Experimental investigation of interference effects for high-speed catamarans

Ocean Engineering 76 (2014) 7 5 - 8 5

ELSEVIER

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Ocean Engineering

j o u r n a l h o m e p a g e : www.elsevier.conn/locate/oceaneng

Experimental investigation of interference effects

for high-speed catamarans

Riccardo Broglia^'* Boris Jacob ^ Stefano Zaghi^ Frederick Stern ^, Angelo Olivieri^

= CNR-INSEAN, National Research Council-I^arine Teclinology Research Institute, via di Vallerano 139, 00128 Rome, Italy '' IIHR-Hydroscience and Engineering, The University of Iowa, Iowa City, IA 52241, USA

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Received 24 February 2013 Accepted 1 December 2013 Available online 22 December 2013 Keywords; High-speed catamaran Wave interference Total resistance Wave field Experimental investigation A B S T R A C T W e h a v e p e r f o r m e d a n 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 a i m e d a t assessing t h e r e l e v a n c e o f h u l l i n t e r f e r e n c e e f f e c t s o n t h e t o t a l r e s i s t a n c e o f a c a t a m a r a n m o d e l a d v a n c i n g i n c a l m w a t e r . To t h i s p u r p o s e , d r a g a n d a t t i t u d e m e a s u r e m e n t s h a v e b e e n c a r r i e d o u t f o r s e v e r a l v a l u e s o f t h e h u l l s e p a r a t i o n , as w e l l as f o r t h e m o n o h u l l , at c h a n g i n g t h e F r o u d e n u m b e r i n t h e r a n g e 0 . 1 - 0 . 8 . D a t a c o n c e r n i n g t h e t w o - d i m e n s i o n a l w a v e f i e l d a r o u n d t h e m o d e l s h a v e also b e e n c o l l e c t e d . O u r r e s u l t s i n d i c a t e t h a t , e x c e p t f o r a n a r r o w r a n g e o f c o n d i t i o n s w h e r e f a v o r a b l e i n t e r f e r e n c e is o b s e r v e d , t h e i n t e r a c t i o n b e t w e e n t h e c a t a m a r a n h u l l s u s u a l l y r e s u l t s i n a m a r k e d increase o f t h e t o t a l r e s i s t a n c e as c o m p a r e d t o t h e m o n o h u l l case. T h i s p e n a l i z a t i o n c a n b e as l a r g e as 30% f o r t h e n a r r o w e s t h u l l s p a c i n g a t Fr ^ 0.5. A t l a r g e r h u l l s p a c i n g s , t h e e f f e c t is a t t e n u a t e d a n d o c c u r s a t s m a l l e r F r o u d e n u m b e r s . I n t e r f e r e n c e e f f e c t s are less e v i d e n t b o t h i n t h e s m a l l a n d l a r g e F r o u d e n u m b e r r a n g e , n a m e l y w h e n Fr < 0.3 a n d F r > 0.7 r e s p e c t i v e l y . I n s i g h t i n t o t h e m e c h a n i s m s u n d e r l y i n g t h i s b e h a v i o r is o b t a i n e d b y e x a m i n i n g t h e r e l a t i o n a l d e p e n d e n c e b e t w e e n t h e r e s i s t a n c e v a r i a t i o n a n d t h e d a t a c o n c e r n i n g a t t i t u d e o f t h e vessel a n d w a v e - f i e l d e l e v a t i o n . © 2 0 1 3 E l s e v i e r L t d . A l l r i g h t s r e s e r v e d .

1. Introduction

M u l t i h u l l vessels are w i d e l y e m p l o y e d i n m i l i t a r y a n d c o m m e r c i a l applications at h i g h o p e r a d o n a l speeds, o w i n g t o c o m p e -d t i v e a-dvantages i n e.g., s t a b i l i t y an-d p a y l o a -d c a p a b i l i t y . As t h e need t o o p t i m i z e t h e p e r f o r m a n c e s o f these ships i n t e r m s o f resistance a n d seakeeping characterisdcs is b e c o m i n g c o m p e l -l i n g , a d d i t i o n a -l e f f o r t s are r e q u i r e d t o characterize i n m o r e d e t a i -l t h e i r salient h y d r o d y n a m i c s features (see, e.g., Faldnsen, 2 0 0 5 ) .

M a n y o f t h e aspects r e l a t e d t o t h e issue o f resistance have already been i n v e s t i g a t e d i n a n u m b e r o f n u m e r i c a l a n d 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 s , w h e r e t h e r o l e o f d e s i g n p a r a -m e t e r s such as d e m i h u U separation a n d d i m e n s i o n , h u l l g e o m e t r y , b e a m - t o - d r a f t r a t i o , w a t e r d e p t h a n d Froude n u m b e r has been c r i t i c a l l y assessed. N o t a b l e e x p e r i m e n t a l w o r k has been c a r r i e d o u t i n p a r t i c u l a r by Insel a n d M o l l a n d ( 1 9 9 2 ) and M o U a n d e t al. (1996), w h e r e t h e i n f l u e n c e o f m a n y o f t h e above factors has been e x a m i n e d a n d tested against t h e o r e t i c a l p r e d i c t i o n s , a n d b y Souto-Iglesias et a l . ( 2 0 0 7 ) a n d SoutoIglesias e t al. (2012), w h e r e resistance m e a s u r e -m e n t s have been c o -m b i n e d w i t h q u a n t i t a t i v e observations o f t h e w a v e p a t t e r n w i t h t h e m a i n p u r p o s e o f e v a l u a t i n g t h e effects associated w i t h d e m i h u l l separation a n d test c o n d i t i o n s ( f i x e d o r f r e e a t t i t u d e o f t h e c a t a m a r a n ) .

• Corresponding author. Tel.; + 3 9 0650299297; fax; -1-39 065070619. £-mai7 address: nccardo.broglia(S>cnrit (R. Broglia).

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F r o m t h e n u m e r i c a l p o i n t o f v i e w , t h e use o f steady a n d unsteady Reynolds Averaged N a v i e r - S t o k e s (RANS) based solvers i n recent studies such as those p e r f o r m e d b y He et a l . (2011), Zaghi et al. (2011), M i l l e r et al. ( 2 0 0 6 ) , M a k i et al. ( 2 0 0 7 ) , S t e r n et al. ( 2 0 0 6 ) a n d Campana et a l . ( 2 0 0 6 ) has a l l o w e d to e x t e n d e a r i i e r results based o n p o t e n t i a l f l o w techniques (e.g., T a r a f d e r a n d Suzuki, 2 0 0 7 ; M o r a e s et al., 2 0 0 4 ; L u g n i et al.. 2 0 0 4 ; Colicchio et al., 2 0 0 5 ) , a n d has a l l o w e d t o p r o v i d e valuable i n s i g h t i n t o t h e c o m p l e x f l o w s t r u c t u r e a r o u n d m u l t i - h u l l vessels.

As a r e s u l t o f t h e above i n v e s t i g a t i o n s , h u l l s e p a r a t i o n has b e e n clearly i d e n t i f i e d as t h e k e y p a r a m e t e r c o n t r o l l i n g c a t a m a r a n resistance i n a n o n t r i v i a l w a y v i a a n u m b e r o f e n t a n g l e d m e c h a n -isms. First o f all, h u l l separation d e t e r m i n e s t h e l e v e l o f n o n - l i n e a r i n t e r a c t i o n b e t w e e n i n d i v i d u a l w a v e systems p r o d u c e d b y each d e m i h u l l , and hence t h e m a g n i t u d e o f t h e w a v e - d r a g c o m p o n e n t . For a g i v e n c a t a m a r a n g e o m e t r y , b o t h d e s t r u c t i v e a n d c o n s t r u c t i v e interferences m a y o c c u r d e p e n d i n g o n t h e value o f t h e Froude n u m b e r , so t h a t i m p r e s s i v e r e d u c t i o n s o f the w a v e resistance can be o b t a i n e d i n p a r t i c u l a r c o n d i t i o n s (see t h e r e m a r k a b l e case r e p o r t e d i n Chen a n d Deo Sharma (1997) c o n c e r n i n g S-shaped catamarans a d v a n c i n g at s u p e r - c r i t i c a l speeds i n s h a l l o w w a t e r s ) . Also, h u l l s e p a r a t i o n affects t h e c a t a m a r a n resistance i n a m o r e i n d i r e c t a n d s u b t i e w a y , since t h e t o p o l o g y o f t h e f r e e - s u r f a c e o r i g i n a t e d f r o m w a v e i n t e r a c t i o n affects i n t u r n t h e f l o w a r o u n d each single h u l l . A t s u f f i c i e n t l y large values o f the w a v e steepness, f o r instance, s t r o n g w a v e - b r e a k i n g can occur a n d i n t e n s e shear-layers can f o r m b e n e a t h t h e f r e e - s u r f a c e . These h i g h - v o r t i c i t y

regions t y p i c a l l y e x t e n d deep i n t o t h e f l u i d a n d i n t e r a c t w i t h the b o u n d a r y layers d e v e l o p i n g over t h e h u l l s , p o s s i b l y l e a d i n g t o t h r e e - d i m e n s i o n a l separations. Note t h a t t h i s process m a y be r e l e v a n t f o r h i g h - s p e e d catamarans e v e n at s m a l l Froude n u m b e r s , due to t h e t y p i c a l l y l o w value o f t h e i r d r a f t .

