Slamming simulation and penetrating wedges at forward speed

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Slamming on f o r c e d o s c i l l a t i n g wedges a t f o r w a r d s p e e d . P a r t I : t e s t r e s u l t s W. Beukelman R e p o r t No.888 May 1991

Delft University of Technology

Ship Hydromechanics Laboratory Mekelvi(eg 2

2628 CD Delft The Netherlands Phone 015 - 78 68 82

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C o n t e n t s A b s t r a c t 1 N o t a t i o n 1 1. I n t r o d u c t i o n 2 2. T e s t d e s c r i p t i o n 3 2.1. Wedges 3 2.2. I n s t r u m e n t a t i o n 5 2.3. T e s t program 6 3. R e s u l t s and a n a l y s i s 7 3.1. I n f l u e n c e o f t r i m a n g l e a 7 3.2. I n f l u e n c e o f d e a d r i s e a n g l e j3 7 3.3. I n f l u e n c e o f v e r t i c a l s p e e d V 8 3.4. I n f l u e n c e o f f o r w a r d s p e e d U 8 3.5. I n f l u e n c e o f t r a n s v e r s e p o s i t i o n 8 3.6. R e v i e w o f t h e r e s u l t s 8 4. C o m p a r i s o n w i t h c a l c u l a t i o n methods 9 4.1. Bottom i m p a c t p r e s s u r e s 9 4.2. F l u i d momentum e x c h a n g e 10 4.3. 3-D Wedge i m p a c t 11 5. C o n c l u s i o n s a n d r e c o m m a n d a t i o n s 12 6. Ackowledgement 12 7. R e f e r e n c e s 13 T a b l e s F i g u r e s

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A b s t r a c t As f o l l o w - u p o f f o r m e r r e s e a r c h f o r c e d v e r t i c a l o s c i l l a t i o n e x p e r i m e n t s h a v e b e e n p e r f o r m e d t o d e t e r m i n e p e a k p r e s s u r e s and r i s e t i m e s f o r f o u r m e t a l wedges w i t h d i f f e r e n t d e a d r i s e a n g l e s . T h e s e p r e s s u r e s h a v e b e e n m e a s u r e d a s f u n c t i o n o f v e r t i c a l o s c i l l a t i o n s p e e d , t r i m a n g l e and f o r w a r d s p e e d . The r e s u l t s i n g e n e r a l show t h a t t h e p e a k v a l u e s o f t h e i m p a c t p r e s s u r e s a r e p r o p o r t i o n a l t o t h e s q u a r e d v a l u e o f t h e v e r t i c a l s p e e d , however, a l s o t h e i n f l u e n c e o f f o r w a r d s p e e d a p p e a r e d t o be s i g n i f i c a n t . I n most o f t h e e x i s t i n g mathema-t i c a l m o d e l s mathema-t h i s r e l a mathema-t i o n h a s u n s u f f i c i e n mathema-t l y a c c o u n mathema-t e d f o r a s shown h e r e . A p p l i c a t i o n o f t h e p r e d i c t i o n m o d e l s i n t h i s r e s p e c t i s u r g e n t l y r e q u i r e d . N o t a t i o n B beam D d i m e n s i o n a l e.g. 3-D = t h r e e d i m e n s i o n a l H d e p t h k p r o p o r t i o n a l i t y c o n s t a n t L l e n g t h m' s e c t i o n a l added mass N' s e c t i o n a l damping p p r e s s u r e s v e r t i c a l d i s p l a c e m e n t (upwards p o s i t i v e ) t t i m e U f o r w a r d s p e e d V v e r t i c a l s p e e d (upwards p o s i t i v e ) y ^ h a l f w i d t h o f t h e submerged c r o s s s e c t i o n on t h e w a t e r -l i n e a t r i m a n g l e (bow up p o s i t i v e ) /3 d e a d r i s e a n g l e p mass d e n s i t y o f f l u i d w c i r c u l a r f r e q u e n c y o f o s c i l l a t i o n

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1. I n t r o d u c t i o n

I t i s w e l l known t h a t f o r n o r m a l s h i p t y p e s v o l u n t a r y s p e e d r e d u c t i o n i s i n g e n e r a l r e q u i r e d t o a v o i d s l a m m i n g .

F o r modern h i g h s p e e d c r a f t , however, s l a m m i n g i s becoming a n i m p o r t a n t p r o b l e m . I n t h i s r e s e a r c h t h e a t t e n t i o n t h e r e f o r e w i l l b e f o c u s s e d a t h i g h f o r w a r d s p e e d s i n r e l a t i o n t o s l a m m i n g . U s u a l c a l c u l a t i o n methods r e l a t e d t o l o w e r s h i p s p e e d s a r e m e n t i o n e d i n [ 1 ] and w i l l be c o n s i d e r e d i n s e c t i o n 4. I n t h e p a s t s l a m m i n g h a s e x p e r i m e n t a l l y b e e n i n v e s t i g a t e d a t t h e 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 o f t h e D e l f t U n i v e r s i t y i n t h e N e t h e r l a n d s by f o r c e d o s c i l l a t i o n o f a s e g m e n t e d s h i p model [ 1 ] . F o r p r e d i c t i o n a c a l c u l a t i o n model b a s e d on s t r i p - t h e o r y and f l u i d momentum e x c h a n g e h a s b e e n d e v e l o p e d s h o w i n g t h a t t h e p e a k p r e s s u r e s a r e p r o p o r t i o n a l l y r e l a t e d t o t h e s q u a r e d v e r t i c a l v e l o c i t y . I n c a s e o f u s u a l m o d e r a t e s h i p s p e e d s t h e a g r e e m e n t b e t w e e n m e a s u r e m e n t s and p r e d i c t i o n s a p p e a r e d t o be r a t h e r s a t i s f a c t o r y . I t s h o u l d be remembered t h a t t h e f o r w a r d s p e e d i s o n l y a c c o u n t e d f o r i n c a s e o f a t r i m a n g l e a s t h e a r i s e o f a v e r t i c a l component o f t h e f o r w a r d v e l o c i t y . The n e x t r e s e a r c h w i t h a f o r c e d o s c i l l a t e d r e c t a n g u l a r and t r i a n g u l a r c y l i n d e r a t h i g h f o r w a r d s p e e d s [2 ( i n D u t c h ) ] showed l e s s r e l i a b l e r e s u l t s .

T h i s m i g h t h a v e b e e n due t o t h e i n f l u e n c e o f a bow wave i n t h e v i c i n i t y o f t h e p r e s s u r e t r a n s d u c e r s , b u t a l s o due t o t h e e l a s t i c i t y o f t h e p o l y e s t e r m a t e r i a l o r / a n d a r e m a i n i n g d e f o r m a t i o n o f t h e bottom r e s u l t i n g i n phenomena a s a i r i n c l u s i o n .

F o r t h i s r e a s o n i t was d e c i d e d t o c o n s t r u c t f o u r s t i f f m e t a l wedges w i t h dead r i s e a n g l e s o f 0, 0.5, 1.0 and 10 d e g r e e s . From p r e c e e d i n g c a l c u l a t i o n s and t e s t s i t v i z . a p p e a r e d t h a t h i g h p e a k p r e s s u r e s o n l y o c c u r w i t h i n s m a l l d e a d r i s e a n g l e s up t o 1 o r 2 d e g r e e . F o r e a c h wedge f i v e p r e s s u r e t r a n s d u c e r s w e r e p l a c e d on a t r a n s v e r s e l i n e n e a r t h e r e a r s i d e o f t h e wedge. W i t h t h o s e wedges o s c i l l a t i n g e x p e r i m e n t s h a v e b e e n c a r r i e d o u t t o d e t e r m i n e maximum s l a m m i n g p r e s s u r e s a n d r i s e t i m e s d e p e n d e n t on v e r t i c a l o s c i l l a t i o n s p e e d , h i g h f o r w a r d s p e e d , t r i m a n g l e (bow up) and d e a d r i s e a n g l e .

