Experimental investigation of water impact on axisymmetric bodies

Applied Ocean Research 31 (2009) 143-156

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Applied Ocean Research

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Experimental investigation of water impact on axisymmetric bodies

G. De Backer^'*, M. Vantorre^ C. Beels^ J. De P r é ^ S. Victor^ J. De Rouck^ C. Blommaert^

W. Van Paepegem ^

^ Department of Civil Engineering, Ghent University, Technologiepark Zwijnaarde 904, B 9052 Zwijnaarde, Belgium '' Department of Materials Science and Engineering. Ghent University, Sint-Pietersnieuwstraat 41, B 9000 Gent, Belgium

A R T I C L E I N F O A B S T R A C T Article history: Received 31 October 2008 Received i n revised f o f m 27 A p r i l 2009 Accepted 15 July 2009 Available online 20 August 2009

Keywords: Slamming Drop tests Point absorbers Wave energy Experimental study

The results of an elaborate experimental investigation on b o t t o m slamming of axisymmetric objects are presented. Drop tests have been performed on a hemisphere and t w o conical shapes w i t h different deadrise angles. The test setup is designed so as to prevent small rotations of the test objects w h i c h cause scatter in the measurement data. The pressure distribution and evolution as well as the body m o t i o n parameters are measured during impact. By means of a high speed camera the water uprise is visualized and the w e t t i n g factor is determined for the cones. The results are compared w i t h a three-dimensional asymptotic theory for axisymmetric rigid bodies w i t h constant entry velocity. The ratio between the registered peak pressures and the asymptotic theory are in accordance w i t h comparable experiments in the literature. The asymptotic theoty, however, is found to be quite conservative, since the measured peak pressure levels appear to be approximately 50% to 75% of the theoretical levels.

® 2009 Elsevier Ltd. All rights reserved.

1. I n t r o d u c t i o n A n e x p e r i m e n t a l t e s t p r o g r a m m e has b e e n e x e c u t e d t o i n v e -stigate b o t t o m s l a m m i n g p h e n o m e n a o n p o i n t absorbers. P o i n t a b s o r b e r s y s t e m s are w a v e e n e r g y c o n v e r t e r s c o n s i s t i n g o f o s c i -l -l a t i n g bodies w i t h h o r i z o n t a -l d i m e n s i o n s t h a t are s m a -l -l c o m p a r e d t o t h e i n c i d e n t w a v e l e n g t h . E x a m p l e s o f p o i n t a b s o r b e r devices are t h e FO^ [1] a n d W a v e Star E n e r g y [ 2 ] . The p o i n t a b s o r b e r b u o y s m o v e a c c o r d i n g t o one o r m o r e degrees o f f r e e d o m (heave, surge, p i t c h , r o l l ) as a response t o i n c o m i n g w a v e s a n d t h e i r k i n e t i c e n e r g y is t r a n s f e r r e d i n t o e l e c t r i c a l e n e r g y e i t h e r d i r e c t l y o r by m e a n s o f a h y d r a u l i c i n t e r m e d i a t e stage. Since t h e b u o y s g e n e r a l l y have a h i g h e r n a t u r a l f r e q u e n c y t h a n t h e d o m i n a n t i n c i d e n t w a v e f r e q u e n c i e s , t h e p o i n t a b s o r b e r response is o f t e n t u n e d t o t h e c h a r a c t e r i s t i c s o f t h e i n c o m i n g w a v e s p e c t r u m b y i n c r e a s i n g t h e s y s t e m i n e r d a o r b y a p p l y i n g l a t c h i n g c o n t r o l [ 3 ] . T h i s enables t h e p o i n t a b s o r b e r t o o p e r a t e closer t o resonance c o n d i t i o n s , w h i c h increases t h e e n e r g y c a p t u r e . H o w e v e r , i t m i g h t cause t h e b u o y s t o rise o u t o f t h e w a t e r w h i c h results i n s l a m m i n g back i n t o t h e w a t e r surface o n r e - e n t r y . This p h e n o m e n o n occurs p a r t i c u l a r l y f o r p o i n t absorbers w i t h a s m a l l d r a f t i n a n e n e r g e t i c w a v e c l i m a t e . S l a m m i n g can be r e d u c e d b y i n f l u e n c i n g t h e c o n t r o l p a r a m e t e r s o f t h e b u o y , i.e. b y i n c r e a s i n g t h e d a m p i n g a n d / o r b y d e t u n i n g

* Corresponding author. Tel.: +32 9 264 54 93; fax: +32 9 264 58 37.

E-mail address: griet.debacker@ugent.be (G. De Backer).

0141-1187/$ - see f r o n t matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.apor.2009.07.003 t h e b u o y . H o w e v e r , t h e s e measures r e s u l t i n p o w e r a b s o r p t i o n losses as s h o w n i n [ 4 , 5 ] . C o n s e q u e n t l y , a c e r t a i n l e v e l o f s l a m m i n g w i l l u s u a l l y be a l l o w e d . For t h i s reason i t is i m p o r t a n t t o k n o w t o w h i c h p r e s s u r e m a g n i t u d e s t h e b o d y is exposed w h e n s l a m m i n g occurs. This p a p e r a i m s t o i n v e s t i g a t e b o t t o m s l a m m i n g o n p o i n t absorbers b y means o f e x p e r i m e n t a l d r o p tests. The r e s u l t s are c o m p a r e d w i t h a n a l y t i c a l results o b t a i n e d b y C h u a n g [ 6 ] a n d F a l t i n s e n a n d Z h a o [ 7 ] . S l a m m i n g p h e n o m e n a have b e e n s t u d i e d o v e r s e v e r a l decades especially i n n a v a l h y d r o d y n a m i c s . P i o n e e r i n g r e s e a r c h has b e e n c a r r i e d o u t b y v o n K a r m a n [ 8 ] a n d W a g n e r [ 9 ] . W a g n e r s t u d i e d t h e w a t e r i m p a c t o n rigid t w o - d i m e n s i o n a l b o d i e s b y a p p r o x i m a t i n g t h e bodies w i t h a f l a t p l a t e a n d t a k i n g i n t o a c c o u n t t h e w a t e r u p r i s e o n t h e b o d y i n a s i m p l i f i e d w a y . Because o f t h e b l u n t b o d y a p p r o a c h , t h e bodies are a s s u m e d to have s m a l l d e a d r i s e angles i n t h e range o f 4 u p t o 2 0 degrees [ 1 0 ] . Zhao a n d F a l t i n s e n p r e s e n t e d n u m e r i c a l results, based o n t h e f i n d i n g s o f D o b r o v o l ' s k a y a [ 11 ] , f o r t w o - d i m e n s i o n a l bodies w i t h deadrise angles b e t w e e n 4 a n d 8 1 degrees [ 1 2 , 1 3 ] . I n s p i r e d b y Zhao's w o r k , M e i et al. [ 1 4 ] d e v e l o p e d a n a n a l y t i c a l s o l u t i o n f o r t h e w a t e r i m p a c t p r o b l e m o f g e n e r a l t w o -d i m e n s i o n a l bo-dies. T h e m a i n -d i f f e r e n c e w i t h t h e W a g n e r m e t h o -d is t h a t t h e exact b o d y b o u n d a r y c o n d i t i o n s are f u l f i l l e d , i n s t e a d o f a p p r o x i m a t i n g t h e b o d y b y a f l a t plate. T h e a d v a n t a g e o f W a g n e r ' s a p p r o x i m a t i o n is t h e a b i l i t y t o use a n a l y t i c a l e x p r e s s i o n s f o r t h e v e l o c i t y p o t e n t i a l . H o w e v e r , w i t h t h e g e n e r a l i z e d W a g n e r m e t h o d , a b r o a d e r range o f ( l o c a l ) deadrise angles can be i n v e s t i g a t e d i n a m o r e accurate w a y .

144 C. De Backer et al./Applied Ocean Research 31 (2009) 143-156 N o m e n c l a t u r e bo w e t r a d i u s at t h e u n d i s t u r b e d f r e e w a t e r surface [ m ] b w e t radius at t h e i m m e d i a t e f r e e w a t e r surface [ m ] s l a m m i n g pressure c o e f f i c i e n t [ - ] c o e f f i c i e n t o f v a r i a t i o n [ - ] w e t t i n g f a c t o r [ - ] F f o r c e [ N ] g g r a v i t a t i o n a l a c c e l e r a t i o n [ m / s ^ ] h d r o p h e i g h t [ m ] h* e q u i v a l e n t d r o p h e i g h t c o r r e s p o n d i n g t o t h e m e a -s u r e d i m p a c t v e l o c i t y [ m ] 'jet j e t h e i g h t [ m ] M b o d y mass [ k g ] i n f i n i t e f r e q u e n c y l i m i t o f t h e a d d e d mass [ k g ] P pressure [ b a r = 10^ Pa] r r a d i a l c o o r d i n a t e [ m ] radius o f h e m i s p h e r e [ m ] Pearson c o r r e l a t i o n c o e f f i c i e n t [ - ] t t i m e [s] U e n t r y v e l o c i t y [ m / s ] z v e r t i c a l c o o r d i n a t e [ m ] yfi deadrise angle [ d e g ] [ r a d ]

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w a t e r e l e v a t i o n at i n t e r s e c t i o n w i t h b o d y [ m ] P mass d e n s i t y o f f l u i d [ k g / m ^ ] A s u b s t a n t i a l a m o u n t o f e x p e r i m e n t a l w o r k has b e e n p e r f o r m e d t o v a l i d a t e t h e a n a l y t i c a l a n d n u m e r i c a l m o d e l s . L i n a n d S h i e h [ 1 5 ] e x p e r i m e n t a l l y i n v e s t i g a t e d t h e pressure c h a r a c t e r i s t i c s o f a c y l i n d e r d u r i n g w a t e r i m p a c t . T h e y also v i s u a l i z e d t h e f l o w p a t t e r n d u r -i n g p e n e t r a t -i o n b y m a k -i n g use o f a d -i g -i t a l -i m a g -i n g s y s t e m a n d a h i g h speed data a c q u i s i t i o n s y s t e m . Zhao a n d F a l t i n s e n [ 1 3 ] perf o r m e d d r o p tests t o s t u d y t w o d i m e n s i o n a l perf l o w s i t u a t i o n s o perf h o r -i z o n t a l w e d g e s d r o p p e d o n t o t h e f r e e w a t e r surface. E x p e r -i m e n t s b y Y e t t o u [ 1 6 ] et a l . consist o f f r e e f a l l d r o p tests o n s y m m e t r i -cal wedges. They i n v e s t i g a t e d t h e i n f l u e n c e o f t h e d r o p h e i g h t , the d e a d r i s e angle a n d t h e mass o f t h e w e d g e a n d c o m p a r e d t h e results w i t h e x i s t i n g m o d e l s f r o m IVlei et a l . [ 14] a n d Zhao a n d F a l t i n -sen [ 1 3 ] .

