Flexible oscillating duct: An approach to a novel propulsor

Applied Ocean Reseaich 36 (2012) 3 6 - 5 0

ELSEVIER

Contents lists available at SciVerse ScienceDirect

Applied Ocean Research

O C E A N

RESEARCH

Flexible Oscillating Duct: An approach to a novel propulsor

Gerasimos Politis*, Vasileios Tsarsitalidis

NTUA, Athens, Greece

A R T I C L E I N F O

Article lustory: Received 2 1 October 2011

Received i n revised f o r m 17 January 2012 Accepted 28 January 2012

Available online 14 March 2012

Keywords:

Biomimetic propulsion Unconventional propulsion Boundary element method Unsteady wake rollup

A B S T R A C T I n s p i r e d b y a j e l l y f i s h , w h e r e a b u l k m u s c l e o s c i l l a t o r y m o t i o n p r o d u c e s t h r u s t , t h e i n i t i a t i v e w a s tal<en t o e x p l o r e t h e p r o p u l s i o n c a p a b i l i t i e s o f a n e w p r o p u l s o r c o n c e p t b a s e d o n a n o s c i l l a t i n g / p u l s a t i n g f l e x -i b l e d u c t . W e n a m e t h -i s d e v -i c e ' F l e x -i b l e O s c -i l l a t -i n g D u c t ' ( F O D ) . I n t h -i s p a p e r o u r p u r p o s e -is t h r e e f o l d . F i r s t l y , t o u n d e r s t a n d t h e FDD's basic t h r u s t p r o d u c i n g m e c h a n i s m s , s e c o n d l y t o p r e s e n t s y s t e m a t i c FOD h y d r o d y n a m i c p e r f o r m a n c e r e s u l t s , n e c e s s a r y f o r i t s o p t i m u m d e s i g n , a n d t h i r d l y t o c o m p a r e t h e p o w -e r i n g p -e r f o r m a n c -e o f s h i p s -e q u i p p -e d w i t h FODs w i t h t h o s -e w i t h c o n v -e n t i o n a l p r o p -e l l -e r s . T o t h i s -e n d , t h -e p r o b l e m o f a n a c t i v e l y d e f o r m i n g FOD, t r a v e l l i n g w i t h a g i v e n v e l o c i t y i n a n i n f i n i t e l y e x t e n d e d f l u i d , is f o r m u l a t e d a n d s o l v e d u s i n g a n i n d i r e c t S o u r c e - D o u b l e t 3D-BE1V1, t o g e t h e r w i t h a t i m e s t e p p i n g a l g o r i t h m c a p a b l e o f t r a c i n g t h e f r e e v o r t e x s h e e t g e o m e t r y . A n o n l i n e a r p r e s s u r e t y p e K u t t a c o n d i t i o n is a p p l i e d at t h e t r a i l i n g e d g e o f t h e FOD. W i t h a m o l l i f i e r based filtering o f t h e i n d u c e d v e l o c i t i e s , t h e u n s t e a d y r o l l u p p a t t e r n , c r e a t e d b y t h e FOD m o t i o n , e m e r g e s . T h e p r o d u c e d p a t t e r n i n d i c a t e s t h a t FOD t h r u s t is p r o d u c e d t h r o u g h a t r a i n o f c o a x i a l v o r t e x r i n g s w i t h c i r c u l a t i o n s o f a l t e r n a t i n g signs. FOD d e s i g n p a r a m e t e r s are t h e n i n t r o d u c e d a n d d i s c u s s e d a n d d e c i s i o n s w e r e t a k e n r e g a r d i n g c r e a t i o n o f a FOD s y s t e m a t i c series. A s p e c i a l d a t a g e n e r a t i o n a l g o r i t h m has b e e n d e v e l o p e d , c a p a b l e o f g e n e r a t i n g t h e u n s t e a d y t h r u s t -p r o d u c i n g m o t i o n s f o r t h e FOD, i n c l u d i n g c h o r d - w i s e flexibility. U s i n g t h i s d a t a - g e n e r a t i o n c o d e , w e f e e d t h e B E M w i t h s y s t e m a t i c g e o m e t r i c a n d m o t i o n d a t a a n d c a l c u l a t e t h e o p e n w a t e r t h r u s t , p o w e r a n d e f f i c i e n c y f o r t h e p r o p o s e d FOD s e r i e s . T h e s o l u t i o n o f t h e FOD o p t i m u m d e s i g n p r o b l e m f o r r e a l s h i p s is t h e n p r e s e n t e d . T h e d e s i g n m e t h o d has b e e n a p p l i e d f o r t h r e e s h i p t y p e s . C o m p a r i s o n s s h o w t h a t t h e FOD is a p r o m i s i n g s y s t e m w i t h p r o p u l s i v e c o e f f i c i e n t s s u p e r i o r t o t h a t o f a c o n v e n t i o n a l p r o p e l l e r . © 2012 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 c o n o m i c a n d ecological reasons, s c a r c i t y o f resources and d e m a n d f o r speed d i c t a t e a n e v e r - g r o w i n g d r i v e f o r e f f i c i e n c y . The a p p l i c a t i o n o f b i o m i m e t i c systems i n p r o p u l s i o n is far f r o m n e w as a t h o u g h t , y e t i t is d i s r e g a r d e d as i m m a t u r e , d e s p i t e t h e f a c t t h a t lias been u t i l i z e d b y b i r d s a n d sea creatures f o r m i l l i o n s o f years, as a r e s u l t o f n a t u r a l s e l e c t i o n since i t is t h e o n l y s y s t e m capa-b l e o f e x t r a c t i n g e n e r g y f r o m the e n v i r o n m e n t a n d t r a n s f o r m i n g i t t o u s e f u l p r o p e l l i n g t h r u s t . I n s o m e o f his w o r k s , t h e p h i l o s o p h e r A r i s t o t l e considers the a n a t o m y a n d l o c o m o t i o n o f s w i m m i n g crea-tures. Leonardo Da V i n c i h a d also m a d e m a n y designs o f b i r d - l i k e f l y i n g m a c h i n e s , w h i c h w o u l d be successful, i f he had t h e today's t e c h n o l o g y .

The m o d e r n h i s t o r y o f b i o m i m e t i c s starts i n 1935 w i t h Gray's paradox, a n d t h e o r e t i c a l d e v e l o p m e n t s s t a r t w i t h t h e w o r k s o f L i g h t h i l l [ 1 1 , 1 2 ] a n d W u [ 2 6 ] , A t h o r o u g h r e v i e w o f those t h e o -ries can be f o u n d i n [ 2 2 ] , Extensive r e v i e w s o f c o m p u t a t i o n a l and

* Corresponding a u t h o r Tel.: +30 2107721041.

E-rttail addresses: polit@central.ntua.gr (G. Politis), vtsars@central.ntua.gr (V. Tsarsitalidis).

0141-1187/$ - see f r o n t matter © 2 0 1 2 Elsevier Ltd. All rights reserved, d o i : 10.1016/j.apor.2012.01.006

e x p e r i m e n t a l w o r k i n b i o m i m e t i c s can be f o u n d i n the papers o f Shyy et al. [ 2 1 ] r e g a r d i n g a e r o d y n a m i c s a n d aeroelasticity, o f T r i -a n t -a f y l l o u e t -a l , [ 2 5 , 2 4 ] r e g -a r d i n g e x p e r i m e n t -a l d e v e l o p m e n t s -a n d o f Rozhdestvensky a n d R y z h o v [ 2 0 ] r e g a r d i n g all t y p e s o f a p p l i c a -tions, even f u l l scale, w i t h a d d i t i o n a l care g i v e n to t h e w o r k d o n e b y eastern scientists (i.e. Russians a n d Japanese). I n t e r e s t i n g i n f o r -m a t i o n is also i n c l u d e d i n t h e books by A l e x a n d e r [ 2 ] , Shyy et al. [ 2 1 ] a n d T a y l o r et a l . [ 2 3 ] . IVlarine b i o m i m e t i c p r o p u l s o r s are also discussed i n t h e b o o k o f Bose [ 3 ] .

