Ocean Engineering 83 (2014) 1-15
ELSEVIER
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Ocean Engineering
j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / o c e a n e n g
Experimental seakeeping assessment of a warped planing hull
model series
E. Begovic'^, C. Bertorello, S. Pennino
University of Naples Federico II, Department of Industrial Engineering, Via Claudia 21, Naples, Italy
(D
CrossMarkA R T I C L E I N F O
Article history:
Received 14 August 2013 Accepted 6 March 2014 Available online 28 March 2014
Keywords:
Warped hull forms Planing hulls Regular wave tests Vertical motions
Vertical accelerations first and second harmonics
•eadrise variation
A B S T R A C T
One monohedral and three warped hard chine planing hull forms were designed at Department of Industrial Engineering, University of Naples. All models have the same transversal section at 0.25 L from the stern w i t h 16.7° deadrise angle. Warped models have deadrise angle linearly varying along the hull length. In previous work the results of an extensive experimental program in calm water were presented. This paper is focused on the seakeeping assessment in regular waves.
Aim of this research is the evaluation of the effect of deadrise angle variation along the hull length on seakeeping characteristics. Regular wave tests have been chosen to get better insight on motion and acceleration properties connected w i t h warped hull forms and to provide a benchmark for CFD of planing hulls.
Models w i t h same load coefficient C^=0A28 were tested at three volumetric Froude numbers Frv=1.92, 2.60 and 3.25. Reported results are nondimensional responses for heave, pitch, and accelerations at 0.5/. and bow. Particular emphasis was given to acceleration analysis by crest to trough method and spectral analysis of first and second harmonics. All results are commented w i t h respect to hull form variation. Uncertainty analysis has been performed according to ITTC 2011 recommendations. © 2014 Elsevier Ltd. All rights reserved.
1. Introduction
E x p e r i m e n t a l studies o f p l a n i n g boat b e h a v i o u r i n waves s t a r t e d w i t h w e l l k n o w n systematic m o d e l tests o f c o n s t a n t d e a d n s e p r i s m a d c h u l l s by Fridsma (1969, 1971). Sixteen m o d e l s w i t h L/B values 4, 5 a n d 6 w i t h t h r e e deadrise angles w e r e t e s t e d i n r e g u l a r ( 1 9 6 9 ) a n d i r r e g u l a r waves (1971) at t h r e e speeds a n d t h r e e HjB r a d o . These w o r k s are t h e m i l e s t o n e s f o r seakeeping c o n s i d e r a t i o n s o f p l a n i n g h u l l s as e x p e r i m e n t a l results a n d as e m p i r i c a l f o r m u l a s d e v e l o p e d by Fridsma (1971), r e v i e w e d i n Savitsky a n d W a r d B r o w n (1976) a n d Savitsky a n d Koebel ( 1 9 9 3 ) . T h e y c o n c e r n added resistance a n d accelerations values at CG and b o w , at d i f f e r e n t speed r e g i m e s . The r e g u l a r w a v e results f o r m o d e l m o t i o n s w e r e p r e s e n t e d as RAOs, w h e r e heave w a s n o n -d i m e n s i o n a l i z e -d by w a v e a m p l i t u -d e a n -d p i t c h b y w a v e slope. M o t i o n s w e r e f o u n d t o have a linear relationship w i t h wave h e i g h t at t h e l o w e s t speed and for either v e r y short o r v e r y l o n g wave lengths. Accelerations at the CG and b o w w e r e generally f o u n d to have a n o n l i n e a r relationship w i t h wave height. A d d e d resistance i n waves has a n o n l i n e a r relationship w i t h wave h e i g h t as generally i t is for
• Corresponding author. Tel.: +39 081 7683708; fax: +39 081 2390380.
E-mail address: begovic@unina,it (E, Begovic).
http://dx.doi.Org/10.1016/j.oceaneng.2014.03.012 0029-8018/© 2014 Elsevier Ltd. All rights reserved.
any ship. Motions and accelerations w e r e f o u n d t o be d e p e n d e n t o n deadrise. Running t r i m angle was f o u n d to have little effect o n motions, b u t a significant i m p a c t o n accelerations.
F r o m t h e tests i n i r r e g u l a r waves (Fridsma, 1971) i t w a s f u r t h e r c o n c l u d e d t h a t m o t i o n s f o l l o w e d a d i s t o r t e d Rayleigh d i s t r i b u t i o n , w h i l e accelerations t e n d e d t o f o l l o w an e x p o n e n t i a l d i s t r i b u t i o n . F u n d a m e n t a l i n f o r m a t i o n r e g a r d i n g the effect o f deadrise w e r e r e p o r t e d i n t h a t w o r k . It w a s f o u n d t h a t deadrise i n f l u e n c e increased w i t h speed; a d d e d resistance decreased w i t h increasing deadrises. M o t i o n s a n d accelerations w e r e n o t i c e a b l y r e d u c e d w i t h increasing deadrises. A final c o n c l u s i o n o f t h e c i t e d w o r k w a s t h a t since RAO responses w e r e n o t i n a g r e e m e n t f o r r e g u l a r a n d i r r e g u l a r seas, v e r y s t r o n g n o n - l i n e a r b e h a v i o u r o f accelerations a n d n o n l i n e a r responses o f m o r i o n w i t h w a v e h e i g h t v a r i a -t i o n i m p l i c a -t e necessi-ty o f u s i n g s-ta-tis-tical analysis o f da-ta i n -t i m e d o m a i n .
Since t h a t , f e w systematic w o r k s o n p l a n i n g h u l l s e a k e e p i n g w e r e p u b l i s h e d a n d all o f t h e m p r e s e n t results i n i r r e g u l a r waves f o l l o w i n g Fridsma's (1971) a p p r o a c h . I n ( 1 9 8 1 ) Z a r n i c k a n d T u r n e r c o n d u c t e d s t u d y i n i r r e g u l a r waves o n several m o n o h e d r a l h u l l s w i t h L / B = 7 a n d o n e m o n o -h e d r a l -h u l l w i t -h L / B = 9 , as e x t e n s i o n o f Fridsma's m o d e l s t o -h i g -h e r LjB r a t i o s at t h r e e speed l e n g t h rarios vjL"-^ = 2, 3 a n d 4 . T h e conclusions o f t h e a u t h o r s c o n f i r m e d s u p e r i o r b e h a v i o u r o f h i g h
2 E. Begovic et al. / Ocean Engineering 83 (2014) 1-15
Nomenclature
RT-CW t o t a l resistance i n c a l m water, N T d r a u g h t , m A w a v e a m p l i t u d e , m TAP d r a u g h t a t a f t p e r p e n d i c u l a r , m a acceleration, m/s^ V speed, m/s B b e a m , m V d i s p l a c e m e n t v o l u m e , m ^ Be b e a m at chine, m VCG v e r t i c a l p o s i t i o n o f t h e c e n t r e o f gravity, mCv Froude n u m b e r based o n b r e a d t h Cy = v/^/gB u{y) s t a n d a r d u n c e r t a i n t y o f variable y, u(y) = y ^ l i u ^ deadrise angle, d e g CA load c o e f f i c i e n t = Q ^A/pgB^
P
s t a n d a r d u n c e r t a i n t y o f variable y, u(y) = y ^ l i u ^ deadrise angle, d e g Fry v o l u m e t n c Froude n u m b e r f r i / = v / \ / g • V ' / ^ acceleration o f gravity, 9.80665 m/s^ A d i s p l a c e m e n t , N g v o l u m e t n c Froude n u m b e r f r i / = v / \ / g • V ' / ^ acceleration o f gravity, 9.80665 m/s^ X w a v e l e n g t h , m H w a v e h e i g h t , m '/3 heave d i s p l a c e m e n t , m HjB w a v e h e i g h t t o b e a m r a t i o n o n - d i m e n s i o n a l heave response HjA w a v e steepness - r a t i o b e t w e e n w a v e h e i g h t a n d '/s p i t c h d i s p l a c e m e n t , d e g w a v e l e n g t h ilslkA n o n - d i m e n s i o n a l p i t c h response k w a v e n u m b e r , r a d / m , k=2njX/'
w a v e d i r e c t i o n w i t h r e s p e c t to t h e b o w , d e g l e n g t h b e t w e e n p e r p e n d i c u l a r s , m OJ w a v e f r e q u e n c y , rad/sLoA l e n g t h over a l l , m COE e n c o u n t e r w a v e frequency, rad/s, cüE^co-kv cos /<
LA-B l e n g t h o f clear p a r t o f m o d e l s , m RMS r o o t m e a n square
LCG l o n g i t u d i n a l p o s i t i o n o f t h e c e n t r e o f g r a v i t y f r o m t r a n s o m , m
deadrise a n g l e a n d no p a r t i c u l a r advantage w a s f o u n d i n h i g h L/B r a t i o f o r m f o r b o w acceleration.
