Hydrodynamic modeling of planing hulls with twist and negative deadrise

Ocean Engineering 82 (2014) 14-19

ELSEVIER

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Ocean Engineering

journal homepage: www.elsevier.conn/locate/oceaneng

Hydrodynamic modeling of planing hulls with twist

and negative deadrise

Konstantin I. Matveev*

School of Mechanical and Materials Engineering. Washington State University, Pullman, WA 99164-2920, USA

(D

CrossMark A R T I C L E I N F O A B S T R A C T Article history: Received 26 August 2013 Accepted 19 February 2014 Available o n l i n e 13 March 2014 Keywords: Planing boat Twisted hull Negative deadrise M e t h o d o f hydrodynamic sources H u l l s o f h a r d - c h i n e p l a n i n g b o a t s o f t e n h a v e c o m p l i c a t e d g e o m e t r i e s . I n t h i s s t u d y , n o n - p r i s m a t i c h u l l s w i t h v a r i a b l e a n d n e g a t i v e d e a d r i s e a n g l e s are c o n s i d e r e d . T h e m e t h o d o f h y d r o d y n a m i c s o u r c e s is a p p l i e d t o m o d e l s t e a d y w a t e r f l o w a r o u n d s u c h h u l l s i n t h e l i n e a r i z e d a p p r o x i m a t i o n at f i n i t e F r o u d e n u m b e r s . T h e s o l u t i o n i n c l u d e s p r e s s u r e d i s t r i b u t i o n o n t h e h u l l s u r f a c e w h i c h d e f i n e s t h e l i f t f o r c e a n d c e n t e r o f p r e s s u r e . For v a l i d a t i o n , m o d e l i n g r e s u l t s are c o m p a r e d w i t h a n a n a l y t i c a l s o l u t i o n f o r a f l a t p l a t e a n d t e s t d a t a o f a h u l l w i t h a c o n s t a n t - d e a d r i s e s e c t i o n . P a r a m e t r i c c a l c u l a t i o n s a r e c a r r i e d o u t f o r a t w i s t e d h u l l a t a f i x e d a t t i t u d e a n d c o m p a r e d w i t h a n e f f e c t i v e l y s i m i l a r p r i s m a t i c p l a n i n g s u r f a c e i n a r a n g e o f F r o u d e n u m b e r s . N o n - p r i s m a t i c h u l l s w i t h v a r i a b l e a s p e c t r a t i o s are a l s o i n v e s t i g a t e d . In a d d i t i o n , r e s u l t s are p r e s e n t e d f o r t w o h u l l s h a v i n g p o s i t i v e a n d n e g a t i v e d e a d r i s e a n g l e s . © 2 0 1 4 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. I n t r o d u c t i o n E f f i c i e n t b o a t m o t i o n a t s u f f i c i e n t l y h i g h speeds o f t e n occurs i n t h e p l a n i n g m o d e . I n t h i s r e g i m e , t h e b o a t m o v e m e n t resembles s k i m m i n g a l o n g t h e w a t e r surface r a t h e r t h a n p l o w i n g t h r o u g h the water, a n d t h e h y d r o d y n a m i c l i f t e x e r t e d o n t h e h u l l b e c o m e s larger t h a n t h e h y d r o s t a t i c l i f t . The occurrence o f t h e p l a n i n g m o d e can be a p p r o x i m a t e l y c h a r a c t e r i z e d b y a c h i e v e m e n t o f h i g h Froude n u m b e r s , e.g., w h e n t h e l e n g t h Froude n u m b e r exceeds a b o u t one o r t h e d i s p l a c e m e n t Froude n u m b e r is g r e a t e r t h a n a b o u t three.

For approximate estimations o f p l a n i n g h u l l hydrodynamics, s i m p l i f i e d methods are c o m m o n l y used, such as semi-empirical correlations f o r prismatic (constant-deadrise or m o n o h e d r a l ) hulls given by Savitsl<y (1964). However, m o s t real boats have m o r e complicated geometry, i n c l u d i n g variable deadrise, steps, a n d so on. The m o t i v a t i o n f o r using t w i s t e d hulls, f o r example, is to i m p r o v e seaworthiness w i t h larger deadrise near the bow, w h i l e keeping h i g h l i f t - d r a g ratio w i t h low-deadrise a f t sections. To account f o r the h u l l twist, several simple methodologies w e r e proposed i n the past (Blount and Fox, 1976; Bertorello and Oliviero, 2006; Savitsl<y et al., 2007), w h i c h usually suggested using an equivalent prismatic h u l l w i t h specially chosen geometrical parameters. Such recommendations, although useful as i n i t i a l approximations, cannot f u l l y account f o r indinsically m o r e c o m p l e x hydrodynamics o f hulls w i t h m o r e c o m -plicated geometries.

* T e t : + 1 509 335 1327; fax: + 1 509 335 4642. E-mail address: matveev@wsu.edu http://dx.doi.Org/10.1016/j.oceaneng.2014.02.021 0029-8018 © 2014 Elsevier Ltd. All rights reserved.