Scope o f t h e p r e s e n t w o r k is t o i n t r o d u c e the FOD, s h o w its f e a -s i b i l i t y a-s a real -s h i p p r o p u l -s o r , e x p l a i n it-s h y d r o - m e c h a n i c -s a n d p r o p o s e a design m e t h o d o l o g y a l o n g w i t h the necessary design d i a g r a m s . A n a t u r a l p a r a d i g m o f a p r o p u l s o r w i t h s i m i l a r t h r u s t p r o d u c t i o n m e c h a n i s m s to t h a t o f t h e FOD, is t h e j e l l y f i s h , j e l l y f i s h f l o w mechanics has been i n v e s t i g a t e d by D a b i r i a n d G h a r i b [ 4 , 5 ] w h o have m a d e m a n y n u m e r i c a l a n d e x p e r i m e n t a l s i m u l a t i o n s a n d have s h o w n the f e a s i b i l i t y o f u s i n g v a r i a b l e d i a m e t e r nozzles as p r o p u l s o r s .

F l o w s i m u l a t i o n s u s i n g B E M is a w e l l - d e v e l o p e d a n d success-f u l m a t h e m a t i c a l / n u m e r i c a l t h e o r y w i t h v e r y g o o d p r e d i c t i o n s success-f o r cases w h e r e l i f t is t h e m a i n m e c h a n i s m o f f o r c e p r o d u c t i o n [ 1 0 ] . The first 3 - D BEM f o r a n a l y z i n g t h e f l o w a r o u n d a r b i t r a r y n o n - l i f t i n g bodies can be a t t r i b u t e d t o Hess a n d S m i t h [ 8 ] . The first 3 - D BEM

C. Politis, V. Tsarsitalidis /Applied Ocean Research 36 (2012) 36-50 37

f o r a n a l y z i n g the steady f l o w a r o u n d a m a r i n e p r o p e l l e r can be a t t r i b u t e d t o Hess and V a l a r e z o [ 9 ] , w h e r e t h e classical Hess a n d S m i t h l i f t i n g f o r m u l a t i o n has b e e n u t i l i z e d f o r t h e r e p r e s e n t a t i o n o f a s t e a d i l y t r a n s l a t i n g a n d r o t a t i n g p r o p e l l e r u s i n g a p r e s c r i b e d w a k e shape to m o d e l t r a i l i n g v o r t e x sheets. I n t h e f o l l o w i n g years, v a r i o u s a l t e r n a t i v e f o r m u l a t i o n s o f the p a n e l m e t h o d have been p r e s e n t e d f o r the s o l u t i o n o f f l o w p r o b l e m s . A b r i e f p r e s e n t a t i o n o f t h e h i s t o r i c a l aspects o f p r o p e l l e r r e l a t e d BEM f o r m u l a t i o n s can be f o u n d i n [ 1 4 ] . This paper also p r e s e n t s t h e f o r m u l a t i o n a n d s o l u t i o n o f the p r o b l e m o f an u n s t e a d i l y m o v i n g p r o p e l l e r u s i n g a M o r i n o -t y p e B E M . The m a i n i n n o v a -t i o n i n -t h i s w o r k w a s -t h e c a l c u l a -t i o n b y t h e code o f t h e d y n a m i c e v o l u t i o n o f the f r e e v o r t e x sheets, e m a n a t -i n g f r o m t h e p r o p e l l e r blades. D u r -i n g t h e p e r -i o d 2 0 0 2 - 2 0 0 8 , th-is code, i n i t i a l l y d e v e l o p e d t o t r e a t p r o p e l l e r p r o b l e m s , w a s e x p a n d e d to i n c l u d e a n y 3 D u n s t e a d y i n c o m p r e s s i b l e n o n v i s c o u s f l o w p r o b -l e m a r o u n d a n a r b i t r a r y s y s t e m o f i n t e r a c t i n g n o n - -l i f t i n g / -l i f t i n g r i g i d / f l e x i b l e bodies [ 1 6 , 1 5 ] , F u r t h e r m o r e , i n [ 1 7 ] , t h e r e w a s a f i r s t a t t e m p t t o s y s t e m a t i c a l l y i n v e s t i g a t e t h e e f f e c t o f f l a p p i n g w i n g p r o p u l s o r g e o m e t r y , o n its o p e n w a t e r p e r f o r m a n c e u s i n g t h e same 3 D B E M code. By f u r t h e r d e v e l o p i n g s p e c i a l i z e d data g e n e r a t i o n codes, t h e y p r e s e n t s i m u l a t i o n s o f b i r d f l i g h t [ 1 8 ] a n d i n [ 1 9 ] a p r e -l i m i n a r y i n v e s t i g a t i o n f o r the FOD w a s p r e s e n t e d , t h a t -l e d to t h e p r e s e n t paper. In t h i s p a p e r a n e w p r o p u l s o r c o n c e p t n a m e d FOD is i n t r o d u c e d . H y d r o d y n a m i c p e r f o r m a n c e s i m u l a t i o n s are t h e n u n d e r t a k e n u s i n g t h e a f o r e m e n t i o n e d 3 D - B E M code, f o r a s y s t e m a t i c series o f FODs, a n d t h e c a l c u l a t e d m e a n t h r u s t a n d p o w e r are p l o t t e d i n d i a g r a m s , p r o p e r f o r FOD d e s i g n . Consistency m a t t e r s l i k e g r i d i n d e p e n d e n c e a n d c o n v e r g e n c e o f the o b t a i n e d results are also e x a m i n e d . S y s t e m a t i c r u n s i n c l u d e v a r i a t i o n o f S t r o u h a l n u m b e r a n d p i t c h angle a m p l i t u d e f o r t h e p r e - s e l e c t e d FOD g e o m e t r i e s o f our series. A d e s i g n m e t h o d , t h a t e m p l o y s t h e p r o d u c e d charts, is i n t r o d u c e d a n d used t o calculate t h e p o w e r i n g p e r f o r m a n c e o f t h r e e d i f f e r e n t c o m m o n s h i p t y p e s : a b u l k c a r r i e r , a passen-ger/car f e r r y and a speedboat, e q u i p p e d w i t h a FOD, as t h e i r m a i n p r o p u l s o r . P o w e r i n g p e r f o r m a n c e c a l c u l a t i o n s have also b e e n m a d e f o r t h e s a m e ships e q u i p p e d w i t h a c o n v e n t i o n a l p r o p e l l e r . The c o m p a r i s o n s h o w s t h a t t h e FOD can p r o d u c e h i g h e f f i c i e n c i e s s u p e -rior to t h a t o b t a i n e d w i t h c o n v e n t i o n a l p r o p e l l e r s , g i v i n g t h u s e n c o u r a g i n g results, so t h a t t e c h n i c a l d i f f i c u l t i e s c o n c e r n i n g its c o n s t r u c t i o n s h o u l d be addressed a n d r e s o l v e d . 2. FOD g e o m e t r y , m o t i o n a n d p a n e l g e n e r a t i o n The s t a r t i n g p o i n t f o r a n u n s t e a d y B E M s i m u l a t i o n o f a flexible b o d y is t h e g e n e r a t i o n o f t h e t i m e d e p e n d e n t p a n e l i n g d e s c r i b i n g t h e g e o m e t r y o f t h e s y s t e m [ 1 5 ] . The FOD t i m e d e p e n d e n t g e o m e t r y is p r o d u c e d as d e s c r i b e d b e l o w : W e s t a r t f r o m a c o n v e n t i o n a l 2 - D f o i l m o v i n g p a r a l l e l to the X - a x i s w i t h v e l o c i t y U, w h i c h p e r f o r m s a h e a v i n g m o t i o n ( a l o n g t h e v e r t i c a l V-axis) w i t h a m p l i t u d e ho a n d a p i t c h i n g m o t i o n w i t h a m p l i t u d e OQ ( m e a s u r e d f r o m t h e X - d i r e c t i o n ) a n d a phase angle \p w i t h respect to t h e h e a v i n g m o t i o n . The p i t c h i n g m o t i o n is p e r -f o r m e d a r o u n d a p r e - s e l e c t e d g i v e n p i t c h i n g axis at distance b -f r o m t h e l e a d i n g edge o f the s e c t i o n . B o t h h e a v i n g a n d p i t c h i n g m o t i o n s are p e r f o r m e d w i t h a n g u l a r v e l o c i t y co = 2nn, w h e r e n is t h e c o r -r e s p o n d i n g f -r e q u e n c y . F u -r t h e -r m o -r e , the c a m b e -r o f t h e 2 - D f o i l p e r f o r m s an u n s t e a d y m o r i o n w i t h a c h o r d - w i s e d i s t r i b u t i o n ydt, u ) ( u d e n o t e s the n o n - d i m e n s i o n a l c h o r d - w i s e p o s i t i o n m e a s u r e d f r o m l e a d i n g edge a n d t d e n o t e s t h e t i m e ) , t a k e n e.g. f r o m t h e NACA series, w i t h a n i n s t a n t a n e o u s m a x i m u m c a m b e r m c ( t ) (expressed as a f r a c t i o n o f c h o r d ) . A n o s c i l l a t i n g c a m b e r m o t i o n can t h e n be o b t a i n e d b y d e c i d i n g a b o u t t h e f o r m o f t h e f u n c t i o n mc(t). There are at least t w o possible r e a s o n a b l e selections f o r mdt): ( a ) select mc(t) s u c h t h a t t h e i n s t a n t a n e o u s e f f e c t i v e ( d e f i n e d w i t h respect t o