K l o s i n s k i a n d W a r d B r o w n (1993a, b ) r e p o r t e d tests i n i r r e g u l a r w a v e s f o r t w o w a r p e d 1/18 scale m o d e l s o f U.S. Coast G u a r d N o t i o n a l Designs o f 110 FT a n d 120 FT W P B H u l l s . T h e f i r s t o n e has iBp/Bc(,max=4.21 a n d later 5.19. H u l l f o r m s are w a r p e d w i t h d e a d -rise v a r i a t i o n f r o m 10" a n d 15" respectively a t t r a n s o m t o 2 2 " at m i d s h i p . I n Klosinski (1993a) t h e effect o f LCG v a r i a t i o n o n m o t i o n s a n d acceleration is r e p o r t e d f o r 36%, 39% a n d 42% o f LBP. The m o s t f o r w a r d p o s i t i o n (42% LBP) is f o u n d t o r e s u l t i n m i n i m u m i m p a c t accelerations a n d a d d e d resistance. I n K l o s i n s k i ( 1 9 9 3 b ) o n l y o n e LCG p o s i t i o n (36% o f LBP) was c o n s i d e r e d f o r t w o m o d e l s . Tests w e r e p e r f o r m e d f o r t h e ship speed o f 10 k n o t s i n i r r e g u l a r head seas w i t h s i g n i f i c a n t w a v e h e i g h t s o f 12.5% o f t h e b e a m a n d at t h r e e speeds c o r r e s p o n d i n g to Cv=T5, 3.0 a n d 4.0 ( f r o m 2 5 to 7 0 k n o t s i n s h i p scale) i n i r r e g u l a r head seas h a v i n g a s i g n i f i c a n t w a v e h e i g h t o f 20% o f t h e b e a m , t h e l o w e s t v a l u e o f those tested b y F r i d s m a (1971). M o d e l s w e r e t o w e d a t c o n s t a n t speed. T i m e d o m a i n " c r e s t - t r o u g h analysis" was c a r r i e d o u t f o r t h e p i t c h , heave, a n d accelerations at five d i f f e r e n t locations. For each signal, t h e m e a n a n d RMS, t h e n u m b e r o f oscillations, t h e average o f t h e crests a n d t r o u g h s , t h e average o f t h e 1 / 3 - h i g h e s t a n d o f t h e 1/10-h i g 1/10-h e s t crests a n d t r o u g 1/10-h s a n d t 1/10-h e e x t r e m e values o f t 1/10-h e crests a n d t r o u g h s w e r e r e p o r t e d . A l l data w e r e scaled to f u l l size. T h e c o n c l u s i o n i n t e r m s o f c o m p a r i s o n a m o n g t w o h u l l s w a s t h a t t h e l o n g e r h u l l w i t h t h e h i g h e r l e n g t h - b e a m r a t i o , t h e 120 f t W P B , has a b o u t 40% less response t o t h e waves. The RMS accelerations a t v a r i o u s l o n g i t u d i n a l locations t h r o u g h o u t t h e b o a t are l i n e a r l y r e l a t e d t o t h e m i d s h i p acceleration.
The c o n c l u s i o n r e p o r t e d i n Klosinski ( 1 9 9 3 a ) c o n c e r n i n g a d d e d resistance w a s t h a t considerable scatter i n t h e a d d e d d r a g values c o u l d i n d i c a t e t h a t t h e c o n c e p t o f added resistance i n p l a n i n g c r a f t s h o u l d be r e v i e w e d a n d d e t e r m i n i s t i c analysis s h o u l d be a p p l i e d . D u r i n g 1990s, t h e m o s t s i g n i f i c a n t w o r k i n p l a n i n g c r a f t s e a k e e p i n g w a s t h e Enlarged Ship Concept (ESC), b y K e u n i n g a n d P i n k s t e r ( 1 9 9 5 , 1997). The w o r k w a s based o n t h e idea to increase t h e s h i p l e n g t h b y a b o u t 25% w h i l s t all o t h e r d e s i g n c h a r a c t e r i s t i c s s u c h as speed, payload, f u n c t i o n a l i t i e s a n d o t h e r j d i m e n s i o n s w e r e k e p t t h e same as m u c h as possible. T h e o p t i m i -z a t i o n o f t h e h u l l g e o m e t r y was c o n c e n t r a t e d i n p a r t i c u l a r a t t h e b o w sections f r o m a h y d r o d y n a m i c p o i n t o f v i e w . A c o n s i d e r a b l e i m p r o v e m e n t i n t h e c a l m w a t e r resistance a n d t h e m o t i o n s o f t h e ship i n a seaway w a s o b t a i n e d . A u t h o r s r e p o r t e d as m u c h as 30% for resistance a n d u p t o 70% i n t h e o p e r a b i l i t y o f a ESC d e s i g n w h e n c o m p a r e d w i t h t h e o r i g i n a l design. A f u r t h e r e x t e n s i o n o f this idea, k n o w n as t h e A X E B o w Concept (ABC) was p r e s e n t e d b y K e u n i n g e t al. ( 2 0 0 2 ) . It c o n s i s t e d i n : fore ship sections w i t h n o flare, v e r t i c a l s t e m a n d v e r y s l e n d e r f o r w a r d p a r t o f t h e h u l l . There was a s i g n i f i c a n t l y increased sheer a n d a d o w n w a r d s s l o p i n g c e n t e r i i n e a t t h e b o w . These m o d i f i c a t i o n s w e r e a i m e d a t r e d u c i n g t h e w a v e e x c i t i n g forces a n d t h e h y d r o d y n a m i c l i f t i n p a r t i c u l a r i n the f o r w a r d e n d o f t h e s h i p w h i l s t m a i n t a i n i n g a d e q u a t e b u o y -ancy. K e u n i n g a n d V a n W a l r e e ( 2 0 0 6 ) a n d K e u n i n g et a l . (2011) f r o m t h e tests c a r r i e d o u t r e p o r t e d a considerable i m p r o v e m e n t i n v e r t i c a l accelerations peaks at t h e b o w w i t h t h i s c o n c e p t w h e n c o m p a r e d w i t h t h e ESC.
In t h e last decade pressure d i s t r i b u t i o n along the p l a n i n g h u l l a n d along t h e transversal sections was studied n u m e r i c a l l y a n d e x p e r i -m e n t a l l y by Rosén a n d Gar-me ( 2 0 0 4 ) , Rosén (2004), Gar-me et al. (2010) , Lewis et al. (2010). These results are i m p o r t a n t for loads d e t e r m i n a t i o n , f o r rules d e v e l o p m e n t and software b e n c h m a r k i n g .
G r i g o r o p o u l o s e f ' a l . ( 2 0 1 0 ) a n d G r i g o r o p o u l o s a n d D a m a l a (2011) p r e s e n t e d s e a k e e p i n g results f o r a s y s t e m a t i c series o f d o u b l e - c h i n e w i d e - t r a n s o m h u l l f o r m w i t h w a r p e d p l a n i n g sur-face. T h e series consist o f five h u l l f o r m s w i t h L/B ratios = 4.00, 4.75, 5.50, 6.25 a n d 7.00. T w o m o d e l s f o r each h u l l f o r m have b e e n c o n s t r u c t e d a n d tested i n c a l m w a t e r at six d i s p l a c e m e n t s , i n c l u d i n g v e r y l i g h t ones. G r i g o r o p o u l o s e t al. p r e s e n t e d t h e e x p e r i m e n t a l results i n r e g u l a r ( 2 0 1 0 ) a n d i n i r r e g u l a r (2011) waves. V e r y useful i n f o r m a t i o n for designer are r e p o r t e d i n relative speed r e g i m e F n = 0 . 3 4 a n d 0.68.