Finally, aside f r o m its b e a r i n g o n the f r e e - s u r f a c e g e o m e t r y , h u l l s e p a r a t i o n d e t e r m i n e s the e x t e n t t o w h i c h t h e f l o w a r o u n d each h u l l is p e r t u r b e d b y t h e a d j a c e n t one, t h u s f i x i n g t h e degree o f a s y m m e t r y o f t h e v e l o c i t y f i e l d a b o u t each h u l l axis. As f a r as the s h i p resistance is concerned, this e f f e c t can be c o n v e n i e n t l y p a r a m e t r i z e d i n t e r m s o f an a p p r o p r i a t e c o r r e c t i o n f a c t o r account-i n g f o r t h e account-increase o f w a t e r v e l o c account-i t y b e t w e e n t h e h u l l s , as d o n e account-i n Insel a n d M o l l a n d ( 1 9 9 2 ) a n d A r m s t r o n g ( 2 0 0 3 ) .

Despite t h e i m p o r t a n t research progress a c h i e v e d i n t h e p r e -v i o u s w o r k s , e x p e r i m e n t a l k n o w l e d g e o n t h e f l o w s t r u c t u r e a r o u n d m u l t i - h u l l vessels s t i l l r e m a i n s i n c o m p l e t e , essentially because m o s t results are r e s t r i c t e d t o g l o b a l q u a n t i t i e s (e.g., overall resistance) a n d t o t h e i r d e p e n d e n c e o n some o f the g e o m e t r i c a l p r o p e r t i e s o f t h e ship. I n t h e p r e s e n t paper, w e w i s h t o p a r t i a l l y f i l l t h i s gap b y c o n t r i b u t i n g t o u n d e r s t a n d i n t e r f e r e n c e effects f o r n o n s t a g g e r e d catamarans w i t h s y m m e t r i c h u l l s a d v a n -c i n g i n -c a l m w a t e r To t h i s purpose, t h e b e h a v i o r o f a -catamaran ( D e l f t 372 m o d e l ) has b e e n i n v e s t i g a t e d i n a t o w i n g basin at c h a n g i n g t h e values o f h u l l separation a n d Froude n u m b e r .

The p a p e r is o r g a n i z e d as f o l l o w s : i n Section 2, t h e details o f t h e e x p e r i m e n t a l s e t u p and o f t h e m e a s u r e m e n t s y s t e m are r e p o r t e d . I n Section 3, data c o n c e r n i n g the t o t a l resistance and t h e a t t i t u d e o f t h e c a t a m a r a n models, as w e l l as t h e t o p o l o g y o f t h e w a v e f i e l d , are p r e s e n t e d i n c o m p a r i s o n w i t h t h e c o r r e s p o n d -i n g -i n f o r m a t -i o n p e r t a -i n -i n g t o t h e m o n o h u l l . The d-iscuss-ion o f t h e set o f results is p r o v i d e d i n Section 5. Finally, Section 6 s u m -marizes s o m e o f t h e m a i n findings o f the w o r k a n d the f u t u r e perspectives.

2. Experimental procedures

2.1. Catamaran model

The e x p e r i m e n t s r e p o r t e d here have been c a r r i e d o u t at CNR-INSEAN i n t h e t o w i n g basin # 1 , w h i c h is a 470 m l o n g , 13.5 m w i d e and 6.5 m deep f a c i l i t y . The m o d e l used i n o u r i n v e s t i g a t i o n is a 3.0 m fiberglass D e l f t 372 catamaran ( v a n ' t Veer, 1998b, 1998a). This t y p i c a l h i g h - s p e e d m u l t i - h u l l vessel has b e e n e x t e n s i v e l y characterized i n a n u m b e r o f recent e x p e r i m e n t a l a n d c o m p u t a -t i o n a l ac-tivi-ties, w h e r e d i f f e r e n -t aspec-ts such as i n -t e r f e r e n c e (He et al., 2 0 1 1 ; Z a g h i et al., 2011), m a n e u v e r a b i l i t y i n deep a s h a l l o w w a t e r ( Z l a t e v et al., 2 0 0 9 ; M i l a n o v et al., 2010, 2011), w a t e r j e t p r o p u l s i o n ( M i l a n o v a n d Georgiev, 2010; M i o z z i , 2011), h u l l f o r m o p t i m i z a t i o n (e.g.. Peri et al., 2012), sea k e e p i n g (Castiglione e t al., 2 0 1 1 ; Bouscasse et al., 2013) a n d a d v a n c e m e n t i n steady d r i f t (Broglia et al., 2012) have b e e n c o n s i d e r e d . A s u m m a r y o f t h e e x p e r i m e n t a l a c t i v i t y c o n d u c t e d w i t h t h i s m o d e l at CNR-INSEAN is p r e s e n t e d i n Broglia et al. (2011).

The m o d e l is d e p i c t e d i n Figs. 1 and 2, a n d t h e r e l e v a n t g e o m e t r i c a l i n f o r m a t i o n is s u m m a r i z e d i n Table 1. N o t e i n p a r t i -cular t h a t t h e n o m i n a l gap b e t w e e n h u l l s is H = 0 . 7 0 m , w h i c h corresponds t o a d i m e n s i o n l e s s h u l l separation H / L = 0.167. This figure c o u l d h o w e v e r be changed b y s l i d i n g t h e h u l l clamps a l o n g t h e c o n n e c t i n g transverse bars (see Zaghi et al., 2 0 1 1 ; B r o g l i a et al., 2011) i n o r d e r t o assess t h e relevance o f i n t e r f e r e n c e p h e n o m e n a . M o r e precisely, c o m p a r a t i v e tests have been p e r f o r m e d at t h e d i f f e r e n t values o f h u l l separation r e p o r t e d i n Table 2, as w e l l as w i t h a m o n o h u l l ( c o r r e s p o n d i n g to H = o o ) .

Each h u l l w a s e q u i p p e d w i t h t u r b u l e n c e s t i m u l a t o r s i n t h e f o r m o f s m a l l c y l i n d r i c a l studs ( 4 m m i n h e i g h t a n d 3 m m i n d i a m e t e r ) spaced 3 0 m m apart f r o m each o t h e r a n d located 7 0 m m b e h i n d t h e b o w edge.

R. Broglia et ai. / Ocean Engineering 76 (2014) 75-85 B H A F T F O R W A R D » B 10 t2 M \6 , 0 20 Fig. 2. Basic geomelrry definition and body plan of the Delft 372 catamaran model.

Table 1

Main dimensions of the Delft 372 catamaran model (CNR-INSEAN model number-2554).

Table 2

Summary of the investigated geometrical configurations.

Feature Symbol Value Units

Length (overall)

i-OA 3.11 m

Length between perpendiculars L 3.00 m

Beam (overall) B 0.94 m

Beam ( d e m i h u l l ) b 0.24 m

Hull separation ( n o m i n a l ) H 0.70 m

Hull gap (nominal) s 0.0 m

Draught _{T 0.15 m}

Wetted surface _{s 1.945}

Displacement _{A 87.07}

l<S

Vertical center o f gravity K G 0.34 m

Longitudinal center o f gravity L C G 1.41 m

Block coefficient _CB 0.403 _

I n t h e discussion o f t h e results b e l o w , w e w i l l r e f e r t o a c o o r d i n a t e system m o v i n g w i t h t h e ship i n w h i c h the x-, y- a n d z- axis have f i x e d o r i e n t a t i o n s a n d p o i n t i n the f r e e - s t r e a m , transverse and v e r t i c a l d i r e c t i o n s respectively (see Fig. 2 ) . The o r i g i n o f t h e c o o r d i n a t e s y s t e m is located i n the s y m m e t r y p l a n e o f t h e m o d e l at the i n t e r s e c t i o n o f t h e s t i l l w a t e r level w i t h t h e f o r w a r d p e r p e n d i c u l a r . A c c o r d i n g l y , t h e sinkage o f the m o d e l is H ( c m ) H/L s/L 50 0.167 0.087 60 0.200 0.120 70 0.233 0.153 80 _{0.267 0.187} 90 0.300 0.220 c o n s i d e r e d as p o s i t i v e w h e n t h e c e n t e r o f g r a v i t y m o v e s u p w a r d , w h i l e t h e t r i m is p o s i t i v e w h e n t h e b o w moves u p w a r d .