A l t h o u g h m u c h less p o p u l a r t h a n t r a d i t i o n a l V-huUs, p l a n i n g boats w i t h negative deadrise angles have been also i m p l e m e n t e d . As n o t e d b y Egorov et al. (1978), such h u l l s are b e n e f i c i a l f o r h i g h e r loadings. For e x a m p l e , t h e y i n d i c a t e d t h a t a t deadrise angles 5° a n d 15° t h e beam-based l i f t c o e f f i c i e n t Q ( d e f i n e d i n Section 3 b e l o w ) s h o u l d ^exceed values o f a b o u t 0.11 a n d 0.09, respectively, f o r the negative-deadrise hulls to p e r f o r m b e t t e r t h a n p o s i t i v e - d e a d r i s e h u l l s at o p t i m a l t r i m c o n d i t i o n s . A t large m a g n i t u d e s o f n e g a t i v e -deadrise angles and especially a t l o w t r i m angles, a f o a m - a i r zone can b e f o r m e d a l o n g t h e boat c e n t e r i i n e e x t e n d i n g d o w n to the t r a n s o m , thus r e s e m b l i n g a t u n n e l - h u l l c o n f i g u r a t i o n . H o w e v e r , t h e p r e s e n t analysis is l i m i t e d to r e l a t i v e l y l o w deadrise angles a n d s u f f i c i e n t l y h i g h t r i m angles w i t h o u t f o r m a t i o n o f m u l t i - p h a s e m i x t u r e s , m o d e l i n g o f w h i c h is v e r y c h a l l e n g i n g .

A m e t h o d o f h y d r o d y n a m i c sources is utilized i n this s t u d y f o r m o d e l i n g t w i s t e d and negativedeadrise p l a n i n g hulls. Tliis m a t h e -matical approach has been previously presented a n d applied f o r p r e d i c t i n g hydrodynamics o f monohedral hulls and n e a r h u l l / h y d r o -foil wave contours (Matveev and Ocld'en, 2009). This m e t h o d allows us to account f o r n o n - t r i v i a l h u l l geometry t h r o u g h appropriate b o u n d a r y conditions on the hull, as w e l l as f o r finite Froude numbers. This technique belongs to a w i d e r class o f boundary e l e m e n t m e t h o d s that have been extensively used f o r c o m p u t a t i o n a l l y e f f i c i e n t m o d e l i n g o f p l a n i n g h u l l hydrodynamics (e.g.. Doctors, 1974; W e l l i c o m e and Jahangeer, 1978; Lai a n d Troesch, 1996; Xie et al., 2 0 0 5 ; Ghassemi and Ghiasi, 2008).

A m a t h e m a t i c a l f o r m u l a t i o n o f t h e c u r r e n t m e t h o d is o u t i i n e d i n t h e n e x t section. Results are p r e s e n t e d f o r t w o v a l i d a t i o n cases i n v o l v i n g t h r e e - d i m e n s i o n a l p r o b l e m s a n d f o r s o m e p a r a m e t r i c v a r i a r i o n s o f n o n - p r i s m a t i c a n d n e g a t i v e - d e a d r i s e h u l l s .

K.I. Matveev/Ocean Engineering 82 (2014) 14-19 15 2. M a t h e m a t i c a l m o d e l To m o d e l h y d r o d y n a m i c s o f t h r e e - d i m e n s i o n a l p l a n i n g h u l l s , a m e t h o d o f h y d r o d y n a m i c s i n g u l a r i t i e s is u t i l i z e d i n t h i s w o r k . T h e w a t e r f l o w a r o u n d t h e h u l l is a s s u m e d i n v i s c i d , i r r o t a t i o n a l a n d steady. A g e n e r a l s c h e m a t i c o f t h e p r o b l e m is s h o w n i n Fig. 1 . T h e B e r n o u l l i e q u a t i o n is a p p l i e d o n t h e w a t e r s u r f a c e . (1) w h e r e Po a n d Uo are t h e p r e s s u r e a n d v e l o c i t y i n t h e f a r u p s t r e a m u n d i s t u r b e d w a t e r s u r f a c e at z = 0, p is t h e w a t e r d e n s i t y , a n d p ( x , y ) a n d U(x,y) are t h e p r e s s u r e a n d v e l o c i t y o n t h e w a t e r s u r f a c e w i t h e l e v a t i o n Z w ( x , y ) . A s s u m i n g s m a l l t r i m angles o f t h e h u l l a n d s u f f i c i e n t l y h i g h F r o u d e n u m b e r s , o n e c a n e x p e c t r e l a t i v e l y s m a l l f l o w d i s t u r b a n c e s caused b y t h e h u l l p r e s e n c e . T h e r e f o r e , t h e w a v e s l o p e s a n d t h e x - a x i s v e l o c i t y p e r t u r b a t i o n u= UX-UQ w i l l be s m a l l as w e l l . T h e n , t h e B e r n o u l l i e q u a t i o n c a n be l i n e a r i z e d a n d w r i t t e n as f o l l o w s . = 0, (2) w h e r e Cp = (p-po)/(p Ul/2) is t h e p r e s s u r e c o e f f i c i e n t ( z e r o o n t h e f r e e w a t e r s u r f a c e ) a n d X = 2n U^/g is t h e w a v e l e n g t h o n t h e u n c o n s t r a i n e d f r e e w a t e r s u r f a c e . The w a t e r f l o w d i s t u r b a n c e i n d u c e d by the h u l l is m o d e l e d here b y h y d r o d y n a m i c sources d i s t r i b u t e d over a h o r i z o n t a l p l a n e at z = 0 (Fig. l b a n d c). A v e l o c i t y p o t e n t i a l o f each source satisfies t h e Laplace e q u a t i o n i n t h e f l u i d d o m a i n . The c o l l o c a t i o n p o i n t s , w h e r e Eq. ( 2 ) is f u l f i l l e d , are s h i f t e d u p s t r e a m f r o m t h e sources. This staggered a r r a n g e m e n t m i n i m i z e s t h e i n f l u e n c e o f t h e d o w n -s t r e a m b o u n d a r y o f t h e n u m e r i c a l d o m a i n ( B e r t r a m , 2 0 0 0 ) . T h e n , t h e x - c o m p o n e n t o f t h e v e l o c i t y p e r t u r b a t i o n can be d e t e r m i n e d f r o m t h e source i n t e n s i t i e s . (3) Chins line Keel line Stagnation line Collocation points ^e~%-.Q-)^JS-y^.Q • X..O-X--0-X-0-K--G «-.*>->^.o-)(--o-;i.-o •s-.e.7Ïf-^ï->;--c>-*-o-*t-o Pressure area