Recently t w o s y s t e m a t i c series w e r e p r e s e n t e d : t h e first o n e b y Soletic (2010) a n d t h e second o n e b y T a u n t o n e t a l . (2010, 2011). Soletic p u b l i s h e d t h e results o f seakeeping tests i n i r r e g u l a r w a v e s o f systematic series b y M e t c a l f e t a l . ( 2 0 0 5 ) a n d K o w a l y s h y n a n d M e t c a l f ( 2 0 0 6 ) r e l a t i v e to f o u r m o d e l s based o n t h e U n i t e d States Coast G u a r d 4 7 - f o o t M o t o r L i f e b o a t ( M L B ) h u l l f o r m . These m o d e l s present w a r p e d b o t t o m a l t h o u g h w i t h s m a l l deadrise v a r i a t i o n p l o n g t h e h u l l ( f r o m 16.6" to 2 2 . 5 " ) a n d o n e m o d e l has 2 0 - 2 5 " y - d e a d r i s e at stern a n d Section 5 r e s p e c t i v e l y T h e v a r y i n g p a r a m e t e r is L/B, w h i l e d i s p l a c e m e n t is constant. T h e w o r k r e p o r t s s i g n i f i c a n t values ( 1 / 3 a n d 1/10) for accelerations, a d d e d resistance, heave a n d p i t c h m o r i o n s i n sea states 2 a n d 3. The i m p o r t a n t p a r t o f Soletic's w o r k is t h e c o n s i d e r a t i o n o f
E Begovic et al. / Ocean Engineering 83 (2014) 1-15 3
acceleration d i s t r i b u t i o n . A u t h o r r e p o r t e d t h a t t h e e x p e r i m e n t a l data does n o t agree w i t h t h e e x p o n e n t i a l d i s t r i b u t i o n . The e x p e r i m e n t a l data are a p p r o x i m a t e l y 100% h i g h e r t h a n t h e c o r r e s p o n d -i n g e x p o n e n t -i a l d -i s t r -i b u t -i o n data. A u t h o r f u r t h e r m o r e c o m p a r e d m e a s u r e d acceleration w i t h Savitsky a n d Koebel (1993) e m p i r i c a l equations to p r e d i c t average LCG a n d b o w accelerations. A g r e e -m e n t b e t w e e n t h e p r e d i c t e d a n d e x p e r i -m e n t a l acceleration data w a s fair. The error t e n d e d t o u n d e r - p r e d i c t LCG accelerations a n d o v e r p r e d i c t b o w accelerations. This t r e n d l e d t o s i g n i f i c a n t d i s -crepancy i n the c o n s i d e r a t i o n o f r a t i o b o w t o LCG a c c e l e r a t i o n ; Soletic r e p o r t s t h a t e x p e r i m e n t a l data t y p i c a l l y r a n g e d f r o m a p p r o x i m a t e l y 1.8 t o 3.0 w h i l e p r e d i c t i o n suggested a b o w t o LCG acceleration r a t i o r a n g i n g f r o m 2.0 to 6.4. T a u n t o n et al. (2010, 2 0 1 1 ) p r e s e n t e d a series o f f o u r m o n o -h e d r a l -h u l l s , w -h e r e L/B r a n g e d f r o m 6.25 to 3.77 c o r r e s p o n d i n g t o L/V^'^ f r o m 8.70 t o 6.25 a n d c o n s t a n t deadrise angle o f 22.5°. For t h i s systematic series t h e r e are resistance data and r i g i d b o d y m o t i o n s a n d accelerations a t t h r e e speeds ( m o d e l speeds = 6, 10 a n d 12 m / s ) i n i r r e g u l a r w a v e s . A l l tested m o d e l s h a d r a d i i o f g y r a t i o n /<55=0.16 L, a l t h o u g h s t a n d a r d ITTC v a l u e for p i t c h r a d i i o f g y r a t i o n is 0.25 L. A u t h o r s d i d n o t g i v e any c o m m e n t o n t h i s . S m a l l r a d i i o f g y r a t i o n m e a n s t h a t all t h e w e i g h t w e r e v e r y c o n c e n t r a t e close t o t h e CG. To o b t a i n v e r y h i g h relative speeds as Frv=7.9, tested b y a u t h o r s , t h e b o a t has to be o f v e r y l i g h t c o n s t r u c t i o n a n d t h e o n l y i m p o r t a n t w e i g h t is the e n g i n e . These factors lead t o r e d u c e d r a d i i of g y r a t i o n . W a v e h e i g h t a n d vessel m o t i o n s w e r e analysed a p p l y i n g d i s t o r t e d Rayleigh d i s t r i b u t i o n , as d e f i n e d i n C a r t w r i g t a n d L o n g u e t - H i g g i n s (1956), w h i l e accelerations w e r e fitted b y G a m m a d i s t r i b u t i o n s . A u t h o r s d i d n o t p r e s e n t any c o m p a r i s o n a m o n g m o d e l s a n d n o c o m m e n t s o n o b t a i n e d results w e r e g i v e n . The p r o c e d u r e o f u s i n g these data t o p r e d i c t V i b r a t i o n Dose V a l u e (VDV) a n d h u m a n p e r f o r m a n c e o n b o a r d a f u l l scale vessel was g i v e n .
Begovic and B e r t o r e l l o ( 2 0 0 9 a, b ) u n d e r t o o k an e x p e r i m e n t a l p r o g r a m focused o n t h e b e h a v i o u r o f a n o n - u n i f o r m deadrise, p l a n i n g h u l l f o r m i n r e g u l a r a n d i r r e g u l a r head sea at t h e T o w i n g Tank o f t h e U n i v e r s i t y o f Naples. A scale m o d e l was tested a t Froude n u m b e r s f r o m 0.9 t o 1.4 i n r e g u l a r w a v e s w i t h A/L r a n g i n g f r o m 1 to 5. Tests i n s t a n d a r d ITTC t w o p a r a m e t e r spectra w e r e also p e r f o r m e d f o r t h r e e d i f f e r e n t speeds.
Begovic and B e r t o r e l l o ( 2 0 1 2 ) p u b l i s h e d d e t a i l e d resistance d e t e r m i n a t i o n o f a w a r p e d b o t t o m m o d e l series i n r e g u l a r w a v e s . This paper presents research c o n c e r n i n g seakeeping c h a r a c t e r i s -tics o f t h e same m o d e l s at t h r e e speeds. Results for heave at CG, p i t c h a n d accelerations at t w o locations are s u m m a r i s e d i n g r a p h i c a l f o r m as a f u n c t i o n o f n o n d i m e n s i o n a l w a v e l e n g t h . To p r o v i d e an easier a n d p r a c t i c a l t o o l at design stage, t h e acceleration results are p r e s e n t e d as a f u n c t i o n o f deadrise v a r i a -tion also. The i n f l u e n c e s o f b o t t o m w a r p i n g o n seakeeping is clearly i d e n t i f i e d t h r o u g h a c c e l e r a t i o n values at m i d s h i p a n d b o w .
The choice to c o n s i d e r s e a k e e p i n g characteristics i n r e g u l a r w a v e s is d u e to p r o v i d e a database f o r s o f t w a r e t e s t i n g a n d t o analyse t h e physics o f p h e n o m e n o n i n a m o r e systematic w a y t h a n b y statistical d i s t r i b u t i o n s . The significance o f t h e p r e s e n t w o r k i n t h e c o n t e x t o f t h e a f o r e m e n t i o n e d researches is t o p r o v i d e a reference o f seakeeping p e r f o r m a n c e for p l a n i n g hulls w i t h w a r p e d b o t t o m . It s h o u l d serve as a f u r t h e r c o n t r i b u t e t o t h e research w o r k d o n e b y F r i d s m a o n p r i s m a t i c h u l l s .
2. Models characteristics
W a r p e d p l a n i n g h u l l f o r m e m b o d i e s t h e c o n c e p t o f h i g h l i f t i n g surface as m u c h as possible s i m i l a r t o flat plate a n d g o o d seakeeping characteristics o b t a i n e d w i t h h i g h deadrise angle o f b o w a n d i n some cases c e n t r a l sections. Deadrise angle v a r i a t i o n
a l o n g t h e h u l l l e n g t h is e m p i r i c a l l y designed w i t h an a f t e r p a r t c h a r a c t e r i z e d b y a l m o s t c o n s t a n t deadrise o f 1 0 1 6 " a n d a p r o -gressive increase up to 2 5 - 3 0 ° or m o r e at b o w sections. The v a r i a t i o n t r e n d , g e n e r a l l y leads t o c u r v e c h i n e p r o f i l e s . A d o u b l e i n f l e x i o n i n c h i n e p r o f i l e is also c o m m o n , l e a d i n g to a character-istic S shaped c h i n e p r o f i l e .
The m o d e l series d e v e l o p e d b y Begovic a n d B e r t o r e l l o (2012), o n t h e basis o f p r e v i o u s w o r k s Begovic a n d B e r t o r e l l o ( 2 0 0 9 ) , Begovic e t al. ( 2 0 0 9 ) a n d B e r t o r e l l o a n d O l i v i e r o ( 2 0 0 7 ) , deals w i t h t h r e e w a r p e d h u l l f o r m s a n d o n e m o n o h e d r a l . W a r p i n g o f hulls is m a t h e m a t i c a l l y d e f i n e d as linear v a r i a t i o n o f c h i n e f r o m t r a n s o m to 0.8 L f r o m s t e r n , w i t h a characteristic section a t 0.25 L w i t h 16.7" deadrise i d e n t i c a l for all t h e f o u r m o d e l s .
The m o n o h e d r a l h u l l has c o n s t a n t deadrise o f 16.7°. The t h r e e w a r p e d h u l l f o r m s c a n n o t be d e f i n e d " p r i s m a t i c " as c h i n e is n o t p a r a l l e l t o t h e keel, a l t h o u g h b o t t o m a n d sides are s t i l l d e v e l o p -able as p l a n e surfaces. A f a i r e d b o w has been fitted t o a l l o w t h e t r a n s i t i o n to p l a n i n g r e g i m e . As Fridsma's m o d e l s t h e y d o n o t have a b o w shape i n f l u e n c i n g t h e seakeeping p r o p e r t i e s .
For each m o d e l s t h e p r i n c i p a l characteristics are r e p o r t e d i n Table 1 a n d h u l l g e o m e t r y is s h o w n i n Fig. 1.
3. Experimental set-up and test matrix
The seakeeping tests w e r e p e r f o r m e d i n the T o w i n g Tank of University o f Naples Federico 11, w h i c h d i m e n s i o n s are 135 m x 9 m X 4.2 m , equipped w i t h a m u l t i f l a p electrically d r i v e n w a v e -m a k e r and t o w i n g carriage w i t h 8 -m/s o f -m a x i -m u -m speed.