The t e s t r e s u l t s i n g e n e r a l showed f o r t h e p e a k v a l u e s o f t h e s l a m m i n g p r e s s u r e s p r o p o r t i o n a l i t y t o t h e s q u a r e d v a l u e o f t h e v e r t i c a l s p e e d and a l s o a s t r o n g i n f l u e n c e o f f o r w a r d s p e e d . T h i s l a s t r e l a t i o n was n o t a c c o u n t e d f o r i n most o f t h e e x i s t i n g c a l c u l a t i o n m o d e l s . I n p a r t 2 a n e x t e n s i o n and improvement o f t h e c a l c u l a t i o n model a s d e s c r i b e d i n [ l ] w i l l be p r e s e n t e d e s p e c i a l l y w i t h r e s p e c t t o t h e i n f l u e n c e o f f o r w a r d s p e e d and 3-D e f f e c t s . M o r e o v e r i t a p p e a r e d from t h e t e s t r e s u l t s t h a t t h e peak p r e s s u r e s i n c r e a s e w i t h t r i m a n g l e , w h i l e t h e r i s e t i m e s d e c r e a s e w i t h t r i m a n g l e , v e r t i c a l a n d f o r w a r d v e l o c i t y . T h e r e i s a l s o o b s e r v e d a s l i g h t r e d u c t i o n o f t h e p e a k p r e s s u r e from t h e c e n t r e l i n e o f t h e wedge t o t h e edge and a

s t r o n g r e d u c t i o n f o r t h e wedge w i t h h i g h e s t d e a d r i s e a n g l e .

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-2. T e s t d e s c r i p t i o n As s a i d b e f o r e i n t h e i n t r o d u c t i o n t h e l a s t r e s e a r c h [ 2 ] r e l a t e d t o s l a m m i n g showed r e s u l t s w i t h a s t r o n g d i s p e r s i o n . T h i s r e s e a r c h h a s b e e n c a r r i e d o u t w i t h a t r i a n g u l a r and r e c t a n g u l a r c y l i n d e r m a n u f a c t u r e d from 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 . The p r e s s u r e t r a n s d u c e r s w e r e p l a c e d a t t h e f o r w a r d p a r t o f t h e model w h i l e t r i m a n g l e s w e r e a d j u s t e d w i t h t h e bow up. From t h e o b s e r v a t i o n s i t a p p e a r e d t h a t a s t r o n g bow wave was a l w a y s p r e s e n t i n v i c i n i t y o f t h e p r e s s u r e t r a n s d u c e r s s o t h a t t h e a c t u a l t r i m a n g l e i n most c a s e s was n o t a c c o r d i n g t o t h e r e q u i r e d n o m i n a l t r i m a n g l e i f one l o o k s a t t h e p o s i t i o n o f t h e w a t e r - l e v e l . M o r e o v e r i t a p p e a r e d t h a t t h e low e l a s t i c i t y o f t h e p o l y e s t e r m a t e r i a l n o t o n l y c a u s e d an a c t u a l , b u t a l s o a p e r m a n e n t d e f o r m a t i o n r e s u l t i n g i n a n o t d e s i r e d a i r i n c l u s i o n . To a v o i d a l l t h e s e o b j e c t i o n s and t o c o n t i n u e t h e e x p e r i m e n -t a l and a n a l y -t i c a l s l a m m i n g r e s e a r c h s -t i f f m e -t a l wedges w e r e c o n s t r u c t e d w i t h t h e p r e s s u r e t r a n s d u c e r s a s a f t a s p o s s i b l e . C a r r y i n g o u t a h e a v i n g m o t i o n i n c l u d i n g a t r i m a n g l e w i t h t h e bow up r e s u l t e d i n a d i r e c t and f i r s t c o n t a c t o f t h a t r e a r p a r t o f t h e wedge i n w h i c h t h e p r e s s u r e t r a n s d u c e r s w e r e mounted, s o t h a t no p r e s s u r e wave was p r e s e n t a t t h e moment o f i m p a c t . 2.1. Wedges F o u r wedges w i t h s m a l l d e a d r i s e a n g l e s w e r e c o n s t r u c t e d a s shown i n f i g u r e 1 and t a b l e 1. The d i m e n s i o n s a r e : L x B x H = 0.50 x 0.25 x 0.25m F i g u r e 1. Wedges c o n s i d e r e d 3

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-T a b l e 1

Wedge n r . Dead r i s e p H a l f beam (|B) mm Wedge n r . mm d e g r e e H a l f beam (|B) mm 1 0 0 125 2 1 0.46 125 3 2.5 1.15 125 4 22 . 0 9.98 125 From f o r m e r r e s e a r c h [ 2 ] i t a p p e a r e d t h a t h i g h p e a k p r e s s u r e s o n l y a r i s e f o r s m a l l d e a d r i s e a n g l e s , up t o 1 o r 2 d e g r e e . F o r t h i s r e a s o n t h e wedges ( e x c e p t n r . 4 ) h a v e s u c h r e s t r i c t e d dead r i s e a n g l e s . Wedge n r . 4 w i t h a d e a d r i s e a n g l e o f 10 d e g r e e s was c h o s e n t o d e m o n s t r a t e t h e f a l l i n t h e p e a k p r e s s u r e s . F i v e p r e s s u r e t r a n s d u c e r s w e r e p l a c e d on a t r a n s v e r s e l i n e 10 mm from t h e a f t edge o f t h e wedge a s d e n o t e d i n f i g u r e 2 s h o w i n g t h e wedge's bottom.

F i g u r e 2

T r a n s d u c e r s n r . 1 and 3 h a v e t h e same d i s t a n c e o f 40 mm t o t h e c e n t r e o f t h e wedges and a r e s u p p o s e d t o show e q u a l p e a k p r e s s u r e s . The p r e s s u r e t r a n s d u c e r s i n a t r a n s v e r s e row s h o u l d show t h e p e a k p r e s s u r e s a s f u n c t i o n o f t h e d i s t a n c e from t h e c e n t r e l i n e .

I t s h o u l d be n o t i c e d t h a t a l t h o u g h t h e s i t u a t i o n o f t h e wedges i s meant t o a p p r o a c h a 2-D c o n d i t i o n 3-D e f f e c t s may p l a y a n i m p o r t a n t r o l e .

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-2.2. I n s t m m e n t a t i o n The wedges w e r e i n s t r u m e n t e d w i t h f i v e m i n i a t u r e s e m i -c o n d u -c t o r p r e s s u r e t r a n s d u -c e r s . T h e s e w e r e p o s i t i o n e d a s i n d i c a t e d i n f i g u r e 2. The m a i n c h a r a c t e r i s t i c s o f t h e s e t r a n s d u c e r s w e r e a s f o l l o w s : - M a n u f a c t u r e : - T y p e : - Range : - A c c e l e r a t i o n s e n s i t i v i t y : - T e m p e r a t u r e d r i f t : - N a t u r a l f r e q u e n c y i n a i r : - D i a m e t e r o f membrane: D r u c k T t d . PDCR42 70kPa ( l O p s i ) .002% o f f u l l s c a l e o u t p u t / g .02% o f FSO/°C >70kHz ~5mm. To r e d u c e t h e r m a l s h o c k a t t h e i n s t a n t s t h a t t h e t r a n s d u c e r t o u c h e d t h e w a t e r , a v e r y t h i n r u b b e r c o a t i n g was a p p l i e d t o t h e membrane. Due t o t h i s c o a t i n g and t h e mass o f t h e w a t e r t h e n a t u r a l f r e q u e n c y was r e d u c e d t o a l o w e r v a l u e . I t was d i f f i c u l t t o a s s e s s t h e amount o f t h i s r e d u c t i o n b u t j u d g i n g from m a n u f a c t u r e r s d a t a and some e x p e r i m e n t s i t a p p e a r e d