Fig. 1 . (a) Planing hull w i t h variable deadrise. (b) Longitudinal cross-section, (c) Top v i e w o f a planing hull. Only a small part o f the numencal domain is shown. Distances between sources are exaggerated.

w h e r e (X|,y() a n d ( x ] , y 5 ) are t h e c o o r d i n a t e s o f t h e c o l l o c a t i o n p o i n t i and t h e source J w i t h i n t e n s i t y qj, a n d r , j = y / ( X i - x j ) ^ - K V i - y j ) ^ is the h o r i z o n t a l distance b e t w e e n these p o i n t s . I n the a n a l y t i c a l i n t e g r a l - d i f f e r e n t i a l a p p r o a c h , there w o u l d be an i n t e g r a l i n t h e Cauchy p r i n c i p a l v a l u e sense instead o f a finite s u m i n Eq. ( 3 ) . T h e l i n e a r i z a t i o n o f t h e k i n e m a t i c b o u n d a r y c o n d i t i o n o n t h e w a t e r surface results i n t h e a d d i t i o n a l e q u a t i o n r e l a t i n g source i n t e n s i t i e s a n d t h e local w a t e r surface slope (IVlatveev, 2 0 0 7 ) , 1 ^ q . ^ i 2 A y V A x , _ , • | ^ ) = - 2 a o ^ ( x , , y , , (4) w h e r e a n d g,- are t h e source s t r e n g t h s o f t h e u p s t r e a m a n d d o w n s t r e a m n e i g h b o r s o f t h e c o l l o c a t i o n p o i n t i, a n d A x a n d A y are t h e i n t e r v a l s b e t w e e n t h e source l o c a t i o n s i n x a n d y d i r e c -tions. O n t h e h u l l surface, the slope is k n o w n ; t h e r e f o r e , t h e source i n t e n s i t i e s can be r e l a t e d t o t h e l o c a l t r i m angle o f the h u l l . T h e l i n e a r s y s t e m o f e q u a t i o n s i n v o l v i n g Eqs. ( 2 ) - ( 4 ) is solved d i r e c t l y f o r t h e w a t e r surface elevations o u t s i d e t h e h u l l , pressure c o e f f i -c i e n t o n t h e h u l l , sour-ce i n t e n s i t i e s , a n d v e l o -c i t y p e r t u r b a t i o n s . The l i f t f o r c e o n t h e h u l l a n d the c e n t e r o f pressure are t h e n d e t e r m i n e d f r o m pressure d i s t r i b u t i o n o n t h e h u l l surface.

One c o m p l i c a t i o n i n the p r o b l e m u n d e r c o n s i d e r a t i o n is t h e i n i t i a l l y u n k n o w n w e t t e d lengths o f t h e h u l l Lw(y), since t h e w a t e r tends t o rise i n f r o n t o f t h e p l a n i n g surface (Fig. 1). In o r d e r t o d e t e r m i n e t h e w a t e r rise, an i t e r a t i v e s o l u t i o n p r o c e d u r e is a p p l i e d . I n t h e b e g i n n i n g , t h e w a t e r rise is n e g l e c t e d , and t h e w e t t e d l e n g t h s are assumed to be e q u a l t o t h e n o m i n a l l e n g t h s (Ln(y) i n Fig. 1). T h e n , a s o l u t i o n is f o u n d f o r t h e w a t e r surface elevations. A r e p r e s e n t a t i v e w a t e r surface c o n t o u r o b t a i n e d a f t e r the first i t e r a t i o n is s h o w n b y a d a s h - d o t t e d l i n e i n Fig. l b . Since i t crosses t h e h u l l surface, the w e t t e d l e n g t h s m u s t be c o r r e c t e d . So i n t h e second i t e r a t i o n , t h e w e t t e d l e n g t h s L2(y) are selected as i n d i c a t e d i n Fig. l b ; t h e i r l e f t b o u n d a r i e s c o r r e s p o n d t o t h e i n t e r s e c t i o n s b e t w e e n t h e h u l l surface a n d t h e w a t e r surface c a l c u l a t e d i n t h e first i t e r a t i o n . A n e w s o l u t i o n o b t a i n e d i n t h e second i t e r a t i o n p r o d u c e s a n o t h e r s h i f t o f t h e w a t e r surface, a n d t h e w e t t e d l e n g t h s have t o be c o r r e c t e d a g a i n . Such i t e r a t i v e c a l c u l a t i o n s are r e p e a t e d u n t i l t h e i n t e r s e c t i o n s b e t w e e n t h e w a t e r a n d h u l l surfaces, as w e l l as w a v e c o n t o u r s , s t o p c h a n g i n g . A d d i t i o n a l l y , t h e w a t e r spray t h a t c a n a p p e a r above t h e w a t e r rise l i n e is n e g l e c t e d i n t h i s m o d e l .