the t o t a l u n d i s t u r b e d f l o w v e l o c i t y ) angle o f attack o f t h e s e c t i o n is a c e r t a i n ( t i m e i n d e p e n d e n t ) f r a c t i o n o f t h e i n s t a n t a n e o u s i d e a l angle o f a t t a c k o f t h e s e c t i o n (i.e. t h e s e c t i o n o p e r a t e s i n a p e r c e n t -age o f its s h o c k f r e e e n t r y at all times), ( b ) select m c ( t ) t o o s c i l l a t e h a r m o n i c a l l y w i t h t h e same f r e q u e n c y a n d phase as t h a t used f o r the p i t c h i n g m o t i o n , a n d a m a x i m u m p r e d e t e r m i n e d ( u s e r d e f i n e d ) v a l u e mo. It s h o u l d be n o t e d t h a t i n b o t h cases u n s t e a d y m o t i o n o f the c a m b e r surface has t h e same f r e q u e n c y a n d t h e same phase w i t h t h a t o f the p i t c h i n g m o t i o n . O n the o t h e r h a n d , o n l y case ( b ) results i n p u r e h a r m o n i c m o t i o n f o r c a m b e r . W i t h t h e p r e v i o u s d i s c u s s i o n i n m i n d , at each t i m e t, t h e f o i l g e o m e t r y a n d p o s i t i o n i n t h e a f o r e m e n t i o n e d XY p l a n e has been d e f i n e d . T a k i n g a n axis L p a r a l l e l t o t h e X - a x i s , at a d i s t a n c e R a l o n g y - a x i s a n d r o t a t i n g t h e f o i l b y 3 6 0 ° a r o u n d L, t h e FOD c o n f i g u r a t i o n at t h i s t i m e s t e p is o b t a i n e d . R d e f i n e s t h e FOD r a d i u s a n d D = 2 R the FOD d i a m e t e r .

Q u a n t i f y i n g t h e p r e v i o u s discussion, the i n s t a n t a n e o u s angle o f a t t a c k a ( t ) o f a s e c t i o n ( 2 - D f o i l ) o f the FOD, w i t h r e s p e c t t o t h e u n d i s t u r b e d flow ( r e s u l t i n g f r o m t h e p a r a l l e l m o v e m e n t a l o n g X -axis a n d t h e h e a v i n g m o t i o n o f the p i t c h -axis p o i n t ) , is g i v e n b y t h e e q u a t i o n :

1 / / I o 2 7 r i i - c o s ( 2 m 7 - f ) \

a ( f ) = Öo s i n ( 2 m i . t - H i A ) ^ t a n - M - y - j ( 1 )

or i n n o n - d i m e n s i o n a l f o r m :

fi(t) = 00 sin(27rn • f + i ^ ) - t a n - ' ( : T • Str • cos{2nn • f ) ) ( 2 )

w h e r e Str d e n o t e s t h e S t r o u h a l n u m b e r d e f i n e d b y :

Str = ^ ^ , / i = 2;io ( 3 )

a n d h d e n o t e s t h e heave h e i g h t .

F u r t h e r m o r e , a s s u m i n g a NACA f o u r d i g i t c a m b e r d i s t r i b u t i o n , the c a m b e r m o t i o n is d e s c r i b e d b y t h e f o l l o w i n g e q u a t i o n s :

yc{t,u) = mc{tf--^{2p-u), 0<u<p ( 4 )

ydt. It) = m c ( t ) f • ~ V (1 u - 2 • p ) , p < u < l ( 5 ) V ( l - P ) / w h e r e mdt) is the m a x i m u m i n s t a n t a n e o u s c a m b e r as a f r a c t i o n o f c h o r d , p is t h e l o c a t i o n o f m a x i m u m c a m b e r (as a f r a c t i o n o f t h e c h o r d ) a n d u is t h e n o n - d i m e n s i o n a l c h o r d - w i s e p o s i t i o n [ 1 ] . A f t e r t h i s , i t is possible t o calculate t h e t i m e e v o l u t i o n o f m a x i m u m c a m b e r f o r t h e cases (a) a n d ( b ) discussed p r e v i o u s l y . For case (a) the i n s t a n t a n e o u s m a x i m u m c a m b e r is g i v e n b y :

ƒ • 0 ( 0 • p 2 ( - p + 1 ) ^ 7 1

' ~ 2 ( p - ( l / 2 ) ) ( p 2 ; T -2p& + &^ sin(!?)) ( 6 ) !? = a c o s ( l - 2 p ) w h e r e a ( t ) is c a l c u l a t e d f r o m ( 2 ) a n d ƒ is a t i m e i n d e p e n d e n t c o r -r e c t i o n f a c t o -r ( e s t i m a t e d h e u -r i s t i c a l l y ) b y w h i c h t h e i n s t a n t a n e o u s c a m b e r p a r t i a l l y r e p r e s e n t s t h e i d e a l c a m b e r . R e l a t i o n ( 6 ) is p r o -d u c e -d b y a p p l y i n g t h e f o r m u l a f o r t h e i -d e a l angle o f a t t a c k t o t h e c a m b e r l i n e ( 4 ) a n d ( 5 ) [ 1 ] . For case ( b ) t h e i n s t a n t a n e o u s m a x i m u m c a m b e r is g i v e n b y : j7ic(t) = mo s i n ( 2 j r i i • t + t/f) ( 7 ) w i t h mo h e u r i s t i c a l l y selected b y t h e designer. H a v i n g i n t r o d u c e d t h e a n a l y t i c a l d e s c r i p t i o n o f b o t h g e o m e t r y a n d m o t i o n o f o u r FOD, t h e c r e a t i o n o f a s u r f a c e p a n e l d i s t r i b u t i o n d e s c r i b i n g t h e FOD at c o n s e c u t i v e t i m e steps is s t r a i g h t f o r w a r d .

