t o be known.

1 ,5 V/AVE oinecnoN — / / = 1 0 0 ° -s-/7=1 15° -*-/J=125° — ^ = 1 3 5 ° 7 * 1 0 - 2 WL WL

F i g . 12b: Mean added wave r e s i s t a n c e f o r T/g/LwL = 4.457 and Fn = 0.35 W i t h t h e c o m p u t a t i o n a l c a p a c i t y o f t o day's p e r s o n a l c o m p u t e r s t h e added r e -s i -s t a n c e r e -s p o n -s e o p e r a t o r o f a y a c h t can be e a s i l y d e t e r m i n e d when t h e l i n e s -p l a n and t h e l o n g i t u d i n a l d i s t r i b u t i o n o f mass a r e g i v e n as shown i n [7] and [ 9 ] . The mean added . r e s i s t a n c e t h e n f o l l o w s f r o m e q u a t i o n ( 2 9 ) . A n o t h e r a p p r o a c h f o r t h i s case, by Reumer [ 3 ] , uses an a p p r o x i m a t i o n o f t h e added r e s i s t a n c e r e s p o n s e o p e r a t o r . For a l l 39 models o f t h e D e l f t S y s t e m a t i c S e r i e s t h e o p e r a t o r has b e e n c a l c u l a t e d f o r a range o f Froude numbers, wave f r e q u e n c i e s , wave d i r e c t i o n s and p i t c h g y r a d i i . U s i n g a l e a s t s q u a r e s p r o c e d u r e t h e r e -s u l t i n g added r e -s i -s t a n c e o p e r a t o r -s c o u l d be e x p r e s s e d i n one p o l y n o m i a l e x p r e s -s i o n : R _AW a,(L„L/Vc^'^) + aj(L„L/Vc^^')^ + + a3(L„L/Vcl/3)3 + a,(%L/BwL) + a 5( % L / B „ L ) 2 + a^(B„L/Tc) + <-p (32) •f a^ Cp + aa Cp^ + The c o e f f i c i e n t s aj^ to ag a r e a f u n c t i o n o f t h e wave d i r e c t i o n , wave f r e q u e n c y and t h e Froude number.

I n F i g . 9 t h e r e s u l t o f (32) i s compared w i t h a d i r e c t c o m p u t a t i o n f o r t h e models 1 and 25, a s s u m i n g a w a t e r l i n e l e n g t h ^WL = 1° m e t e r s , = 165 d e g r e e s (15 d e g r e e s o f f t h e bow) and Fn = 0.25.

The methods d e s c r i b e d above may be used t o a n a l y s e t h e r e l a t i v e i m p o r t a n c e o f t h e mean added r e s i s t a n c e o f a s a i l i n g y a c h t i n _ a seaway by i n c l u d i n g t h e c a l -c u l a t e d Rp^i^ i n a v e l o -c i t y p r e d i -c t i o n . 4. R e f e r e n c e s [1] G e r r i t s m a , J . and J.A. K e u n i n g , P e r f o r m a n c e o f l i g h t - and heavy d i s p l a c e m e n t s a i l i n g y a c h t s i n waves. The Second Tampa Bay S a i l i n g Y a c h t Symposium, S t . P e t e r s -b u r g , F l o r i d a 1988. [2] Monhaupt, A.,ITC, C o m p a r a t i v e s t u d y o f d i f f e r e n t p o l y n o m i a l f o r m u l a t i o n s f o r t h e r e s i d u a r y r e s i s -t a n c e o f -t h e S y s -t e m a -t i c D e l f -t S e r i e s model 1 t o 28.

[3] Reumer, J.G., Een o n t w e r p v o o r een e e n v o u d i g e p o l y n o o m b e n a d e r i n g v a n de toegevoegde w e e r s t a n d v a n z e i l j a c h t e n i n g o l v e n . T e c h n i s c h e U n i v e r s i t e i t D e l f t A f s t u d e e r w e r J t , R a p p o r t n r . 874-S, 1991. [4] G e r r i t s m a , J . and G. Moeyes, The s e a k e e p i n g p e r f o r m a n c e and s t e e r i n g p r o p e r t i e s o f s a i l i n g y a c h t s , 3 r d HISWA Symposium 1973, Amsterdam. [5] G e r r i t s m a , J . , G. Moeyes and R. O n n i n k , T e s t r e s u l t s o f a s y s -t e m a -t i c y a c h -t h u l l s e r i e s , 5 -t h HISWA Symposium,1977, TUnsterdam. [6] G e r r i t s m a , J. , R.Onnink and

A. V e r s l u i s , Geometry, r e s i s t a n c e and s t a b i l i t y o f t h e D e l f t S y s t e -m a t i c Y a c h t H u l l S e r i e s ,

7 t h HISWA Symp., 1981, Amsterdam. [7] G e r r i t s m a , J. and J.A. K e u n i n g ,

P e r f o r m a n c e o f l i g h t - and heavy d i s p l a c e m e n t s a i l i n g y a c h t s i n waves, 2nd Tampa Bay S a i l i n g Y a c h t Symposium, 1988, S t . P e t e r s -b u r g , F l o r i d a . [8] G e r r i t s m a , J . , J.A. K e u n i n g and R. O n n i n k , The D e l f t S y s t e m a t i c Y a c h t H u l l S e r i e s I I e x p e r i m e n t s , 10 t h Chesapealce S a i l i n g Y a c h t Symposium, 1991, A n n a p o l i s . [9] G e r r i t s m a , J . and W. Beukelman, 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 waves o f a f a s t -c a r g o s h i p . 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 . 1 9 , Nr. 217, 1972.