2.2. Attitude and resistance measurements

A t t i t u d e a n d resistance tests have b e e n p e r f o r m e d w i t h t h e c a t a m a r a n m o d e l s attached to t h e c a r n a g e o f t h e b a s i n i n such a w a y t h a t rigid m o t i o n s w e r e a l l o w e d o n l y i n t h e v e r t i c a l plane, i.e., t h e m o d e l s w e r e f r e e to heave a n d p i t c h w h i l e y a w was n o t p e r m i t t e d . For m o n o h u l l tests, an a d d i t i o n a l c o n s t r a i n i n g m e c h a n -i s m w a s used t o -i n h -i b -i t r o l l -i n g d -i s p l a c e m e n t s . T h e catamaran m o d e l s w e r e t o w e d h o r i z o n t a l l y w i t h a p u l l e y a c t i n g t h r o u g h t h e c e n t e r o f g r a v i t y . The t o t a l resistance Rt w a s i n f e r r e d f r o m the t e n s i o n i n t h e cable by means o f a O m e g a LC 214 v a r i a b l e -r e l u c t a n c e d i s p l a c e m e n t t -r a n s d u c e -r Resistance time-se-ries w e -r e

Fig. 3. Scliematic drawing indicating the wave probe locations.

c o l l e c t e d at a s a m p l i n g rate o f 2 0 0 H z w i t h a 1 6 - b i t A / D board, and t h e a c q u i r e d data w e r e t i m e - a v e r a g e d t o p r o v i d e m e a n values f o r each r u n . T h e v e l o c i t y o f t h e t o w i n g carriage w a s measured b y m e a n s o f a n encoder w i t h u n c e r t a i n t y b e t t e r t h a n 1 m m / s over t h e e n t i r e range o f velocities. The sinkage a n d t r i m o f t h e m o d e l s w e r e c o m p u t e d f r o m t h e m e a s u r e m e n t s o f t h e v e r t i c a l d i s p l a c e m e n t s o f t w o reference p o i n t s o f t h e vessel, as r e g i s t e r e d b y t w o p o t e n t i -o m e t e r s . The reference p -o i n t s w e r e l-ocated -o n t h e l -o n g i t u d i n a l axis o f t h e c o n n e c t i n g beams o f t h e catamaran, respectively at x = 0 . 4 6 9 m a n d at x = (0.469-1-2.512) m (see Fig. 1).

T h e u n c e r t a i n t y has been e s t i m a t e d f o r b o t h resistance a n d a t t i t u d e measurements. Precision i n d i c e s ( C o l e m a n a n d Steele, 2 0 0 9 ) f o r sinkage, t r i m a n d t o t a l resistance c o e f f i c i e n t have been e s t i m a t e d f r o m r e p e a t a b i l i t y tests o n selected cases ( n a m e l y , at F r = 0 . 5 0 , 0.60 a n d 0.75, a n d H / L = 0.167 a n d 0.233, a n d f o r the m o n o h u l l Broglia et al., 2011), a n d w e r e f o u n d t o be less t h a n 6.0%, 1.6% and 0.5% o f t h e m a x i m u m m e a s u r e d v a l u e . W h e n t h e bias e r r o r s are i n c l u d e d , the u p p e r b o u n d s f o r t h e u n c e r t a i n t y o n t h e m e a s u r e m e n t s are respectively equal t o 6.9%, 2.0% a n d 1.0%.

Table 3

Wave probe location as a function o f hull separation H.

H ( c m ) Series # 1 Series # 2 y / ( s / 2 ) y (cm) y / ( s / 2 ) y ( c m ) 1.00 13.00 1.00 13.00 0.77 10.00 0.77 10,00 0,50 6.50 0,50 6,50 0.23 3.00 0.23 3,00 0.57 13.00 1.00 13,00 0.43 10.00 0.77 10,00 0.28 6.50 0.50 6.50 0.13 3.00 0.23 3.00 0.39 13.00 1.00 13.00 0.30 10.00 0.77 10.00 0.20 6.50 0.50 6.50 0.09 3.00 0.23 3.00 2.3. Wave-field measurements I n o r d e r to characterize t h e w a v e f i e l d a r o u n d t h e m o d e l s , the d i s p l a c e m e n t o f t h e free-surface f r o m its u n d i s t u r b e d reference l e v e l w a s m e a s u r e d r e d u n d a n t l y w i t h t w o series o f s t a t i o n a r y capacitance probes. Each p r o b e r e g i s t e r e d t h e t i m e - h i s t o r y o f t h e w a v e e l e v a t i o n at t h e p r o b e l o c a t i o n . T h e sensors w e r e designed at t h e Electronics L a b o r a t o r y o f CNR-INSEAN, a n d c o n s i s t e d i n a n i n s u l a t e d electric w i r e s t r e t c h e d v e r t i c a l l y across t h e a i r - w a t e r i n t e r f a c e b y means o f a s m a l l suspended w e i g h t . T h e electrical capacitance o f the p r o b e depends o n t h e s u b m e r g e d p o r t i o n o f the w i r e , so t h a t the d i s p l a c e m e n t o f t h e f r e e - s u r f a c e w i t h respect t o t h e s t i l l w a t e r l i n e c o u l d be r e t r i e v e d b y m e a n s o f a static c a l i b r a t i o n p e r f o r m e d p r i o r t o each test r u n . Extensive character-i z a t character-i o n o f t h e p r o b e b e h a v character-i o r has s h o w n t h a t t h e w a v e h e character-i g h t c o u l d be d e t e r m i n e d w i t h a n accuracy b e t t e r t h a n 1 m m a n d a d y n a m i c response b e t t e r t h a n 50 Hz, (see e.g. O l i v i e r i et al., 2001). T h e m a x i m u m g l o b a l u n c e r t a i n t y w h i c h accounts also f o r statis-t i c a l d i s p e r s i o n is a b o u statis-t 3% o f statis-t h e range.

T h e p o s i t i o n o f t h e probes is s h o w n s c h e m a t i c a l l y i n Fig. 3. A f i r s t series o f sensors w e r e p o s i t i o n e d o n a transverse rack a t t a c h e d t o t h e basin w a l l i n o r d e r t o m o n i t o r t h e w a v e p a t t e r n i n t h e r e g i o n e x t e r n a l to t h e c a t a m a r a n . A second series o f probes w e r e used instead t o d e t e r m i n e t h e e l e v a t i o n f i e l d i n t h e spatial

d o m a i n c o n f i n e d by t h e t w o h u l l s . It consisted i n a set o f sensors a r r a n g e d i n 2 transversal r o w s a n d s u p p o r t e d b y a p y l o n r e s t i n g o n t h e b o t t o m o f t h e basin, as o r i g i n a l l y p r o p o s e d b y Souto-Iglesias et a l . ( 2 0 0 7 ) . The p o s i t i o n o f t h e probes c o u l d b e a d j u s t e d i n d e p e n d e n t l y f r o m each o t h e r so t h a t , w h e n t h e s e p a r a t i o n b e t w e e n h u l l s was changed, e i t h e r t h e absolute d i s t a n c e y o f each sensor f r o m t h e h u l l side was k e p t f i x e d ( f o r probes b e l o n g i n g t o t h e f i r s t series) o r t h e d i m e n s i o n l e s s distance y / ( s / 2 ) c o u l d r e m a i n c o n s t a n t ( f o r probes o f the second series). The sensor a r r a n g e m e n t f o r each e x p e r i m e n t a l c o n d i t i o n is s u m m a r i z e d i n Table 3.

3. Calm-water tests: results

C a l m - w a t e r tests, c o n s i s t i n g o f d r a g and a t t i t u d e m e a s u r e m e n t s , have been p e r f o r m e d at c h a n g i n g the Froude n u m b e r Fr=U/(gLf'^ i n t h e range f r = 0 . 1 - 0 . 8 w i t h i n c r e m e n t s AFr = 0.05. For each value o f t h e Froude number, several g e o m e t r i c a l c o n f i g u r a t i o n s o f the c a t a m a r a n - c o r r e s p o n d i n g t o the values o f h u l l s e p a r a t i o n H listed i n Table 2 - have b e e n investigated i n a d d i t i o n to t h e m o n o h u l l case. Table 4 p r o v i d e s a s u m m a r y o f t h e r e l e v a n t f l o w parameters f o r t h e ensemble o f test c o n d i t i o n s .

R. Broglia et al. / Ocean Engineering 76 (2014) 75-85 79

Table 4

Main flow parameters computed at each experimental condition.

Fr _{U(mls) Re} 0.10-0.80 0.07-0.54 0.542-4.339 1.429 X 10'5-1.143 X 10' K.l. = 0 167 V t R = 0 200 7 H.L = 0.233 ^ H I = 0.267 .3 H.L=0.30ö « 2xM;nDliul

Fig. 4. Calm-water tests: The total resistance Rr as a f u n c t i o n of Froude number Fr for each hull separation H. Note that the resistance data have been reseated to the standard reference temperature 1 = 1 5 ' in order to c o m p l y w i t h the ITTC procedures.