A l l m o d e l s w e r e ballasted to achieve a w e i g h t v e r y close t o 3 2 0 N. Exact values are r e p o r t e d i n Table 1 . A l l m o d e l s w e r e t r i m m e d to 1.66° ( s t e r n d o w n ) i n o r d e r to o b t a i n r u n n i n g t r i m o f a b o u t 4° at p l a n i n g speeds. This set u p w a s tested i n c a l m w a t e r a n d p r e s e n t e d i n Begovic a n d B e r t o r e l l o ( 2 0 1 2 ) i n t h e range o f F r v = 0 . 6 5 0 - 4 . 5 1 8 . Seakeeping tests w e r e p e r f o r m e d a t t h r e e speeds - 3.4, 4.6 and 5.75 m/s. It w a s seen f r o m c a l m w a t e r tests t h a t a t speed 4.6 ( F r v = 2 . 6 0 ) all m o d e l s have p e r f e c t flow separa-tion. M a x i m u m realistic speed for t e s t i n g i n w a v e s is 5.75 (Frv,=3.25) a n d t h e l o w e s t speed o f 3.4 ( F r v = 1.92) is t h e speed w h e r e p l a n i n g begins. To get i n s i g h t a b o u t t h e speed r e g i m e Froude n u m b e r s ( v o l u m e t r i c a n d based o n c h i n e b e a m ) , Taylor n u m b e r s {vlL°-^) and ship speed i n k n o t s f o r t w o possible m o d e l scales are g i v e n i n Table 2. Reported scale r a t i o are g i v e n as a reference o f t w o ships i n f u l l size t o w h i c h t h i s h u l l f o r m s c o u l d be suitable. The first scale r a t i o (10) is r e l a t i v e to a p l e a s u r e boat w i t h i n l i m i t l e n g t h f o r t h e a p p l i c a t i o n o f 9 4 / 2 5 EC D i r e c t i v e , w h i l e t h e second scale r a t i o ( 2 0 ) refers t o a t y p i c a l s m a l l fast f e r r y for 2 8 0 passengers. I n Table 3, t h e values o f s i g n i f i c a n t p a r a m e t e r s f r o m c a l m w a t e r tests f o r tested speeds are r e p o r t e d .
For t h e seakeeping tests t h e v e r t i c a l centre o f g r a v i t y a n d the r a d i i o f g y r a t i o n have been m e a s u r e d b y t h e i n e r t i a l balance s h o w n i n Fig. 2. M e a n periods o f r o l l a n d p i t c h have been d e t e r m i n e d o n the basis o f 100 oscillations m e a s u r e d b y the a c c e l e r o m e t e r Cross B o w CXL04GP3-R-AL at s a m p l i n g f r e q u e n c y
Table 1
Models principal characteristics.
MONO W A R P l WARP 2 W A R P S LoA (m) 1.900 1.900 1.900 1.900 LA-B (m) 1.500 1.500 1.500 1.500 B = B f (m) 0.424 0.424 • 0.424 0.424 TAP (m) 0,096 0.106 0.110 0,108 A (N) 319.7 320.4 319.7 318.5 CA 0.428 0.429 0.428 0.426 P (deg) 16.7 14.3-23.8 11.6-30.1 9,1-35.8
4 £. Begovic et al. / Ocean Engineering 83 (2014) 1-15
Fig. 1. Geometries of hull models.
Table 2
Models' velocities test matrix.
All models V (m/s) 3 . 4 4 , 6 5 , 7 5 Frv 1 , 9 2 2 , 6 0 3 , 2 5 Cv 0 . 9 4 2 1 , 2 7 5 1 , 5 9 4 vIL"-^ (kn/ft''^) 2 , 6 4 7 3 . 5 8 2 4 , 4 7 7 " (l^n)sca/e Ral/o=]0 2 0 . 9 2 8 , 3 3 5 . 4 (kn)5cQfe Rati'o = 20 2 9 , 6 4 0 , 0 5 0 , 0 Table 3
Values of significant parameters from calm water tests.
u (m/s) MONO W A R P 1 W A R P 2 W A R P 3
3 . 4 Running trim (deg) 3 . 9 7 2 4 , 0 5 5 4 , 1 7 9 4 , 1 1 2
Sinkage (mm) - 3 , 6 4 8 0 , 6 1 3 4 , 9 6 5 5 , 6 6 8 RT-CW (N) 3 8 , 7 5 6 4 0 , 9 7 2 4 5 , 1 0 1 4 7 . 4 6 4
4 . 6 Running trim (deg) 4 . 1 7 4 4 , 1 1 9 4 , 1 4 5 3 . 6 7 9
Sinkage (mm) 9 , 7 0 5 1 2 . 7 6 2 2 , 5 0 2 6 . 8 8
Rr-cw (N) 4 6 . 4 7 4 4 8 , 7 6 8 5 0 , 7 4 0 5 1 . 5 3 4 5 , 7 5 Running trim (deg) 4 . 0 2 4 3 , 6 9 6 3 . 2 2 1 2 , 6 4 6
Sinkage (mm) 1 7 . 4 2 5 2 2 , 2 9 6 2 9 . 6 0 1 3 5 . 7 7 8 RT-CW ( N ) 5 2 , 0 4 4 5 4 , 3 1 9 5 5 , 5 3 5 5 7 , 1 1 4
o f 1000 Hz a n d repeated t h r e e d m e s . L o n g i t u d i n a l a n d t r a n s v e r s a l r a d i i o f g y r a t i o n s have been d e t e r m i n e d in air. O b t a i n e d values are r e p o r t e d i n Table 4 . The LCG p o s i t i o n for each m o d e l is set f r o m tests i n c a l m w a t e r w i t h a i m to assure the t r i m o f a b o u t 4 " at speed, l e a d i n g to 1.66" astern as static t r i m . This a p p r o a c h represents a realistic scenario f o r a p l a n i n g h u l l i n service.
Seakeeping tests w e r e p e r f o r m e d at c o n s t a n t speed w i t h m o d e l s r e s t r a i n e d t o sway, r o l l and y a w . M o d e l speed, heave, p i t c h , a d d e d resistance w e r e measured at t h e p o i n t o f 53.5 c m f r o m t h e s t e r n as can be seen f r o m Fig. 3. Heave is recalculated t o t h e l o n g i t u d i n a l c e n t r e o f g r a v i t y LCG p o s i t i o n r e p o r t e d f o r each m o d e l i n Table 4.
The e n c o u n t e r w a v e was m e a s u r e d by t w o u l t r a s o u n d gauges BAUMER U N D K 301U6103/S14, at t w o positions, t h e f i r s t a t 3.97 m f o r w a r d f r o m t h e t o w i n g p o i n t a n d the second at t h e same p o s i t i o n o f t h e t o w i n g p o i n t laterally i n the tank. V e r t i c a l accel-erations w e r e m e a s u r e d by Cross B o w CXL04GP3-R-AL acceler-o m e t e r s at b acceler-o w (1.62 m f r acceler-o m t r a n s acceler-o m c acceler-o r r e s p acceler-o n d i n g tacceler-o O.SSLOA) a n d close t o m i d s h i p (0.72 m f r o m t r a n s o m c o r r e s p o n d i n g to
Fig. 2. Inertial balance.
Table 4
Models' inertial properties.
MONO WARP 1 WARP 2 WARP 3
LCG (m) 0 , 6 9 7 0 . 6 6 0 0 . 6 0 9 0 . 5 8 6 LCG (% Lo/i) 3 6 , 7 3 4 , 7 3 2 . 1 3 0 , 8 VCG (m) 0 . 1 4 3 0 . 1 5 2 0 . 1 5 5 0 , 1 5 6 k44 (m) 0 . 1 2 8 1 0 . 1 2 8 5 0 . 1 2 8 6 0 , 1 1 6 6 kAA (% Be) 3 0 , 2 3 0 , 3 3 0 , 3 0 . 2 7 5 fc55 (m) 0 , 5 8 3 3 0 , 5 7 1 9 0 , 5 4 9 1 0 , 5 1 8 7 kss (% LOA) 3 0 . 7 3 0 . 1 2 8 . 9 2 7 , 3
0.38Lo/i). The same p o s i t i o n s f o r all m o d e l s w e r e chosen f o r a m o r e c o n g r u e n t i n v e s t i g a t i o n o f h u l l f o r m effect. T h e l o n g i t u d i n a l centre o f g r a v i t y for t h e same static t r i m c o n d i t i o n shifts a b o u t 10 c m l o n g i t u d i n a l l y c o r r e s p o n d i n g to a p p r o x i m a t e l y 5% o f
LQA-A l l data w e r e s a m p l e d a t the f r e q u e n c y o f 5 0 0 H z . LQA-Accelerations w e r e n o t f i l t e r e d d u r i n g data a c q u i s i t i o n , o n l y i n data analysis.