M o s t studies have f o c u s e d o n t w o d i m e n s i o n a l i m p a c t p r o b -l e m s since s -l a m m i n g o n ships has been a m a j o r c o n c e r n . H o w e v e r , t h e r e is a n e e d f o r t h r e e - d i m e n s i o n a l s o l u t i o n s because real i m p a c t p h e n o m e n a are t h r e e d i m e n s i o n a l . I n t h i s paper, v e r t i c a l s l a m m i n g o f t h r e e - d i m e n s i o n a l objects, m o r e s p e c i f i c a x i s y m m e t r i c bodies, is c o n s i d e r e d . Early studies i n t h i s area have b e e n p u b l i s h e d by S h i f f m a n a n d Spencer [ 1 7 , 1 8 ] . They i n v e s t i g a t e d v e r t i c a l s l a m m i n g p h e n o m e n a o n spheres a n d cones a n a l y t i c a l l y b y a p p r o x i m a t i n g t h e bodies as a lens a n d an e l l i p s o i d a n d p r e s e n t e d s o l u t i o n s f o r the i m p a c t f o r c e o n these a x i s y m m e t r i c o b j e c t s . W a g n e r ' s t h e o r y has b e e n e x t e n d e d t o a x i s y m m e t r i c bodies b y C h u a n g [ 6 ] a n d F a l t i n -sen a n d Zhao [ 7 ] . I n t h e case o f a x i s y m m e t r i c objects, t h e b o d y shape is a p p r o x i m a t e d w i t h a g r o w i n g flat disc analogous t o W a g -ner's flat p l a t e a p p r o x i m a t i o n f o r t w o - d i m e n s i o n a l shapes. Based o n t h i s p r i n c i p l e , C h u a n g [ 6 ] d e v e l o p e d an a n a l y t i c a l e x p r e s s i o n f o r t h e pressure d i s t r i b u t i o n o n a cone w i t h s m a l l deadrise an-gle. I n 1997 Faltinsen a n d Zhao [ 7 ] p r e s e n t e d an a s y m p t o t i c t h e o r y f o r w a t e r e n t r y o f h e m i s p h e r e s a n d cones w i t h s m a l l ( l o c a l ) d e a d -rise angles based o n t h e a s s u m p t i o n s b e h i n d t h e W a g n e r t h e o r y . A n o t h e r i m p o r t a n t c o n t r i b u t o r t o a x i s y m m e t r i c s l a m m i n g p r o b -l e m s is M i -l o h [ 1 9 - 2 1 ] w h o d e v e -l o p e d a n a -l y t i c a -l expressions f o r t h e s l a m m i n g forces o n a x i s y m m e t r i c bodies. One o f the m a i n d i f -ferences b e t w e e n his w o r k a n d W a g n e r ' s t h e o r y is t h a t t h e b o d y b o u n d a r y c o n d i t i o n s are s a t i s f i e d e x a c t l y o n t h e a c t u a l b o d y sur-face i n s t e a d o f o n a flat disc.

I n 2 0 0 3 B a t t i s t i n a n d l a f r a t i [ 2 2 ] n u m e r i c a l l y s t u d i e d i m p a c t loads a n d pressure d i s t r i b u t i o n s o n t w o - d i m e n s i o n a l a n d ax-i s y m m e t r ax-i c bodax-ies. T w o years later Faltax-insen a n d C h e z h ax-i a n [ 2 3 ] p r e s e n t e d a g e n e r a l i z e d W a g n e r m e t h o d f o r t h r e e - d i m e n s i o n a l s l a m m i n g based o n t h e a p p r o a c h p r e s e n t e d b y Zhao e t a l . [ 1 3 ] f o r t w o d i m e n s i o n a l w a t e r i m p a c t p r o b l e m s . To v a l i d a t e t h e n u m e r i -cal s i m u l a t i o n s , t h e y p e r f o r m e d d r o p tests o n a t h r e e - d i m e n s i o n a l s h i p l i k e c o m p o s i t e s t r u c t u r e f r o m w h i c h t h e y o b t a i n e d several f o r c e m e a s u r e m e n t s . Peseux, C o r n e t a n d D o n g u y [ 2 4 ] s o l v e d t h e t h r e e - d i m e n s i o n a l W a g n e r p r o b l e m n u m e r i c a l l y f o r b o t h r i g i d a n d d e f o r m a b l e bodies. T h e n u m e r i c a l m o d e l is v a l i d a t e d w i t h a n i n t e r e s t i n g e x p e r i m e n t a l i n v e s t i g a t i o n c o n s i s t i n g o f d r o p tests o f c o n -ical shapes w i t h s m a l l deadrise angles ( 6 ° — 1 0 ° — 1 4 ° ) . K i m a n d H o n g [ 2 5 ] n u m e r i c a l l y s t u d i e d t h e i m p a c t o f a r b i t r a r y t h r e e -d i m e n s i o n a l bo-dies w i t h a n e x t e n -d e -d v o n K a r m a n a n -d an e x t e n -d e -d W a g n e r a p p r o a c h , i n c l u d i n g t h e presence o f i n c o m i n g w a v e s . T h e y also p r e s e n t e d e x p e r i m e n t a l results o n t h e i m p a c t loads d u r i n g w a t e r e n t r y o f t h r e e - d i m e n s i o n a l s t r u c t u r e s .

V e r y f e w e x p e r i m e n t s are available f o r v a l i d a t i o n o f t h e o r e t ical pressure p r e d i c t i o n s f o r a x i s y m m e t r i c bodies. I n 1 9 6 1 N i s e -w a n g e r [ 2 6 ] p e r f o r m e d d r o p tests o n a l u m i n i u m h e m i s p h e r e s a n d m e a s u r e d pressure d i s t r i b u t i o n s w i t h s e l f m a d e p r e s s u r e t r a n s -ducers. For c o n i c a l shapes, e x p e r i m e n t a l research has been c a r r i e d o u t b y C h u a n g a n d M i l n e [ 2 7 ] i n 1971 a n d m o r e r e c e n t l y b y Peseux et a l . as m e n t i o n e d above. I n t h e f o r m e r s t u d y i m p a c t pressures are m e a s u r e d o n cone shapes w i t h s m a l l deadrise angles v a r y i n g f r o m 1° to 1 5 ° . P o i n t absorbers w i t h a c o n i c a l shape are v e r y l i k e l y t o h a v e l a r g e r deadrise angles ( > 2 0 ° ) . I n t h i s p a p e r t h e r e s u l t s o f n e w i m p a c t e x p e r i m e n t s o n a h e m i s p h e r e a n d o n c o n e shapes w i t h l a r g e r d e a d r i s e angles are presented.

2. E x p e r i m e n t a l d e s i g n

Z J . Test setup and test objects

Table 1 s h o w s t h e t h r e e d i f f e r e n t bodies t h a t h a v e b e e n t e s t e d : a h e m i s p h e r e a n d t w o cones w i t h deadrise angles o f 2 0 ° a n d 4 5 ° . The m o d e l s are m a d e f r o m p o l y u r e t h a n e a n d have a large t h i c k n e s s f r o m 30 m m t o 5 0 m m . As m e n t i o n e d i n Table 1 t h e d i a m e t e r o f t h e o b j e c t s is 0.30 m , w h i c h is s u f f i c i e n t t o r e d u c e s u r f a c e t e n s i o n e f f e c t s . The bodies are d r o p p e d i n a w a t e r basin w i t h h o r i z o n t a l d i m e n s i o n s o f 1.20 m b y 1.00 m a n d a h e i g h t o f 1.25 m . T w e l v e d i f f e r e n t d r o p h e i g h t s b e t w e e n 0.05 m a n d 2.00 m h a v e been e v a l u a t e d , c o r r e s p o n d i n g t o i m p a c t v e l o c i t i e s o f 1.0 m/s a n d 6.3 m/s. A r e a l i s t i c s t r o k e f o r a p o i n t absorber b u o y is a b o u t 5 t o 10 m . D e p e n d e n t o n t h e c o n t r o l parameters, a f r e e f a l l o f 2 m can be c o n s i d e r e d as an e x t r e m e case. S m a l l e r d r o p h e i g h t s w i l l o c c u r m o r e f r e q u e n t l y and are t h e r e f o r e r e l e v a n t as w e l l . Because o f reasons o f s i m i l i t u d e , t h e cone shape tests can be c o n s i d e r e d as f u l l - s c a l e tests, a p a r t f r o m t h e f a c t t h a t t h e masses are n o t c o r r e c t l y scaled. I n case o f t h e h e m i s p h e r e , t h e r e s u l t s f r o m t h e s m a l l e s t d r o p h e i g h t s (0.05 m - 0 . 2 0 m ) need t o be u p s c a l e d t o p r o t o t y p e values, a c c o r d i n g t o t h e d i m e n s i o n s o f a f u l l - s c a l e b o d y . E x p e c t e d scaling e f f e c t s m i g h t arise f r o m surface t e n s i o n a n d v i s c o u s e f f e c t s . For c o m p l e t e n e s s , the tests w i t h t h e h e m i s p h e r e are also p e r f o r m e d f o r l a r g e r d r o p h e i g h t s .