p l a u s i b l e t h a t t h e b a n d w i d t h was s t i l l h i g h e r t h e n 35kHz. A s s u m i n g i n a d d i t i o n t h a t t h e - n o n - s p e c i f i e d - r e l a t i v e damping f a c t o r was l o w e r t h e n 1 ( t y p i c a l v a l u e s f o r p r e s s u r e t r a n s d u -c e r s r a n g e from .5 t o .7) t h e r i s e t i m e -c a n be -c a l -c u l a t e d t o be l e s s t h e n 16;us. R i s e t i m e i s d e f i n e d h e r e a s t h e t i m e i t t a k e s t h e t r a n s d u c e r o u t p u t t o r i s e from 10% t o 90% o f i t s f i n a l v a l u e i n r e s p o n s e t o a s t e p p r e s s u r e c h a n g e . As t h e r i s e t i m e s o f t h e s l a m m i n g p u l s e s f e l l i n t h e r a n g e o f .2ms t o I2._5ms t h e r i s e t i m e o f t h e p r e s s u r e t r a n s d u c e r s was not a l i m i t i n g f a c t o r , n o r d i d i t s e v e n t u a l ( m o d e r a t e ) o v e r s h o o t i n f l u e n c e t h e r e s u l t s . The s l a m m i n g p e a k s w e r e r e c o r d e d on a n i n s t r u m e n t a t i o n r e c o r -d e r an-d a U V - r e c o r -d e r . The b a n -d w i -d t h o f t h e U V - r e c o r -d e r was t o o s m a l l t o r e c o r d t h e s l a m m i n g p u l s e s a c c u r a t e l y and was o n l y u s e d f o r c o n t r o l p u r p o s e s d u r i n g t h e e x p e r i m e n t .

The a c t u a l p r o c e s s i n g o f t h e d a t a was done a f t e r t h e e x p e r i -ment. The d a t a - t a p e s were r e p l a y e d and t h e s l a m m i n g p u l s e s

f e d t o a d i g i t a l o s c i l l o s c o p e .

From e a c h measurement r u n f i v e p u l s e s from e a c h t r a n s d u c e r w e r e c a t c h e d and t h e r i s e t i m e and m a g n i t u d e o f them m e a s u r e d . The r e s u l t i n g v a l u e s o f t h e f i v e p u l s e s w e r e a v e r a g e d t o r e d u c e t h e l a r g e f l u c t u a t i o n s t h a t p l a g u e t h i s t y p e o f e x p e r i m e n t . As r i s e t i m e was t a k e n t h e t i m e i n t e r v a l b e t w e e n t h e 0% s t a r t v a l u e o f t h e p u l s e and t h e f i r s t ( l o c a l ) maximum v a l u e o f i t . The m a g n i t u d e was t h e g l o b a l maximum o f t h e p u l s e . Some s e l e c t e d p u l s e s w e r e f e d t o a c o m p u t e r v i a t h e s c o p e ' s i n t e r f a c e and p l o t t e d .

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-2.3. T e s t p r o g r a m E a c h wedge was f o r c e d o s c i l l a t e d a s h e a v i n g m o t i o n i n v e r t i c a l d i r e c t i o n w i t h a n a d j u s t e d t r i m a n g l e i n s u c h a way t h a t t h e a v e r a g e p o s i t i o n o f t h e t r a n s d u c e r s was s i t u a t e d i n t h e z e r o p o s i t i o n o f t h e h a r m o n i c m o t i o n and t h e s t i l l w a t e r l e v e l . T h i s means t h a t t h e t r a n s d u c e r s h i t t h e w a t e r s u r f a c e w i t h maximum o s c i l l a t i o n s p e e d . I f t h e v e r t i c a l d i s p l a c e m e n t o f o s c i l l a t i o n i s c h a r a c t e r i z e d a s s = s ^ C O S w t ( 1 ) w i t h : w = o s c i l l a t i o n f r e q u e n c y s ^ = a m p l i t u d e o f o s c i l l a t i o n i t f o l l o w s t h a t t h e maximum v e r t i c a l s p e e d becomes V = s = -ws^ ( 2 )

The f o l l o w i n g program h a s b e e n p e r f o r m e d f o r e a c h wedge:

1. One o s c i l l a t i o n - f r e q u e n c y w i t h t h r e e d i f f e r e n t a m p l i t u d e s d e l i v e r e d t h r e e v e r t i c a l s p e e d s v i z : w = 12 r a d / s w i t h s ^ = 0.02m V = 0.24 m/s w = 12 r a d / s w i t h s ^ = 0.04m V = 0.48 m/s w = 12 r a d / s w i t h s ^ = 0.06m V = 0.72 m/s 2. F i v e t r i m ancfles w i t h bow up ( p o s i t i v e ) w e r e c o n s i d e r e d v i z : a = 0, 0.5, 1.0, 2.0 and 2.5 d e g r e e s and f o r wedge n r . l a l s o a = 3 d e g r e e s . 3. T h r e e f o r w a r d s p e e d s w e r e a d j u s t e d v i z : U = 1.0 m/s U = 2.0 m/s U = 3.0 m/s By means o f t h e s e e x p e r i m e n t s i t was p o s s i b l e t o m e a s u r e p e a k p r e s s u r e s and r i s e t i m e s a s f u n c t i o n o f v e r t i c a l s p e e d , t r i m a n g l e , f o r w a r d s p e e d , dead r i s e a n g l e and t r a n s v e r s e p o s i t i o n .

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I t s h o u l d be r e m a r k e d t h a t f o r a l l t r i m c o n d i t i o n s e x c e p t a = 0° no i n f l u e n c e o f t h e f o r w a r d p a r t o f t h e wedge was p r e s e n t . The m e a s u r e d d a t a a r e p r e s e n t e d i n t h e t a b l e s 2 t o 5 f o r e a c h wedge. A c c u r a t e d e t e r m i n a t i o n o f p e a k p r e s s u r e s and r i s e t i m e s was n o t a l w a y s p o s s i b l e e s p e c i a l l y i n t h e c a s e o f 1 m/s f o r w a r d s p e e d s h o w i n g m o s t l y l o w p r e s s u r e v a l u e s and u n c e r t a i n r i s e t i m e s . I n t h e t a b l e s 2 t o 5 p e a k p r e s s u r e v a l u e s b e l o w I k P a a r e n o t r e p o r t e d . A l l p r e s e n t e d d a t a a r e a v e r a g e v a l u e s o f f i v e o s c i l l a t i o n s . 3. R e s u l t s a n d a n a l y s i s From t h e t e s t r e s u l t s a s p r e s e n t e d i n t h e t a b l e s 2 t o 5 and t h e f i g u r e s 3 t o 6 t h e f o l l o w i n g t e n d e n c i e s may be d e r i v e d f o r t h e p e a k p r e s s u r e s and r i s e t i m e s w i t h r e s p e c t t o t h e p a r a m e t e r s c o n s i d e r e d : 3.1. I n f l u e n c e o f t r i m a n g l e a S e e f i g u r e 3.1, 3.2, 3.3 and 3.4 P e a k p r e s s u r e : I n g e n e r a l i n c r e a s e w i t h t r i m a n g l e f o r dead r i s e a n g l e /3 > 0.5 d e g r e e , b u t w i t h a l m o s t no v a r i a t i o n a t t h e maximum d e a d r i s e a n g l e /3 = 10 d e g r e e s . F o r dead r i s e a n g l e /3 < 0.5 d e g r e e a s t r o n g v a r i a t i o n w i t h t r i m a n g l e may be o b s e r v e d w i t h minimum v a l u e s a t a b o u t a = 0.5 d e g r e e t r i m a n g l e . A l m o s t e q u a l and s m a l l v a l u e s f o r t h e p e a k p r e s s u r e s c o u l d be e s t a b l i s h e d f o r : V = 0.24 m/s and U = 1, 2, 3 m/s V = 0.48 m/s and U = 1 m/s V = 0.72 m/s and U = 1 m/s R i s e t i m e : A s t r o n g r e d u c t i o n i f t h e t r i m a n g l e a i n c r e a s e s . 3.2. I n f l u e n c e o f d e a d r i s e a n g l e p S e e f i g u r e 4.1, 4.2, 4.3 and 4.4. P e a k p r e s s u r e : A s t r o n g i n c r e a s e from ^ = 0 up t o yS = 1.15 d e g r e e f o l l o w e d by a s t r o n g r e d u c t i o n a t = 10 d e g r e e s . T h i s t e n d e n c y h a s b e e n c o n f i r m e d by t h e e x p e r i m e n t s o f Chuang [ 1 5 ] . R i s e t i m e : I n f l u e n c e o f /3 on r i s e t i m e s i s r a t h e r s m a l l .