Fig. 1 s h o w s the panel d i s c r e t i z a t i o n o f t h e FOD at n i n e time instances, c o r r e s p o n d i n g to t h e m a x i m u m , m i n i m u m a n d i n f l e c t i o n p o i n t p o s i t i o n s o f a FOD s e c t i o n i n one p e r i o d . Fig. 2 s h o w s t h e

38 G. Poliüs, V. Tsarsitalidis / Applied Ocean Research 36 (2012) 36-50

Fig. 1. Panel discretization at selected time steps.

s e c t i o n o f ttie FOD w i t t i a plane c o n t a i n i n g t l i e X-axis f o r ttie same t i m e instances.

W i t t i ttie FOD p a n e l i n g i n t i m e I m o w n , t l i e u n s t e a d y BEM code [15] can be a p p l i e d t o calculate the FOD u n s t e a d y forces, energy r e q u i r e m e n t s a n d f r e e shear layer e v o l u t i o n .

3. C a l c u l a t i o n of forces, m o m e n t s , p o w e r a n d e f f i c i e n c y

I n t h e case o f a f l e x i b l e b o d y , e n e r g y is t r a n s f e r r e d t h r o u g h f l e x i b i l i t y i n a p o i n t - w i s e m a n n e r . The u n s t e a d y BEM code uses a specialized p r o c e d u r e f o r t h e c a l c u l a t i o n o f the p o w e r p r o v i d e d to t h e FOD m e c h a n i s m , t e r m e d DHP ( D e l i v e r e d Horse P o w e r ) a n d the p o w e r d e v e l o p e d b y the FOD t o p r o p e l t h e s h i p , t e r m e d EHP ( E f f e c -t i v e Horse P o w e r ) . The c o r r e s p o n d i n g m e -t h o d o l o g y has already been a p p e a r e d in [ 1 7 ] and w i l l be repeated here f o r c o m p l e t e n e s s . Let A a p o i n t b e l o n g i n g t o t h e surface S o f a b o d y o f o u r s y s t e m . Let F ( t ) d e n o t e the t o t a l f o r c e e x e r t e d b y t h e f l u i d t o t h e b o d y at A.

Y

Fig. 2. XY section of ttie nine positions of the FOD.

This f o r c e is a s u m o f pressure forces n o r m a l to b o d y d e n o t e d b y P ( f ) a n d viscous forces t a n g e n t i a l to b o d y d e n o t e d b y D(t)- Thus:

F{t)=:P{t) + D{t) ( 8 ) Let also ï)(t) d e n o t e the v e l o c i t y o f A w i t h respect t o an i n e r t i a

r e f e r e n c e f r a m e . Consider also a n i n s t a n t a n e o u s parallel v e c t o r f i e l d d ( t ) w i t h d a u n i t v e c t o r . A c c o r d i n g to a w e l l - k n o w n v e c t o r i d e n t i t y ( i t i s t h e r e s u l t of: I^A xB) x C = B{A • C) - C(A • B) b y s e t t i n g A = C = d.B = ll),

v = {d-ïi)-d + id xv)xd ( 9 ) The p o w e r done f r o m the f l u i d t o t h e b o d y at A is g i v e n b y :

p o w ( t ) = F ( t ) . m = Cm • d m m • m ) +F{t)-mt)xm)xd{t)) ( 1 0 ) a n d t h e n e t p o w e r f r o m a s y s t e m o f f l e x i b l e bodies (S denotes t h e t o t a l s y s t e m b o u n d i n g surface) is g i v e n b y : n e f - p o w ( r ) = / Fit) • v{t) • dS = / ( F ( t ) • d ( t ) ) ( d ( t ) • i } ( f ) ) • dS + [ F ( f ) { ( d ( r ) x P ( f ) ) x d ( f ) ) . d S ( 1 1 ) In p r o p u l s i o n p r o b l e m s t h e r e is a l w a y s a p r e f e r a b l e i n s t a n t a -neous d i r e c t i o n a l o n g w h i c h t h e s y s t e m m o v e s . For t h e FOD case, t h i s is t h e X-axis d i r e c t i o n . Take d a l o n g t h i s d i r e c t i o n . T h e n t h e f i r s t t e r m i n t h e rhs ( r i g h t h a n d side) o f Eq. ( 1 1 ) is t h e i n s t a n t a -neous EHP a n d the second t e r m is t h e i n s t a n t a n e o u s DHP. The r a t i o EHP/DHP defines t h e i n s t a n t a n e o u s e f f i c i e n c y . For the a p p l i c a t i o n o f the p r e v i o u s f o r m u l a s t h e p r e s s u r e forces o n e l e m e n t c e n t r o i d s can be c a l c u l a t e d f r o m t h e u n s t e a d y B e r n o u l l i e q u a t i o n [ 1 5 ] . The code also contains a s i m p l e s u b r o u t i n e f o r t h e c a l c u l a t i o n o f viscous forces u s i n g an e l e m e n t a l surface d r a g c o e f f i c i e n t , w h i c h depends o n t h e local Reynolds n u m b e r w i t h a d r a g c o r r e c t i o n f o r an e s t i -m a t e d sectional angle o f attack greater t h a t t h e i d e a l .

4. T h e o r e t i c a l f o r m u l a t i o n a n d s o l u t i o n o f the F O D p r o p u l s o r design p r o b l e m

FOD p r o p u l s o r d e s i g n p r o b l e m consists i n finding t h e p r o p u l s o r g e o m e t r i c a n d m o t i o n characteristics b y w h i c h i t can p r o p e l a g i v e n s h i p w i t h a g i v e n s h i p speed. F r o m all possible design s o l u t i o n s , sat-i s f y sat-i n g t h e c o n s t r a sat-i n o f a g sat-i v e n s h sat-i p speed; t h e r e sat-is a n o p t sat-i m u m one, w h i c h succeeds t h i s w i t h a m i n i m u m d e l i v e r e d p o w e r . A l t h o u g h t h e o p t i m u m p r o p u l s o r p r o b l e m is a p r o b l e m o f m u t u a l p r o p u l -s o r / -s t e r n o p t i m i z a t i o n , i n m o -s t ca-se-s w e o p t i m i z e t h e p r o p u l -s o r a s s u m i n g that t h e h u l l / s t e r n g e o m e t r y is g i v e n . D e v e l o p m e n t o f a design t h e o r y f o r a n e w p r o p u l s o r , r e q u i r e s decisions r e g a r d i n g the i n d e p e n d e n t g e o m e t r i c / m o t i o n variables c o n t r o l l i n g the t h r u s t p r o d u c t i o n a n d t h e e n e r g y a b s o r p t i o n . For-t u n a For-t e l y , For-t h e p r o p o s e d FOD s y s For-t e m is a g e n e r a l i z a For-t i o n o f For-t h e w e l l - d e v e l o p e d flapping w i n g . Thus, t h e c o r r e s p o n d i n g decisions are s t r a i g h t - f o r w a r d as f o l l o w s :

I n d e p e n d e n t variables b y w h i c h t h e state o f t h e FOD is d e f i n e d u n i q u e l y can be d e c o m p o s e d i n t w o g r o u p s . G r o u p A c o n t a i n s the g e o m e t r i c variables and G r o u p B c o n t a i n s t h e m o t i o n r e l a t e d v a r i -ables. Group A: A s s u m i n g g e o m e t r i c a l s i m i l a r i t y , a m i n i m u m set o f p a r a m e t e r s d e f i n i n g flexible v i b r a t i n g d u c t p r o p u l s o r g e o m e t r y is t h e s e c t i o n c h o r d c, the c h o r d - w i s e p o s i t i o n o f t h e p i t c h i n g axis b, t h e d u c t m e a n radius R ( d e f i n e d at t h e p o s i t i o n o f s e c t i o n a l p i t c h i n g axis) or e q u i v a l e n t l y the d u c t m e a n d i a m e t e r D. In n o n - d i m e n s i o n a l f o r m , FOD g e o m e t r y is c h a r a c t e r i z e d by: (R/c, bjc). I t s h o u l d be n o t e d t h a t t h e f o i l s e c t i o n g e o m e t r y ( t h i c k n e s s a n d c a m b e r f o r m s )

G. Politis, V. Tsarsitalidis/ApiJlied Ocean Research 36 (2012) 36-50 ₃₉

has been i n t e n t i o n a l l y n o t i n t r o d u c e d as a m a i n g e o m e t r i c p a r a m -eter f o r the FOD, since i t m a i n l y c o n t r o l s p h e n o m e n a l i k e viscous pressure drag, as w e l l as s e p a r a t i o n (at larger angles o f a t t a c k ) , a n d n o t t h e FOD t h r u s t and p o w e r (i.e. exerts a s e c o n d a r y e f f e c t o n t h r u s t a n d p o w e r ) .