I n t h i s paper, t h e results o f an i m p r o v e d test s e t u p are p r e -s e n t e d . I n i t i a l l y t h e te-st-s w e r e c a r r i e d o u t w i t h o u t a n y g u i d i n g s t r u c t u r e . A l t h o u g h t h e test objects w e r e b a l a n c e d p r e c i s e l y , t h e scatter i n t h e m e a s u r e d data a p p e a r e d to be s i g n i f i c a n t . I n o r d e r t o p r e v e n t s m a l l r o t a t i o n s o f t h e floaters w h i l e f a l l i n g d o w n , t h e s e t u p w a s e q u i p p e d w i t h a g u i d i n g s y s t e m c o n s i s t i n g o f tightened steel w i r e s [ 5 ] . The results discussed i n t h i s paper, h o w e v e r , are o b t a i n e d f r o m a test s e t u p w i t h a n i m p r o v e d g u i d i n g s y s t e m . The t i g h t e n e d steel rods are r e p l a c e d by a r a i l m o u n t e d o n s t i f f a l u m i n i u m p r o f i l e s . The test bodies are a t t a c h e d t o a p r o f i l e s t r u c t u r e e q u i p p e d

C. De Backer et al. / Applied Ocean Research 31 (2009) 143-156 145 g u i d i n g system: p r o f i l e s w i t h r a i l wheels Table 1

Test object characteristics.

II

II

H fl

. II

/ B a l l a s t ^_ ^ . a b

Fig. 1 . Schematic view o f the experimental test setup [ m m ] .

Fig. 2. Picture o f t h e experimental test setup.

w i t h w h e e l s , r o l l i n g d o w n t h e r a i l as s h o w n i n Fig. 1. W i t h t h i s t e s t s e t u p t h e v e r d c a l i t y o f t l i e i m p a c t i n g o b j e c t is assured a n d t h e tests are v e r y w e l l r e p r o d u c i b l e . The masses m e n t i o n e d i n Table 1 c o r r e s p o n d t o the t o t a l f a l l i n g mass, i.e. t h e s u m o f t h e mass o f t h e p o l y u r e t h a n e bodies a n d t h e a l u m i n i u m carriage. The d r o p h e i g h t ,

h, is l i m i t e d to 2 m , c o m p a r e d t o 4 m f o r t h e o r i g i n a l test s e t u p .

A 10 m m plexiglass sheet is i n s t a l l e d i n the b a s i n w h i c h a l l o w s t o film t h e i m p a c t p h e n o m e n a . A p i c t u r e o f t h e t e s t s e t u p is g i v e n i n Fig. 2.

Test objects (dimensions in mm) Characteristics

1 4ÓT ,5f) 1 g

Hemisphere

Local deadrise angles: 7 . 7 ° and 18.4° Radius: 0.15 m Material thickness: 0.05 m Mass: 11.5 kg Cone Deadrise angle: 2 0 ° Max. radius: 0.15 m Material thickness: 0.03 m Mass: 9.8 kg Cone Deadrise angle: 4 5 ° Max. radius: 0.15 m Material thickness: 0.03 m Mass: 10.2 kg 1 40 1 60 Table 2 Sensor characteristics.

Sensor Measurement range Resonance frequency (kHz)

A07 3.45 bar > 250

K30, K31 2 bar > 150

Shock accelerometer 500 g > 54

2.2. Instrumentation

2.2.1. Pressure sensors and shock accelerometer

T h e p r e s s u r e t i m e h i s t o r y , t h e p o s i t i o n a n d d e c e l e r a t i o n o f t h e b o d y w e r e r e c o r d e d d u r i n g i m p a c t . T h r e e h i g h f r e q u e n c y p i e z o e l e c t r i c p r e s s u r e sensors w e r e used. One ICP p r e s s u r e s e n s o r ( A 0 7 ) has a b u i l t - i n m i c r o e l e c t r o n i c a m p l i f i e r w h i l e t w o o t h e r h i g h f r e q u e n c y pressure sensors ( K 3 0 , K 3 1 ) h a v e e x t e r n a l a m p l i f i e r s . T h e m e a s u r e m e n t range f o r these devices is 3.45 b a r a n d 2 bar, r e s p e c t i v e l y . The pressure cells have a s m a l l d i a p h r a g m o f 5.5 m m a n d a v e r y h i g h r e s o n a n c e f r e q u e n c y , see Table 2. C o n s e q u e n t l y t h e sensors are v e r y w e l l s u i t e d f o r m e a s u r i n g i m p a c t p h e n o m e n a . T h e sensors are flush-mounted at a h o r i z o n t a l d i s t a n c e o f 0.04 m a n d 0.09 m , r e s p e c t i v e l y f r o m t h e s y m m e t r y axis, as i l l u s t r a t e d i n Table 1. T h e d e c e l e r a t i o n o f t h e o b j e c t d u r i n g i m p a c t w a s m e a s u r e d b y a s h o c k a c c e l e r o m e t e r w i t h a m e a s u r e m e n t r a n g e u p t o 5 0 0 g a n d a r e s o n a n c e f r e q u e n c y o f 5 4 k H z .

Fig. 3 s h o w s t h e c o n f i g u r a t i o n o f t h e p r e s s u r e cells. T h e first t h r e e c o n f i g u r a t i o n s ( a - c ) r e p r e s e n t t h e sensor p o s i t i o n s f o r t h e h e m i s p h e r e . T h e sensors i n Fig. 3(a) are m o u n t e d o n t w o o p p o s i t e m e r i d i a n s i n o r d e r to e v a l u a t e t h e v e r d c a l i t y o f t h e p e n e t r a t i o n . W i t h t h e c o n f i g u r a t i o n i n Fig. 3 ( b ) a c o m p a r i s o n b e t w e e n t h e t w o l o c a l d e a d r i s e angles can be m a d e a n d i n Fig. 3(c) t h e s a m p l i n g f r e q u e n c y is increased u p t o 100 k H z f o r o n e p r e s s u r e sensor a n d t h e s h o c k a c c e l e r o m e t e r . I n Fig. 3 ( d ) a n d ( e ) t h e c o n f i g u r a t i o n o f t h e p r e s s u r e sensors is g i v e n f o r t h e 2 0 ° cone. I n each c o n f i g u r a t i o n t w o d i f f e r e n t p r e s s u r e sensors are m o u n t e d o n m e r i d i a n s close t o each o t h e r , a l l o w i n g f o r t h e assessment o f t h e d i f f e r e n t sensors. Fig. 3 ( f ) s h o w s t h e p r e s s u r e sensor c o n f i g u r a t i o n i n case o f t h e 4 5 ° cone, w h i c h is s i m i l a r t o Fig. 3(a) c o m b i n e d w i t h ( b ) . Each case has b e e n t e s t e d at least t h r e e t i m e s f o r e v e r y d r o p h e i g h t , v a r y i n g b e t w e e n 0.05 m a n d 2 m .

A s a m p l i n g f r e q u e n c y (SF) o f at least 30 k H z w a s u s e d f o r r e c o r d i n g . Such h i g h s a m p l i n g f r e q u e n c i e s are r e q u i r e d , since t h e

1 4 6 _{G. De Backer et al./Applied Ocean Research 31 (2009) 143-156} 3 0 0 3 0 0 Hemisphere 3 0 0 3 0 0 300 Cone 2 0 ° Cone 4 5 °

Fig. 3. Pressure sensor positions [ m m ] for tlie hemisphere: (a) Sensors K30A and K31A - SF = 30 kHz, (b) Sensors K30B, K31B and A07B - SF = 30 kHz, (c) Sensor K31C - SF = 100 kHz, f o r the 2 0 ° cone : (d) Sensors K30A, K31A and A07A - SF = 30 kHz, (e) Sensors K30B and A07B - S F = 30 kHz, for the 4 5 ° cone: ( f ) Sensors K30, K31 and A07 - SF = 30 kHz. 4 3.5 3 is 2.5 £ 1.5 1 0.5 0 1 0.03 Asymptotic theory - 1 = 0.002 s - h = 2 m 0.035 0.04 r [ m ] 0.045

Fig. 4. Theoretical pressure d i s t r i b u t i o n as a f u n c t i o n of r for a cone w i t h deadrise angle 2 0 ° and drop height 2 m .

p r e s s u r e peal<s o c c u r i n a v e r y s m a l l t i m e i n t e r v a l ( o r d e r o f m a g n i -t u d e m i l l i s e c o n d s ) . For -t h e same r e a s o n -t h e r e s o n a n c e f r e q u e n c y o f t h e sensors s h o u l d be h i g h e n o u g h . A s m a l l pressure cell d i -a p h r -a g m -are-a is necess-ary since t h e pressure pe-aks -are -also v e r y m u c h l o c a l i z e d i n space as w e l l , as can be seen i n Fig. 4, s h o w i n g the t h e o r e t i c a l l y p r e d i c t e d p r e s s u r e d i s t r i b u t i o n a c c o r d i n g t o a s y m p -t o -t i c -t h e o r y a-t -t = 0 . 0 0 2 s f o r a cone w i -t h deadrise angle 2 0 ° a n d d r o p h e i g h t 2 m .

T a b l e s

Influence of pressure sensor diameter: estimated deviations f r o m peak pressure for drop heights of 1 m and 4 m .