3.3. Total resistance coefficient

Let us f i r s t discuss t h e b e h a v i o r o f t h e m o d e l resistance, i n t e r m s o f b o t h t o t a l resistance Rj a n d n o n - d i m e n s i o n a l c o e f f i c i e n t

CT ( d e f i n e d as the r a t i o o f t h e t o t a l resistance RT to the reference

q u a n t i t y ^pU'^S). The t o t a l resistance a n d t h e resistance c o e f f i c i e n t are s h o w n versus Froude n u m b e r i n Fig. 4 a n d i n Fig. 5 f o r the d i f f e r e n t g e o m e t r i c a l c o n f i g u r a t i o n s . The results c l e a r l y i n d i c a t e t h a t large d i f f e r e n c e s b e t w e e n t h e m o n o h u l l a n d t h e catamarans e x i s t over t h e e n t i r e range o f Froude n u m b e r s , i.e., i n t e r f e r e n c e effects are always present. Such e f f e c t s are p a r t i c u l a r l y r e l e v a n t f o r Froude n u m b e r s i n t h e range b e t w e e n 0.35 a n d 0.7, w h e r e t h e resistance c o e f f i c i e n t o f t h e c a t a m a r a n is f o u n d to l a r g e l y exceed t h e c o r r e s p o n d i n g v a l u e o f t h e m o n o h u l l . D i f f e r e n c e s are o n the c o n t r a r y less m a r k e d i n the l o w Froude n u m b e r range ( n a m e l y , f o r F r < 0.3) as w e l l as at t h e e x t r e m e e n d o f the i n v e s t i g a t e d speed i n t e r v a l (i.e., f o r f r > 0.7).

From the analysis o f the v a r i a t i o n o f the t o t a l resistance RT, s h o w n i n Fig. 4 ( f o r comparison purposes, the resistance f o r the m o n o h u l l is m u l t i p l y by t w o ) , i t can be clearly seen t h a t f o r the m o n o h u l l the resistance m o n o t o n i c a l l y increases w i t h t h e advancement speed, whereas, the presence o f h u m p s are e v i d e n t f o r the catamaran c o n f i g u r a t i o n .

The analysis o f t h e v a r i a t i o n o f t h e resistance c o e f f i c i e n t f o r t h e catamarans shows t h a t a l l t h e c o r r e s p o n d i n g Cr curves e x h i b i t a local m i n i m u m at F r s » 0.35 a n d t w o d i s t i n c t local m a x i m a , located r e s p e c t i v e l y a t Fr^ 0.3 a n d at Fr^ 0.5. T h e c o m p a r i s o n b e t w e e n t h e curves evidences t h a t h u l l s p a c i n g has l i t t i e i n f l u e n c e o n t h e m a g n i t u d e a n d o n the l o c a t i o n o f b o t h t h e m i n i m u m a n d o f t h e first m a x i m u m . O n the o t h e r h a n d , t h e f e a t u r e s o f t h e second m a x i m u m at Fr =s 0.50 are s t r o n g l y a f f e c t e d b y the specific v a l u e o f t h e h u l l separation. I n fact, t h e i n t e n s i t y o f t h i s peak increases c o n s i d e r a b l y as the separation is r e d u c e d , w h i l e its p o s i t i o n s h i f t s t o l a r g e r values o f t h e Froude n u m b e r N o t e also t h a t the b e h a v i o r o f t h e resistance c o e f f i c i e n t o f t h e m o n o h u l l is q u a l i t a t i v e l y s i m i l a r t o t h a t observed f o r t h e catamarans, y e t t h e v a r i a t i o n s i n CT are less p r o n o u n c e d o w i n g t o t h e s m a l l e r elevations o f the associated w a v e p a t t e r n . H,l = 0 2C^) K1.= 0.233 H.l- = 0.26r H,l = 0,3-30 ' • • : \ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Fr

Fig. 5. Calm-water tests: The resistance coefficient C j as a f u n c t i o n of Froude n u m b e r Fr f o r each hull separation hi.

-10 . n i = 0 20o • : H/L = 0233 - MIL = 0 257 — H,L = 03CO —• MSnOhLTl 0 0.1 0.2 0.4 0.5 O.S 0.7 0.8 0.<, Fr 0.1 0.2 0.3 0.4 0.5 0.6 Fr 0.7 O.f

Fig. 6. Calm-water tests: The sinkage (n) and the t r i m ( r ) o f the investigated catamarans as a f u n c t i o n o f Froude number f r (top and b o t t o m panel, respectively).

3.2. Sinkage and trim

The sinkage a n d t r i m o f t h e m o d e l s v a r y w i t h F r o u d e n u m b e r a n d h u l l s e p a r a t i o n as d i s p l a y e d i n Fig. 6 ( t o p a n d b o t t o m panel, r e s p e c t i v e l y ) . As f o r the catamarans, o n e can o b s e r v e t h a t t h e c e n t e r o f g r a v i t y o f t h e m o d e l s moves p r o g r e s s i v e l y d o w n w a r d as t h e Froude n u m b e r increases, u n t i l a m i n i m u m is a t t a i n e d at Fr«= 0.45 ( t o p p a n e l ) . I n this range o f r e l a t i v e l y l o w a d v a n c e m e n t speeds, t h e sinkage rr o f t h e catamarans is m u c h l a r g e r t h a n t h e c o r r e s p o n d i n g value o f t h e m o n o h u l l , a n d is e s s e n t i a l l y u n a f f e c t e d b y h u l l s e p a r a t i o n . This t e n d e n c y is r e v e r t e d a t l a r g e r speeds, because t h e c a t a m a r a n m o d e l s start t o m o v e u p t o w a r d s t h e i r o r i g i n a l s u b m e r s i o n level, w h i l e at t h e same time 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 t h e m o n o h u l l is m u c h less sensitive t o v a r i a t i o n s i n Fr. This results i n a larger sinkage o f t h e m o n o h u l l c o n f i g u r a t i o n w i t h respect t o catamarans f o r F r > 0.5 ( n o t e also t h a t p o s i t i v e values o f t h e sinkage can be detected a t n a r r o w h u l l spacings f o r t h e catamarans w h e n t h e Froude n u m b e r a p p r o a c h e s 0.75).

The b e h a v i o r o f t h e t r i m ( b o t t o m p a n e l o f Fig. 6 ) s h o w s t h a t t h i s q u a n t i t y takes o n n e g l i g i b l e values f o r all i n v e s t i g a t e d c o n f i g -u r a t i o n s , p r o v i d e d t h a t t h e Fro-ude n -u m b e r is less t h a n 0.35. A b o v e t h i s p o i n t - w h i c h corresponds t o t h e sharp increase o f t h e resistance c o e f f i c i e n t observed i n Fig. 5 - t h e t r i m o f c a t a m a r a n m o d e l s d r o p s s u d d e n l y a n d t h e d y n a m i c a l b e h a v i o r o f s i n g l e - a n d m u l t i - h u l l c o n f i g u r a t i o n s s t a r t to d i f f e r As f o r t h e m o n o h u l l , a

- e H,1. = C1167 RT. = 0 20O - 0 H:t = 0 233 - A K'L = 0.267 - e KL = 0 300 O 0.1 0.2 0.3 0.4 0.5 0.6 . 0.7 0.8 0.9 Fr

Fig. 7. Calm water tests: Interference f a c t o r / f a s a f u n c t i o n o f Froude number Fr for each hull separation H.

g e n t l e m o n o t o n i c increase o f t h e t r i m is o b s e r v e d as t h e advance-m e n t speed increases. O n t h e contrary, t h e t r i advance-m i n c r e a s i n g steeply f o r the c a t a m a r a n u n t i l a m a x i m u m is reached. The p r o p e r t i e s o f t h i s p e a k s t r o n g l y d e p e n d o n h u l l separation. As t h e s e p a r a t i o n H is d i m i n i s h e d , t h e m a g n i t u d e is greatly a m p l i f i e d , a n d at t h e same t i m e its p o s i t i o n s h i f t s t o larger values o f t h e Froude n u m b e r . Beyond t h e peak, t h e t r i m o f t h e catamarans g r a d u a l l y decreases a n d e v e n t u a l l y levels o f f at a n a m p l i t u d e s i m i l a r t o t h a t o f t h e m o n o h u l l .

The a t t i t u d e o f t h e catamaran, as w e l l as t h e Cr curves, is s t r o n g l y r e l a t e d t o t h e features o f the w a v e p a t t e r n i n t h e r e g i o n b e t w e e n t h e h u l l s : t h i s r e l a t i o n s h i p w i l l be discussed i n m o r e d e t a i l i n s u b s e q u e n t sections. 3.3. Interference factor T h e e f f e c t s i n d u c e d o n t h e t o t a l resistance b y changes i n h u l l s e p a r a t i o n are b e t t e r a p p r e c i a t e d b y c o n s i d e r i n g t h e i n t e r f e r e n c e f a c t o r lp, w h i c h is d e f i n e d as the r e l a t i v e d i f f e r e n c e b e t w e e n t h e resistance c o e f f i c i e n t s o f t h e c a t a m a r a n a n d o f t h e m o n o h u l l , a c c o r d i n g t o -(M) 2R m I n this e x p r e s s i o n , superscripts M a n d C r e f e r r e s p e c t i v e l y t o t h e m o n o h u l l a n d t o t h e c a t a m a r a n . The v a r i a t i o n o f t h e i n t e r f e r e n c e f a c t o r w i t h Froude n u m b e r is s h o w n p a r a m e t r i c a l l y i n Fig. 7 as a f u n c t i o n o f h u l l s e p a r a t i o n . The p l o t c l e a r l y evidences t h a t t h e i n t e r f e r e n c e f a c t o r is g e n e r a l l y positive, i n d i c a t i n g t h a t t h e d r a g e x p e r i e n c e d b y t h e c a t a m a r a n is a l m o s t a l w a y s larger t h a n t w i c e t h e d r a g o f t h e m o n o h u l l . I n t h e range o f Froude n u m b e r s 0.32 < Fr < 0.37, h o w e v e r , t h e p a r a m e t e r Ip c o m p u t e d f o r t h e largest separations, H / L > 0.233, appears t o be s l i g h t l y negative, suggesting f a v o r a b l e i n t e r f e r e n c e effects. T h i s v e l o c i t y i n t e r v a l corresponds t o t h e range o f Froude n u m b e r s w h e r e m i n i m a o f t h e resistance c o e f f i c i e n t Cj are o b s e r v e d (see Fig. 5 ) . N o t e also that, i n t h i s n a r r o w range o f c o n d i t i o n s , t h e i n f l u e n c e o f h u l l gap is c r i t i c a l , as a s m a l l change i n t h e s e p a r a t i o n H leads to considerable fluctuations o f t h e i n t e r -ference f a c t o r .