Since Fridsma's conclusions i r r e g u l a r w a v e tests have been c o n s i d e r e d v e r y u s e f u l . One reason is statistical analysis o f data due to t h e n o n l i n e a r i t i e s i n response a n d a n o t h e r o n e is designer practice to consider 1/JV h i g h e s t values o f m o t i o n s a n d accelera-tions. On t h e o t h e r h a n d , w h e n s p e a k i n g o n n o n l i n e a r i t y , i r r e g u l a r sea a n d statistical analysis does n o t p e r m i t t o see t h e physics o f responses a n d w h e r e t h e n o n - l i n e a r i t y are c o m i n g f r o m . This is especially t r u e w h e n t h e results have to be r e p o r t e d t o t h e
£ Begovic et al. / Ocean Engineering 83 (2014) 1-15 5
lead to n o n - l i n e a r craft response, a n d have larger influence o n accelerations t h a n o n m o t i o n s as p o i n t e d o u t by Fridsma (1969).
For each m o d e l t i m e d o m a i n and f r e q u e n c y d o m a i n analysis o f m e a s u r e d data are p e r f o r m e d . T i m e d o m a i n analysis consist o f c r e s t - t o - t r o u g h analysis o f recorded time h i s t o r y . Frequency d o m a i n analysis is p e r f o r m e d t a k i n g i n t o c o n s i d e r a t i o n first a n d second h a r m o n i c . The second h a r m o n i c RAO is d e f i n e d as second h a r m o n i c o f response d i v i d e d b y first h a r m o n i c o f w a v e . Even at h i g h e s t speed and longest w a v e , s u f f i c i e n t n u m b e r s o f e n c o u n t e r waves ( a b o u t 10) are c o n s i d e r e d . A f t e r careful c o n s i d e r a t i o n o f spectral analysis f o r m o t i o n s , i t w a s seen t h a t t h e c o n t r i b u t i o n o f second h a r m o n i c is n o t m o r e t h a n 5% at h i g h e s t speed and i n resonance frequencies. Therefore the results g i v e n f o r heave and p i t c h at d i f f e r e n t velocities are based o n time d o m a i n analysis only. As e x p e c t e d for accelerations t h e c o n t r i b u t i o n o f second h a r m o n i c is s i g n i f i c a n t and f o r all m o d e l s at each speed m e a s u r e d
Fig. 3. Experimental set up, accelerations are g i v e n as t i m e d o m a i n (crest t o t r o u g h ) a n d
spectral analysis o f 1st a n d 2 n d "RAO" f o r each r u n .
Table 5
Regular waves matrix.
/ ( H z ) 0) (rad/s) k (rad/m) X(m) A (mm) HIX
1.00 6,283 4,026 1.561 0.821 16 0,020 0.90 5,655 3,261 1.927 1.014 20 0,021 0,80 5.027 2.576 2.439 1.284 20 0.016 0,70 4,398 1,973 3,185 1.676 20 0,013 0.65 4.084 1,701 3.694 1,944 32 0,017 0,60 3.770 1.449 4.335 ' 2,282 32 0,015 0,55 3.456 1.218 5,160 2,716 35 0.014 0,50 3,142 1,006 6.243 3,286 35 0,011 0.45 2,827 0,815 7.708 4,057 45 0,012 0,40 2,513 0,644 9,755 5,134 45 0,009 4.J. Monohedral hull
In Figs. 4 and 5 heave and p i t c h responses calculated f r o m crest-t o - crest-t r o u g h analysis for each crest-tescrest-ted speed and i n Fig. 6 crest-the accelera-tions close to m i d s h i p are given. In Figs. 7 a n d 8 acceleraaccelera-tions measured at b o w are reported. The values are presented n o n -d i m e n s i o n a l l y as ratio ajg. From Figs. 6 - 8 i t can be note-d t h a t c r e s t - t r o u g h analysis gives the values a p p r o x i m a t e l y equal to the sum o f the first and the second h a r m o n i c ; this is n o t true for frequencies close to the resonance. From t h e comparison of Figs. 6 a n d 8 i t can be noted t h a t c o n t r i b u t i o n o f the second h a r m o n i c is higher at b o w t h a n in the central position. F r o m Fig. 8 i t can be noted the i m p o r t a n c e of second h a r m o n i c as the speed increases.
c o m p a r i s o n a m o n g d i f f e r e n t h u l l s ; i t has t o be n o t e d t h a t n o n e o f t h e e x p e r i m e n t a l w o r k s p r e v i o u s l y d e s c r i b e d t r i e d t o give i n s i g h t o n t h i s aspect.
T h e r e f o r e regular w a v e tests w e r e chosen to s t u d y the increase o f n o n l i n e a r i t i e s i n acceleration responses due o n l y t o increase o f m o d e l speed even i n s m a l l a m p l i t u d e w a v e s , t o d i s t i n g u i s h b e t t e r t h e effect o f b o t t o m w a r p i n g o n seakeeping characteristics a n d finally t o p r o v i d e b e n c h m a r k f o r s o f t w a r e t u n i n g .
The considered wave lengths range f r o m AILOA=0-8 u p to 5.13. The longest wave ( A = 9 . 7 5 5 m ) is the u p p e r l i m i t to get a reasonable n u m b e r o f encounters. H i g h frequency waves are chosen w i t h H/ A = l / 5 0 b u t i t was n o t possible to keep this ratio constant due to v i o l e n t w a t e r on deck events. The tests w e r e p e r f o r m e d w i t h H / A = 1/75 and 1/100. A s u m m a r y o f significant parameters is given i n Table 5.
4. Results presentation
The c o m m o n l y used r e p r e s e n t a t i o n o f s e a k e e p i n g results for d i s p l a c e m e n t ship is based o n l i n e a r t h e o r y a s s u m p t i o n s . This m e a n s t h a t d i a g r a m s o f Response A m p l i t u d e Operators (RAO) as a f u n c t i o n o f n o n d i m e n s i o n a l e n c o u n t e r f r e q u e n c y are g i v e n w h e r e RAO are d e t e r m i n e d as t h e first h a r m o n i c o f t h e response a n d first h a r m o n i c o f t h e r e g u l a r waves. These results can b e m u l t i p l i e d by any spectra a n d p r o v i d i n g m e a n i n g f u l values. The m o s t l i m i t i n g aspect o f t h i s p r o c e d u r e is the l i n e a r a s s u m p t i o n o f m o t i o n s , i.e. r e l a t i v e l y s m a l l d i s p l a c e m e n t s caused b y s m a l l w a v e a m p l i t u d e s .
The m a i n difference b e t w e e n l o w speed displacement ship and p l a n i n g h u l l seakeeping is strong variations o f w e t t e d area o f n o n -vertical h u l l surface w h i c h results f r o m change o f t r i m and sinkage o f the p l a n i n g vessel due to wave i n d u c e d m o t i o n s . These p h e n o m e n a
4.2. Model WARP I
I n Figs. 9 a n d 10 heave a n d p i t c h responses o b t a i n e d f r o m c r e s t - t r o u g h analysis for each tested speed are g i v e n . Accelera-tions are g i v e n i n Figs. 1 1 - 1 3 . A g a i n , i t can be n o t e d t h e h i g h e r c o n t r i b u t i o n o f second h a r m o n i c at b o w t h a n at 0.38LOA C o m p a r -i n g o n l y c r e s t - t r o u g h results f o r WARP 1 a n d M O N O (F-igs. 7 a n d 12), i t can be n o t e d t h a t w a r p e d h u l l has l o w e r values o f v e r t i c a l b o w accelerations.
4.3. Model WARP 2
I n Figs. 14 a n d 15 heave a n d p i t c h responses o b t a i n e d f r o m c r e s t - t r o u g h analysis f o r each tested speed are g i v e n . It can be
l a / A O
•
A O X A X o X A X A O A v=3,4 m/s X v=4.6 m/s O v=5.75 m/ss
A v=3,4 m/s X v=4.6 m/s O v=5.75 m/s o ! 8\
6 E. Begovic et al. / Ocean Engineering S3 (2014) 1 -15 2,00 1,75 1,50 1,25 1.00 0,75 0.50 0.25 0.00 0 O 0
*
^ ^
' i
A g ^ 0 A .Jj X A O 0*
^ ^
' i
A g ^ 0 A .Jj A v=3.4 m/s X v=4.6 m/s O v=5.75 m/s 0*
8 - é_... A v=3.4 m/s X v=4.6 m/s O v=5.75 m/s XI L 2 3 4 Fig. 5. Pitch RAO for model IVIONO.0 1 2 3 4 5
Fig. 6 . Accelerations for model MONO at 0.38LOA.
2.00 1.80 1.60 1,40 ] 1.20 1.00 0.80 0.60 0.40 0.20 0.00 aeow^ 9 -»• v=3.4 m/s ••X- v=4.6 m/s v=5.75 m/s — -»• v=3.4 m/s ••X- v=4.6 m/s v=5.75 m/s —
- . / / ; «
•• x —
\ 'x \ \ -»• v=3.4 m/s ••X- v=4.6 m/s v=5.75 m/s — \ ^ ,. / X • * '•• x —
\ 'x \ \ 1} i T,
«
i t"
x,vr" ^^^*Ss^ - - t g XI L 1 1 A 5 6Fig. 7. Accelerations for model MONO at bow from crest to trough analysis.
seen t h a t some p o i n t s w e r e repeated, i n o r d e r t o evaluate u n c e r t a i n t i e s . Accelerations are g i v e n i n Figs. 1 6 - 1 8 . C o m p a r i n g m o d e l WARP 2 w i t h WARP 1 and M O N O , i t can be p o i n t e d o u t t h a t o n l o n g e s t waves WARP 2 has h i g h e r heave a n d p i t c h response. Accelerations at r e s o n a n t frequencies are m u c h l o w e r w h i l e at l o n g e s t waves are s o m e w h a t higher.