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3.3. I n f l u e n c e o f v e r t i c a l s p e e d V See f i g u r e 5.1, 5.2, 5.3 and 5.4. P e a k p r e s s u r e : S t r o n g , a l m o s t q u a d r a t i c , i n c r e a s e w i t h v e r t i c a l s p e e d V, e s p e c i a l l y a t ;8 = 0 d e g r e e . S m a l l i n c r e a s e f o r /3 = 10 d e g r e e . R i s e t i m e : S i g n i f i c a n t r e d u c t i o n w i t h i n c r e a s e o f v e r t i c a l s p e e d V. 3.4. I n f l u e n c e o f f o r w a r d s p e e d U See f i g u r e 6.1, 6.2, 6.3 and 6.4. P e a k p r e s s u r e : I n many c a s e s a s t r o n g i n c r e a s e w i t h s o m e t i m e s q u a d r a t i c c h a r a c t e r c o u l d be e s t a b l i s h e d w i t h r e s p e c t t o t h e f o r w a r d s p e e d U. R i s e t i m e : R a t h e r s t r o n g r e d u c t i o n w i t h i n c r e a s e o f f o r w a r d s p e e d U. 3.5. I n f l u e n c e o f t r a n s v e r s e p o s i t i o n See t a b l e s 2 t o 5. P e a k p r e s s u r e : M o d e r a t e r e d u c t i o n t o t h e edge and t o t h e c e n t r e l i n e w i t h a n optimum v a l u e a t t r a n s d u c e r n r . 1 - 3 . R i s e t i m e : M o d e r a t e r e d u c t i o n t o t h e edge. 3.6. R e v i e w o f t h e r e s u l t s Some r e m a r k a b l e t e n d e n c i e s c o u l d be s e e n from t h e t e s t r e s u l t s . I n g e n e r a l t h e h i g h e s t p e a k p r e s s u r e s o c c u r a t t h e l o w e s t dead r i s e a n c f l e s , b u t n o t r e s u l t i n g i n e x t r e m e maximum v a l u e s f o r z e r o dead r i s e a n g l e a s one m i g h t be e x p e c t e d The e x p e r i m e n t s c o n f i r m e d t h e w e l l k n o w n r e l a t i o n t h a t t h e p e a k v a l u e s o f t h e s l a m m i n g p r e s s u r e s a r e p r o p o r t i o n a l t o t h e s q u a r e d v a l u e o f t h e v e r t i c a l s p e e d . M o r e o v e r a v e r y s t r o n g i n f l u e n c e c o u l d be e s t a b l i s h e d w i t h r e s p e c t t o t h e f o r w a r d s p e e d . T h i s l a s t r e l a t i o n was n o t a c c o u n t e d f o r i n most o f t h e e x i s t i n g c a l c u l a t i o n m o d e l s . T h i s i n f l u e n c e m i g h t be i m p o r t a n t e x p e c i a l l y f o r h i g h f o r w a r d s p e e d s a s a l r e a d y m e n t i o n e d by Watanabe [ 3 ] , Takemoto e t a l [ 4 ] and Beukelman [ 2 ] . 8

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-I n t h e p r e d i c t i o n model a s p r e s e n t e d i n [ 1 ] t h e f o r w a r d s p e e d i n f l u e n c e h a s o n l y b e e n i n t r o d u c e d w i t h t h e v e r t i c a l compo-n e compo-n t o f t h i s s p e e d i compo-n t h e c a s e o f a t r i m a compo-n g l e . I t s h o u l d be n o t i c e d from t h e m e a s u r e d r e s u l t s t h a t a l s o f o r z e r o t r i m a n g l e s i g n i f i c a n t f o r w a r d s p e e d i n f l u e n c e was o b s e r v e d . The e x p e r i m e n t a l r e s u l t s g e n e r a l l y show a f l u c t u a t i n g i n c r e a s e o f t h e p e a k p r e s s u r e s w i t h t r i m a n g l e (bow u p ) . I n t r a n s v e r s e l i n e t h e s e p r e s s u r e s r e d u c e t o t h e c e n t r e l i n e and t h e edge o f t h e wedge.

The g e n e r a l p i c t u r e f o r t h e r i s e t i m e s i s a d e c r e a s e w i t h t r i m a n g l e , v e r t i c a l and f o r w a r d v e l o c i t y . See t a b l e 2 t o 5. 4. C o m p a r i s o n w i t h c a l c u l a t i o n methods 4.1. Bottom i m p a c t p r e s s u r e s Most e x i s t i n g c a l c u l a t i o n methods f o r b o t t o m i m p a c t p r e s s u r e s [ 1, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16] d e t e r m i n e t h e s e p r e s s u r e s r e l a t e d t o t h e s q u a r e d v e r t i c a l v e l o c i t y o n l y a s : p = k p V 2 (3) w i t h : k = p r o p o r t i o n a l i t y c o n s t a n t p = mass d e n s i t y V = v e r t i c a l v e l o c i t y I n t h e c a s e o f f o r c e d h e a v e o s c i l l a t i o n a r o u n d t h e w a t e r -s u r f a c e w i t h v e r t i c a l d i -s p l a c e m e n t S = Sa C O S w t (4) t h i s i m p a c t p r e s s u r e may a l s o be e x p r e s s e d a s p = k p s 2 (5)

The p r o p o r t i o n a l i t y c o n s t a n t k f o r wedges and c o n e s may be d e t e r m i n e d by t h e Wagner wedge i m p a c t t h e o r y [ 1 7 ] , t h e Chuang cone i m p a c t t h e o r y [ 1 8 ] and NRSDC d r o p t e s t s o f wedges and c o n e s [ 1 9 ] .

Some o f t h e above m e n t i o n e d c a l c u l a t i o n methods [ 1 , 14, 16] i n t r o d u c e t h e f o r w a r d s p e e d i n f l u e n c e a s t h e v e r t i c a l compo-n e compo-n t o f t h e f o r w a r d s p e e d i compo-n c a s e o f a t r i m a compo-n g l e . I t m i g h t be r e m a r k e d h e r e t h a t i n t h e p r e s e n t r e s e a r c h f o r w a r d s p e e d i n f l u e n c e was a l s o o b s e r v e d f o r z e r o t r i m a n g l e . T h e r e i s a s c a t t e r i n g i n t h e v a l u e o f t h e p r o p o r -t i o n a l i -t y c o n s -t a n -t k f o r -t h e m e n -t i o n e d c a l c u l a -t i o n m e -t h o d s . Some o f them u s e k ~ 60 among w h i c h e.g. T a k e z a w a [ 6 ] . C a l c u l a t i o n s a c c o r d i n g t o t h i s method a r e shown i n t h e f i g u r e s 5.2 r e l a t e d t o wedge n r . 2 = 0.46 d e g r e e s ) w i t h t r i m a n g l e a = 0.5 and 2.5 d e g r e e s .