Group B: FOD m o t i o n is d e f i n e d b y : ( i ) t h e FOD v e l o c i t y o f

advance U, ( i i ) t h e a m p l i t u d e ho o f a s i n u s o i d a l h e a v i n g m o t i o n o f each s e c t i o n n o r m a l to t h e v e l o c i t y o f advance Lf, ( i i i ) the a m p l i t u d e do o f a s i n u s o i d a l p i t c h i n g m o t i o n , ( i v ) the a m p l i t u d e mdt) o f t h e c a m b e r d e f o r m a t i o n , ( v ) t h e f r e q u e n c y n ( c o m m o n to b o t h h e a v -i n g , p -i t c h -i n g m o t -i o n s a n d c a m b e r d e f o r m a t -i o n s ) , a n d ( v -i ) t h e phase angle ij/ b e t w e e n h e a v i n g a n d p i t c h i n g / c a m b e r m o t i o n s . A s s u m i n g t h a t t h e c a m b e r d e f o r m a t i o n has b e e n selected as i n (a)-Section 1, t h e m o t i o n o f FOD is c o m p l e t e l y d e f i n e d b y the f o l l o w i n g d i m e n -s i o n a l v a r i a b l e -s : (U, ho, 9o,f, n, xj/). I n ca-se c a m b e r ha-s been -selected as i n ( b ) - S e c t i o n 1, the m o t i o n o f FOD is d e f i n e d b y : (Lf, ho, 9o, mo, n, r j f ) . I n t h e sequel, mo, ƒ are o m i t t e d f o r the sake o f s i m p l e r f o r m a l -i s m , -i n t r o d u c -i n g -i t o n l y w h e n n e e d e d . U s -i n g d -i m e n s -i o n a l analys-is, t h e n o n - d i m e n s i o n a l p a r a m e t e r s c o n t r o l l i n g FOD p e r f o r m a n c e can be o b t a i n e d : (OQ, hole, Str, f ) . A s s u m i n g f u r t h e r t h a t \jf=m°, t h r e e n o n - d i m e n s i o n a l p a r a m e t e r s c o n t r o l l i n g t h e FOD m o t i o n r e m a i n : (Ö0, hole Str).

R e c a p i t u l a t i n g t h e discussion u n d e r ( A ) a n d (B), the FOD p e r -f o r m a n c e is c o n t r o l l e d b y t h e -f o l l o w i n g g e o m e t r i c a n d m o t i o n p a r a m e t e r s : {OQ, Str, hole, Rjc, bjc). N o t i c e t h a t t h e f o l l o w i n g c o n -c e p t u a l -c o u p l i n g b e t w e e n n o n - d i m e n s i o n a l p a r a m e t e r s used i n t h e FOD a n d t h e c o n v e n t i o n a l p r o p e l l e r can b e e n m a d e (P/D denotes t h e p r o p e l l e r p i t c h r a t i o , AEIAQ denotes p r o p e l l e r blade area r a t i o , D u n d e r t h e ( p r o p e l l e r ) c o l u m n denotes p r o p e l l e r d i a m e t e r , CQJR d e n o t e s blade c h o r d at 70% o f p r o p e l l e r r a d i u s ) : FOD R P r o p e l l e r P Po.7R A f Ao ( 1 2 )

A w e l l - p o s e d p r o p u l s i o n p r o b l e m f o r a FOD p r o p u l s o r can be set as f o l l o w s :

Calculate the t i m e d e p e n d e n t o p e n w a t e r FOD p e r f o r m a n c e u s i n g the BEM code f o r a range o f t h e p a r a m e t e r s {9o, Str) a s s u m -i n g g -i v e n {hole, Rjc, bjc). Calculate t h e n t h e p e r -i o d - m e a n values f o r t h r u s t a n d d e l i v e r e d p o w e r a n d d e n o t e t h e m b y : T a n d DHP, respec-t i v e l y . FOD p e r f o r m a n c e can respec-t h e n be expressed b y respec-the f o l l o w i n g n o n - d i m e n s i o n a l ( m e a n ) t h r u s t a n d ( d e l i v e r e d ) p o w e r c o e f f i c i e n t s : CT C p 0.5pU^S DHP O S p Ï P S CT h , s t , , , , -( R ho b ( 1 3 ) ( 1 4 )

w h e r e S denotes t h e m e a n FOD disc surface (=7r R2). I n self-p r o self-p u l s i o n c o n d i t i o n s , w e a s s u m e t h a t a T a y l o r w a k e f r a c t i o n w is d e f i n e d b y :

U = V ( l - w) ( 1 5 )

w h e r e Vis the ship speed. F u r t h e r m o r e a r e l a t i v e r o t a t i v e e f f i c i e n c y r]R is d e f i n e d b y : >1R DHP D H p ; ( 1 6 ) w h e r e DHPg denotes t h e ( p e r i o d - m e a n ) p o w e r d e l i v e r e d to t h e FOD i n s e l f - p r o p u l s i o n c o n d i t i o n s . A s s u m i n g f u r t h e r t h a t a ' t h r u s t e q u a l i z a t i o n m e t h o d ' has b e e n used f o r t h e d e t e r m i n a t i o n o f FOD-h u l l i n t e r a c t i o n c o e f f i c i e n t s w , t, i]^ ( w FOD-h e r e t denotes t FOD-h e t FOD-h r u s t d e d u c t i o n f a c t o r ) , t h e FOD t h r u s t a n d p o w e r , i n t h e s e l f - p r o p u l s i o n c o n d i t i o n s b e c o m e s : rB = T = 0 . 5 p ( \ / ( l - w ) ) ' S CT Ö, DHPB)JR = 0 . 5p( \ / ( 1 - w ) ) ^ S . C p (^0, n • 2ho R ho b V ( l - ^ w ) ' c ' T ' c n • 2ho R liQ b c ' c ( 1 7 ) ( 1 8 ) 1/(1 - w ) ' For a s e l f p r o p e l l e d h u l l , m o v i n g w i t h v e l o c i t y V, t h e s u r r o u n d -i n g flu-id -i n t e r a c t s w -i t h t h e h u l l d e v e l o p -i n g a res-istance f o r c e : ' ^ o ( ^ / ( l - 1 ) , w h e r e Ro(^) denotes t h e h u l l t o w i n g resistance. A h u l l can also p u l l an o b j e c t w i t h a f o r c e F(case o f a t u g - b o a t or a t r a w l e r ) . T h e n the FOD t h r u s t , u n d e r s e l f - p r o p u l s i o n c o n d i t i o n s , is g i v e n b y :

T a. F ( 1 9 )

A s s u m i n g t h a t V, 9o, c, R, ho, b, w , t, Ï;R are k n o w n p a r a m e t e r s , Eqs. ( 1 7 ) ( 19) b e c o m e a n o n l i n e a r s y s t e m o f t h r e e algebraic e q u a -t i o n s w i -t h -t h r e e u n k n o w n s : (T, D H P B , n). T h i s s y s -t e m can be s o l v e d f o r a range o f s h i p speeds: V e ( V i , V2) and p i t c h angles: ÖQ e ( ö o _ a , 9o-b). Thus, t h e t o t a l i t y o f d e s i g n s o l u t i o n s f o r the g i v e n s h i p is o b t a i n e d :

DHPs(\/, f^o). n{V, 9o) *- Igiven : s h i p , c, R, ho, w, t, ( 2 0 )

The c o n t e n t o f Eq. ( 2 0 ) can be r e p r e s e n t e d i n a 2 D DHPg n d i a -g r a m i n t h e f o r m o f p a r a m e t r i c curves o f c o n s t a n t V a n d c o n s t a n t (?o-N o t i c e t h a t t h i s p r e s e n t a t i o n is s i m i l a r t o t h a t used i n c o n v e n t i o n a l p r o p e l l e r s , w h e r e t h e p r o p e l l e r p i t c h r a t i o P/D is t a k i n g t h e place o f (?o U s i n g t h i s p r e s e n t a t i o n , w e can finally e x t r a c t t h e r e q u i r e d o p t i -m u -m FOD b y s e l e c t i n g t h e characteristics ( g e o -m e t r i c a n d -m o t i o n ) w h i c h r e q u i r e t h e m i n i m u m DHPg f o r the g i v e n s h i p speed V.