Sensor diameter ( m m ) h = 1 m (%) ft = 4 m (%)

5.5 10.8 13.9

19 30.5 34.2

I n e a r i i e r i n v e s t i g a t i o n s , sensors w i t h l a r g e r d i a m e t e r s have s o m e t i m e s b e e n used, w i t h values u p t o 19 m m i n [ 1 6 ] . I n t h a t case t h e pressure peaks m i g h t have a s m a l l e r s p a t i a l e x t e n t t h a n t h e sensor area. Even p r e s s u r e cells w i t h d i a m e t e r 5.5 m m m i g h t m e a s u r e a space-averaged pressure, w h i c h is s l i g h t i y d i f f e r e n t f r o m t h e p e a k pressure. T h e p r e s s u r e d i s t r i b u t i o n is p a r t i c u l a r l y m o r e p e a k e d w h e n t h e ( l o c a l ) deadrise angle is s m a l l a n d t h e i m p a c t v e l o c i t y h i g h . A s s u m i n g t h a t a pressure cell r e g i s t e r s t h e space-averaged pressure w h e n s u b j e c t to a n o n - u n i f o r m p r e s s u r e d i s t r i b u t i o n , t h e d e v i a t i o n b e t w e e n t h e p e a k pressure a n d t h e sensor r e c o r d c a n be d e t e r m i n e d . I n [ 2 3 ] , F a l t i n s e n e s t i m a t e d t h a t t h e t h e o r e t i c a l peak pressure is at m a x i m u m 11% h i g h e r t h a n t h e space-averaged pressure, m e a s u r e d b y a sensor w i t h a d i a m e t e r o f 4 m m . D e v i a t i o n s o f t h e same m a g n i t u d e can be d e r i v e d , based o n t h e t h e o r e t i c a l l y p r e d i c t e d pressure d i s t r i b u t i o n b y t h e t h r e e -d i m e n s i o n a l a s y m p t o t i c t h e o r y . For pressure cells w i t h -d i a m e t e r 5.5 m m i t is e s t i m a t e d w i t h t h e l a t t e r m e t h o d t h a t t h e m e a s u r e d pressure o n a cone w i t h d e a d r i s e angle 2 0 ° deviates b e t w e e n 10% a n d 14% f r o m t h e peak pressure f o r d r o p h e i g h t s o f 1 m a n d 4 m . I n a s i m i l a r w a y as above, i t is e x p e c t e d t h a t a pressure sensor w i t h a d i a m e t e r o f 19 m m , w o u l d u n d e r e s t i m a t e t h e peak p r e s s u r e w i t h m o r e t h a n 30% f o r the same case o f a cone w i t h deadrise a n g l e 2 0 ° , as s h o w n i n Table 3.

C. De Backer et al. /Applied Ocean Research 31 (2009) 143-156 147

2.2.2. High speed camera

A h i g h speed c a m e r a w a s u s e d t o r e c o r d t h e p e n e t r a t i o n o f t h e i m p a c t i n g bodies as a f u n c t i o n o f t i m e . The camera p r o v i d e d i n f o r m a t i o n o n t h e w a t e r u p r i s e a l o n g t h e b o d y a n d o n the p o s i t i o n a n d v e l o c i t y o f t h e i m p a c t i n g b o d y . For t h i s p u r p o s e a m a r k e r t r a c k i n g t e c h n i q u e has b e e n a p p l i e d . The h i g h speed camera is able to d e l i v e r images u p t o 2 5 0 0 0 0 f r a m e s per second ( f p s ) a n d has f u l l m e g a p i x e l r e s o l u t i o n at 3 0 0 0 f p s . I n t h i s test case, i t has b e e n used at 5 0 0 0 u p t o 18 0 0 0 f p s , d e p e n d e n t o n the desired p i x e l r e s o l u t i o n . Because o f t h e h i g h f r a m e rate, t h e camera s h u t t e r t i m e is e x t r e m e l y s h o r t . I n o r d e r t o o v e r c o m e l o w i l l u m i n a t i o n a n d to a v o i d i n t e r f e r e n c e w i t h t h e g n d f r e q u e n c y , special f l i c k e r f r e e l i g h t s have b e e n used. T w o lasers are m o u n t e d o n t o p o f t h e basin a n d serve as a t r i g g e r f o r the data a c q u i s i t i o n s y s t e m . W h e n t h e d r o p p e d o b j e c t s i n t e r s e c t t h e laser beams, t h e r e c o r d i n g o f t h e pressure sensors, a c c e l e r o m e t e r a n d c a m e r a s i g n a l starts a u t o m a t i c a l l y .

3. A n a l y t i c a l f o r m u l a t i o n

The e x p e r i m e n t a l results are c o m p a r e d w i t h e x i s t i n g a s y m p -t o -t i c s o l u -t i o n s based o n -t h e classical W a g n e r m e -t h o d e x -t e n d e d -t o a x i s y m m e t r i c bodies, as i t w a s p r o p o s e d b y C h u a n g [ 6 ] a n d F a l t i n -sen et a l . [ 7 ] . Despite t h e i n t e r e s t i n g w o r k t h a t has a l r e a d y been c a r r i e d o u t i n the f i e l d o f w a t e r i m p a c t , W a g n e r ' s m e t h o d is e v e n n o w a d a y s s t i l l v e r y v a l u a b l e , since i t p r o d u c e s a n a l y t i c a l f o r m u -las t h a t are easy t o h a n d l e a n d g i v e a v e i y g o o d first i n s i g h t i n t o t h e p r o b l e m . The f l u i d flow is d e s c r i b e d b y p o t e n t i a l t h e o r y a n d a c o n s t a n t e n t r y v e l o c i t y U is a s s u m e d . The i n i t i a l time i n s t a n t fo is d e f i n e d as t h e time w h e r e the b o d y touches t h e c a l m w a t e r s u r face. A t a t i m e t , t h e p e n e t r a t i o n d e p t h r e l a t i v e t o t h e c a l m w a -t e r surface (z = 0) equals U-t a n d -t h e c o r r e s p o n d i n g i n s -t a n -t a n e o u s r a d i u s at t h e w e t s e c t i o n o f t h e cone is fao(t), as s h o w n i n Fig. 5. T h e i n s t a n t a n e o u s radius b ( t ) at t h e i n t e r s e c t i o n p o i n t b e t w e e n t h e b o d y a n d the w a t e r is f o u n d b y i n t e g r a t i n g t h e v e r t i c a l v e l o c -i t y o f t h e w a t e r part-icles at z = 0. For a cone shape t h -i s results -i n

b(t) = 4Ut/ {n t a n / 3 ) [ 7 ] . I t s h o u l d be m e n t i o n e d t h a t Fig. 5 gives

a s i m p l i f i e d p r e s e n t a t i o n o f t h e w a t e r u p r i s e , since i n r e a l i t y a j e t flow occurs w h i c h m i g h t e n d i n a spray, d e p e n d i n g o n the c o n v e x -i t y o f t h e o b j e c t .

The pressure o n a cone shape w i t h deadrise angle ;S, at a c e r t a i n d i s t a n c e r f r o m t h e s y m m e t r y axis, is expressed b y : 1 9 Vcone = - p L f 1 -6 4 j r ^ t a n ^ ^ 16

_iwrY

( 1 )

Eq. ( 1 ) is c o m p o s e d o f t h r e e t e r m s . The first t e r m expresses t h e s t a g n a t i o n pressure. The s e c o n d t e r m is a consequence o f t h e p e r m a n e n t flow a r o u n d t h e disc a n d t h e t h i r d t e r m accounts f o r t h e e x p a n s i o n o f the disc, r e p r e s e n t i n g t h e e f f e c t o f t h e n o n - s t a t i o n a r y b e h a v i o u r o f the f l o w a r o u n d t h e disc. As m e n t i o n e d b e f o r e , t h e b l u n t b o d y a s s u m p t i o n i n W a g n e r ' s m e t h o d i m p l i e s t h a t bodies s h o u l d have s m a l l local deadrise angles. I n t h e l i t e r a t u r e , i t is s t a t e d t h a t t h e classical W a g n e r t h e o r y gives q u i t e accurate results f o r w e d g e s w i t h deadrise angles i n t h e range o f 4 to 2 0 degrees [ 1 3 ] . W h e n deadrise angles are s m a l l e r t h a n 4 degrees, a n air c u s h i o n is f o r m e d , w h i c h reduces t h e pressure o n t h e s t r u c t u r e a n d as a r e s u l t , W a g n e r t h e o r y o v e r e s t i m a t e s t h e pressure b y a large m a r g i n .

For a h e m i s p h e r e t h e r e l a t i o n s h i p b e t w e e n t h e p e n e t r a t i o n d e p t h a n d i n s t a n t w e t radius b is n o t as s t r a i g h t f o r w a r d as i t is f o r a cone shape. Faltinsen a n d Zhao [7 ] suggested a q u a d r a t i c r e l a t i o n b e t w e e n Ut a n d b w h i c h is o n l y v a l i d f o r s m a l l s u b m e r g e n c e s

Fig. 5. Cone penetrating t h r o u g h originally calm water: clarification o f parameters.