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-From t h e s e f i g u r e s i t m i g h t be o b v i o u s t h a t t h e c a l c u l a t e d f o r w a r d s p e e d i n f l u e n c e i s t o o s m a l l compared t o t h e m e a s u r e m e n t s , w h i l e t h e c a l c u l a t e d v a l u e s show a n o v e r e s t i m a -t i o n f o r -t h e l o w e r f o r w a r d s p e e d s . W i t h r e s p e c t t o t h e r e m a i n i n g c a l c u l a t i o n methods [ 1 , 7, 13, 14] i t h a s b e e n e s t a b l i s h e d t h a t t h e p r e d i c t i o n s f o r t h e low dead r i s e a n g l e s show v e r y h i g h v a l u e s f o r t h e i m p a c t p r e s s u r e s . F o r t h i s r e a s o n t h e r e s u l t s w e r e n o t p r e s e n t e d i n t h e r e l a t e d f i g u r e s . I n c a s e o f wedge n r . 4 w i t h t h e h i g h e s t d e a d r i s e a n g l e (/3 = 9.98 d e g r e e s ) t h e computed r e s u l t s a c c o r d i n g t o M.D. O c h i [ 7 ] a r e p r e s e n t e d i n f i g u r e 5.4 s h o w i n g 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 m e a s u r e m e n t s a t t h e s m a l l e s t t r i m a n g l e (a = 0.5 d e g r e e s ) 4.2. F l u i d momentum e x c h a n g e A n o t h e r method w i t h a r e s t r i c t e d f o r w a r d s p e e d i n f l u e n c e i s t h a t o f Beukelman [ 1 ] . The i m p a c t p r e s s u r e i s d e t e r m i n e d on b a s e o f f l u i d momentum e x c h a n g e and s t r i p t h e o r y r e s u l t i n g i n t h e f o l l o w i n g e x p r e s s i o n : P -2yw ( N s + dm d s s 2 + m s ) (6) w i t h : N ' = s e c t i o n a l damping m' = s e c t i o n a l added mass y ^ = h a l f w i d t h o f t h e submerged c r o s s s e c t i o n on t h e w a t e r l i n e s = COSwt = v e r t i c a l d i s p l a c e m e n t The f i r s t t e r m w i t h s e c t i o n a l damping i s o f m i n o r i m p o r t a n c e and may be n e g l e c t e d . I n c a s e o f f o r w a r d s p e e d w i t h a t r i m a n g l e a (bow up) t h e t o t a l v e r t i c a l s p e e d w i l l b e : = s - U s i n a ₍₇₎ The same o b j e c t i o n s a s m e n t i o n e d b e f o r e h o l d s f o r t h i s t y p e o f f o r w a r d s p e e d i n f l u e n c e : i t i s t o o r e s t r i c t e d a n d z e r o f o r z e r o t r i m a n g l e . An e x t e n s i o n o f t h i s method w i l l be p r e s e n t e d i n p a r t 2 t a k i n g i n t o a c c o u n t more s i g n i f i c a n t f o r w a r d s p e e d i n f l u e n c e and 3-D e f f e c t s . 10

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-The c a l c u l a t e d r e s u l t s g e n e r a l l y show t h e same t e n d e n c y a s m e n t i o n e d e a r l i e r : v e r y h i g h p e a k v a l u e s f o r t h e wedges w i t h t h e low d e a d r i s e a n g l e s . So c a l c u l a t e d r e s u l t s a r e o n l y p r e s e n t e d i n f i g u r e 5.4 f o r wedge n r . 4 w i t h t h e h i g h e s t d e a d r i s e a n g l e (/9 = 9.98 d e g r . ) . I t i s s t r i k i n g t h a t f o r t h i s c a s e t o o low c a l c u l a t e d v a l u e s a r e shown. 4.3. 3-D wedge i m p a c t I t i s a l s o w o r t h w h i l e t o m e n t i o n s e p a r a t e l y t h e c a l c u l a t i o n method o f S t a v o v y - C h u a n g [ 1 4 ] and Chuang ( 1 9 7 3 ) [ 1 6 ] t o d e t e r m i n e i m p a c t p r e s s u r e s f o r t h r e e d i m e n s i o n a l m o d e l s a t h i g h f o r w a r d s p e e d .

The p r e s s u r e t h a t a c t s n o r m a l t o t h e h u l l b o t t o m i n t h e s l a m m i n g a r e a may be s e p a r a t e d i n t o two c o m p o n e n t s :

1. The i m p a c t p r e s s u r e pj_ due t o t h e v e l o c i t y component o f t h e c r a f t n o r m a l t o t h e w a t e r s u r f a c e .

2. The p l a n i n g p r e s s u r e pp due t o t h e v e l o c i t y component o f t h e c r a f t t a n g e n t i a l t o t h e w a t e r s u r f a c e .

The p l a n i n g p r e s s u r e a c t i n g n o r m a l t o t h e h u l l b o t t o m i s :

Pp = IpV^cosySeh ( 8 )

w i t h /3eh = e f f e c t i v e i m p a c t a n g l e i n t h e h o r i z o n t a l l o n g i t u d i n a l p l a n e .

The t o t a l p r e s s u r e due t o v e l o c i t y components o f t h e c r a f t b o t h n o r m a l and t a n g e n t t o t h e wave s u r f a c e i s t h e r e f o r e : ( P t = P i + Pp ( 9 ) F o r low s p e e d s t h e p l a n i n g p r e s s u r e i s u s u a l l y s m a l l c o m p a r e d t o t h e i m p a c t p r e s s u r e . Making u s e o f f o r o u r c a s e e x t r a p o l a t e d 3-D p r e d i c t i o n s a s p r e s e n t e d i n Chuangs r e p o r t [ 1 6 ] ( f i g u r e s 8^, 8^, 12, 13, 21) one c a l c u l a t e d v a l u e c o u l d be p r e s e n t e d i n f i g u r e 5.4 f o r wedge n r . 4 (^9 = 9.98 d e g r e e s ) s h o w i n g 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 t e s t r e s u l t s . N e v e r t h e l e s s i t r e m a i n s r e m a r k a b l e t h a t Chuang f o u n d by e x p e r i m e n t and 3-D c a l c u l a t i o n t h a t a c h a n g e i n f o r w a r d v e l o c i t y w i l l n o t a l t e r t h e m a g n i t u d e o f i m p a c t p r e s s u r e i f t h e v e r t i c a l v e l o c i t i e s r e m a i n t h e same. 11

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-5. C o n c l u s i o n s and r e c o m m e n d a t i o n s A n a l y s i s o f t h e t e s t r e s u l t s f o r t h e wedges c o n s i d e r e d and c o m p a r i s o n w i t h e x i s t i n g c a l c u l a t i o n methods d e l i v e r t h e f o l l o w i n g c o n c l u s i o n s and r e c o m m e n d a t i o n s : 1. The m e a s u r e d p e a k p r e s s u r e s show a c l e a r p r o p o r t i o -n a l i t y t o t h e s q u a r e d v a l u e s o f t h e v e r t i c a l s p e e d , b u t a l s o a s i g n i f i c a n t i n f l u e n c e o f t h e r e l a t i v e l y h i g h f o r w a r d s p e e d e v e n a t z e r o t r i m a n g l e . 2. P e a k p r e s s u r e s g e n e r a l l y a l s o d e m o n s t r a t e a f l u c t u a t i n g i n c r e a s e w i t h t r i m a n g l e (bow up) e s p e c i a l l y a t t h e l o w e r d e a d r i s e a n g l e s . A s t r o n g i n c r e a s e o f t h e s e p r e s s u r e s w i t h dead r i s e a n g l e c o u l d be e s t a b l i s h e d up t o 1.15 d e g r e e d e a d r i s e a n g l e . F o r 10 d e g r e e a s t r o n g p r e s s u r e r e d u c t i o n was o b s e r v e d . F o r an a c c u r a t e d e t e r m i n a t i o n o f t h e i n f l u e n c e o f d e a d r i s e a n g l e on t h e p e a k p r e s s u r e s more t e s t s w i t h a d d i t i o n a l d e a d r i s e a n g l e s a r e r e q u i r e d . 3. I n t r a n s v e r s e l i n e o f t h e wedges r e d u c t i o n o f t h e p e a k p r e s s u r e s t o t h e c e n t r e l i n e and t h e edge c o u l d be o b s e r v e d . 4. R i s e t i m e s o f t h e p e a k p r e s s u r e s d e c r e a s e r a t h e r s t r o n g l y w i t h t r i m a n g l e , v e r t i c a l and f o r w a r d v e l o c i t y , b u t t h e i n f l u e n c e o f d e a d r i s e a n g l e may be n e g l e c t e d i n t h i s r e s p e c t .