5. D e c i s i o n s r e g a r d i n g G e o m e t r i c a n d F l o w / m o t i o n v a r i a b l e s f o r t h e p r o p o s e d F O D series.

To p r o c e e d to a series based design process f o r a FOD, decisions have to be t a k e n , o n t h e c o r r e s p o n d i n g g e o m e t r i c a n d flow r e l a t e d p a r a m e t e r s . I n l i n e w i t h t h e discussion o f t h e p r e v i o u s s e c t i o n , t h e series s h o u l d c o n s i s t o f f o u r d i f f e r e n t FOD g e o m e t r i e s , t e r m e d i n the sequel as Cases 1-4, as f o l l o w s :

Case 1: hole = 1.0 w i t h NACA 0 0 1 2 s e c t i o n ( n o c a m b e r ) .

Case 2 : holc=\.Q w i t h NACA 6 4 1 2 s e c t i o n a n d t i m e d e p e n d e n t m a x i m u m c a m b e r a c c o r d i n g t o Eq. ( 7 ) .

Case 3: /7o/c= 1.5 w i t h NACA 0 0 1 2 s e c t i o n ( n o c a m b e r ) ,

Case 4 : / i o / c = 1 . 5 w i t h NACA 6412 section a n d t i m e d e p e n d e n t m a x i m u m c a m b e r a c c o r d i n g to Eq. ( 7 ) .

O t h e r series g e o m e t r i c data have b e e n selected c o n s t a n t , as f o l -l o w s : mo = 0.04 i n r e -l a t i o n ( 7 ) a n d R/c = 3 ( w h e r e R is t h e m e a n FOD radius m e a s u r e d at t h e p o s i t i o n o f p i t c h i n g axis w h i c h have been selected at 1 /3 c h o r d f r o m l e a d i n g edge, i.e. b/c = 1 / 3 ) .

R e g a r d i n g flow/motion r e l a t e d variables the S t r o u h a l n u m b e r has b e e n selected i n t h e range: Str=0.\5~0.7. U s i n g o u r p r e v i o u s experience, g a i n e d f r o m flapping w i n g s , this s e l e c t i o n is e x p e c t e d to c o n t a i n t h e r e g i o n o f FOD m a x i m u m h y d r o d y n a m i c e f f i c i e n c y . F i n a l l y the r a n g e o f t h e p i t c h angle has b e e n selected f r o m 5 ° to a m a x i m u m value, w h i c h depends o n the S t r o u h a l n u m b e r . This m a x i m u m v a l u e o f t h e p i t c h angle has b e e n p r o p e r l y selected t o r e s u l t i n a t h r u s t p r o d u c i n g FOD. 6. T r a n s i e n t F O D p e r f o r m a n c e a n d s e l e c t i o n of s i m u l a t i o n p e r i o d The m a i n d i f f e r e n c e b e t w e e n a t r a d i t i o n a l p r o p e l l e r a n d a FOD is t h a t t h e FOD p r o d u c e s a p e r i o d m e a n t h r u s t as a r e s u l t o f a h i g h l y u n s t e a d y i n s t a n t a n e o u s t h r u s t . The s i m u l a t i o n m e t h o d i n h a n d can

40 C. Politis. V. Tsarsitalidis/ AppUed Ocean Research 36 (2012) 36-50

20 10 60

Fig. 6. Time e v o l u t i o n of thrust for {Case 3, Str=0.56, theta = 23.5}.

Fig. 3. Time evolution of t h r u s t for {Case 1, Str=0.29, theta = 23.54}.

7. W a k e v i s u a l i z a t i o n s • p r o d u c e s t h r u s t

U n d e r s t a n d i n g h o w t h e F O D

Fig. 4. Time evolution of t h r u s t for {Case l , S f r = 0 . 4 2 , theta = 24.27).

p r o d u c e t h i s t i m e d e p e n d e n t t h r u s t b u t , since i t is a t i m e s t e p p i n g m e t h o d , i n i t i a l c o n d i t i o n s o n FOD m o t i o n have t o be i m p o s e d . A b u r s t s t a r t i n g FOD is used as the s t a r t i n g c o n d i t i o n . I n t h i s case a t r a n s i e n t p h e n o m e n o n occurs. Thus t h e m e a n p e r i o d values f o r t h r u s t or p o w e r have to be c a l c u l a t e d a f t e r the passage o f this i n i t i a l t r a n s i e n t p h e n o m e n o n . To care f o r this, t i m e d o m a i n s i m u l a t i o n s have b e e n p e r f o r m e d f o r t w o periods and f o r t h e f o l l o w i n g s e l e c t i o n o f FOD state v a r i a b l e s ( g e o m e t r y a n d m o t i o n ) :

(I) {Case l , S t r = 0,29, t h e t a = 2 3 . 5 4 } (II) {Case 1, S t r = 0 . 4 2 , t h e t a = 2 4 . 2 7 } (III) {Case 2, S t r = 0 . 5 6 , t h e t a = 2 3 . 5 0 } ( I V ) {Case 3, S t r = 0 . 4 2 , t h e t a = 2 4 . 2 7 }

Results f o r the t i m e d e p e n d e n t FOD t h r u s t f o r c e Fx are s h o w n i n Figs. 3 - 6 r e s p e c t i v e l y w h e r e p denotes the f l u i d d e n s i t y . F r o m those f i g u r e s w e c o n c l u d e that, f o r t h e selected parameters, t h e t r a n s i e n t p h e n o m e n o n is l i m i t e d t o the f e w i n i t i a l t i m e steps a f t e r the b u r s t start. Thus w e can s a f e l y use t h e 2 n d p e r i o d o f s i m u l a -t i o n , -t o calcula-te -t h e m e a n FOD -t h r u s -t a n d p o w e r -t o be used i n o u r d e s i g n charts. N o t i c e t h a t a c c o r d i n g to o u r t h r u s t s i g n c o n v e n t i o n s , a n e g a t i v e t h r u s t f o r c e Fx is a p r o p e l l i n g f o r c e , c o n v e n t i o n w h i c h is used i n all f u t u r e charts s i m i l a r to Fig. 3.