4.4. Model WARP 3
I n Figs. 19 a n d 2 0 heave a n d p i t c h responses calculated f r o m c r e s t - t r o u g h analysis f o r each tested speed are g i v e n . Accelera-t i o n s are g i v e n i n Figs. 2 1 - 2 3 . S i m i l a r Accelera-to Accelera-t h e b e h a v i o u r o f WARP 2,
0 1 2 3 4 5 Fig. 8. Accelerations for MONO at bow from spectral analysis.
2,00 1.75 1.50 1.25 1.00 0.75 0.50 0.25 0.00 2.00 1.75 1.50 1.25 1.00 0.75 0.50 • 0.25 ' 0.00 •
'13/A
0 0 • A _ ^ * X A X A X 0 A A 8 A v=3,4m/s X v=4.6 m/s 0 v=5.75 m/s S•
-A v=3,4m/s X v=4.6 m/s 0 v=5.75 m/s*!
X I I0
1 ;Fig. 9. H( 'ave RAO for
4 ! model WARP 1, > 6 'I5 / kA 0 Q 0 X ^ X 1^.. _., . .„ _ _„ .„ ^ A A A A A 0 A v=3.4 m/s X v=4,6 m/s 0 v=5,75 m/s X ^ 0 A v=3.4 m/s X v=4,6 m/s 0 v=5,75 m/s X 0 4 A v=3.4 m/s X v=4,6 m/s 0 v=5,75 m/s 0
1
1 2 3 4Fig. 10. Pitch RAO for model WARP 1,
t h i s m o d e l has h i g h e r heave a n d p i t c h at t h e longest waves a n d s i g n i f i c a n t l y l o w e r v e r t i c a l a c c e l e r a t i o n at b o w .
5. Comparison of hull forms
For easier c o m p a r i s o n o f o b t a i n e d results w i t h respect to t h e h u l l f o r m , t h e results o f f o u r m o d e l s are p r e s e n t e d f o r each tested speed a n d f o r each response. A t Figs. 2 4 a n d 25 heave a n d p i t c h at speed 3.4 m/s are g i v e n . I t can be seen no s i g n i f i c a n t d i f f e r e n c e a m o n g h u l l s u p to AILQA a p p r o x i m a t e l y 2.2. For l o n g e r waves t h e
E. Begovic et al. / Ocean Engineering 83 (2014) 1-15 7 0 . 8 0 0 . 7 0 0 . 6 0 0 . 5 0 0 . 4 0 0 . 3 0 0 . 2 0 0 . 1 0 0 . 0 0 2 . 0 0 1.80 1.60 1.40 1.20 1.00 0 . 8 0 0 . 6 0 0 . 4 0 0 . 2 0 0 . 0 0 ao.38LOA' 9 - O - 1sl@v=3.4frVs " O - 1st@v=4.6m/s - e - 1 s l @ v = 5 . 7 5 m / s - O .2nd@v=3.4m/s " O - 2nd@v=4.6rn/s - B - 2 n d @ v = 5 . 7 5 m / s - » • A C C _ 0 . 5 L ( g v = 3 . 4 A C C _ 0 . 5 L @ V = 4 . 6 - » - A C C _ 0 . 5 L @ v = 5 . 7 5 1 2 3 4 5
Fig. 11. Accelerations at 0.38Lo^ for model WARP 1.
• v = 3 . 4 m / s • • » • v = 4 . 6 m / s - « - v = 5 . 7 5 m / s • v = 3 . 4 m / s • • » • v = 4 . 6 m / s - « - v = 5 . 7 5 m / s 1 ƒ ! t • v = 3 . 4 m / s • • » • v = 4 . 6 m / s - « - v = 5 . 7 5 m / s
\
^•k^
if\
\ ' V \ \ V . . «... •»•. :
XI L 0 1 2 3 4 5 6Fig. 12. Accelerations for model WARP 1 at bow from crest to trough analysis.
1.00 0 . 9 0 0.80 0.70 0 . 6 0 0 . 5 0 0 . 4 0 0.30 0 . 2 0 0 . 1 0 0 . 0 0 a B o w / 9 1 H > 1 s t @ v = 3 . 4 m / s - B - 2 n d @ v = 3 . 4 m / s - O - 1 s t @ v = 4 . 6 m / s " Q - 2 n d @ v = 4 . 6 m / s - e - 1 s t @ 5 . 7 5 r T i/s - e - 2 n d @ v = 5 . 7 5 m / s IP ^s^/:^:s^.ji .."^jn ' \ \ \ •'• Cf" i • 3.^ ^ ^ -..i-.j— ' a ''•a " 0 I' ~ - -EJ U •—'^Z'Z 0 1 2 3 4 6 6
Fig. 13. Accelerations for model WARP 1 at bow from spectral analysis.
2 . 0 0 1.7S 1.50 1.25 1.00 0 . 7 5 0 . 5 0 0 . 2 5 0 . 0 0 2 . 0 0 1.75 1.50 1.25 1.00 0 J 5 0 . 5 0 0 . 2 5 0 . 0 0 0 . 8 0 0 . 7 0 0 . 6 0 0 . 5 0 0 . 4 0 0 . 3 0 0 . 2 0 0 . 1 0 0 . 0 0 1 3 / A A X
' 8
X 0 e : X a Xh
0 X -A A X' 8
X 0 e : X i v = 3 . 4 m / s X v = 4 . 6 m / s 0 v = 5 . 7 5 m / sL
X °
i v = 3 . 4 m / s X v = 4 . 6 m / s 0 v = 5 . 7 5 m / s 0 _ „ . . „ . . . . « L i v = 3 . 4 m / s X v = 4 . 6 m / s 0 v = 5 . 7 5 m / s 1 2 3 4Fig. 14. Heave RAO for model WARP 2 .
A v = 3 . 4 m / s
X v = 4 . 6 m / s
O v = 5 , 7 5 m / s
^ / L
1 2 3 4
Fig. 15. Pitch RAO for model WARP 2 .
A ' 9
-o- I s l g Jv=3.'3m/s - B -2ncJ@v-3.4m/s A C C _ 0 . 5 L @ v = 3 . 4 ••©• 1st€ 3v=4.6 ••Q- 2nd@v=4.6rn/s ACC„0.5L@v='1.6 - e - i s n g 3v=5.75m/s - G - 2 n d @ v = 5 . 7 5 n V s - » - A C C _ 0 . 5 L @ v = 5 . 7 5
1 2 3 4 5
Fig. 16. Accelerations for model WARP 2 at 0.38LOA.
differences are v i s i b l e b u t t h e y are always w i t h i n 15%. As r e g a r d accelerations close t o t h e m i d s h i p , expressed n o n - d i m e n s i o n a l l y as {ajgy^LoAlA) i n Fig. 26 i t can be n o t e d t h a t M O N O a n d WARP 1 have l o w e r acceleration t h a n WARP 2 a n d WARP 3. This r e s u l t is affected by the d i f f e r e n t LCG p o s i t i o n o f t h e m o d e l s due t o t h e c o n s t a n t static t r i m . WARP 2 a n d WARP 3 have LCG p o s i t i o n a l m o s t 5% LQA d i s t a n t f r o m t h e m e a s u r i n g p o i n t w h i l e for M O N O a n d WARP 1 this p o s i t i o n is a l m o s t c o i n c i d e n t w i t h LCG. For t h i s reason for WARP 2 a n d WARP 3 accelerations are m o r e affected by p i t c h . Nevertheless t h e choice t o c o m p a r e accelerations at t h e same l o n g i t u d i n a l p o s i t i o n f o r all m o d e l s appears effective a n d
s o u n d as design reference. It s h o u l d be f u r t h e r c o m m e n t e d t h a t some r e p e a t e d p o i n t s are g i v e n f o r WARP 2 a n d W A R P 3 and t h e d i f f e r e n c e s a m o n g t h e m are a b o u t t h e same as differences a m o n g m o d e l s . For t h e b o w accelerations g i v e n i n Fig. 27, t h e same t r e n d s are o b s e r v e d , M O N O and WARP 1 have s o m e w h a t l o w e r values i n all f r e q u e n c y range. Considering t h e heave at h i g h e r speed g i v e n i n Fig. 28 for v=4.6 m/s a n d Fig. 3 2 f o r v = 5 . 7 5 m/s i t can be n o t e d n o s i g n i f i c a n t difference a m o n g m o d e l s u p to i/LoA a p p r o x i m a t e l y 2.2. M o r e w a r p e d hulls present h i g h e r heave b u t t o t a l difference is w i t h i n 15%. As regard p i t c h , g i v e n i n Fig. 2 9 f o r v=4.6 m/s and i n Fig. 3 3 f o r ^ = 5 . 7 5 m/s, t h e same . t r e n d o f heave is v a l i d , b u t t h e
8 £ Begovic ec al. / Ocean Engineering 83 (2014) 1-15 2.00 1.80 1.60 1.40 1.20 1.00 0.80 0.60 0.40 0.20 0.00 ^BOW / 9 — ^ — — ^ — v=3.4m;s "K- v=4.6 m/s -m- v=5.75 m/s
\
v=3.4m;s "K- v=4.6 m/s -m- v=5.75 m/s / . i L . \ \ •j^ . X/r ^jir^ -
* \ v X/ \X '
•X
j 1 XI L 0 1 2 3 4 5 6 Fig. 17. Accelerations for model WARP 2 at bow from crest to trough analysis.1.00 0.90 0.80 0.70 0.60 0.50 0.30 0.20 0.10 0.00 ^BOW / 9 - O 1st@v=3.4m/s - Q •2nd@v=3.4m/s " O - 1sl@v=4.6m/s " Q - 2nd@v=4.6m's - e - 1sl@v=5.75m/s - B - 2 n d @ v = 5 . 7 5 r W s - O 1st@v=3.4m/s - Q •2nd@v=3.4m/s " O - 1sl@v=4.6m/s " Q - 2nd@v=4.6m's - e - 1sl@v=5.75m/s - B - 2 n d @ v = 5 . 7 5 r W s • o... ^ j! gj D
JQ—A '
/ ^ a ' / \ L o - • ' \ \ 'o. \ '••Q. f T / 7 : A \ 1 a ._ °__J__JB- '°; '^'^^^
& V J3 -^ B - - o ••• - • 0 1 2 3 4 5 6 Fig. 18. Accelerations for model WARP 2 at bow from spectral analysis.2.00 1.75 1.50 1.25 1.00 0.75 0.50 0.25 0.00 i l j / A
0-- - : . .