5. Most e x i s t i n g c a l c u l a t i o n methods show t o o h i g h

p r e s s u r e p r e d i c t i o n s and i n a l l c a s e s t h e f o r w a r d s p e e d i n f l u e n c e i s v e r y p o o r l y r e p r e s e n t e d . 6. E x t e n s i v e f u r t h e r r e s e a r c h t o t h e i n f l u e n c e o f f o r w a r d s p e e d , 3-D and s u r f a c e e f f e c t s i s r e q u i r e d . 6. Acknowledgment S p e c i a l t h a n k s a r e due t o t h e v a r i o u s members o f t h e s t a f f o f t h e 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 o f t h e 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 f o r t h e i r a s s i s t e n c e i n c o n s t r u c t i n g t h e wedges, r u n n i n g and w o r k i n g o u t t h e s l a m m i n g e x p e r i m e n t s . P a r t i c u l a r y Dr. D. Radev o f t h e B u l g a r i a n S h i p H y d r o d y n a m i c C e n t r e s h o u l d be m e n t i o n e d , who a s r e s e a r c h - f e l l o w i n D e l f t c o o p e r a t e d i n a n a l y s i n g t h e t e s t r e s u l t s . H i s v a l u a b l e m a i n work, t h e e x t e n s i o n and a p p l i c a t i o n o f an e x i s t i n g p r e d i c t i o n model, w i l l be p r e s e n t e d i n p a r t 2. 12

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-7. R e f e r e n c e s [ 1 ] Beukelman, W., •Bottom i m p a c t p r e s s u r e s due t o f o r c e d o s c i l l a t i o n ' , 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 , Volume 27, No.3 09, May 1980. [ 2 ] Beukelman, W., 'Slamming p r e s s u r e s on t h e c y l i n d e r s u r f a c e s due t o f o r c e d o s c i l l a t i o n ' . R e p o r t Nr.728, 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 , November 1986 ( i n D u t c h ) . [ 3 ] Watanabe, I . , ' E f f e c t s o f t h e t h r e e d i m e n s i o n a l i t y o f s h i p h u l l on wave i m p a c t p r e s s u r e ' , J o u r n a l o f t h e 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 , pp. 163-174, Volume 162, 1 9 8 7 .

[ 4 ] Takemoto, H., H a s h i z u m e , Y. and Oka, S.,

' F u l l s c a l e m e a s u r e m e n t s o f wave i m p a c t l o a d s a n d h u l l r e s p o n s e o f a s h i p i n w a v e s ' , J o u r n a l o f t h e 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 , Volume 158, 1 9 8 5 . [ 5 ] O c h i , M.K. a n d M o t t e r , L . E . , ' P r e d i c t i o n o f s l a m m i n g c h a r a c t e r i s t i c s and h u l l r e s p o n s e s f o r s h i p d e s i g n ' , SNAME 8 1 , 1973, pp. 144-176. [ 6 ] T a k e z a w a , S. a n d Hasegawa, S., 'On t h e c h a r a c t e r i s t i c s o f w a t e r i m p a c t p r e s s u r e s a c t i n g on a h u l l s u r f a c e among w a v e s ' . J o u r n a l o f t h e 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 , Volume 13, 1 9 7 5 . [ 7 ] O c h i , M a r g a r e t D. a n d B o n i l l a - N o r a t , J . , ' P r e s s u r e - v e l o c i t y r e l a t i o n s h i p i n i m p a c t o f a s h i p model d r o p p e d o n t o t h e w a t e r s u r f a c e a n d i n s l a m m i n g i n w a v e s ' , NSRDC, R e p o r t 3153, J u n e 1 9 7 0 . [ 8 ] O c h i , M.K., ' P r e d i c t i o n o f o c c u r r e n c e and s e v e r i t y o f s h i p s l a m m i n g a t s e a ' , 5 t h Symposium on N a v a l H y d r o d y n a m i c s , B e r g e n , Norway, 1 9 6 4 . [ 9 ] T a s a i , F . , 'A s t u d y on t h e s e a k e e p i n g q u a l i t i e s o f f u l l s h i p s ' . R e p o r t o f R e s e a r c h I n s t i t u t e f o r A p p l i e d M e c h a n i c s , J a p a n , Volume X V I , No.55, 1 9 6 8 . [ 1 0 ] O c h i , M.K., 'Model e x p e r i m e n t s on s h i p s t r e n g t h a n d s l a m m i n g i n r e g u l a r w a v e s ' . T r a n s . SNAME, Volume 66, 1 9 5 8 . [ 1 1 ] H a g i w a r a , K. a n d Y u h a r a , T.,

'Fundamental s t u d y on wave i m p a c t l o a d s on s h i p bow' ( 1 s t r e p o r t ) , S e l e c t e d p a p e r s from J o u r n a l o f t h e 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 , Volume 14, 1 9 7 6 .

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[ 1 2 ] Chuang, S . L . , ' E x p e r i m e n t s on f l a t - b o t t o m s l a m m i n g ' . J o u r n a l o f S h i p R e s e a r c h , March 1966, pp. 10-17. [ 1 3 ] K a p l a n , P. and M a l a k h o f f , A., 'Hard s t r u c t u r e s l a m m i n g o f S E S c r a f t i n w a v e s ' , AIAA/SNAME A d v a n c e d M a r i n e V e h i c l e s C o n f e r e n c e , S a n D i e g o , A p r i l 1978. [ 1 4 ] S t a v o v y , A l e x a n d e r B. and Chuang, S . L . , ' A n a l y t i c a l d e t e r m i n a t i o n o f s l a m m i n g p r e s s u r e s f o r h i g h - s p e e d v e h i c l e s i n w a v e s ' . J o u r n a l o f S h i p R e s e a r c h , December 1976, pp. 190-198. [ 1 5 ] Chuang, S . L . , ' E x p e r i m e n t s on s l a m m i n g o f wedge-shaped b o d i e s ' . J o u r n a l o f S h i p r e s e a r c h , S e p t e m b e r 1967, pp. 190-198. [ 1 6 ] Chuang, S . L . , 'Slamming t e s t s o f t h r e e - d i m e n s i o n a l m o d e l s i n c a l m w a t e r and w a v e s ' , NSRDC, R e p o r t 4095, S e p t e m b e r 1 9 7 3 . [ 1 7 ] Wagner, V.H., 'Über s t o s z - und G l e i t v o r g a n g e a n d e r O b e r f l a o h e v o n Flüssigkeiten', Z e i t s c h r i f t für Angewandte M a t h e m a t i k und M e c h a n i k , Volume 12, No.4, pp. 193-215, A u g u s t 1 9 3 2 . [ 1 8 ] Chuang, S . L . , ' T h e o r e t i c a l i n v e s t i g a t i o n s on s l a m m i n g o f C o n e - s h a p e d b o d i e s ' . J o u r n a l o f S h i p R e s e a r c h , V o l . 1 3 , No.4, 1 9 6 9 . [ 1 9 ] Chuang, S . L . a n d M i l n e , D.T., 'Drop t e s t s o f c o n e s t o i n v e s t i g a t e t h e t h r e e d i m e n -s i o n a l e f f e c t -s o f -s l a m m i n g ' , NSRDC, R e p o r t 3543, 1 9 7 1 . 14