(Ut/R < 1 / 5 ) : b = V^RUt. The pressure o n a n i m p a c t i n g

h e m i s p h e r e w i t h r a d i u s R, at a distance r f r o m t h e s y m m e t r y axis, is expressed as f o l l o w s :

Phemisphere = 2^^^ 1 - (2)

The m e a s u r e d p e n e t r a t i o n a n d a c c e l e r a t i o n w i l l be c o m p a r e d w i t h t h e o r e t i c a l values t h a t are based o n t h e c o m p u t a t i o n o f t h e h y d r o d y n a m i c i m p a c t f o r c e , F3, a c t i n g o n a b o d y p e n e t r a t i n g t h e f r e e w a t e r surface. This f o r c e is c a l c u l a t e d i n t w o w a y s . By m a k i n g use o f the a d d e d mass t h e o r e m ( A M ) , F3 can be expressed as:

d{Ma,,U) d h diMo33 dz

(3) d t ^'dt2 d t d t

w h e r e Ma^^ is t h e h i g h f r e q u e n c y l i m i t o f t h e a d d e d mass. The second t e r m i n Eq. ( 3 ) can also be c o m p u t e d b y i n t e g r a t i o n o f t h e pressures g i v e n i n Eqs. ( 1 ) a n d ( 2 ) . This w i l l be r e f e r r e d t o as t h e pressure i n t e g r a t i o n (PI) m e t h o d . W h e n F3 is k n o w n , t h e accelera-t i o n aaccelera-t each accelera-t i m e saccelera-tep is d e r i v e d and accelera-t h e v e l o c i accelera-t y a n d p e n e accelera-t r a accelera-t i o n d e p t h are o b t a i n e d b y n u m e r i c a l i n t e g r a t i o n o f t h e a c c e l e r a t i o n . 4. E x p e r i m e n t a l test r e s u l t s

4 . 1 . Water uprise and impact velocity

Fig. 6 s h o w s a selected n u m b e r o f images o f a h e m i s p h e r e p e n e t r a t i n g t h e f r e e w a t e r surface, d r o p p e d f r o m 1 m . A s o f t w a r e p r o g r a m recognizes t h e p a t t e r n o f t h e m a r k e r a n d d e t e r m i n e s its c o o r d i n a t e s at each t i m e step. C o n s e q u e n t l y t h e p o s i t i o n o f the b o d y is k n o w n as a f u n c t i o n o f time a n d t h e v e l o c i t y can be d e t e r m i n e d . The p i c t u r e s c l e a r i y s h o w t h e w a t e r u p r i s e a l o n g t h e h e m i s p h e r e . The j e t flow is q u i c k l y d e t a c h e d f r o m t h e b o d y s u r f a c e e n d i n g u p i n a spray. This p h e n o m e n o n has also b e e n o b s e r v e d f o r c y l i n d e r s b y G r e e n h o w a n d L i n i n [ 2 8 ] a n d [ 2 9 ] . Figs. 7 a n d 8 s h o w c a m e r a i m a g e s o f t h e i m p a c t i n g cones f o r a d r o p h e i g h t o f 1 m . The c r e a t i o n a n d p r o p a g a t i o n o f a j e t a l o n g t h e cone s u r f a c e can be c l e a r l y seen a n d m e a s u r e d . F r o m t h e p h o t o g r a p h s o f t h e cones t h e r a t i o C„, can be d e -t e r m i n e d a n d c o m p a r e d w i -t h -t h e o r e -t i c a l values. The C„, f a c -t o r is d e f i n e d as t h e r a t i o b e t w e e n the h e i g h t s o f t h e i m m e d i a t e a n d u n d i s t u r b e d f r e e w a t e r surfaces m e a s u r e d f r o m t h e b o t t o m p o i n t o f t h e f a l l i n g o b j e c t :

c.. = . +

i

^

w i t h f t h e z c o o r d i n a t e o f t h e i n t e r s e c t i o n p o i n t b e t w e e n t h e o b -j e c t a n d t h e f r e e w a t e r surface, see Fig. 5. W h e n flow s e p a r a t i o n occurs above f , as i n t h e case o f the h e m i s p h e r e , Cy, has t h e p h y s -ical m e a n i n g o f a w e t t i n g f a c t o r . H o w e v e r , i n t h e case o f a cone a t h i n j e t flow m i g h t o c c u r above this i n t e r s e c t i o n p o i n t as o b s e r v e d i n Figs. 7 a n d 8. The w e t t i n g f a c t o r C„; f o r a cone w i t h a t t a c h e d j e t flow can t h e n be d e f i n e d as:

C„ 1 +

? ( b , 0 + /jer

Ut

_bo

Ut

(5) w h e r e /jet is t h e h e i g h t o f t h e j e t . C o n s i d e r i n g t h e o u t e r flow

G. De Backer et al./Applied Ocean Research 31 (2009) 143-156 149

Fig. 8. Cone ( f t

t . = 0.028 s.

4 5 ° ) penetrating the w a t e r - (a) t = 0.000 s, (b) f = 0.004 s, (c) t = 0.008 s, (d) t = 0.012 s, (e) f = 0.016 s, ( f ) t = 0.020 s, (g) f = 0.024 s, (h)

d o m a i n , Faltinsen et al. [ 7 ] f o u n d a r a t i o b/bo e q u a l t o 4/7t f o r cones based o n W a g n e r ' s b l u n t b o d y a p p r o a c h . By m a t c h i n g t h e o u t e r t h r e e - d i m e n s i o n a l s o l u t i o n f o r a x i s y m m e t r i c f l o w w i t h t h e i n n e r t w o - d i m e n s i o n a l j e t f l o w s o l u t i o n b y W a g n e r , F a l t i n s e n d e s c r i b e d t h e j e t f l o w d u r i n g w a t e r e n t r y o f a cone. Based o n F a l t i n sen's c o n s i d e r a t i o n s , t h e h e i g h t o f t h e j e t is f o u n d t o be —cos/?, r e -s u l t i n g i n a w e t d n g f a c t o r C „ e q u a l t o ^ ( 1 -j-co-s/S) f o r a cone w i t h a t t a c h e d j e t f l o w . The f o r m u l a b y F a l t i n s e n et a l . [ 7 ] is s l i g h t l y d i f f e r e n t f r o m the l a t t e r , p r o b a b l y d u e to a t y p i n g e r r o r i n [ 7 ] . I n n u -m e r i c a l -m o d e l s t h a t s a t i s f y t h e r e a l b o d y b o u n d a r y c o n d i d o n s , t h e d e s c r i p t i o n o f the j e t f l o w can be v e r y c o m p l e x . Zhao a n d F a l t i n -sen [ 1 2 ] d e v e l o p e d a n u m e r i c a l m o d e l t h a t s i g n i f i c a n t l y s i m p l i f i e s t h e d e s c r i p t i o n o f t h e j e t f l o w . This a p p r o a c h has been a d o p t e d b y B a t t i s d n a n d l a f r a t i [ 2 2 ] w h o d e t e r m i n e d t h e w a t e r surface e l e -v a t i o n n u m e r i c a l l y f o r a x i s y m m e t r i c bodies, a m o n g t h e m a cone w i t h d e a d r i s e angle 3 0 ° . H o w e v e r , t h e j e t s are t r u n c a t e d at t h e t o p , w h i c h m a k e s i t i m p o s s i b l e t o d e r i v e t h e c o r r e c t w e t d n g f a c t o r . Figs. 9 a n d 10 i l l u s t r a t e t h e w e t t i n g f a c t o r as a f u n c t i o n o f p e n e t r a -t i o n d e p -t h f o r -three d i f f e r e n -t d r o p h e i g h -t s f o r -t h e 4 5 ° cone a n d t h e 2 0 ° cone, r e s p e c t i v e l y . The v a l u e o f C„, is r e l a t i v e l y c o n s t a n t d u r i n g p e n e t r a t i o n , a l t h o u g h i n b o t h cases s l i g h t l y h i g h e r v a l u e s are m e a s u r e d f o r s m a l l p e n e t r a t i o n d e p t h s . F u r t h e r m o r e t h e i n f l u -ence o f t h e d r o p h e i g h t appears t o be n o t v e r y s i g n i f i c a n t a n d a s m a l l e r w e t t i n g f a c t o r is f o u n d f o r the h i g h e s t d e a d r i s e angle. O n average t h e m e a s u r e d values are 19% a n d 23% s m a l l e r t h a n t h e v a l -ues f o u n d b y F a l t i n s e n e t al. f o r t h e 4 5 ° a n d 2 0 ° cone, r e s p e c t i v e l y . For t h e h e m i s p h e r e i t is n o t possible t o d e r i v e t h e w e t t i n g f a c t o r b y m e a n s o f t h e c a m e r a images, since i t is d i f f i c u l t t o c o r r e c t l y d e t e r -m i n e t h e i n t e r s e c t i o n p o i n t b e t w e e n t h e f r e e w a t e r s u r f a c e a n d t h e b o d y , d u e to t h e d i s t u r b i n g e f f e c t o f t h e t h r e e - d i m e n s i o n a l spray. I n o r d e r t o b e t t e r v i s u a l i z e t h e f l o w s e p a r a t i o n at t h e h e m i s p h e r e , i t w o u l d be necessary t o create a l i g h t sheet t h r o u g h t h e s y m m e t r y axis o f t h e h e m i s p h e r e b y m e a n s o f a s t r o n g laser. I n t h a t case t h e w a t e r s p r a y p a r t i c l e s i n f r o n t o f t h e h e m i s p h e r e are n o t i l l u m i n a t e d a n d d o n o t d i s t u r b the m e a s u r e m e n t .

Fig. 11 i l l u s t r a t e s t h e v e l o c i t y d u r i n g t h e i n i t i a l i m p a c t stage d e t e r m i n e d b y t h e h i g h speed camera as a f u n c t i o n o f t h e e n t r y d e p t h . For each shape t h r e e i n i t i a l v e l o c i t i e s , Lfo, are c o n s i d e r e d :

ISO

O 2.5

1.5

0.5

C. De Backer et al./Applied Ocean Research 31 (2009) 143-156 6 | O * O d : * oa * o <?P : * o O h = 0.50 m - p = 45° • h = 1 . 0 0 m - p = 45° * h = 1 . 7 5 m - p = 45° C^ = 4(1 + cos (7t/4))/jt 0 0.02 0.04 0.06 0.08 0.1 Ut [m]

Fig. 9. W e t t i n g factor as a f u n c t i o n o f penetration depth on the 4 5 ° cone.

O 2.5 1.5 0.5 , 9 o h = 0.50 m - P = 20° • h = 1.00 m - P = 20°

*

h = 1.75 m - P = 20° - C w = 4(1 + cos (;t/9))/7c 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 U t [ m ]

Fig. 10. W e t t i n g factor as a f u n c t i o n o f penetration d e p t h on the 2 0 ° cone.

Uo < 3 m / s , (Jo « 4 m / s a n d UQ > 4.6 m / s . A l t h o u g h t h e mass o f t h e h e m i s p h e r e is t h e largest o f t h e t h r e e t e s t e d objects, t h e v e l o c i t y decrease d u r i n g t h e i n i t i a l stage o f t h e i m p a c t is m o s t p r o n o u n c e d f o r t h i s shape. This is especially t h e case f o r h i g h e r d r o p h e i g h t s , c o r r e s p o n d i n g t o larger values o f UQ a n d c o n s e q u e n t l y h i g h e r i m p a c t forces.