Fig. 7 s h o w s t h a t i n t e r f e r e n c e effects - t o g e t h e r w i t h t h e associated d r a g p e n a l i z a t i o n - are o f p a r t i c u l a r relevance w h e n t h e Froude n u m b e r is larger t h a n Fr ^ 0.4. A c t u a l l y , b e y o n d t h i s l i m i t , t h e i n t e r f e r e n c e f a c t o r increases s h a r p l y a n d a s t r o n g d e p e n d e n c y o n t h e specific h u l l s e p a r a t i o n is apparent. T h e m a x i m u m l e v e l o f i n t e r f e r e n c e is a c h i e v e d w i t h t h e n a r r o w e s t h u l l s p a c i n g ( H / L = 0.167) a t Fr^ 0.5, w h e r e t h e i n t e r f e r e n c e f a c t o r c a n be as large as 0.33. A t l a r g e r separations, t h e p e a k i n t e r a c t i o n reduces a n d occurs at l o w e r a d v a n c e m e n t speeds.

n I 0.012 0.006 0.000 -0.006 I -0.012 0.10 0.00 -0.10

Fig. 8. Dimensionless free-surface elevation (;;) around the different models at F r = 0 . 3 0 . From top to bottom, the panels correspond respectively to: H / L = 0.167, H / L = 0.233, H/L = 0.300 and H/L = oo (monohull).

Fig. 9. Photograph of the wave field around the H/L = 0.167 catamaran at Fr=0.30.

I n p a r t i c u l a r , at H / L = 0.233, t h e m a x i m u m i n t e r f e r e n c e c o e f f i c i e n t is / F ! « 0 . 2 4 a n d is a t t a i n e d at F r a ; 0.48, w h i l e a t H / L = 0.300 t h e peak v a l u e Ip ^ 0.21 occurs at Fr f« 0.44. Finally, Fig. 7 c o n f i r m s t h a t t h e i n t e r a c t i o n b e t w e e n t h e h u l l s is v e r y w e a k at l o w speeds (i.e., w h e n Fr < 0 . 3 0 ) as w e l l as a t t h e e x t r e m e e n d o f t h e i n v e s t i g a t e d Froude n u m b e r i n t e r v a l (i.e., f o r Fr > 0.7). N o t e t h a t t h e observed b e h a v i o r at large Froude n u m b e r s is i n d e e d consistent w i t h t h e v i e w t h a t t h e i n d i v i d u a l w a v e systems p r o d u c e d b y each d e m i h u l l are i n c r e a s i n g l y c o n f i n e d as t h e v e l o c i t y increases, so t h a t m u t u a l i n t e r a c t i o n s are a t t e n u a t e d .

4. Wave-field characterization

The w a v e - f i e l d s g e n e r a t e d b y t h e c a t a m a r a n m o d e l s at several Froude n u m b e r s are d i s p l a y e d as c o n t o u r lines o f t h e n o n -d i m e n s i o n a l free-surface e l e v a t i o n // = / i / L i n Figs. 8, 11 a n -d 15, w h i c h c o r r e s p o n d r e s p e c t i v e l y t o Fr = 0.3, 0.5 a n d 0.75. The panels i n each figure r e f e r t o d i f f e r e n t values o f t h e h u l l s e p a r a t i o n ( H increases f r o m t o p t o b o t t o m ) . T h e w a v e field a r o u n d t h e m o n o h u l l is also r e p o r t e d i n each figure t o serve as a r e f e r e n c e .

R. Broglia et al. / Ocean Engineering 76 (2014) 75-85 81

Fig. 10. Photograph of the wave field around the H / L = 0.300 catamaran at F r = 0 . 3 0 . 0.10 0.00 -0.10 t O 0.25 0.5 0.75 1 1.25 1.5 1.75 2 X/L

Fig. 11. Dimensionless free-surface elevation (i;) around the d i f f e r e n t models at F r = 0 . 5 0 . From t o p to bottom, the panels correspond respectively to: H / L = 0.167,

H/L = 0.233, H/L = 0.300 and H/L = oo ( m o n o h u l l ) .

4 . J . Fr=0.3

A t l o w Froutie n u m b e r s , F r = 0 . 3 , t h e b o t t o m p a n e l o f Fig. 8 s h o w s t h a t t h e w a v e a m p l i t u d e g e n e r a t e d b y t h e m o n o h u l l is v e r y w e a k , as t h e n o n d i m e n s i o n a l f r e e s u r f a c e e l e v a t i o n ;/ is e v e r y -w h e r e less t h a n 0.01. Some d i f f e r e n c e s b e t -w e e n t h e m o n o h u l l and t h e c a t a m a r a n m o d e l s - i n d u c e d b y i n t e r f e r e n c e effects - are n o n e t h e l e s s discernible, especially at t h e n a r r o w e s t h u l l gap. I n o r d e r t o g a i n a q u a l i t a t i v e f e e l i n g o f the effects associated w i t h h u l l i n t e r a c t i o n , representative p h o t o g r a p h s o f t h e w a v e p a t t e r n t a k e n w i t h a camera l o o k i n g f o r w a r d f r o m astern are r e p o r t e d i n Figs. 9 a n d 10 f o r t h e t w o e x t r e m e separations ( n a m e l y , f o r

H/L = 0.167 a n d f o r H / L = 0.300). This i n f o r m a t i o n can be u t i l i z e d

i n c o m b i n a t i o n w i t h t h e m o r e q u a n t i t a t i v e i n s i g h t p r o v i d e d b y t h e analysis o f t h e e l e v a t i o n c o n t o u r i n Fig. 8.

A t H / L = 0.167, one can observe i n t h e t o p p a n e l o f Fig. 8 t h a t t h e w a v e f i e l d o n t h e o u t e r side o f t h e c a t a m a r a n is o n l y s l i g h t i y p e r t u r b e d w i t h respect t o the m o n o h u l l case. I n t e r f e r e n c e effects are o n t h e o t h e r h a n d clearly v i s i b l e i n t h e r e g i o n b e t w e e n the h u l l s . For instance, t h e w a v e e l e v a t i o n m e a s u r e d a t x / L = 0.25 is t h r e e t i m e s as large as at t h e c o r r e s p o n d i n g p o i n t o n t h e e x t e r n a l side. I n t h e i n n e r r e g i o n , t h e i n t e r a c t i o n b e t w e e n h u l l s results i n a r o u g h l y t w o - d i m e n s i o n a l w a v e f i e l d w h i c h is c h a r a c t e r i z e d b y a

Fig. 12. Photograph o f the wave field around the H / L = 0.167 catamaran at Fr=0.50.

Fig. 13. Photograph o f the wave field around the H / l = 0.233 catamaran at f r = 0 . 5 0 .

series o f p r o n o u n c e d crests a n d t r o u g h s as w e l l as b y a r o o s t e r t a i l e x t e n d i n g a s t e r n f r o m x/L 1 to x / L 1.25. These f e a t u r e s can be clearly d i s t i n g u i s h e d also i n t h e p i c t u r e o f Fig. 9.

As t h e h u l l s p a c i n g is increased t o H / L = 0.233, t h e w a v e f i e l d does n o t change s i g n i f i c a n t l y o n t h e o u t e r side, w h i l e a c o m p l e x t h r e e - d i m e n s i o n a l p a t t e r n develops i n t h e i n n e r r e g i o n . Here, t h e w a v e s y s t e m i n v o l v e s a m a r k e d t r o u g h o n t h e c a t a m a r a n center-l i n e ( center-l o c a t e d a p p r o x i m a t e center-l y a m i d s h i p s ) as w e center-l center-l as a n a r r o w a n d intense peak c e n t e r e d a t x / L ^ 0.5. A t t h e same t i m e , a n e x t e n d e d depressure r e g i o n appears near the stern. The r o o s t e r t a i l is c o n f i n e d t o a zone a r o u n d 1 < x / L < 1.25, as a t s m a l l e r h u l l separation, b u t t h e o v e r a l l p e r t u r b a t i o n n o w e x t e n d s t o m u c h larger d o w n s t r e a m distances. F i n a l l y at t h e largest h u l l gap, H / L = 0.300, t h e c h a r a c t e r o f t h e w a v e p a t t e r n is s u b s t a n t i a l l y preserved, b u t t h e m a g n i t u d e o f t h e f r e e s u r f a c e d e f o r m a t i o n appears s i g n i f i c a n t l y r e d u c e d e v e r y -w h e r e . Also, b o t h t h e -w a v e peak o n t h e l o n g i t u d i n a l axis a n d t h e depressure r e g i o n astern are observed t o s h i f t s l i g h t l y d o w n -s t r e a m . Farther a w a y f r o m t h e m o d e l , i.e., f o r x / L > l , t h e w a v e field again q u a l i t a t i v e l y resembles t h a t observed a t H / L = 0.233.