T h e c o n d u c t e d m e s h i n d e p e n d e n c e studies suggested t h e f o l -l o w i n g r e c o m m e n d a t i o n s f o r s e -l e c t i n g d i m e n s i o n s o f n u m e r i c a -l cells a n d a size o f the e n t i r e d o m a i n . T h e distances b e t w e e n sources o n t h e h u l l i n b o t h x - a n d y - d i r e c t i o n s s h o u l d be chosen as m i n ( B / 1 2 , L w / 1 2 , /1/30), w h e r e B a n d are t h e b e a m a n d t h e m e a n n o m i n a l w e t t e d l e n g t h o f t h e h u l l , respectively, a n d A is t h e w a v e l e n g t h o n t h e u n c o n s t r a i n e d f r e e w a t e r surface. Outside o f t h e h u l l . A x can g r a d u a l l y increase t o w a r d s the f r o n t a n d b a c k b o u n d a r i e s o f t h e n u m e r i c a l d o m a i n ( t o m a k e c o m p u t a t i o n s m o r e e f f i c i e n t ) , b u t i t is capped by t h e v a l u e A / 3 0 . T h e d i s t a n c e s i n f r o n t and o n t h e sides o f t h e h u l l u p t o t h e d o m a i n b o u n d a r i e s are chosen as 2B; a n d t h e distance f r o m t h e h u l l t r a n s o m t o t h e d o w n s t r e a m d o m a i n b o u n d a r y is t a k e n as m a x ( 1 2 B , O . U ) . Select-i n g fSelect-iner cells or larger d o m a Select-i n does n o t lead t o s Select-i g n Select-i f Select-i c a n t changes i n c a l c u l a t e d h y d r o d y n a m i c p r o p e r t i e s o f t h e h u l l .

3. R e s u l t s

3.J. Validation cases

The first v a l i d a t i o n is c a r r i e d o u t f o r a flat p l a n i n g surface w i t h a finite aspect r a t i o . W a n g and Rispin ( 1 9 7 1 ) o b t a i n e d a n a n a l y t i c a l

16 K.I. Matveev / Ocean Engineering 82 (2014) 14-19

Fig. 2. Distribution o f pressure coefficient (normalized by attack angle) along (a) midspan and (b) 90-perecnt span o f a flat plate. Filled circles, results of the current method; open circles, solutions o f W a n g and Rispin (1971).

Fig. 3. (a) Lift coefficient and (b) mean w e t t e d length o f a nearly monohedral hull. Filled circles, results of the current method; open circles, test data o f Begovlc and Bertorello (2012).

s o l u t i o n f o r such a plate m o v i n g at h i g h b u t f i n i t e Froude n u m b e r s . T h e y p r e s e n t e d the l o n g i t u d i n a l pressure d i s t r i b u t i o n s at the plate m i d s p a n ( c e n t e r i i n e ) a n d 9 0 - p e r c e n t span. A c o m p a r i s o n o f t h e i r s o l u t i o n w i t h results o f t h e p r e s e n t n u m e r i c a l m e t h o d is s h o w n i n Fig. 2 f o r t h e b e a m Froude n u m b e r Fre = L f o / ^ / g B = 2.24 a n d t h e m e a n w e t t e d aspect r a r i o L „ m / B = l . The a g r e e m e n t b e t w e e n results o f t w o m e t h o d s is reasonably g o o d . It s h o u l d be k e p t i n m i n d t h a t the a n a l y t i c a l s o l u t i o n w a s o b t a i n e d u n d e r a d d i r i o n a l l i m i t a r i o n s , such as a c o n s t a n t w e t t e d l e n g t h across span a n d a finite e x p a n s i o n i n the s o l u t i o n u p to t e r m s o f o r d e r o f Fr-'*.

The second v a l i d a t i o n is conducted f o r a m o n o h e d r a l h u l l . W h i l e s i m p l e f u n c t i o n a l correlations f o r h y d r o d y n a m i c properties o f such h u l l s exist (e.g., Savitsky, 1964), t h e y m a y produce results d e v i a t i n g s i g n i f i c a n t l y f r o m actual test data f o r specific h u l l s and test c o n d i t i o n s due to a w i d e range o f h u l l parameters a n d speed regimes covered b y these correlations (Payne, 1995; P e m b e r t o n e t al„ 2 0 0 1 ) . The a p p r o x i m a t e a g r e e m e n t b e t w e e n t h e present m e t h o d a n d Savitsky's equations has been p r e v i o u s l y d e m o n s t r a t e d (IVlatveev a n d Ockfen, 2 0 0 9 ) . Here, a c o m p a r i s o n is made w i t h r e c e n t l y r e p o r t e d e x p e r i m e n t a l data by Begovic and Bertorello (2012). T h e i r tested h u l l s w e r e f r e e to heave a n d p i t c h , and t h e h u l l sinkages and t r i m angles w e r e r e p o r t e d . One o f the s h j d i e d h u l l s had a l o n g section w i t h t h e c o n s t a n t deadrise angle ^ = 1 6 . 7 ° a n d a f a i r e d b o w piece. The beam-based l i f t c o e f f i c i e n t , Ci =