For a b e t t e r u n d e r s t a n d i n g o f the u n d e r l y i n g p h y s i c a l m e c h -anisms o f FOD's t h r u s t p r o d u c t i o n , the w a k e f r e e shear l a y e r is p l o t t e d , f o r a range o f t h e FOD state variables, r e p r e s e n t a t i v e o f o u r series d e s c r i b e d i n Section 5. M o r e s p e c i f i c a l l y w e p r e s e n t t w o g r o u p s o f results. For t h e f i r s t g r o u p w e have selected a d e f i n i t e c o m b i n a t i o n o f s t a t e v a r i a b l e s : { C a s e l , S t r = 0 . 4 2 , t h e t a = 2 4 . 2 7 } a n d p r e s e n t instances o f the w a k e e v o l u t i o n at the f o l l o w i n g six c o n -secutive t i m e steps: { T / 2 , T, 3T/2, 2T, 51/2. 3T} w h e r e T d e n o t e s t h e s i m u l a t i o n p e r i o d . Fig. 7 s h o w s t h e FOD t h r u s t as a f u n c t i o n o f t i m e , w i t h v e r t i c a l bars a t the p o i n t s w h e r e w a k e snapshots have b e e n taken. Figs. 8 ( p e r s p e c t i v e v i e w ) a n d 9 ( t o p v i e w ) s h o w t h e FOD and c o r r e s p o n d i n g w a k e snapshots, at t h e selected t i m e steps. The FOD surface a n d t h e f r e e v o r t e x sheet surface o n t h o s e f i g u r e s h a v e been c o l o r e d a c c o r d i n g t o t h e i r surface d i p o l e d i s t r i b u t i o n i n t e n -sity. N o t i c e t h a t c o n s t a n t d i p o l e lines c o i n c i d e w i t h s u r f a c e v o r t e x lines. By u s i n g e i t h e r the last p r o p e r t y or t h e d e f o r m a t i o n p a t t e r n s of t h e f r e e v o r t e x sheets, a n u m b e r o f s t r o n g r i n g v o r t i c e s i n t h e w a k e o f t h e FOD are m a d e recognizable. Those r i n g v o r t i c e s p r o -duce series o f o b l i q u e j e t f l o w s b y w h i c h t h e FOD p r o d u c e s t h r u s t . Fig. 10 s h o w s a slice w i t h a c o n s t a n t - Y p l a n e o f t h e FOD f r e e v o r t e x sheet f o r t h e six t i m e steps considered, w h i l e Fig. 11 c o n c e n t r a t e s o n t h e slice a t f = 3T. Fig. 11 also c o n t a i n s a r t i s t i c add-ons, s h o w i n g the t r a i n o f r i n g v o r t i c e s ( c u r v e d a r r o w s ) a n d c o r r e s p o n d i n g j e t s ( s t r a i g h t a r r o w s ) b y w h i c h t h e FOD p r o d u c e s t h r u s t . M o r e s p e c i f -ically t h e s t r a i g h t a r r o w s are the r e s u l t s o f t h e i n d u c e d v e l o c i t i e s p r o d u c e d b y t h e r i n g v o r t i c e s . N o t i c e also the a n a l o g y o f t h e v o r -tex p i c t u r e s h o w n i n t h e slice o f Fig. 11 w i t h t h e reverse K a r m a n v o r t e x street w a k e a p p e a r i n g i n a q u a t i c a n i m a l a n d / o r b i r d f l a p -p i n g f o i l -p r o -p u l s o r s [ 2 3 ] . By s t u d y i n g t h e e v o l u t i o n o f t h e FOD at c o n s e c u t i v e g e o m e t r i c p o s i t i o n s , i t is s h o w n t h a t t h e m a x i m u m angle o f a t t a c k d u r i n g FOD c o n t r a c t i o n a n d d u r i n g FOD e x p a n s i o n occurs at a s y m m e t r i c FOD g e o m e t r i e s . M o r e s p e c i f i c a l l y , d u r i n g the e x p a n s i o n phase o f the FOD m o t i o n , t h e l e a d i n g edge d i a m e t e r is greater t h a n the c o r r e s p o n d i n g d i a m e t e r d u r i n g t h e c o n t r a c t i o n phase. As a r e s u l t t h e local m a x i m u m o f t h r u s t forces Fx d u r i n g

-0,5

t

/ \

' \

_/

_/

_/

/

\

J \

/

\

\ 1 ^

J

\

Fig. 5. Time evolution of t h r u s t for {Case 2, Str=0.42, theta = 24.27),

Its

Fig. 7. Time e v o l u t i o n of FOD thrust for {Case 1 , S f r = 0.42, theta = 24.27).The vertical lines indicate the times o f t h e snapshots taken.

C. Poliüs. V. Tsarsitalidis / Applied Ocean Research 36 (2012) 36-50 41

Fig. 8. Wake visualizations for (Case 1, Str=0.42. theta = 24.27}, at {1/2, T, 3T/2, 21, 5T/2, 3T), respectively. Perspective v i e w .

FOD e x p a n s i o n is greater t h a n t h e c o r r e s p o n d i n g local m a x i m u m d u r i n g FOD c o n t r a c t i o n . Fig. 7.

For t h e s e l e c t i o n o f t h e second g r o u p o f w a k e v i s u a l i z a t i o n s , w e d e c i d e t o trace t h r o u g h the g e o m e t r y o f a l l f o u r d e s i g n cases (i.e. Cases 14, S e c t i o n 5), s e l e c t i n g f o u r d i f f e r e n t S t r o u h a l n u m -bers f o r e a c h case: S t r = { 0 . 1 5 , 0.29, 0.42, 0,56} a n d t w o d i f f e r e n t p i t c h a n g l e s f o r each S t r o u h a l n u m b e r . The t w o selected p i t c h -angles are p r o p e r l y S t r o u h a l d e p e n d e n t , i n o r d e r to r e p r e s e n t a l o w e r FOD l o a d i n g a n d a h i g h e r FOD l o a d i n g f o r each g i v e n S t r o u h a l . Figs. 1 2 - 2 7 p r e s e n t 7 = 0 slices o f the f r e e v o r t e x sheet g e o m e t r y at t = 2 1 , f o r all the s i x t e e n c o m b i n a t i o n s o f g e o m e t r y a n d S t r o u h a l

n u m b e r . Each f i g u r e c o n t a i n s t w o w a k e slices, t h e f i r s t ( d a s h e d l i n e ) c o r r e s p o n d s to t h e h i g h e r FOD l o a d i n g ( s m a l l e r p i t c h - a n g l e ) w h i l e t h e second ( s o l i d l i n e ) to t h e l i g h t e r l o a d i n g (greater p i t c h a n g l e ) .

F r o m those figures w e observe t h a t f o r a s m a l l e r OQ ( w h i c h m e a n s a h i g h e r angle o f a t t a c k ) a w i d e r w a k e is d e v e l o p e d . This p r e -s u m e -s -s t r o n g e r i n d u c e d v e l o c i t i e -s a n d t h u -s h i g h e r l o a d i n g . Al-so, b y a w a k e i n s p e c t i o n b e t w e e n t h e figures, i t is s h o w n t h a t i n c r e a s i n g the S t r o u h a l n u m b e r the l o a d i n g is i n c r e a s e d . R e g a r d i n g t h e e f f e c t o f a n o n - z e r o c a m b e r (Cases 2 a n d 4 ) t h e w a k e s u r v e y / c o m p a r i s o n s h o w s o n l y s l i g h t d i f f e r e n c e s w i t h t h e n o n - c a m b e r case f o r s i m i l a r S t r o u h a l a n d p i t c h angle c o n d i t i o n s .

42 G. Politis. V. Tsarsitalidis I Applied Ocean Research 36 (2012) 36-50

Fig. 10. Wake visualizations for (Case 1, Str=0.42, theta = 24.27), at ( 7 / 2 , 7 37/2,27,57/2, 37), respectively. Slice w i t h t h e X Z plane.