4.2. Fr=0.5

A t this v a l u e o f t h e Froude n u m b e r , d i f f e r e n c e s b e t w e e n t h e m o n o h u l l a n d t h e H / L = 0.167 c a t a m a r a n are c l e a r l y d e t e c t e d e v e n o n t h e o u t e r side o f t h e vessel, w h e r e f r e e s u r f a c e p e r t u r b a -t i o n s are g e n e r a l l y f o u n d -t o be a m p l i f i e d (see Fig. 11). For ins-tance, t h e b o w w a v e is c o n s i d e r a b l y accentuated a n d a m a r k e d d e p r e s -sure i n v o l v e s a s i g n i f i c a n t f r a c t i o n o f t h e ship m o d e l a s t e r n . I n t h e

Fig. 14. Pliotograpii o f the wave field around the H / L = 0.300 catamaran at Fr=0.50.

O 0.25 0.5 0.75 1 1.25 1.5 1.75 2

O 0.25 0.5 0.75 1 1.25 1.5 1.75 2 x/L

Fig. 15. Dimensionless free-surface elevation (;;) around the different models at F r = 0 . 7 5 . From top to bottom, the panels correspond respectively to: H / L = 0.167, H / L = 0.233, H / L = 0.300 and H / L = oo ( m o n o h u l l ) . i n n e r r e g i o n , t h e w a v e f i e l d is characterized b y a n a r r o w a n d e l o n g a t e d peak o r i g i n a t e d f r o m t h e s u p e r p o s i t i o n o f t h e t w o d i v e r g i n g b o w crests. T h e p e r t u r b a t i o n focalizes o n t h e l o n g -i t u d -i n a l ax-is a t x » ^ 0.3, w h e r e t h e presence o f a r e g -i o n o f -i n c -i p -i e n t w a v e b r e a k i n g is revealed b y t h e f o r m a t i o n o f d r o p l e t s o b s e r v e d i n t h e p h o t o g r a p h o f Fig. 12. A m a j o r depressure zone s p a n n i n g t h e e n t i r e transverse gap extends a x i a l l y f r o m t o X R JI. This r e g i o n t e r m i n a t e s i n a U-shaped f o r m w h i c h leaves t h e s t e r n d r y . Farther d o w n s t r e a m , t h e free-surface is d o m i n a t e d by a h i g h l y c o n v o l u t e d r o o s t e r t a i l s t r e t c h i n g f r o m x^^.0 t o x f » 1 . 2 . T h i s p r o n o u n c e d r o o s t e r t a i l consists i n a sheet-like j e t w h i c h is characterized b y a n active d r o p f o r m a t i o n process. The i n t e n s e t u r b u l e n t a c t i v i t y i n t h e w a k e f l o w , e v i d e n t i n t h e p i c t u r e o f Fig. 12, is also n o t e w o r t h y .

As t h e h u l l s e p a r a t i o n is increased t o H / L = 0.233, no s i g n i f i -c a n t -change o-c-curs i n t h e e x t e r n a l p a r t o f the d o m a i n , w h i l e t h e w a v e p a t t e r n i n t h e i n t e r n a l r e g i o n g r a d u a l l y evolves t o w a r d a t h r e e - d i m e n s i o n a l o r g a n i z a t i o n (see Fig. 11). Here, t h e f r e e - s u r f a c e appears w r i n k l e d a n d c o r r u g a t e d , a l t h o u g h b r e a k i n g seems t o be i n h i b i t e d d u e t o t h e a c t i o n o f surface t e n s i o n , Fig. 13. O f f t h e s t e r n , t h e w a v e f r o n t o r i g i n a t e d b y t h e h u l l i n t e r a c t i o n e x h i b i t s a c u s p -like ( V ) shape a n d is characterized b y the f o r m a t i o n o f c a p i l l a r y waves, see Fig. 13. W a k e t u r b u l e n c e effects are q u a l i t a t i v e l y s i m i l a r

Fig. 16. Photograph of the wave field around the H / L = 0.167 catamaran at Fr=0.75.

to t h e H / L = 0.167 case, since n o m a j o r change is e x p e c t e d i n t h e b o u n d a r y layer p r o p e r t i e s . A n a r r o w peak is v i s i b l e a m i d s h i p s o n t h e c a t a m a r a n axis. The depressure r e g i o n b e t w e e n t h e h u l l s extends f r o m x/L ^0.7 to x » = 1 . 0 a l t h o u g h its effects a p p e a r t o be s l i g h t l y a t t e n u a t e d as c o m p a r e d t o s m a l l e r h u l l gap. O n t h e contrary, t h e s t e r n is v i s i b l y m o r e dry, w h i l e t h e rooster t a i l has b r o a d e n e d a n d d r i f t e d f a r t h e r d o w n s t r e a m .

F i n a l l y at t h e largest h u l l spacing, H / L = 0.300, i n t e r f e r e n c e effects b e t w e e n a d j a c e n t w a v e systems are s u b s t a n t i a l l y r e d u c e d . The t o p o l o g y o f the w a v e f i e l d q u a l i t a t i v e l y resembles t h a t observed at H / L = 0.233, y e t t h e i n t e n s i t i e s are reduced. As i t can be see i n Fig. 11 a n d i n Fig. 14, a f u r t h e r d o w n s t r e a m d i s p l a c e m e n t o f b o t h t h e depressure r e g i o n a n d t h e rooster t a i l takes also place.

4.3. Fr=0.75

The e l e v a t i o n c o n t o u r s h o w n i n Fig. 15 indicates t h a t d e p a r -tures o f t h e w a v e p a t t e r n f r o m t h e m o n o h u l l s i t u a t i o n can be s t i l l observed i n t h e e x t e r n a l r e g i o n , especially at r e d u c e d values o f t h e h u l l gap. A t H / L = 0.167, f o r instance, t h e scars i m p r i n t e d o n the f r e e - s u r f a c e b y b o w - w a v e b r e a k i n g effects ( O l i v i e r i et al., 2 0 0 7 ) are clearly v i s i b l e , as i n d i c a t e d by the d i s r u p t u r e o f the w a v e f r o n t d e t e c t e d a t x = 0 . 2 5 - 0 . 6 , y = 0.2-0.3 i n Fig. 16. This f e a t u r e is cleariy v i s i b l e also at l a r g e r separations, a n d is o n t h e c o n t r a r y h a r d l y d i s c e r n i b l e f o r t h e m o n o h u l l . Quite u n d e r s t a n d i n g l y i n t e r -ference e f f e c t s are accentuated i n t h e i n n e r r e g i o n . A t H / L = 0.167, a r e g i o n o f elevated f r e e - s u r f a c e extends o n t h e l o n g i t u d i n a l axis over a s u b s t a n t i a l p o r t i o n o f t h e ship m o d e l ( r a n g i n g f r o m x ^ 0.3 t o X ~ 0.6). The p i c t u r e i n Fig. 16 cleariy s h o w s t h a t t h e crest o f t h i s r e g i o n consists i n a sheet-like j e t w i t h v i g o r o u s d r o p f o r m a t i o n . Intense small-scale t u r b u l e n c e is also e v i d e n t o n t h e f r e e - s u r f a c e . The depressure zone extends f r o m x RJ 0.8 t o w e l l b e y o n d t h e s t e r n ( n a m e l y t o x ~ 1.2). S i m i l a r l y t o t h e F r = 0 . 5 case, t h e s t e r n is l e f t c o m p l e t e l y d r y . I n t h e d o w n s t r e a m w a v e p a t t e r n , t h e p r o m i n e n t rooster t a i l f o r m i n g i n t h e r e g i o n 1.3;<x;<1.6 d u e to t h e super-p o s i t i o n o f t h e i n t e r n a l s t e r n waves takes t h e f o r m o f a n a l m o s t c o n t i n u o u s t u r b u l e n t j e t .

As t h e separation increases t o H / L = 0.233, t h e effects described above p r o g r e s s i v e l y attenuate. For instance, t h e w a v e peak i n t h e i n n e r r e g i o n reduces a n d moves b e y o n d x R* 0.5, w h i l e t h e i n t e n s i t y o f t h e depressure i n t h e r e g i o n 0.9;<x;<1.2 is also s i g n i f i c a n t l y lessened. O n t h e c o n t r a r y , t h e i n t e n s i t y o f t h e rooster t a i l ( s t i l l c h a r a c t e r i z e d b y a n intense small-scale t u r b u l e n c e o n t h e f r e e surface) increases a n d moves d o w n s t r e a m , o f f t h e f i e l d o f v i e w o f t h e camera i n Fig. 17.