F/(0.5pUlB^), w h e r e F is t h e l i f t force, and t h e n o r m a l i z e d m e a n

w e t t e d l e n g t h o f this h u l l are s h o w n i n Fig. 3. The a g r e e m e n t b e t w e e n e x p e r i m e n t a l a n d m o d e l i n g results is satisfactory, g i v i n g t h a t a s m a l l p a r t o f t h e w e t t e d area e x t e n d e d b e y o n d a constant-deadrise p o r t i o n o n the tested h u l l . I n t h e same paper o f Begovic a n d Bertorello (2012), results w e r e also presented f o r o t h e r hulls w i t h l i n e a r l y variable-deadrise sections. U n f o r t u n a t e l y , m o d e l i n g o f those h u l l s cannot be p r e s e n t l y a c c o m p l i s h e d since r e p o r t e d e x p e r i m e n t a l c o n d i t i o n s i n v o l v e d 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 w e t t e d area o n the c u r v e d b o w , w h o s e g e o m e t r y w a s n o t specified.

3.2. Variable-deadrise hull

To d e m o n s t r a t e c a l c u l a t i o n s o f h y d r o d y n a m i c effects o f a v a r i a b l e - d e a d r i s e h u l l , a t w i s t e d h u l l is c o n s i d e r e d here. The t r i m angles o f b u t t o c k lines (these angles are k e p t s m a l l d u e t o l i n e a r i t y

a s s u m p t i o n ) are assumed to g r a d u a l l y increase i n t h e sidewise d i r e c t i o n f r o m t h e keel w i t h t h e rate \dr/dy\, so t h a t the c h i n e b u t t o c k lines have t r i m angles larger t h a n t h a t o f t h e keel l i n e b y

Ar=\dT/dy\B/2. This t w i s t also leads t o t h e d e a d r i s e increase

t o w a r d s t h e b o w . Namely, t h e v a r i a t i o n o f t h e local deadrise angle c a n be expressed as f o l l o w s .

/ ? ( - X) = arctan ^ t a n ( ^ o ) + (5)

w h e r e /?o is the h u l l deadrise at t r a n s o m a n d t h e s i g n m i n u s i n f r o n t o f X is used here d u e ' t h e chosen d o w n s t r e a m d i r e c t i o n o f t h e X - a x i s ' ( F i g . 1). H u l l s w i t h i n c r e a s i n g deadrise angles t o w a r d s t h e b o w are v e r y c o m m o n i n practice, since h i g h e r d e a d r i s e at t h e b o w results i n b e t t e r seakeeping, w h i l e flatter s t e r n sections i m p r o v e t h e h u l l l i f t - d r a g r a t i o .

I n t h e p r e s e n t n u m e r i c a l e x a m p l e , t h e deadrise at t h e h u l l t r a n s o m is selected t o be zero, /}Q = 0, w h e r e a s t h e t r i m i n c r e m e n t b e t w e e n t h e keel a n d c h i n e lines is chosen as f o u r degrees, A T = 4 ° . A l t h o u g h t h e l o n g i t u d i n a l deadrise increase is n o n l i n e a r a c c o r d i n g t o Eq. (5), its a c t u a l v a r i a t i o n is v e r y close t o a l i n e a r f o r m , as one can see i n Fig. 4 . The p a r a m e t r i c h y d r o d y n a m i c c a l c u l a t i o n s are c a r r i e d o u t h e r e f o r a fixed h u l l a t t i t u d e w i t h t h e k e e l t r i m « ^ = 3 ° a n d t h e n o r m a l i z e d w e t t e d k e e l l e n g t h Lfc„/B = 2. The b e a m Froude n u m b e r is v a r i e d b e t w e e n 1 a n d 5.

As i n d i c a t e d p r e v i o u s l y , boat designers are o f t e n i n t e r e s t e d i n i d e n t i f y i n g a n e f f e c t i v e p r i s m a t i c h u l l t h a t has h y d r o d y n a m i c characteristics s i m i l a r t o t h o s e o f a h u l l w i t h v a r i a b l e deadrise. For e x a m p l e , f o r such a m o n o h e d r a l h u l l Savitsky e t a l . ( 2 0 0 7 ) suggested u s i n g a deadrise angle at t h e s e c t i o n c o n t a i n i n g t h e c e n t e r o f g r a v i t y o f t h e n o n - p r i s m a t i c h u l l , w h i l e t h e e f f e c t i v e t r i m angle s h o u l d be chosen at t h e 1 / 4 b u t t o c k l i n e o f t h e n o n p r i s m a t i c h u l l . H o w e v e r , n o i n d i c a t i o n w a s g i v e n a b o u t an e f f e c -t i v e sinkage o f a p r i s m a -t i c h u l l . T w o s i m p l e o p -t i o n s w o u l d be -t o use e i t h e r t h e same keel s u b m e r g e n c e at t r a n s o m f o r b o t h h u l l s o r to k e e p t h e same w e t t e d keel l e n g t h s . O u r p r e l i m i n a r y c a l c u l a t i o n s s h o w e d t h a t a m o n o h e d r a l h u l l w i t h such a t t i t u d e s w i l l have e i t h e r s i g n i f i c a n t l y l o w e r o r h i g h e r l i f t c o e f f i c i e n t t h a n t h a t o f t h e t w i s t e d h u l l . T h e r e f o r e , a b e t t e r a g r e e m e n t can be e x p e c t e d w h e n a n i n t e r m e d i a t e sinkage is selected f o r a n e q u i v a l e n t p r i s m a t i c h u l l