For t h e 4 5 ° c o n e t h e s l a m m i n g f o r c e is so s m a l l t h a t t h e i m p a c t v e l o c i t y r e m a i n s q u i t e c o n s t a n t . N o t e t h a t t h e v e l o c i t y increases f o r b o t h cone shapes i n p a r t i c u l a r f o r s m a l l values o f UQ. I m m e d i a t e l y a f t e r c o n t a c t i n g the w a t e r surface, the i m p a c t forces o n t h e cones are s t i l l r a t h e r s m a l l c o m p a r e d t o t h e g r a v i t y f o r c e . For t h i s r e a s o n t h e i m p a c t v e l o c i t y f i r s t b u i l d s u p f o r a v e r y s h o r t p e r i o d o f t i m e b e f o r e s t a r t i n g t o decrease. The g r a p h i l l u s t r a t e s t h a t t h e a s s u m p -tion o f a c o n s t a n t e n t r y v e l o c i t y can be b e t t e r j u s t i f i e d f o r s m a l l e r i n i t i a l v e l o c i t i e s UQ. The r e c o r d e d v e l o c i t y t i m e h i s t o r y is s h o r t f o r t h e 2 0 ° cone, because t h e m a r k e r p a t t e r n b e c o m e s q u i c k l y u n c l e a r d u e to t h e w a t e r uprise. Longer v e l o c i t y t i m e h i s t o r i e s are o b t a i n e d w i t h t h e a c c e l e r o m e t e r , as w i l l be i l l u s t r a t e d i n S e c t i o n 4.2.

4.2. Pressure distribution, impact velocity and deceleration

4.2.1. Hemisphere

Figs. 12 a n d 13 s h o w the pressure c o e f f i c i e n t Cp = P/(0.5PUQ) o n the h e m i s p h e r e as a f u n c t i o n o f t i m e f o r a d r o p h e i g h t o f 1 m at 5.5 ' Hemisphere • Cone 20° • Cone 4 5 ° 0.01 0.02 0.03 0.04 Entry depth [m] 0.05 0.06

Fig. 11. Velocity measured by the high speed camera f o r three d i f f e r e n t initial impact velocities f o r each shape.

r = 0 . 0 4 m a n d r = 0.09 m , r e s p e c t i v e l y . T h e i n i t i a l t i m e is d e f i n e d as t h e m o m e n t w h e r e t h e b o t t o m o f the h e m i s p h e r e t o u c h e s t h e w a t e r s u r f a c e . T h e p r e s s u r e m e a s u r e m e n t s are c o m p a r e d w i t h t h e a s y m p t o t i c s o l u t i o n s , a s s u m i n g a c o n s t a n t e n t r y v e l o c i t y . T h e f i g u r e i n d i c a t e s t h a t t h e a s y m p t o t i c t h e o r y o v e r e s t i m a t e s t h e pressures s i g n i f i c a n t l y , p a r t i c u l a r l y f o r s m a l l local d e a d r i s e angles. This w a s also o b s e r v e d i n t h e e x p e r i m e n t s o f L i n a n d S h i e h [ 1 5 ] f o r a c y l i n d e r . T h e p r e s s u r e p r o f i l e s i n d i c a t e t h a t s m a l l e r local d e a d r i s e angles l e a d t o h i g h e r pressures w h i c h h a v e a s h o r t e r d u r a t i o n i n t i m e . T h e rising t i m e o f t h e f i r s t p r e s s u r e p e a k (Fig. 12) is o n l y 0.2 m s . D u e t o the decrease i n v e l o c i t y , t h e time intei-val b e t w e e n t h e m e a s u r e d p r e s s u r e peaks is l a r g e r t h a n b e t w e e n t h e t h e o r e t i c a l l y p r e d i c t e d peaks. F u r t h e r m o r e i t can be n o t e d t h a t t h e pressure d i s t r i b u t i o n o f t h e f o u r sensors at r = 0 . 0 4 m o b t a i n e d f r o m t h e t h r e e d i f f e r e n t test c o n f i g u r a t i o n s i n Fig. 3 ( a - c ) c o i n c i d e v e r y w e l l . T h i s i m p l i e s f i r s t l y t h a t t h e h e m i s p h e r e p e n e t r a t e d p e r f e c t l y a l o n g a v e r t i c a l l i n e a n d s e c o n d l y t h a t a s a m p l i n g f r e q u e n c y o f 30 k H z is s u f f i c i e n t l y large since n o h i g h e r p e a k has b e e n r e g i s t e r e d at 100 k H z . Figs. 1 4 - 1 6 s h o w t h e m e a s u r e d a n d t h e o r e t i c a l a c c e l e r a t i o n , v e l o c i t y a n d e n t r y d e p t h , r e s p e c t i v e l y . T h e t h e o r e t i c a l values are based o n t h e pressure i n t e g r a t i o n m e t h o d (PI) a n d a d d e d mass m e t h o d ( A M ) as e x p l a i n e d i n S e c t i o n 3. T h e p r e s e n t e d v e l o c i t y a n d p o s i t i o n d a t a f r o m t h e h i g h speed c a m e r a are m e a s u r e d at 1 8 0 0 0 f p s f o r t h e t h r e e shapes. The a c c e l e r a t i o n s i g n a l i n Fig. 14 is d i s t u r b e d b y a h i g h f r e q u e n c y noise, p r o b a b l y o r i g i n a t i n g f r o m o s c i l l a t i o n s o f t h e h o r i z o n t a l a l u m i n i u m b e a m since t h e noise w a s n o t r e g i s t e r e d i n the o r i g i n a l s e t u p . N e v e r t h e l e s s t h e a c c e l e r o m e t e r s i g n a l is s t i l l v a l u a b l e , as can be seen i n Fig. 15. T h e v e l o c i t y , based o n n u m e r i c a l i n t e g r a t i o n o f t h e a c c e l e r o m e t e r s i g n a l c o i n c i d e s v e r y w e l l w i t h t h e v e l o c i t y d e r i v e d f r o m t h e h i g h s p e e d c a m e r a images. The t h e o r e t i c a l v e l o c i t i e s d r o p m o r e q u i c k l y , w h i c h is d u e t o t h e f a c t t h a t t h e forces a n d c o n s e q u e n t l y t h e accelerations are o v e r e s t i m a t e d b y b o t h m e t h o d s . T h e m e a s u r e d i n i t i a l v e l o c i t y is 4.0 m/s, w h e r e a s t h e c a l c u l a t e d speed Uo =

y/2gh w o u l d be 4.4 m / s . This d i f f e r e n c e can be a t t r i b u t e d m a i n l y

to f r i c t i o n i n the g u i d i n g s y s t e m . For this reason a l l t h e o r e t i c a l v a l u e s are c a l c u l a t e d based o n t h e m e a s u r e d i n i t i a l s p e e d . N o t e t h e v e r y s h o r t time s p a n o f 12 m s i n t h e p l o t s . I n t h i s t i m e s p a n t h e h e m i s p h e r e has r e a c h e d a s u b m e r g e n c e o f a b o u t R/3 (see Fig. 16) a n d t h e r e l e v a n t i m p a c t p h e n o m e n a have o c c u r r e d .

O 35 30 25 20 15 10 5 O - 5 1 1 ,,: Exp - K30A - r = 4 cm • E x p - K 3 1 A - r = 4 c m 1 1 i E x p - K 3 1 B - r = 4 c m E x p - K 3 1 C - r = 4 c m Asymptotic theory - r = 4 cm • } " i' i' E x p - K 3 1 B - r = 4 c m E x p - K 3 1 C - r = 4 c m Asymptotic theory - r = 4 cm • } " i' i'

\

: : , . :

1 ;

J

V • V v ^ ^ ' ^ -: •

G. De Backer el al. /Applied Ocean Research 31 (2009) 143-156 600 I

151

0.002 0.004 0.006 0.008 Time [s]

Fig. 12. Measured and calculated pressure distribution on the hemisphere at r 0.04 m for Uo = 4.0m/s. CL O 3 5 30 25 20 15 10 Exp - A07B - r = 9 cm • Exp - K30B - r = 9 cm •Asymptotic theory - r = 9 cm 0 0.002 0.004 0.006 0.008 0.01 0.012 Time [s]

Fig. 13. Measured and calculated pressure distribution on the hemisphere at r = 0.09 m for Uo = 4.0 m / s .

4.2.2 Cone 2 0 °

Fig. 17 s h o w s t h e m e a s u r e d a n d c a l c u l a t e d pressure d i s t r i b u -t i o n o n -t h e 2 0 ° cone f o r a m e a s u r e d i m p a c -t v e l o c i -t y Üq = 3 . 8 5 m / s . It c a n be n o d c e d t h a t t h e pressures m e a s u r e d w i t h t h e d i f f e r -e n t s-ensor typ-es c o r r -e s p o n d v -e r y w -e l l i n b o t h s-ensor p o s i t i o n s r = 0 . 0 4 m a n d r = 0 . 0 9 m . A c c o r d i n g t o t h e a s y m p t o t i c t h e -o r y , t h e p e a k pressure l e v e l d-oes n -o t change a l -o n g t h e -o b j e c t . I n t h e e x p e r i m e n t s t h e s e c o n d pressure peak is s l i g h t l y larger t h a n t h e f i r s t o n e . O n average o v e r a l l t h e tests, t h e d i f f e r e n c e i n peak pressure b e t w e e n t h e t w o p o s i t i o n s is 3.8%. This p h e n o m e n o n w a s also o b s e r v e d b y Peseux et a l . [ 2 4 ] w i t h e v e n m o r e p r o n o u n c e d d i f -ferences f o r cones w i t h s m a l l e r deadrise angles ( 1 4 ° - 1 0 ° - 6 ° ) . T h e reason f o r t h i s t r e n d is n o t e n t i r e l y clear. I t c o u l d p o s s i b l y be a t t r i b u t e d t o m o u n t i n g p r o b l e m s d u e t o the s m a l l r a d i u s o f c u r v a -t u r e a-t r = 0 . 0 4 m c o m p a r e d -t o r = 0.09 m . The sensors, h a v i n g a f l a t m e m b r a n e area, d i s t u r b t h e g e o m e t r y o f t h e cone m o r e at Exp (Accelerometer) Theoretical (AM) Theoretical (PI) -600 0 0.002 0.004 0.006 0.008 0.01 0.012 Time [s]

Fig. 14. Measured and calculated acceleration on the hemisphere.