R. Broglia et at. / Ocean Engineering 76 (2014) 75-S5 83

Fig. 17. Pliotograph o f the wave field around the H / i = 0.233 catamaran at Fr=0.75.

Fig. 18. Photograph o f the wave field around the H / i = 0.300 catamaran at Fr=0.75,

A t t h e U t m o s t h u i l gap, H/L = 0,300, t h e d o w n s t r e a m displacem e n t o f t h e s t r u c t u r e s o r i g i n a t e d f r o displacem t h e i n t e r a c t i o n o f i n d i v i -d u a l w a v e systems is even m o r e evi-dent. The r e g i o n o f m e a s u r a b l e f r e e - s u r f a c e depressure moves o f f t h e s t e r n a n d t h e r o o s t e r t a i l involves a b r o a d e r r e g i o n , w h i c h is h o w e v e r located f a r t h e r a w a y f r o m t h e m o d e l . T h i s results i n the s y m m e t r y b e t w e e n i n n e r a n d o u t e r regions b e i n g b r o k e n o n l y i n a localized zone o v e r t h e l o n g i t u d i n a l axis, as revealed b y t h e w h i t e - w a t e r r e g i o n v i s i b l e i n t h e p i c t u r e o f Fig. 18.

5. Concluding discussion

The results presented i n t h e previous sections c l e a r l y i n d i c a t e t h a t t h e i n t e r a c t i o n b e t w e e n the d e m i - h u l l s o f a c a t a m a r a n d e p e n d s s t r o n g l y o n b o t h t h e h u l l separation H and t h e advance-m e n t speed ( n a advance-m e l y o n t h e Froude n u advance-m b e r Fr). This i n t e r a c t i o n has a p r o f o u n d i m p a c t o n b o t h the value a n d t h e v a r i a t i o n o f t h e t o t a l resistance c o e f f i c i e n t Cj.

The specific features o f the resistance curves, w h i c h w e r e o u t -l i n e d i n Section 3.1, can be better understood a t t h i s p o i n t by c o n s i d e r i n g the Cr data s h o w n i n Fig. 5 i n c o n j u n c t i o n w i t h t h e a d d i t i o n a l results c o n c e r n i n g t h e v a r i a t i o n o f sinkage and t r i m (Fig. 6 ) a n d t h e properties o f the wave field (Figs. 8 - 1 5 ) . The resistance c o e f f i c i e n t o f t h e investigated c o n f i g u r a t i o n s was typically f o u n d t o e x h i b i t a series o f peaks and troughs (as observed also i n M o l l a n d et al., 1996; Moraes et al., 2 0 0 4 ; Souto-Iglesias e t al., 2 0 0 7 ) . A first local m a x i m u m is observed at F r » j 0.30 f o r b o t h t h e

m o n o h u l l and the catamarans (except the one w i t h t h e n a r r o w e s t separation). U p o n increasing t h e Froude number, t h e resistance coefficient Cr i n i t i a l l y decreases a n d goes t h r o u g h a m i n i m u m a t FrRJ 0.35. Here, a m o d e s t degree o f favorable interference is obtained f o r the large-separation cases, i.e f o r H / L > 0.233 c m . Beyond this value o f t h e Froude number, large levels o f u n f a v o r a b l e interference are observed and t h e resistance coefficient Cr increases sharply up to a m a x i m u m w h o s e propert:ies sti-ongly d e p e n d o n t h e value o f the h u l l separation. S i m i l a r trends, a l t h o u g h w i t h s l i g h t l y d i f f e r e n t levels o f interference, w e r e observed f o r o t h e r catamaran shapes (see e.g., Souto-Iglesias et al., 2007, 2012).

A first p o i n t t o be n o t e d here concerns t h e r e m a r k a b l e s i m i l a r i t y b e t w e e n t h e behaviors observed f o r t h e resistance c o e f f i c i e n t and t h e t r i m o f t h e catamarans. I n p a r t i c u l a r , t h e p o s i t i o n o f t h e first m a x i m u m i n t h e Cj p l o t correlates v e r y w e l l w i t h t h e ( w e a k ) local m a x i m u m observed i n t h e t r i m curves o f t h e large- a n d m e d i u m - s e p a r a t i o n catamarans ( n o appreciable change i n the t r i m is d e t e c t e d f o r t h e H / L = 0.167 c a t a m a r a n u n t i l Fr exceeds 0.35). S i m i l a r l y , t h e absolute m a x i m u m o f Cr is reached a t a p p r o x i m a t e l y t h e same values o f t h e Froude n u m b e r w h e r e m a x i m u m d e v i a t i o n s o f t r i m a n d sinkage f r o m t h e i r h y d r o s t a t i c reference levels take place. For t h e H / L = 0.167 c a t a m a r a n , f o r instance, t h e m a x i m u m Cr a n d t h e m a x i m u m i n t e r f e r e n c e f a c t o r occur at Fr«= 0.5, w h i l e m a x i m u m t r i m a n d sinkage are at F r R j 0.6 and Fr 0.4 respectively. The same o b s e r v a t i o n applies to l a r g e r separations, i.e., t h e m a x i m u m values o f Cj are consistently observed a p p r o x i m a t e l y h a l f w a y b e t w e e n t h e positions w h e r e t h e t r i m a n d sinkage are largest. This s t r i k i n g coincidence can be explained by c o n s i d e r i n g the e v o l u t i o n o f t h e w a v e p a t t e r n at c h a n g i n g Froude n u m b e r , w h i c h is displayed i n Figs. 8 - 1 5 . Let us first consider t h e case at F r = 0.3. For t h e n a r r o w e s t separation catamaran, H / L = 0.167, t h e l i m i t e d gap b e t w e e n t h e h u l l s induces i n t h e i n n e r region a quasi t w o - d i m e n s i o n a l wave p a t t e r n (Fig. 8 ) . The w a v e system consists o f a n a l m o s t u n p e r t u r b e d interface at t h e bow, f o l l o w e d by a peak at x / L RJ 0.25 a n d a b y a t r o u g h a t x / L 0.4. The w a v e elevation t h e n vanishes at x / L RJ 0.5, a n d secondary peaks and t r o u g h s are again observed a t x/L^O.7 and X/ LR JO . S. Even-t u a l l y Even-t h e w a v e h e i g h Even-t is again zero aEven-t Even-t h e sEven-tern. This r e g u l a r a l t e r n a t i o n o f peaks a n d t r o u g h s does n o t e v i d e n t l y force the catamaran to t r i m , and t h e r e f o r e results i n a s m o o t h v a r i a t i o n o f t h e t r i m and Cj curves f o r the H / f = 0.167 catamaran. This o b s e r v a t i o n accounts f o r t h e absence o f t h e local m a x i m u m w h i c h is o t h e r w i s e detected i n p r o x i m i t y o f F r = 0 . 3 f o r t h e larger h u l l separations. A c t u a l l y at H / L = 0.233 a n d H / L = 0.300, t h e i n t e r n a l w a v e f i e l d is d o m i n a t e d by the intense depressure a r o u n d the stern region, w h i c h is l i k e l y t o i n d u c e a b o w - u p c o n f i g u r a t i o n o f t h e catamaran. This e f f e c t p r e s u m a b l y c o m b i n e s w i t h t h e t e n d e n c y o f the m o d e l to sink, a n d results i n a c o m p a r a t i v e l y large v a r i a t i o n o f t h e resistance c o e f f i c i e n t a r o u n d F r ^ j 0 . 3 , as i n d i c a t e d b y the m a x i m a detected a t t h i s p o s i t i o n f o r these t w o separations.

A t h i g h e r Froude n u m b e r s , F r = 0.5, t h e s t r u c t u r e o f t h e w a v e -p a t t e r n changes -p r o f o u n d l y (Fig. 11). A t t h e n a r r o w e s t se-paration, the w a v e system i n t h e i n t e r h u l l r e g i o n is s t i l l a p p r o x i m a t e l y t w o -d i m e n s i o n a l , b u t t h e t y p i c a l w a v e l e n g t h is t w i c e as large as at F r = 0 . 3 . The w a v e e l e v a t i o n is p o s i t i v e i n t h e f o r e b o d y r e g i o n a n d negative i n the a f t b o d y Z e r o - e l e v a t i o n occurs respectively at t h e bow, at the stern a n d a m i d s h i p s . As a consequence, a b o w - u p a t t i t u d e is observed f o r t h e H / L = 0.167 m o d e l . As t h e separation increases, the w a v e f i e l d r e m a i n s q u a l i t a t i v e l y similar, y e t w i t h r e d u c e d a m p l i t u d e s o f t h e w a v e f i e l d a n d w i t h an o v e r a l l d o w n -s t r e a m -s h i f t o f t h e d o m i n a n t -s t r u c t u r e -s . The-se effect-s c o u l d p l a u s i b l y e x p l a i n w h y a m u c h m o r e e v i d e n t increase o f t h e t r i m is registered f o r t h e n a r r o w e s t c a t a m a r a n . Note also t h a t t h e intense depressure w h i c h extends a r o u n d t h e a f t b o d y at H / L = ^ 0.167 p r o b a b l y c o n t r i b u t e s t o t h e l a r g e r sinkage as c o m p a r e d t o w i d e r c o n f i g u r a t i o n s ( H / L = 0.233 a n d H / L = 0.300).