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-T a b l e 2

WEDGE NR.l 0 = 0 DEGREES PRESSURE p kPa RISETIME t MS V TRIM ANGL a DEGR U TR ANSDUCER NR. TRANSDUCER NR. M/s TRIM ANGL a DEGR M/ 3 1 2 3 4 5 1 2 3 4 5 0 1 2 3 2. 2. i 2.( i 1.; ) 2. ) 2.. L 2.; i l . f i 1.. i 1.. •) 2. ) l . f I 2. ) 1. 2 1. 3 1.. ' 2. ) 2. ) 1.7 3 1.8 0.5 1 2 3 2.( l . f ) 2.1 i l.£ 2.( 2.; -1.; 1.7 _ - - -0.24 1.0 1 2 3 2.5 1.; 1.4 1.4 2.C 1.4 -1.3 1.7 1.2 -- -0.24 2.0 1 2 3 2.; 5.C 3.3 2.2 2.1 2.8 2.1 3.7 4.2 0.8 3.5 2.2 2.1 2.4 - -2.5 1 2 3 6.2 2.0 3.6 3.1 2.7 7.C l . S 3.4 2.2 1.9 2.5 1.5 2.4 2.7 - -- - -3.0 1 2 3 1.2 1.9 4.9 1.6 2.3 1.8 3.7 3.4 1.1 1.5 1.7 2.2 3.3 - - - - -0 1 2 3 5.5 12.6 4.9 10.6 5.8 12.8 6.5 14.1 2.3 4.3 1.9 1.1 2.2 1.2 1.9 1.2 2.2 1.1 1.9 1.4 0.5 1 2 3 4.4 5.1 4.0 5.4 4.1 4.9 1.9 2.4 3.9 1.3 1.1 1.3 0.9 1.0 1.0 1.2 1.0 1.3 0.48 1.0 1 2 3 5.5 7.6 5.7 5.6 6.3 7.5 9.8 5.9 3.4 4.5 12.9 7.2 10.2 7.5 12.7 7.5 10.4 6:4 7.8 0.48 2.0 1 2 3 2.1 13.0 13.3 1.5 6.2 10.0 2.2 12.9 11.3 3.5 12.5 1.0 1.3 4.1 2.2 6.9 5.8 7.1 5.8 7.4 5.6 6.1 -2.5 1 2 3 4.6 12.7 13.9 2.4 6.7 9.2 4.1 8.8 15.4 2.1 8.4 12.1 6.3 6.1 5.3 5.6 5.2 5.7 6.1 5.5 4.4 4.6 4.6 3.0 1 2 3 4.5 11.7 12.5 3.1 10.6 8.9 5.7 11.0 10.8 2.6 10.2 11.2 7.5 6.7 6.0 5.7 6.5 6.6 6.7 5.9 5.5 5.9 5.1 4.4 0 1 2 3 7.9 28.4 7.2 33.1 6.3 32.9 6.5 31.9 7.3 29.8 1.9 0.6 2.5 1.1 2.0 0.6 1.8 0.7 0.8 0.8 0.5 1 2 3 9.3 9.1 9.4 7.3 10.4 L4.5 L2.3 14.1 8.3 5.0 .72 -1.0 1 2 3 8.9 L5.3 : t l . l _{L2.4 4.9 2.6 2.3 2.4 1.2 1.1} 2.0 1 2 ] 3 2 7.5 ] 2.6 ] 5.8 ] 8.1 2 8.3 0.7 ] 9.3 ] 4.9 ] 1.7 2.3 0.5 0.3 0.8 0.6 0.6 0.3 0.4 0.5 0.7 0.6 2.5 1 2 2 3 3 2.9 1.3 2 4.1 3 2.8 7.7 2 5.2 3 2.5 3.4 1 3.1 2 3.8 0.1 1 3.8 2 2.3 2.6 3.7 3.4 0.9 0.2 2.7 0.7 0.4 3.7 0.4 0.2 3.3 0.6 0.2 2.3 1.1 0.3 3.0 1 2 2 3 2 1.5 2.2 2 9.4 2 2.0 9.7 2 9.5 3 2.2 1.9 1 1.0 2 2.3 5.2 1 4.3 2 1.4 2.6 2.0 8.8 0.2 0.2 3.1 0.2 0.3 4.7 0.2 0.2 5.3 0.2 0.1 5.1 0.2 0.2

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T a b l e 3

WEDGE NR.2 /8 = 0.A6 DEGREES PRESSURE P kPa RISETIME t MS V TRIM ANGLE a DEGR. U TRANSDUCER NR. TRANSDUCER NR. M/s TRIM ANGLE a DEGR. M/s 1 2 3 4 1 2 3 4 5 0 1 2 3 4.1 2.6 4.0 2.0 3.8 2.3 3.1 2.0 1.8 1.6 - -- - -0.5 1 2 3 3.5 1.4 2.7 2.9 2.0 1.8 1.3 1.7 -- -- -0.24 1.0 1 2 3 3.8 2.0 1.5 1.7 3.6 1.8 2.9 1.4 2.0 1.4 - - - - -0.24 2.0 1 2 3 1.9 3.7 4.2 3.9 2.0 2.4 1.5 3.6 3.2 2.4 5.7 2.5 3.1 3.6 - -- -2.5 1 2 3 6.6 2.6 4.8 2.4 2.7 6.5 2.8 4.1 3.0 2.5 3.3 1.9 3.2 2.9 7.6 - 7.2 - -3.0 1 2 3 _ -

;

-- - -- - -0 1 2 3 4.7 12.7 4.5 12.3 4.6 14.8 3.5 11.5 9.2 3.6 4.1 5.7 4.0 4.6 0.5 1 2 3 8.5 9.0 4.5 6.1 7.8 9.9 4.5 7.9 3.4 0.9 13.9 9.0 15.8 6.8 13.8 7.7 7.7 -0.A8 1.0 1 2 3 1.5 5.5 8.3 1.4 5.0 6.3 1.7 6.2 9.3 2.1 5.2 8.4 1.5 4.6 5.5 12.1 8.3 11.7 8.1 11.6 8.2 7.7 7.2 4.4 7.1 0.A8 2.0 1 2 3 2.4 14.6 20.9 1.6 10.3 13.0 2.5 17.2 16.6 3.1 9.0 16.9 3.1 7.4 6.9 6.1 6.5 5.6 7.6 5.9 5.9 5.5 5.8 2.5 1 2 3 4.2 16.1 14.1 3.7 12.7 8.5 4.8 13.7 15.9 2.3 13.1 13.9 3.3 8.0 6.5 6.9 6.2 6.7 6.3 6.4 6.0 5.8 6.5 3.0 1 2 3 _

-;

-

- - - -0 1 2 3 1.6 4.2 35.6 1.2 3.3 32.8 1.4 4.3 33.6 1.7 4.4 26.3 9.3 21.2 1.6 1.0 1.6 1.0 1.4 0.6 1.5 0.7 1.0 0.9 0.5 1 2 3 3.8 15.3 9.5 15.0 8.8 15.5 4.7 14 .6 8.9 16.0 1.9 0.8 5.4 0.7 5.5 0.6 1.6 0.7 0.6 0.7 0.72 1.0 1 2 3 1.1 14.3 13.0 1.6 15.6 1.4 13.3 1.3 8.0 7.8 8.6 7.6 1.5 3.4 0.72 2.0 1 2 3 1.4 18.1 18.2 15.9 17.7 1.2 18.1 17.5 2.1 9.7 18.5 3.0 7.4 0.5 0.4 0.8 0.5 0.8 0.5 0.6 0.5 1.0 2.5 1 2 3 4.3 18.4 33.8 2.8 15.9 33.7 4.2 18.2 31.4 3.8 14.5 24.7 2.5 6.8 19.1 0.5 0.3 0.6 0.4 0.5 0.2 0.5 0.3 0.5 0.3 3.0 1 2 3 - - -_ -_ _