4.5

„ 4

3.1

I

2.5

O Exp (High Speed Camera) • Exp (Accelerometer) •= = ° Theoretical (AM)

• •• - Theoretical (PI)

—1 _•

O Exp (High Speed Camera) • Exp (Accelerometer) •= = ° Theoretical (AM) • •• - Theoretical (PI) 1 1 i i 1 0 0.002 0.004 0.006 0.008 0.01 0.012 Time [s]

Fig. 15. Measured and calculated velocity on the hemisphere.

0.06

0.05

O Exp (High Speed Camera) • Exp (Accelerometer) • - - T h e o r e t i c a l (AM) " ° Theoretical (PI)

0 0.002 0.004 0.006 0.008 0.01 0.012 Time [s]

Fig. 16. Measured and calculated position o n the hemisphere.

a s m a l l e r r a d i u s o f c u r v a t u r e and t h i s m i g h t s l i g h t l y i n f l u e n c e t h e pressure m e a s u r e m e n t .

I n Fig. 18 a q u i t e h i g h d e c e l e r a t i o n peak o f a b o u t - 1 0 0 m / s ^ can be n o t i c e d , w h i c h results i n a n o n - n e g l i g i b l e v e l o c i t y decrease (Fig. 19). As i n t h e case o f t h e h e m i s p h e r e , t h e t h e o r y is r a t h e r

152 G. De Backer et nl. / Applied Oceaa Research 31 (2009) 143-156 ' Exp - K30A - r = 4 cm • Exp - A 0 7 A - r = 9 cm - E x p - K 3 1 A - r = 9 c m • Exp - A07B - r = 4 cm ' Exp - K30B - r = 4 cm •Asymptotic theory - r = 4 cm • Asymptotic theory - r = 9 cm Ü 35 30 25 20 15 10 5 0 0.002 0.004 0.006 0.008 0.01 0.012 Time [s]

Fig. 17. IMeasured and calculated pressure d i s t r i b u t i o n on cone ( f i = 2 0 ° ) f o r

Va = 3.85 m / s . 100 50 c % - 5 0

I

- 1 0 0 -150 -200 • Exp (Accelerometer) - - - Theoretical (AM) • " - • T h e o r e t i c a l (PI)

^

0.002 0.004 0.006 0.008 Time [s] 0.01 0.012

Fig. 18. Measured and calculated acceleration on cone (fi = 2 0 ° ) .

c o n s e r v a t i v e , especially t h e a d d e d mass m e t h o d . The h e i g h t o f t h e tested cone shape is 0.055 m , w h i c h m e a n s i t is a l m o s t c o m p l e t e l y s u b m e r g e d a f t e r 12 ms (Fig. 2 0 ) .

4.2.3. Cone 45°

Figs. 2 1 2 4 s h o w the pressure d i s t r i b u d o n , a c c e l e r a t i o n , v e l o c -i t y a n d e n t r y d e p t h f o r t h e 4 5 ° cone w -i t h an -i m p a c t v e l o c -i t y o f 4.05 m / s . A l t h o u g h the classical W a g n e r p r i n c i p l e assumes s m a l l deadrise angles, a q u i t e g o o d c o r r e s p o n d e n c e is f o u n d b e t w e e n t h e o r y a n d e x p e r i m e n t s f o r t h e f i r s t sensor p o s i t i o n . H o w e v e r , the peak at the second sensor p o s i t i o n seems t o be s i g n i f i c a n t l y s m a l l e r t h a n t h e f i r s t peak w h e r e a s t h e t h e o r y p r e d i c t s t h e same values because o f t h e s i m i l a r i t y o f the p r o b l e m . The d i s c r e p a n c y b e t w e e n t h e t w o sensor p o s i t i o n s has b e e n o b s e r v e d f o r a l l i m p a c t v e l o c i t i e s a n d is o n average 35%. This pressure d r o p c a n n o t be e x p l a i n e d b y a s m a l l e r i n s t a n t a n e o u s v e l o c i t y , since t h e v e l o c i t y d u r -i n g the second peak -is a b o u t t h e same v a l u e as d u r -i n g t h e f -i r s t peak. H o w e v e r , t h e a c c e l e r o m e t e r measures a s m a l l a c c e l e r a t i o n ( d u r -i n g the f -i r s t 10 m s ) f o l l o w e d b y a d e c e l e r a t -i o n . The -i n f l u e n c e o f t h i s a c c e l e r a t i o n a n d d e c e l e r a t i o n o n t h e pressure is n o t t a k e n i n t o

4.5

2,5

O Exp (High Speed Camera) • Exp (Accelerometer) • - - Theoretical (AM)

• •Theoretical (PI)

0 0.002 0.004 0.006 0.008 0.01 0.012 Time [s]

Fig. 19. Measured and calculated velocity on cone (fi = 2 0 ° ) .

0.06 0.05 0.04 Q. (D •o C O •.^ 2

£

CL 0.03 0.02 0.01 0

O Exp (High Speed Camera) • Exp (Accelerometer) - - - Theoretical (AM) '•=••-Theoretical (PI)

0 0.002 0.004 0.006 0.008 0.01 0.012 Time [s]

Fig. 20. Measured and calculated position on cone (/J = 2 0 ° ) .

a c c o u n t b y t h e a s y m p t o t i c t h e o r y . A s s u m i n g a u n i f o r m pressure d i s t r i b u t i o n o r i g i n a t i n g f r o m t h e p a r t o f t h e i m p a c t f o r c e p r o p o r tional to t h e a c c e l e r a t i o n ( M ^ j j ^ ) , i t is e s t i m a t e d t h a t t h i s c o n -t r i b u -t i o n -t o -t h e pressure is b e -t w e e n 5% a n d 15% o f -t h e m e a s u r e d pressure, w h i c h is r a t h e r s m a l l a n d does n o t e x p l a i n t h e pressure d r o p . A s m a l l t i m e s h i f t o f 0.5 ms is o b s e r v e d b e t w e e n t h e pressure signals o f sensor 1(30 a n d K 3 1 . As t h i s c o r r e s p o n d s t o a v e r t i -cal distance o f 2.0 m m , w h i c h is a f r a c t i o n o f t h e sensor d i a m e t e r o f 5.5 m m , t h i s s h i f t m i g h t be caused b y i m p e r f e c t i o n s i n the sensor m o u n t i n g .

The d e c e l e r a t i o n , v e l o c i t y a n d p e n e t r a t i o n are w e l l p r e d i c t e d b y t h e a n a l y t i c a l approaches f o r s m a l l e n t r y d e p t h s , since t h e pressures c o r r e s p o n d w e l l w i t h t h e e x p e r i m e n t s i n t h i s case. The d e c e l e r a t i o n peak is —25 m / s ^ , w h i c h is o n l y o n e q u a r t e r o f the peak m e a s u r e d f o r t h e cone 2 0 ° .

For t h i s range o f i m p a c t v e l o c i t i e s the t h e o r e t i c a l a s s u m p t i o n o f a c o n s t a n t i m p a c t v e l o c i t y is acceptable f o r t h e 4 5 ° cone a n d the h e m i s p h e r e . The 2 0 ° cone experiences t h e largest v e l o c i t y d r o p , w h i c h is s t i l l s m a l l e r t h a n 20% a f t e r a l m o s t c o m p l e t e s u b m e r g e n c e .

4.3. Comparison between shapes

Fig. 2 5 - 2 6 s h o w t h e s l a m m i n g pressure c o e f f i c i e n t as a f u n c t i o n o f t h e d i m e n s i o n l e s s e n t r y d e p t h Uot/R at r/R = 0.267,

G. De Backer et al./Applied Ocean Research 31 (2009] 143-156 153 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 O -0.5 1 r Exp - K30 - r = 4 cm Exp - K31 - r = 4 cm — • Exp - AO? - r = 9 cm Asymptotic theory - r = 4 cm ' Asymptotic theory - r = 9 cm \dk : ; ' ,..; . K .

: I J

^' ' ^ ' V . - w i : f : : i j f : ^ ^ ^ ' ' ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ 0.15 0 0.005 0.01 0.015 0.02 0.025 0.03 Time [s]

g. 21. Measured and calculated pressure d i s t r i b u t i o n on cone (/J = 4 5 ° ) for = 4.05 m / s . 50

Ï

s -50 -100 • Exp (Accelerometer) - - - Theoretical (AM) - ' " • Theoretical (PI) 0.005 0.01 0.015 0.02 Time [s] 0.025 0.03

Fig. 22. Measured and calculated acceleration on cone (ji = 4 5 ° ) .

4.5 „ 4

i

t

3-5

5

3 2.5

O Exp (High Speed Camera) • Exp (Accelerometer) • - -Theoretical (AM)

Theoretical (PI)

0 0.005 0.01 0.015 0.02 0.025 0.03 Time [s]

Fig. 23. Measured and calculated velocity o n cone ( f i = 4 5 ° ) .