O X * o X X A O 4 v=3.4 m/s X v=4.6 m/s o v=5.75 m/s X A0
X 4 v=3.4 m/s X v=4.6 m/s o v=5.75 m/s0
8 4 v=3.4 m/s X v=4.6 m/s o v=5.75 m/sa '
0 1 2 3 4 5 6 Fig. 19. Heave RAO for model WARP 3,m a x i m u m d i f f e r e n c e is w i t l i i n 20%. For acceleration c o m p a r i s o n at b o t h p o s i t i o n s at h i g h e r speed, t h e appreciable effect o f w a r p i n g is s h o w n i n Figs. 30, 3 1 , 3 4 a n d 35, w h e r e i t is clearly v i s i b l e t h a t t h e m o r e w a r p e d the h u l l is, the l o w e r the acceleration w i l l be. T h e differences a m o n g m o d e l s i n t h i s case are f r o m 30% at c e n t r a l p o s i t i o n at 4.6 m/s u p t o 50% at b o w at 5.75 m/s.
I n Fig. 3 6 t h e r a t i o b e t w e e n accelerations at b o w a n d at 0.38Lo/i for a l l m o d e l s a n d for all speeds is given. I t is w o r t h t o c o m m e n t t h a t a t speeds o f 4.6 a n d 5.75 for all m o d e l s t h e r a t i o asow/Qo.ssi. is a l w a y s b e t w e e n 2 a n d 3. It can be seen t h a t at t h e l o w e s t speed a n d at v e r y s h o r t waves t h e r a t i o OBow/ao.ssr is h i g h e r for all m o d e l s a n d i t is i n s i d e range 4 a n d 5. This d i a g r a m f u r t h e r 2.00 1.75 1.50 1.25 1.00 0.75 0.50 0.25 0.00 0.80 0.70 0.60 0.60 0.40 0.30 0,20 0.10 0,00 2,00 1.80 1.60 1.40 1.20 1.00 O.BO 0.60 0.40 0.20 0.00 11,/kA X A ê X A v=3.4 m/s X v=4.6 m/s O v=5.75 m/s J . / L 1 2 3 4
Fig. 20. Pitch RAO for model WARP 3.
- O 1sl@v=3.4m/s • O - 1sl@v=4.6m/s - e - 1 s t @ v = 5 . 7 5 m / s - B •2nd@v=3.4m/s ••Q- 2nd@v=4.6m/s - B - 2 n d @ v = 5 . 7 5 r r L / s -tr A C C _ 0 . 5 L @ V = 3 . 4 A C C _ 0 . 5 L @ v = 4 . 6 - » - A C C _ 0 . 5 L @ v = 5 . 7 5 1 2 3 4 5 Fig. 21. Accelerations for model WARP 3 at O.SSLoa
^BOW / 9 -Xr v=3,4 m/s ..«£• v=4,6 m/s - * - v = 5 , 7 5 m/s -Xr v=3,4 m/s ..«£• v=4,6 m/s - * - v = 5 , 7 5 m/s
/
^
. . , / f . / f..A i .
V
*•••„ ^ \ a» - , _
•X X XI L 0 1 2 3 4 5 6 Fig. 22. Accelerations for model WARP 3 at bow from crest to trough analysis,c o n f i r m s t h e results o f Soletic's w o r k a n d p o i n t s o u t t h a t t h e c o m m o n l y used e m p i r i c a l formulas d o n o t describe w e l l t h e p h e -n o m e -n o -n . Fig. 36 also i-ndicates t h a t the ratio o f acceleratio-n o-nboard w i l l n o t depend particularly o n f o r m , i t can be n o t e d t h a t t h e curves are g r o u p e d according to speed and i t is possible to appreciate differences at various speeds, b u t n o t a m o n g various forms.
6. Nonlinearities in accelerations responses
To g e t better i n s i g h t o n n o n l i n e a r i t i e s i n accelerations f o r m o n o h e d r a l m o d e l a t speed v = 4 . 6 m/s a n d / l / L o / i = 1 . 9 4 , m e a s u r e d
E. Begovic et al. / Ocean Engineering 83 (2014) 1-15 1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 - O 1sl@v=3.4m/s - • -2nd@v=3.4m's ••O- 1sl@v=4.6m/s " D - 2 n d @ v ^ . 6 m / s - O 1sl@v=3.4m/s - • -2nd@v=3.4m's ••O- 1sl@v=4.6m/s " D - 2 n d @ v ^ . 6 m / s - e - 1 s l @ v = 5 . 7 5 m / s - Q - 2 r d@v=5.75m/s —-__-_..J?._i.„ / 'h / - J.. V /
\"
, ® .\
{ / ' r 0 V \ \ J u / -É^^^ Q._ •••••Q...^...„,.. 0,.^ . . . .. ^ -O B' I • a 0 1 2 3 4 5 Fig. 23. Accelerations for model WARP 3 at bow from spectral analysis.1 , / A
s i
: J O WARP 1 • WARP 2 a WARP 3 X MONO 1 2 3 4 5 Fig. 24. Heave RAO comparison at i'=3.4 m/s.1.0
0,4
0.2
0 1 2 3 4 5 Fig. 25. Pitch RAO comparison at 11=3.4 m/s.
Xlt. % / l < A A 0 A A " «,
s
A A 3 XL
\
X A _ H O WARP 1 • WARP 2 A WARP 3 X MONO O WARP 1 • WARP 2 A WARP 3 X MONO*
O WARP 1 • WARP 2 A WARP 3 X MONO 1•
3 W L 10 0.8 0.6 LOA'A A A6 |
S "
o A A6 |
S "
o O WARP 1 O WARP 2 A WARPS X MONO•
A ' 0 • X >' i
Ê
O WARP 1 O WARP 2 A WARPS X MONO•
A ' 0 • X >' i
Ê
&
1
È
\ 1 2 3 4 5 Fig. 26. Acceleration comparison at 0.38LOA (i' = 3.4 m/s).( a e o w / g r k A/A A é X a A 0-^ X a A 0-^ O WARP 1 O WARP 2 A WARP 3 X MONO
•
*
X«
O WARP 1 O WARP 2 A WARP 3 X MONO 1 » . ... .. _.. ^. . W L 0 1 2 ,Fig. 27. Acceleration compa
1 4 5 £ ison at bow (i<=3,4 m/s).
A j n A
g
• • O X As
X -n
X è X 0 O WARP 1 • WARP 2 A .WARP 3 X MONO6
O WARP 1 • WARP 2 A .WARP 3 X MONO A 5 * i X / l 1 2 3 4 5 Fig. 28. Heave RAO comparison at i'=4,6 m/s.e n c o u n t e r w a v e , heave, p i t c h a n d accelerations are r e p o r t e d i n Fig. 37. I n Fig. 3 8 spectral analysis o f r e c o r d e d data is g i v e n . I t can be a p p r e c i a t e d t h e s m a l l second h a r m o n i c i n p i t c h a n d v e r y huge second a n d even t h i r d h a r m o n i c o f accelerations. S i m i l a r b e h a v i o r has b e e n observed f o r all m o d e l s , t h a t is w h y t h e results are r e p o r t e d f o r m o n o h e d r a l only.