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T a b l e 4

WEDGE NR.3 = 1.15 DEGREES PRESSURE p kPa RISETIME t MS V M/S TRIM ANGLE a DEGR. U M/S TRANSDUCER NR. TRANSDUCER NR. V M/S TRIM ANGLE a DEGR. U M/S 1 2 3 4 5 1 2 3 4 5 0.24 0 1 2 3 1.8 3.0 2.0 1.3 2.4 1.7 1.9 2.5 1.7 2.2 - - -- -0.24 0.5 1 2 3 1.9 2.0 1.1 1.8 1.7 1.7 2.7 -- - - - -0.24 1.0 1 2 3 1.8 3.2 4.0 2.3 2.2 1.4 3.0 2.3 1.4 3.8 3.0 2.6 - -- - -0.24 2.0 1 2 3 1.3 2.7 4.3 3.3 1.4 2.1 1.6 4.1 2.9 2.1 5.0 2.6 3.2 -- - - -0.24 2.5 1 2 3 2.7 3.4 4.9 1.5 3.1 2.4 3.8 1.7 4.1 2.6 2.3 5.2 2.2 3.8 3.4 - -- -0.24 3.0 1 2 3 - - - -0.48 0 1 2 3 4.3 14.8 4.3 11.0 4.7 12.7 4.1 10.3 3.4 5.7 4.2 4.0 4.4 4.0 4.3 0.48 0.5 1 2 3 5.2 15.1 2.9 9.6 5.4 13.1 5.4 9.8 3.4 7.7 4.8 6.0 5.1 10.5 6.3 6.2 0.48 1.0 1 2 3 1.6 7.3 20.3 1.0 6.2 10.7 1.2 5.9 13.5 2.6 5.9 11.3 2.8 6.0 10.5 4.3 9.5 5.3 11.6 5.5 11.8 6.0 7.4 0.48 2.0 1 2 3 16.2 13.5 7.3 8.6 12.8 8.2 15.7 13.2 3.6 6.2 6.2 5.0 5.9 4.8 5.9 5.5 6.1 7.0 6.1 0.48 2.5 1 2 3 4.4 12.4 14.4 1.7 9.4 15.8 2.8 13.6 11.2 1.8 9.8 14.3 2.2 4.7 6.1 5.4 5.3 5.8 4.9 5.4 5.4 5.8 -0.48 3.0 1 2 3 _ - - -- - - -0.72 0 1 2 3 9.4 35.3 7.6 32.4 9.1 25.2 8.8 31.1 6.3 16.0 7.3 1.1 6.8 0.8 7.8 0.8 8.1 0.4 5.1 0.6 0.72 0.5 1 2 3 1.8 9.7 35.8 1.6 11.0 27.7 1.9 11.8 29.4 1.7 9.0 29.9 8.1 9.2 0.4 10.4 0.7 10.8 0.7 7.6 0.5 0.5 0.72 1.0 1 2 3 14.3 36.0 12.4 27.5 17.3 33.3 14.6 28.1 6.8 9.4 0.3 9.0 0.5 9.6 0.2 6.2 0.5 0.7 0.72 2.0 1 2 3 2.7 18.3 31.8 2.6 15.4 24.5 2.4 21.9 24.0 3.2 15.2 33.3 2.4 5.5 17.7 7.8 0.2 6.8 0.3 6.6 0.2 7.2 0.2 4.5 0.4 2.5 1 2 3 4.4 28.8 32.4 2.8 20.9 26.6 3.4 28.5 30.7 3.1 16.7 26.8 2.3 4.5 24.3 6.5 0.1 5.7 0.3 6.2 0.1 6.5 0.2 2.2 0.2 3.0 1 2 3 _ _ _ ~ _ _ -

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-T a b l e 5

WEDGE NR.A p = 9.98 DEGREES PRESSURE P kPa RISETIME t MS V TRIM ANGLE a DEGR. U TRANSDUCER NR. TRANSDUCER NR. M/s TRIM ANGLE a DEGR. M/S 1 2 3 4 5 1 2 3 4 5 0 1 2 3 1.6 1.2 1.6 1.6 1.5 -- - -0.5 1 2 3 1.1 1.5 2.8 1.8 1.2 1.4 1.3 - - - -0.24 1.0 1 2 3 1.9 1.7 4.2 1.4 2.0 1.6 1.3 1.5 1.3 - - - -0.24 2.0 1 2 3 1.2 1.6 1.2 1.2 1.8 1.2 1.8 1.1 - - - -2.5 1 2 3 4.1 1.2 2.0 2.4 1.5 4.4 1.4 1.8 10.4 1.5 7.5 1.1 -

\

- - -3.0 1 2 3 - " _ --

-; -;

-0 1 2 3 5.2 3.5 4.4 1.6 2.5 3.3 7.2 4.1 3.9 1.6 4.1 4.1 4.7 6.4 3.6 2.1 -2.1 - -0.5 1 2 3 2.8 4.3 4.4 3.9 1.7 3.2 3.2 3.8 4.4 2.4 4.0 4.1 1.7 4.2 4.2 -- - -0.48 1.0 1 2 3 3.8 4.2 4.2 5.1 2.2 3.2 3.6 4.0 4.9 2.1 3.8 4.7 2.3 5.6 4.5 - - - 0.9 0.48 2.0 1 2 3 4.1 4.3 4.3 2.5 2.8 3.8 4.6 4.0 4.4 4.5 3.8 4.5 1.9 4.1 3.6 - -- - -2.5 1 2 3 4.1 4.5 2.1 3.5 3.9 4.4 3.8 4.4 3.6 4.1 - - -- -3.0 1 2 3 - - -

_;

- - - -0 1 2 3 1.6 10.0 9.1 1.3 5.7 10.0 1.6 9.1 9.5 3.9 10.6 8.7 4.8 7.8 9.1 0.6 0.4 0.7 0.3 0.6 0.4 0.7 0.4 2.7 0.5 0.5 1 2 3 2.5 8.5 9.4 1.5 5.4 8.9 2.9 8.6 9.5 4.5 9.9 8.4 8.7 8.6 0.9 0.4 0.6 0.3 0.5 0.4 0.7 0.5 0.6 0.72 1.0 1 2 3 2.3 8.6 10.7 1.7 4.4 10.1 2.4 9.5 9.5 8.6 10.6 9.0 7.1 7.3 0.6 0.5 0.8 0.3 0.6 0.4 7.6 0.6 0.6 2.2 0.7 0.72 2.0 1 2 3 6.6 8.5 9.5 2.2 5.8 9.6 3.9 8.8 10.3 5.6 9.8 7.8 10.0 9.3 8.2 8.1 0.7 0.6 0.7 0.3 0.5 0.5 3.8 0.8 0.5 0.8 1.1 0.7 2.5 1 2 3 6.9 7.8 10.5 7.3 6.6 8.9 5.3 8.2 9.5 3.0 8.5 7.7 8.7 8.2 10.2 4.8 0.6 0.5 6.4 0.4 0.4 0.6 0.5 0.7 0.6 0.9 0.9 0.7 3.0 1 2 3 _ -_ - -_ --

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30 20 [ k L i 10 6 = 1 . 1 5 " U = 2.00M/S V = 0 . 2 4 M/S

r

6 = 1 . 1 5 " U = 2.00 M/S V=0.48M/S 30 B = 1.15" Ö A fl <3. V I U = 3.00M/S V = 0 . 4 8 M/S -V v . _ . F i g u r e 3.3. Peak p r e s s u r e a s f u n c t i o n o f t r i m a n g l e a.

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30^ 20 6 = 10° G A Q O V U=2.00M/S V=0.72M/S 30 P [ k P a ] 10 H 6 = 10° 5 Q O ^ D = 3.00M/S V=0.24M/S F i g u r e 3.4. Peak p r e s s u r e a s f u n c t i o n o f t r i m a n g l e a.

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30 20\ P [kPa 1 0 Ö A Q <i> V 0 = 0.5° U = IM/S U = 2M/S U=3M/S M.D. OCHI 0.24 0.48 V [M/S] 0.72 30 20 P k P a 1 0 B = 10.0° ~Ö A Q O ^ 0 = 2.5° — U = 1 M / S U = 2M/S U = 3M/S M.D. OCHI TEST 'O' ^CHUANG, 1973 ^ ( y = 2 . 5 M/s) BEUKELMAN, 1980 0.72 F i g u r e 5.4. Peak p r e s s u r e as f u n c t i o n o f v e r t i c a l v e l o c i t y V.

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F i g u r e 6.2. Peak p r e s s u r e as f u n c t i o n o f f o r w a r d s p e e d U.

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0 1 2 3 0 1 2 3 U [M/S] ^ D [M/S F i g u r e 6.3. Peak p r e s s u r e as f u n c t i o n o f f o r w a r d s p e e d U. F i g u r e 6.4. Peak p r e s s u r e a s f u n c t i o n o f f o r w a r d s p e e d U.