Q . 0 •a c •2 s 0.1 S 0.05 0

© Exp (High Speed Camera) • Exp (Accelerometer) - - - Theoretical (AM) • " Theoretical (PI) i j i 1 0 0.005 0.01 0.015 0.02 0.025 0.03 Time [s]

Fig. 24. Measured and calculated position on cone (/i = 4 5 ° ) . 35

30 25 20

-Asymptotic theory hemisphere - • Asymptotic theory cone 20° — Asymptotic theory cone 4 5 °

Exp hemisphere - K31A Exp cone 2 0 ° - K30A Exp cone 4 5 ° - K30

Ugt/R[-]

Fig. 25. Slamming pressure coefficient at r / i ? = 0.267.

r e s p e c t i v e l y r / i ? = 0.300. A l t h o u g h , t h e a n a l y t i c a l s o l u t i o n is q u i t e c o n s e r v a t i v e i n p r e d i c t i n g the peak levels, the g l o b a l p r e s s u r e d i s t r i b u t i o n f i t s t h e e x p e r i m e n t s q u i t e w e l l . I n t h e b o t t o m area, t h e h e m i s p h e r e is s u b j e c t t o m u c h h i g h e r s l a m m i n g c o e f f i c i e n t s t h a n the cones. For v e r y s m a l l r - v a l u e s , the local d e a d r i s e angle o f t h e h e m i s p h e r e t e n d s t o zero a n d v e r y h i g h i m p a c t pressures m a y occur. M a t e r i a l designers s h o u l d pay s p e c i a l a t t e n t i o n t o t h i s z o n e . For l a r g e r values o f rJK t h e s l a m m i n g c o e f f i c i e n t o n t h e h e m i s p h e r e d r o p s r a p i d l y , w h i c h is n o t the case f o r t h e cones. N o t e i n Fig. 26 t h a t t h e p e a k v a l u e o f t h e h e m i s p h e r e is s m a l l e r t h a n f o r t h e 2 0 ° cone, w h e r e a s t h e l o c a l deadrise angle o f the f o r m e r is o n l y 1 8 . 4 ° .

4.4. Peak pressure

M a t e r i a l d e s i g n e r s are o f t e n i n t e r e s t e d i n m a x i m u m pressures. Figs. 2 7 - 3 0 g i v e t h e m a x i m u m pressures as a f u n c t i o n o f t h e e q u i v a l e n t d r o p h e i g h t , h*, w h i c h c o r r e s p o n d s t o t h e d r o p h e i g h t c a l c u l a t e d f r o m the m e a s u r e d i m p a c t v e l o c i t y . The use o f t h i s e q u i v a l e n t d r o p h e i g h t m a k e s i t possible t o c o m p a r e t h e m e a s u r e m e n t results w i t h o t h e r research results. Since the m a x i m u m p r e s -sure is p r o p o r t i o n a l t o the d r o p h e i g h t , a l i n e a r least squares f i t t i n g (LSF) has b e e n a d o p t e d . The v a l u e o f t h e s q u a r e d Pearson c o r r e l a -tion c o e f f i c i e n t , R^, is a l w a y s v e r y close to o n e , i n d i c a t i n g a h i g h l i n e a r c o r r e l a t i o n b e t w e e n t h e d i f f e r e n t d a t a p o i n t s o f each test

154 G. De Backer et al. / Applied Ocean Research 31 (2009) 143-156

O

0.2 0.3 0.4 U^t/R [ - ]

Fig. 26. Slamming pressure coefficient at r/R = 0.300.

Table 4

Coefficient of variation for the hemisphere and cone 2 0 ° , drop height = 1 m .

series. Tlie average d e v i a t i o n b e t w e e n t h e m e a s u r e d a n d a n a l y t -ical peak pressure levels can be easily assessed f r o m t h e g r a p h s . For t h e h e m i s p h e r e , t h e m e a s u r e d peak values are r e s p e c t i v e l y 58% a n d 55% o f the W a g n e r peak values, f o r t h e f i r s t a n d second sensor p o s i t i o n . For t h e 2 0 ° c o n e t h e r a t i o s are 66% a n d 68% r e s p e c t i v e l y a n d f o r the 4 5 ° cone 73% a n d 48%. The r a t i o b e t w e e n Chuang's e x -p e r i m e n t s [ 2 7 ] a n d a s y m -p t o t i c t h e o r y is 27% a n d 86% f o r a cone w i t h deadrise angle 3 ° a n d 1 5 ° , r e s p e c t i v e l y . I n [ 2 4 ] a n u m e r i c a l s o l u t i o n o f the W a g n e r t h r e e - d i m e n s i o n a l p r o b l e m is suggested a n d e v a l u a t e d b y e x p e r i m e n t s o n cone shapes w i t h deadrise a n gles 6 ° , 1 0 ° a n d 1 4 ° . The rarios b e t w e e n t h e e x p e r i m e n t s a n d n u -m e r i c a l s o l u t i o n are o n average 53%, 67% a n d 76%, r e s p e c t i v e l y a n d c o n s e q u e n t l y c o m p a r a b l e t o t h e r a d o s f o u n d i n t h i s paper. N i s e w a n g e r [ 2 6 ] f o u n d pressure peaks o n h e m i s p h e r e s t h a t are closer t o t h e a s y m p t o t i c t h e o r y levels u s i n g pressure t r a n s d u c e r s w i t h a d i a p h r a g m o f 6.4 m m . G e n e r a l l y t h e b l u n t b o d y a p p r o a c h is f o u n d t o be c o n s e r v a t i v e . This is c o n s i d e r e d as t h e m a i n reason f o r t h e discrepancies b e t w e e n e x p e r i m e n t s a n d t h e o r y . M i n o r d i f f e r -ences are a t t r i b u t e d t o t h e cell m e m b r a n e d i a m e t e r , w h i c h s h o u l d be as s m a l l as possible. The a s s u m p t i o n o f a c o n s t a n t e n t r y v e l o c -i t y m -i g h t also have a s m a l l -i n f l u e n c e , d e p e n d -i n g o n t h e shape a n d mass o f t h e b o d y . F u r t h e r m o r e t h e t h e o r y assumes r i g i d bodies, a c o n d i t i o n w h i c h is s e l d o m f u l f i l l e d i n practice. D e f o r m a b l e bodies m i g h t e x p e r i e n c e s i g n i f i c a n t l y s m a l l e r pressure as d e m o n s t r a t e d i n [ 2 4 ] .

I n o r d e r t o e v a l u a t e t h e r e p r o d u c i b i l i t y o f t h e tests, t h e h e m i -sphere a n d t h e 2 0 ° cone w e r e each d r o p p e d t e n t i m e s f r o m a d r o p h e i g h t o f 1 m . The sensor p o s i t i o n s c o r r e s p o n d t o t h e c o n f i g u r a -t i o n s i n Fig. 3(b) a n d ( d ) f o r -t h e h e m i s p h e r e and cone, r e s p e c -t i v e l y . Table 4 s h o w s t h e c o e f f i c i e n t o f v a r i a t i o n C„ t h e r a t i o o f the s t a n -d a r -d -d e v i a t i o n t o t h e m e a n - o f t h e m e a s u r e -d peak pressures. For sensor A 0 7 a n d K 3 1 t h e r e l a t i v e s p r e a d i n g o f t h e peak levels t o t h e m e a n is e x t r e m e l y s m a l l . This i n d i c a t e s t h a t these sensors m e a -sure v e r y a c c u r a t e l y a n d t h e tests are w e l l r e p r o d u c i b l e . The l a r g e r s p r e a d i n g f o u n d f o r sensor K 3 0 s h o u l d be a t t r i b u t e d to i n a c c u r a -cies o f t h e sensor i t s e l f b 3 • Exp - K30A - r = 4 cm 0 E x p - K 3 1 A - r = 4 c m ° E x p - K 3 1 B - r = 4 c m X E x p - K 3 1 C - r = 4 c m - - LSF K30A - p = 1.852 h - R " = 0.99 LSF K31A - p = 1.761 h - R " = 0.99 - - LSF K31B - p = 1.867 h - R " = 0.99 LSF K31C - p = 1.912 h - R " = 1.00 Asymptotic theory - r = 4 cm - p = 3.202 h

Coefficient of variation A07(%) K30(%) 1(31 (%) Fig. 27.

Hemisphere 0.66 8.48 0.44

Cone 20 0.93 12.22 1.25

0.5 1 Equivalent drop height [m]

Fig. 27. Peak pressure as a f u n c t i o n of drop height on hemisphere at r = 0.04 m.

1.2 £ 0.6 Ü. 0.4 0.2 • Exp - K30B - r = 9 cm • Exp - A07B - r = 9 cm LSF K30B - p = 0.452 h - R « = 0.98 LSF A07B - p = 0.325 h - R " = 0.99 Asymptotic theory - r = 9 cm - p = 0.711 h 0 0.5 1

Equivalent drop height [m]

Fig. 28. Peak pressure as a f u n c t i o n of drop height on hemisphere at r = 0.09 m.

5. C o n c l u s i o n

S l a m m i n g p h e n o m e n a o n a x i s y m m e t r i c bodies have b e e n ex-p e r i m e n t a l l y s t u d i e d by m e a n s o f d r o ex-p tests. A h e m i s ex-p h e r e a n d t w o cone shapes w i t h deadrise angle 2 0 ° a n d 4 5 ° are d r o p p e d o n t o i n i t i a l l y c a l m w a t e r . The w a t e r surface e l e v a t i o n is v i s u a l -i z e d w -i t h a h -i g h speed camera. A l o n g t h e h e m -i s p h e r e t h e w a t e r u p r i s e q u i c k l y ends i n a spray, w h e r e a s a j e t is a t i a c h e d t o t h e b o d y o f t h e cone shapes. The w e t i i n g f a c t o r is d e t e r m i n e d f o r t h e cones a n d is a b o u t one fifth s m a l l e r t h a n t h e v a l u e p r e d i c t e d b y m a t c h i n g t h e o u t e r t h r e e d i m e n s i o n a l flow w i t h W a g n e r ' s t w o -d i m e n s i o n a l j e t flow m o -d e l as -d e s c r i b e -d b y F a l t i n s e n i n [ 7 ] . The