K.I. Matveev / Ocean Engineering 82 (2014) 14-19 17

as f o l l o w s ,

lio

" 2 Ok J (6)

w h e r e he/f and ho are the k e e l s u b m e r g e n c e s at t r a n s o m ( w i t h respect to the u n d i s t u r b e d w a t e r surface) f o r t h e p r i s m a d c a n d t w i s t e d h u l l s , respectively, is t h e t r i m angle at k e e l o f t h e t w i s t e d h u l l , and Oeff is the e f f e c t i v e t r i m angle f o r the p r i s m a t i c h u l l (aeff = + A T / 2 i n the p r e s e n t e x a m p l e ) . T h e f o r m o f Eq. ( 6 ) is selected t o p r o v i d e b e t t e r a p p r o x i m a t i o n t h a n the m e n d o n e d above o p t i o n s . H o w e v e r , t h i s choice c a n n o t be c o n s i d e r e d p e r f e c t , since n o a d d i t i o n a l e f f o r t s w e r e m a d e t o i n v e s t i g a t e h o w t o d e f i n e t h e o p t i m a l e f f e c t i v e s u b m e r g e n c e , w h i c h m a y also d e p e n d o n o t h e r h u l l a n d speed characteristics.

H y d r o d y n a m i c p r o p e r t i e s o f the chosen t w i s t e d h u l l a n d the " e q u i v a l e n t " p r i s m a t i c h u l l have been c a l c u l a t e d u s i n g t h e n u m e r -ical m e t h o d o u t l i n e d i n the p r e v i o u s s e c t i o n . Results are p r e s e n t e d i n Fig. 5 f o r the beam-based l i f t c o e f f i c i e n t , Q = F/(0.5pUoB^), n o r m a l i z e d centers o f pressure ( f r o m t r a n s o m ) , m e a n w e t t e d lengths, a n d the l i f t - d r a g rario, LDR=Ci/Co, w h e r e the e f f e c t i v e d r a g c o e f f i c i e n t Co is d e f i n e d as f o l l o w s .

(7) w h e r e Cp is the local pressure c o e f f i c i e n t o n t h e h u l l surface, A is t h e h o r i z o n t a l p r o j e c t i o n o f t h e pressure area, a is t h e local t r i m ,

Av, is t h e w e t t e d area ( e x c l u d i n g t h a t c o v e r e d b y spray), a n d Cf is

t h e e f f e c t i v e s k i n - f r i c t i o n c o e f f i c i e n t . T h e f o r m s i m i l a r to Eq. ( 7 ) is c o m m o n l y used f o r r o u g h e s t i m a t i o n s o f t h e p l a n i n g h u l l drag. For s i m p l i c i t y , t h e w h i s k e r spray d r a g a n a l y z e d b y Savitsky e t al. ( 2 0 0 7 ) is e x c l u d e d f r o m t h e p r e s e n t c o n s i d e r a t i o n , w h e r e a s a fixed b u t realistic value is assumed f o r t h e f r i c t i o n c o e f f i c i e n t , C / = 0.004. It s h o u l d be also n o t e d t h a t due to a change o f the c e n t e r o f pressure o f the t w i s t e d h u l l w i t h Froude n u m b e r , t h e c o r r e s p o n d -i n g deadr-ise angle o f the " e q u -i v a l e n t " p r -i s m a t -i c h u l l also var-ies. I n t h e c o n s i d e r e d e x a m p l e , i t changes f r o m 6.6° t o 9.3° f o r Froude n u m b e r v a r i a t i o n f r o m 1 t o 5.

General t r e n d s o f h y d r o d y n a m i c p r o p e r t i e s i n Fig. 5 appear as expected. T h e l i f t c o e f f i c i e n t d r o p s w i t h i n c r e a s i n g Froude n u m b e r due to r e d u c t i o n o f h y d r o s t a t i c c o n t r i b u t i o n to t h e l i f t at h i g h speeds (Fig. 5a); and a s i m i l a r r e d u c t i o n is n o t i c e a b l e f o r t h e l i f t - d r a g r a t i o (Fig. 5 d ) . The center o f pressure moves f o r w a r d , w h i l e the m e a n w e t t e d l e n g t h changes l i t t i e f o r a t w i s t e d h u l l . Due to the a p p r o p r i a t e selection o f the m o n o h e d r a l h u l l sinkage, m o s t o f its h y d r o d y n a m i c p r o p e r t i e s f o l l o w those o f the n o n - p r i s m a t i c h u l l q u i t e closely. The m e a n w e t t e d l e n g t h o f t h e c o n s t a n t - d e a d r i s e h u l l decreases w i t h increasing Froude n u m b e r due t o increase o f d e a d -rise; h o w e v e r , t h i s v a r i a t i o n is o n l y a f e w percent (Fig. 5c).