Finally, a t t h e largest Froude n u m b e r , F r = 0 . 7 5 , t h e m a r k e d depressure b e t w e e n the h u l l s o f t h e n a r r o w e s t c a t a m a r a n ( H / L = 0 . 1 6 7 ) affects t h e e n t i r e stern r e g i o n , w h i l e t h e w a v e peak m o v e s a p p r o x i m a t e l y a m i d s h i p s (Fig. 15). This t o p o l o g y o f t h e f r e e - s u r f a c e a r g u a b l y explains t h e t e n d e n c y t o w a r d s m a l l e r values o f b o t h t h e sinkage and t r i m w h i c h is o b s e r v e d at t h i s Froude n u m b e r . T h e sinkage o f t h e s m a l l - s e p a r a t i o n c a t a m a r a n is i n f a c t p o s i t i v e (i.e., t h e m o d e l is less submersed t h a n i t is a t r e s t ) a n d the t r i m d i f f e r s v e r y l i t t l e f r o m t h a t o f t h e m o n o h u l l . Also, a t larger h u l l separations, t h e catamarans d i s p l a y v a n i s h i n g l y s m a l l values o f sinkage a n d t r i m a m p l i t u d e s w h i c h are v e r y close t o t h a t o f t h e m o n o h u l l . T h i s d i f f e r e n t b e h a v i o r can be u n d e r s t o o d b y n o t i n g t h a t t h e w a v e e l e v a d o n p l o t i n Fig. 15 i n d i c a t e s t h a t t h e i n d i v i d u a l w a v e p a t t e r n s generated b y t h e h u l l s at H / L = 0.233 a n d H / L = 0.300 are a l m o s t i n d e p e n d e n t , so t h a t l a r g e r - s e p a r a t i o n catamar-ans e x p e r i e n c e i n d e e d v e r y s m a l l i n t e r f e r e n c e effects.

6. Conclusions

I n t h i s w o r k , t h e i n t e r f e r e n c e effects b e t w e e n a d j a c e n t h u l l s o f a D e l f t 372 c a t a m a r a n m o d e l have been i n v e s t i g a t e d e x p e r i m e n -t a l l y a-t c h a n g i n g -t h e h u l l separa-tion a n d -t h e Froude n u m b e r . B o -t h g l o b a l q u a n t i t i e s ( n a m e l y , t o t a l resistance, sinkage a n d t r i m ) a n d local p r o p e r t i e s o f t h e w a v e f i e l d have been e x a m i n e d .

The e f f e c t s o f t h e i n t e r a c t i o n w e r e h a r d l y n o t i c e a b l e at l o w speeds, n a m e l y f o r F r < 0.3. A t larger v e l o c i t i e s , h i g h levels o f u n f a v o r a b l e i n t e r f e r e n c e w e r e instead observed, p a r t i c u l a r l y i n p r o x i m i t y o f F r = 0.5. The m a x i m u m l e v e l o f i n t e r f e r e n c e was m e a s u r e d f o r the smallest separation ( H / L = 0.166) at Fr^ 0.5. Here, t h e t o t a l resistance c o e f f i c i e n t Cr w a s f o u n d t o increase b y m o r e t h a n 30% w i t h respect t o t h e m o n o h u l l reference. A t larger separations, t h e peak i n t e r a c t i o n o c c u r r e d a t s m a l l e r values o f t h e Froude n u m b e r , a n d its i n t e n s i t y was a t t e n u a t e d . For instance, at H / L = 0.300, t h e m a x i m u m increase o f Cr w i t h respect t o t h e m o n o h u l l a m o u n t e d t o a p p r o x i m a t e l y 20% at Fr 0.44.

I n t e r f e r e n c e effects w e r e f o u n d to be i n t i m a t e l y c o n n e c t e d w i t h t h e b e h a v i o r o f the sinkage a n d ( i n p a r t i c u l a r ) o f t h e t r i m o f t h e m o d e l . The m e c h a n i s m s w h i c h c o n t r o l t h e change i n resis-tance e x p e r i e n c e d b y the c a t a m a r a n w i t h respect t o t h e s i n g l e - h u l l case can be s u m m a r i z e d as f o l l o w s :

• First, t h e m o d i f i c a t i o n s o f the f r e e s u r f a c e d u e t o t h e i n t e r -ference o f t h e i n d i v i d u a l w a v e systems i n t h e i n t e r - h u l l r e g i o n cause t h e t r i m a n d t h e sinkage o f t h e c a t a m a r a n m o d e l t o change. T h i s e f f e c t leads to a v a r i a t i o n ( g e n e r a l l y , a n increase) o f t h e f o r m resistance i n response t o t h e m o d i f i e d a t t i t u d e o f the c a t a m a r a n .

• I n t u r n , t h e changed a t t i t u d e o f the c a t a m a r a n m o d i f i e s the w a v e p a t t e r n , possibly r e s u l t i n g i n a f u r t h e r v a r i a t i o n o f the w a v e - m a k i n g resistance (as revealed i n p a r t i c u l a r b y t h e h i g h e r w a v e e l e v a t i o n i n t h e r e g i o n e x t e r n a l t o t h e c a t a m a r a n ) .

This t w o f o l d e f f e c t u l t i m a t e l y leads to t h e s i g n i f i c a n t changes o b s e r v e d i n t h e m a g n i t u d e o f t h e t o t a l resistance, p a r t i c u l a r l y at s m a l l h u l l separations a n d at i n t e r m e d i a t e values o f t h e Froude n u m b e r . A c t u a l l y , at l o w speeds, t h e w a v e e l e v a t i o n is t o o s m a l l t o p r o d u c e a s i g n i f i c a n t e f f e c t o n Cj, even a t the n a r r o w e s t h u l l gap w h e r e i n t e r f e r e n c e b e t w e e n the w a v e systems c o u l d be v e r y relevant. A t h i g h speeds, o n t h e o t h e r h a n d , t h e i n d i v i d u a l w a v e systems are v e r y d i v e r g i n g : the s u p e r p o s i t i o n b e t w e e n t h e m is t h u s s u b s t a n t i a l l y reduced a n d t h e m u l t i h u l l c o n f i g u r a t i o n behaves as a c o m b i n a t i o n o f a l m o s t n o n - i n t e r a c t i n g vessels. I n c o n c l u s i o n , t h e largest i n t e r f e r e n c e effects are o b s e r v e d w h e n the s e p a r a t i o n is s m a l l e n o u g h t o r e s u l t i n s u f f i c i e n t l y i n t e n s e i n t e r -a c t i o n o f t h e w -a v e systems -and -a t t h e s-ame t i m e t h e w -a v e

e l e v a t i o n is s u f f i c i e n t l y large t o produce appreciable changes i n t h e c a t a m a r a n t r i m . B o t h c o n d i t i o n s need to be f u l f i l l e d t o observe n o n - v a n i s h i n g values o f t h e i n t e r f e r e n c e c o e f f i c i e n t . The above considerations are s t r i c t l y r e l a t e d t o this p a r t i c u l a r c a t a m a r a n . For o t h e r h u l l shapes, d i f f e r e n t scenarios can be observed, as i n S o u t o -Iglesias et a l . ( 2 0 0 7 ) w h e r e t h e effects o n the f r e e surface d u e t o t r i m v a r i a t i o n s are o f secondary i m p o r t a n c e .

As a final r e m a r k , w e n o t e t h a t f u r t h e r i n s i g h t i n t o t h e w a v e -i n t e r f e r e n c e p h e n o m e n o n c o u l d be ga-ined b y q u a n t -i f y -i n g e x a c t l y t h e i n d i v i d u a l c o n t r i b u t i o n s t o t h e resistance v a r i a t i o n (based, e.g. o n the analysis o f t h e o u t e r w a v e - f i e l d s t r u c t u r e ) . This issue w i l l be e x p l o r e d i n d e t a i l i n a f o r t h c o m i n g paper b y u s i n g a c o m b i n a t i o n o f b o t h e x p e r i m e n t a l a n d n u m e r i c a l i n f o r m a t i o n .

Acloiowledgements

The investigation r e p o r t e d here is part o f a w i d e r c o o p e r a t i o n b e t w e e n the Iowa Institute o f Hydraulic Research (IIHR, U n i v e r s i t y o f I o w a ) and CNR-INSEAN, a i m e d at t h e characterization o f the behavior o f fast catamarans i n c a l m a n d r o u g h seas. Financial s u p p o r t f r o m t h e U.S. Office o f Naval Research t h r o u g h ONR N00014-08-1-1037 G r a n t (Dr. L. Patrick Purtell supervisor) is gratefully acknowledged.

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