I t has t o be h i g h l i g h t e d t h a t t h e o b t a i n e d h a r m o n i c s are d u e t o t h e e l e m e n t a r y m o t i o n s c o m p o s i t i o n , a n d as expected t h e i r i m p o r t a n c e is h i g h e r i n b o w accelerations t h a n a t c e n t r a l p o s i t i o n . F u r t h e r m o r e , as i t w a s o b s e r v e d i n resonance r e g i o n t h e
acceleration a m p l i t u d e o b t a i n e d f r o m crest t o t r o u g h analysis is s i g n i f i c a n t l y h i g h e r t h a n t h e s u m o f t h e f i r s t t w o h a r m o n i c s . To evaluate h o w m a n y s u p e r h a r m o n i c s are s i g n i f i c a n t , f o r a l l m o d e l s , the f i r s t f o u r h a r m o n i c s are analysed a t t h e same w a v e f r e q u e n c y o f / l / L o / i = 1 . 9 4 , f o r t w o velocities a n d t w o p o s i t i o n s . These results are s u m m a r i s e d i n Figs. 3 9 - 4 2 .
The 1st h a r m o n i c is s i m i l a r f o r all m o d e l s b u t t h e h i g h e r . o r d e r h a r m o n i c s f o r m o n o h e d r a l h u l l haye h i g h e r values w i t h respect t o w a r p e d hulls. The t r e n d can be appreciated: t h e m o r e t h e h u l l is w a r p e d , t h e l o w e r is t h e c o n t r i b u t i o n o f higher o r d e r h a r m o n i c s .
1 0 16 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 n , / k A A ft.. ^ ^
9
ii
X •—8
O WARP 1
• WARP 2
A WARP 3
X MONO XO WARP 1
• WARP 2
A WARP 3
X MONO ÉO WARP 1
• WARP 2
A WARP 3
X MONOi
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1E. Begovic et al. / Ocean Engineering 83 (2014) 1-15
1.6 • 1.4 1.2 1.0 0.8 0.6 0.4
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X %0 WARP 1
• WARP 2
A WARP 3
X MONO•
0 WARP 1
• WARP 2
A WARP 3
X MONO•
0 WARP 1
• WARP 2
A WARP 3
X MONOè
XI L 1 2 3 4 5 Fig. 29. Pitch ftAO comparison at f = 4 . 6 m/s.1 2 3 4 5 Fig. 32. Heave RAO comparison at i/=5.75 m/s.
25 (ao.38LOA'9)* L O A / A
* •
* 6— J
O WARPl
• WARP 2
A WARP 3
X MONO X / l 1 2 3 4 5 Fig. 3D. Acceleration comparison at 0.3SLOA {I'=4.6 m/s).90 80 70 60 50 40 30 20 10 0 1 2 3 4 5 Fig. 31. Accelerations comparison at bow (i;=4.6 m/s).
0 WARPl
• WARP 2
A WARP 3
X MONO 00 WARPl
• WARP 2
A WARP 3
X MONO IC * X § d•
it0 WARPl
• WARP 2
A WARP 3
X MONO X M n > A•
0 WARPl
• WARP 2
A WARP 3
X MONO "^
A X A*
EB1
..É .. j X/ L 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0115/kA
A a 0 . _ i^
a m X•
0 X0 WARP 1
• WARP 2
A WARP 3
X MONO X d0 WARP 1
• WARP 2
A WARP 3
X MONO X n0 WARP 1
• WARP 2
A WARP 3
X MONO0 WARP 1
• WARP 2
A WARP 3
X MONO g -X\ ^
XIL 1 2 3 4 5 Fig. 33. Pitch RAO comparison at i' = 5.75 m/s.50 45 40 35 30 25 20 15 10 5 0 (ao.asLoA'S)* I-HA/A
a 8 A — 5
n8 "
0WARP 1
aWARP 2
AWARP 3
X MONO XII 1 2 3 4 5 Fig. 34. Acceleration comparison at 0.38LOA ( I ' = 5.75 m/s).This t r e n d is even m o r e visible a t speed o f 5 . 7 5 m/s, s h o w n i n Figs. 4 1 a n d 4 2 .
As p r e v i o u s l y c o m m e n t e d , v e r t i c a l accelerations at 0 . 3 8 L O A i n case o f t h e WARP 2 a n d WARP 3 are s i g n i f i c a n d y affected b y p i t c h a n d this means t h a t i n Figs. 4 0 a n d 4 2 t h e d i f f e r e n c e i n f a v o u r o f w a r p e d h u l l f o r m s w o u l d be h i g h e r i f t h e m e a s u r e m e n t s w o u l d have been d o n e at each CG p o s i t i o n .
The i m p o r t a n c e o f these results lies i n t o d e t e r m i n i s t i c assess-m e n t o f accelerations r a t h e r t h a n statistical one, i n d i c a t i n g t h e
reasons o f occurrence o f h i g h e r o r d e r h a r m o n i c s a n d g i v i n g b o t h values: frequency a n d a m p l i t u d e . A m o r e c o m p l e t e i n f o r m a t i o n a b o u t these last p a r a m e t e r s can be useful f o r loads d e t e r m i n a t i o n .
7 .
Influence of deadrise variation on accelerations
This analysis concerns accelerations o n l y as t h e y are t h e c o m m o n l y used p a r a m e t e r b y designers a n d classification
E. Begovic et al. / Ocean Engineering 83 (2014) 1-15
societies w h e n c o n s i d e r i n g sealceeping b e h a v i o u r o f HSC i n r o u g h sea. T h e influences o f b o t t o m w a r p i n g o n sealceeping is clearly i d e n t i f i e d so t h a t t h e m a i n goal o f this w o r k is achieved.
140 80 20 (aBow'g)* L OA/A ;
X
i;X
X ^
X
0 D _ X Ö X B ;X
i;X
X ^
X
0 D _ X Ö X B 1 0 W A R P l • W A R P 2 A W A R P 3 X MONO 0 I0
> • S ^ A • A A 0 W A R P l • W A R P 2 A W A R P 3 X MONO Xfi
X•
0 1 2 3 4 5 6 Fig. 35. AcceleraUon comparison at bow (i/=5.75 m/s).5.0
^ B O W ' ^0.38LOA
-;>-~MONO_v=3.4m/s - & - M O N O _ v = 4 . 6 m / s MONO. v=5.75 m/s - « - W 1 _ v = 3 . 4 mis - ^ W 1 _ v = 4 . 6 m/s W1_v = 5.75 m/s - e - W 2 _ v = 3 . 4 m/s - ö - W 2 _ v = 4.6 mis - O W 2 „ v = 5.75 m/s - A - W 3 _ v = 3 . 4 m/s - : . . " W 3 _ v ^ 4 . 6 m / s W3_v = 5.75 m/s
0 1 2 3 4 - 5
Fig. 36. Comparison of Qgow/aosst ratios.
Acceleration results are reviewed t h r o u g h the deadrise variation along t h e h u l l . Nondimensional deadrise variation is defined as
° -(tg(fi,)-tg(p^))
21A _
This is chine profile inclination and i t is zero for m o n o h e d r a l h u l l . For the models considered the chine has been i n t e n t i o n a l l y designed as a straight line. I n t h e general case the chine line is curved a n d this results i n deeper angles at b o w and flatter deadrise a t stern, as can be seen i n Fig. 4 3 . For a sound application o f the presented results the deadrise angles at the inflexion points should be considered as w e l l as t h e i r distance accordingly.
To avoid zero as reference value, n o n d i m e n s i o n a l deadrise v a r i a t i o n is d e f i n e d as A/?/L:
hp
L 1
-B
2LA
<tg(P,)~tg{P^))
A c c o r d i n g t o this d e f i n i t i o n , values f o r M O N O is 1, for WARP 1 is 0 . 9 7 3 6 , f o r WARP 2 is 0 . 9 4 7 0 a n d f o r W A R P 3 is 0 . 9 2 0 9 .
N o n - d i m e n s i o n a l accelerations at b o w a n d close t o m i d s h i p f o r t h r e e speed values c a n be r e p o r t e d f o r each w a v e l e n g t h . I n Figs. 4 4 - 4 8 t h e y are s h o w n f o r A/LOA = 1 - 2 8 , 1 . 9 4 , 2 . 2 8 , 3 . 2 9 a n d 5 . 1 3 respectively as t h e m o s t significant.
F r o m Figs. 4 4 - 4 8 i t can be seen t h a t t h e i n f l u e n c e o f deadrise angle can be substantial at h i g h speed a n d a t XjloA u p t o a p p r o x i m a t e l y 2 . 5 , ie. range o f responses resonance. A t v e r y l o n g waves t h e effect o f h u l l f o r m is m a r g i n a l a n d r e l a t i v e speed is t h e o n l y s i g n i f i c a n t p a r a m e t e r t o w h i c h accelerations are related to. A t t h e l o w e s t investigated speed value t h e i n f l u e n c e o f deadrise v a r i a t i o n is n o t appreciable a n d differences o b s e r v e d are w i t h i n e x p e r i m e n t a l uncertainties.
F r o m these results w a r p e d h u l l f o r m appears as a n i n t e r e s t i n g design feature over vjL°-^=3.5 f o r any sea c o n d i t i o n a n d especially for w a v e l e n g t h o f the same o r d e r o f ship l e n g t h .
8.
Repeatability and uncertainy analysis
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