T h e d i s t r i b u t i o n s o f the pressure c o e f f i c i e n t s f o r b o t h hulls at F r B = 3 are s h o w n i n Fig. 6 a l o n g t w o l o n g i t u d i n a l sections: one is close t o t h e k e e l a n d t h e o t h e r is near t h e c h i n e . I n t h e case o f the p r i s m a t i c h u l l , t h e near-keel pressure increase m o n o t o n i c a l l y t o w a r d s t h e b o w , w h i l e t h e r e is a local m i n i m u m n o t i c e a b l e f o r the t w i s t e d h u l l data close to t h e s t a g n a t i o n p o i n t I n t h e r e g i o n n e a r the c h i n e , b o t h pressure d i s t r i b u t i o n s s h o w a s m a l l decrease at a b o u t t w o - t h i r d s o f t h e w e t t e d l e n g t h t o w a r d s t h e b o w , w h i l e r e c o v e r i n g h i g h values near t h e u p s t r e a m ends o f these sections. M a g n i t u d e s o f Cp f o r t h e t w i s t e d are l o w e r n e a r t h e keel a n d h i g h e r close t h e c h i n e w h e n c o m p a r e d w i t h Cp f o r t h e p r i s m a t i c h u l l . This h a p p e n s because t h e t w i s t e d h u l l has l o w e r t r i m at t h e keel a n d h i g h e r t r i m at t h e c h i n e i n c o m p a r i s o n w i t h t h e " e q u i v a l e n t " m o n o h e d r a l h u l l .

| x | / l

2 | x | / B

Fig. 4. Variation of the deadrise angle along the considered twisted hull.

Fig. 6. Longitudinal distnhutions of pressure coefficient at (a) 4-perecnt span o f f the keel and (b) 46-percent span o f f the keel. Filled circles, twisted h u l l : open circles, "equivalent" monohedral hull.

1.5

CÜ

1.45

1.4

1.35

Q

0

2

4

IS K.I. Matveev / Ocean Engineering 82 (2014) 14-19

0.2

0.15

0.1

0.05

0

? A A I

0.85

0.8

0.75

0.7

A A A

• • •

0

Fr, B

2 4

10

9

8

7

6

5

A A 4. D Fr, B

Fig. 7. (a) Lift coefficient, (b) center o f pressure, (c) mean w e t t e d length, and ( d ) l i f t - d r a g ratio. Circles, Li„/B = 2: squares, Li„/B = 4; and triangles, Lt„/B = 1.

Fig. 8. Cross-sections o f hulls w i t h (a) positive and (b) negative deadrise.

F r „

' B " B ' ' B ' ' B

Fig. 9. (a) Lift coefficient, (b) center of pressure, (c) mean wetted length, and (d) l i f t - d r a g ratio. Filled circles, negative-deadrise h u l l ; open circles, positive-deadrise hull.

H y d r o d y n a m i c p r o p e r t i e s have b e e n also calculated f o r t h e same t w i s t e d h u l l a n d t r i m angle as i n t h e p r e v i o u s e x a m p l e a n d variable w e t t e d keel l e n g t h ( o r v a r i a b l e h u l l sinkage). Results o f these calculations are p r e s e n t e d i n Fig. 7. For easier c o m p a r i s o n i n t h i s p a r a m e t r i c v a r i a t i o n , i t is m o r e m e a n i n g f u l to n o r m a l i z e t h e c e n t e r o f pressure a n d t h e m e a n w e t t e d l e n g t h b y t h e w e t t e d keel l e n g t h a n d m o d i f y a d e f i n i d o n o f t h e l i f t c o e f f i c i e n t b y i n c l u d i n g t h i s l e n g t h , Q , =F/(0.5pUlBLk„). Results i n Fig. 7a i n d i c a t e t h a t at h i g h Froude n u m b e r s larger Cu is p r o d u c e d b y t h e h u l l w i t h l o w e r l e n g t h - t o - b e a m r a t i o , w h e r e a s at t h e l o w e s t Frg larger is achieved o n t h e h u l l w i t h h i g h e r tfe„/B. This d u e to p r e v a l e n c e o f t h e h y d r o d y n a m i c l i f t a t h i g h Frg a n d t h e h y d r o s t a t i c l i f t a t l o w Frg. H u l l s w i t h h i g h e r Lh„/B g e n e r a t e m o r e h y d r o s t a t i c l i f t , a n d h u l l s w i t h l o w e r tfc„/B are m o r e e f f e c t i v e i n p r o d u c i n g h y d r o d y n a m i c l i f t . S i m i l a r t r e n d s are o b s e r v e d f o r t h e l i f t - d r a g r a t i o (Fig. 7 d ) . The n o r m a l i z e d values o f t h e c e n t e r o f pressure a n d the m e a n w e t t e d l e n g t h increase w i t h d e c r e a s i n g t h e l e n g t h - t o - b e a m L^„/B.

3.3. Negative-deadrise hull

The p r e s e n t n u m e r i c a l m e t h o d can be also a p p l i e d t o m o d e l o t h e r v a r i a t i o n s i n p l a n i n g h u l l p a r a m e t e r s . I n t h i s section, a c o m p a r i s o n is g i v e n f o r t w o p r i s m a t i c h u l l s h a v i n g p o s i t i v e a n d n e g a t i v e ( o r i n v e r t e d ) deadrise angles (Fig. 8 ) . T h e absolute value o f deadrise is selected to be 8 ° , t h e t r i m angle o f b o t h h u l l s is 4 ° , a n d t h e m e a n n o m i n a l w e t t e d l e n g t h ( a t t h e 1 / 4 - b u t t o c k l i n e ) is c h o s e n as t w o beams, Lwn/B = 2. The w a t e r f l o w does n o t usually separate f r o m v e r t i c a l sides o f t h e n e g a t i v e - d e a d r i s e h u l l (Fig. 8 b ) , w h i c h leads t o increased w e t t e d surface a n d f r i c t i o n drag. H o w -ever, w i t h a use o f r e l a t i v e l y s m a l l spray d e f l e c t o r s o n t h e outside h u l l surface near t h e keel l i n e , i t is possible t o achieve f l o w s e p a r a t i o n a n d keep t h e side surfaces d r y ( E g o r o v et al., 1978). I n t h e f o l l o w i n g example, t h e side surface o f t h e n e g a t i v e - d e a d r i s e h u l l is a s s u m e d t o be d r y a n d is n o t a c c o u n t e d f o r i n d e t e r m i n i n g t h e l i f t - d r a g r a t i o .

K.I. Matveev / Ocean Engineering 82 (2014) 14-19 19 0.4 0.3 0.2 0.1 0.5 1.5 2.5 | x | / B | x | / B

Fig. 10. Longitudinal distnbudons o f pressure coefficient at (a) 4-perecnt span o f f the l<eel and (b) 46-percent span o f f the keel. Filled circles, negative-deadrise h u l l ; open circles, posidve-deadrise h u l l .

Calculated h y d r o d y n a m i c p r o p e r t i e s o f these h u l l s are s h o w n i n Fig. 9, w h e r e a s t h e l o n g i t u d i n a l d i s t r i b u t i o n s o f pressure c o e f f i c i e n t s near t h e keel a n d chine at Frg = 3 are g i v e n i n Fig. 10. U n d e r t h e c h o s e n c o n d i t i o n s , the l i f t c o e f f i c i e n t a n d t h e l i f t - d r a g r a t i o o f b o t h h u l l s are p r a c t i c a l l y the same (Fig. 9a a n d d ) . The centers o f pressure a n d t h e m e a n w e t t e d lengths are s l i g h t l y smaller o n t h e h u l l w i t h n e g a t i v e deadrise (Fig. 9b a n d c). The pressure c o e f f i c i e n t s r e f l e c t t h e differences b e t w e e n t w o h u l l s w i t h o p p o s i t e deadrise angles (Fig, 10). Near t h e keel, the w e t t e d l e n g t h o n the n e g a t i v e - d e a d n s e h u l l is s h o r t e r a n d Cp is s m a l l e r over m o s t o f this l e n g t h i n c o m p a r i s o n w i t h those values o n t h e positive-deadrise h u l l . Near t h e chine, results s h o w o p p o s i t e trends. 4. C o n c l u d i n g r e m a r k s The f l e x i b i l i t y o f t h e m e t h o d o f h y d r o d y n a m i c sources f o r p l a n i n g h y d r o d y n a m i c s has b e e n d e m o n s t r a t e d t h o u g h m o d e l i n g of h u l l s w i t h t w i s t a n d n e g a t i v e deadrise. V a l i d a t i o n e x a m p l e s s h o w e d a g o o d a g r e e m e n t w i t h an analytical s o l u t i o n a n d a v a i l -able test data. S i m p l i c i t y o f i m p l e m e n t a t i o n o f this m e t h o d a n d its c o m p u t a t i o n a l e f f i c i e n c y can e q u i p boat designers w i t h a fast c o m p u t a t i o n a l t o o l c o n v e n i e n t f o r i n i t i a l p a r a m e t r i c d e s i g n o f p l a n i n g boats.

Sample results o f h y d r o d y n a m i c calculations have b e e n p r e -sented f o r a t w i s t e d h u l l i n a range o f Froude n u m b e r s , as w e l l as

f o r a p r i s m a t i c h u l l w i t h s i m i l a r characteristics. A suggestion was m a d e o n h o w to select a sinkage f o r t h e " e q u i v a l e n t " m o n o h e d r a l h u l l . The i n f l u e n c e o f t h e w e t t e d l e n g t h o r sinkage o n h y d r o -d y n a m i c p r o p e r t i e s o f a t w i s t e -d h u l l w a s i n v e s t i g a t e -d .

M o d e l i n g o f a h u l l h a v i n g negative deadrise has b e e n s h o w n as w e l l . By e x c l u d i n g the side w e t t e d s u r f a c e o f such a h u l l , i t is s h o w n t h a t i t s l i f t and d r a g characterisrics are v e r y s i m i l a r t o those p r o p e r t i e s o f a h u l l w i t h p o s i t i v e d e a d r i s e a n d t h e same sinkage, w h i l e s o m e differences w e r e d e t e c t e d i n t h e center o f pressure a n d m e a n w e t t e d l e n g t h .

O b t a i n i n g e x p e r i m e n t a l data i n t h e f u t u r e f o r r e l a t i v e l y s i m p l e b u t n o n - p r i s m a t i c p l a n i n g h u l l s w i l l be u s e f u l f o r a d d i t i o n a l v a l i d a t i o n o f t h e present c o m p u t a t i o n a l a p p r o a c h . This m e t h o d can be also e x t e n d e d f u r t h e r to i n c l u d e e v e n m o r e c o m p l i c a t e d g e o m e t r i e s o f p l a n i n g hulls, such as steps.

A c l a i o w l e d g m e n t

This m a t e r i a l is based u p o n w o r k s u p p o r t e d b y t h e N a t i o n a l Science F o u n d a t i o n u n d e r G r a n t n o . 1 0 2 6 2 6 4 .

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