8. T h e o p e n w a t e r p e r f o r m a n c e d i a g r a m s f o r o u r FOD s e r i e s

S y s t e m a t i c u n s t e a d y BEIM s i m u l a t i o n s have b e e n p e r f o r m e d w i t h t h e selected FOD series d e s c r i b e d i n Section 5. In all s i m u l a -t i o n s a c h o r d o f c = 0.15 m has been selec-ted. V a r i a -t i o n o f S -t r o u h a l n u m b e r has b e e n a c h i e v e d b y c h a n g i n g t h e f r e q u e n c y o f t h e FOD o s c i l l a t i o n w h i l e t h e c o r r e s p o n d i n g t r a n s l a t i o n a l v e l o c i t y has been h e l d c o n s t a n t a n d e q u a l to V = 2 . 0 m / s . This results to a c o n s t a n t R e y n o l d s n u m b e r (based o n t r a n s l a t i o n a l v e l o c i t y ; Re = ( L / - c ) / v ) e q u a l t o Re = 0.263 • 10^. C o r r e s p o n d i n g Reynolds n u m b e r s based o n t h e m a x i m u m u n d i s t u r b e d f l o w v e l o c i t y are S t r o u h a l d e p e n -d e n t a c c o r -d i n g to t h e r e l a t i o n ; Rest;- = ( ( U • c)/v)\/1 4- (JT • S t r f , o r Restr = 0.291 •10'^ at S t r = 0 , 1 5 and Restr = 0 . 6 3 6 • 10^ at S'tr=0.7

V o r t e x C o r e

J

-1 z

Vortex direction

Induced jet

Fig. 11. Slice of the FOD wake at t = 3 7 f ü r { C a s e 1, Str = 0.42, theta = 24.27). Artistic add-ons s h o w i n g the train o f ring vortices and corresponding jets by w h i c h the FOD produces thrust.

( a s s u m e d k i n e m a t i c v i s c o s i t y y = 1.139 • 10"^ m ^ / s ) . M e a n t h r u s t a n d p o w e r have t h e n b e e n c a l c u l a t e d b y r u n n i n g the BEM code for t w o t i m e p e r i o d s a n d c a l c u l a t i n g t h e m e a n values o f t h e u n s t e a d y forces o v e r t h e second p e r i o d (see Section 6 ) . The results are p r e s e n t e d i n t h e f o r m o f CJ-OQ d i a g r a m s ( w h e r e 00 t h e t a i n d i a g r a m s ) (Figs. 2 8 , 3 0 , 3 2 a n d 3 4 ) a n d Cp-t^o d i a g r a m s (Figs. 2 9 , 3 1 , 3 3 a n d 3 5 ) w i t h p a r a m e t e r t h e S t r o u h a l n u m b e r ( t h i c k l i n e i n t h e d i a g r a m s ) .

G. Politis. V. Tsarsitalidis / Applied Ocean Research 36 (2012) 3 6 - 5 0 43

Fig. 13. Case 1, Wakes for 5tr=0.29 and theta = 5 (dashed) and 23.54 (solid).

Figs. 28, 30, 3 2 and 3 4 c o n t a i n a d d i t i o n a l l y i n p a r a m e t r i c f o r m ( t h i n l i n e s ) the o p e n w a t e r e f f i c i e n c y i] o f the FOD:

Also, Figs. 29, 3 1 , 33 a n d 35 c o n t a i n a d d i t i o n a l l y i n p a r a m e t r i c f o r m t h e amax angle ( t h i n l i n e s ) d e f i n e d as the m a x i m u m v a l u e o f a( t), r e l a t i o n (2), over one p e r i o d . This last i n f o r m a t i o n is v e r y u s e f u l f o r t h e designer i n o r d e r t o a v o i d m a x i m u m angles w i t h a p o t e n -t i a l d a n g e r f o r s e p a r a -t i n g f l o w (e.g. grea-ter -t h a n 2 0 ° ) , p h e n o m e n o n w h i c h is n o t m o d e l e d b y o u r m e t h o d a n d c o n s e q u e n t l y o u r BEM s i m u l a t i o n s can r e s u l t i n q u e s t i o n a b l e a n d / o r i n c o r r e c t p r e d i c t i o n s i n t h a t r e g i o n . z

Fig. 14. Case 1, Wakes for Str=0.42 and theta = 14.63 (solid) and 43.53 (dashed).

Fig. 15. Case 1, Wakes for S f r = 0 . 5 6 and theta = 14.25 (dashed) and 32.75 (solid).

I n t e r e s t i n g c o n c l u s i o n s d r a w n f r o m those f i g u r e s are t h e f o l l o w -i n g : ( a ) t h e r e -is a r e l a t -i v e l y w -i d e r e g -i o n o f m a x -i m u m h y d r o d y n a m -i c FOD e f f i c i e n c y w h i c h is a c h i e v e d at a m a x i m u m angle o f a t t a c k less t h a n 1 5 ° i.e. at t h e r e g i o n o f f l o w w i t h o u t e x p e c t e d s e p a r a t i o n , ( b ) FODs w i t h a d d i t i o n a l u n s t e a d y c a m b e r m o t i o n (Cases 2 a n d 4 ) increase b o t h the p r o d u c e d t h r u s t a n d t h e e f f i c i e n c y . It s h o u l d be also n o t e d t h a t s y s t e m a t i c i n s p e c t i o n s o f t h e c a l c u l a t e d p r e s -sure d i s t r i b u t i o n s gave no i n d i c a t i o n o f local pres-sure less t h a n t h e c o r r e s p o n d i n g v a p o r pressure i n t h e r e g i o n a r o u n d o p t i m u m p e r f o r m a n c e . As a r e s u l t n o c a v i t a t i o n is e x p e c t e d i n t h a t r e g i o n .

Figs. 2 8 - 3 5 can be used t o select a n o p t i m u m FOD f o r a g i v e n s h i p u s i n g a h a n d c a l c u l a t o r . For e x a m p l e , a s s u m i n g t h a t a s h i p is g i v e n w i t h a d e s i g n speed o f V k n o t s , t h e p r o b l e m o f d e s i g n -i n g a Case 1 FOD (-i.e. select -its o p t -i m u m g e o m e t r y w -i t h t h e

44 G. Politis, V. Tsarsitalidis /Applied Ocean Research 36 (2012) 36-50

X

Fig. 17. Case 2, Wakes f o r Str=0.29 and theta = 5 (dashed) and 23.54 (sohd).

c o r r e s p o n d i n g r e v o l u t i o n s a n d r e q u i r e d DHP) can be s o l v e d as f o l l o w s : (a) w i t h t h e d e s i g n speed k n o w n , the s h i p resistance and, f r o m Eq. ( 1 9 ) , t h e p r o p e l l e r t h r u s t a n d CT can be c a l c u -l a t e d ; ( b ) w i t h t h i s Cj d r a w a h o r i z o n t a -l -l i n e i n Fig. 28 a n d f i n d t h e i n t e r s e c t i o n o f t h i s h o r i z o n t a l l i n e w i t h the v a r i o u s c o n s t a n t S t r o u h a l lines, let ((^o. Sfr),-, / = 1, ihtr d e n o t e the Hstr i n t e r s e c t i o n p o i n t s ; ( c ) f r o m each S t r o u h a l n u m b e r t h e f r e q u e n c y o f the p r o p u l -sor m o t i o n can be f o u n d : ni = S t r i - V / ( 2 ;io); ( d ) use Fig. 29 t o f i n d Cpi f o r t h e p o i n t s {OQ, S f r ) , , i = 1, list,-, f r o m t h e Cp,,- f i n d t h e r e q u i r e d o p e n w a t e r p o w e r a n d use ( 1 6 ) t o f i n d D H P g , ; ( e ) f r o m t h e c a l c u l a t e d DHPB,,-, / = 1, Ustr select t h a t w i t h m i n i m u m r e q u i r e d D H P B .

2

Fig. 18. Case 2, Wakes for Str = 0.42 and theta = 14.63 (dashed) and 43.53 (solid).

Fig. 19. Case 2, Wakes for S(r= 0.56 and theta = 14.25 (dashed) and 32.75 (solid).

9. G r i d i n d e p e n d e n c e o f the c a l c u l a t e d o p e n w a t e r d i a g r a m s In o r d e r to analyze t h e e f f e c t o f t h e g r i d d e n s i t y t o t h e c a l c u l a t e d o p e n w a t e r FOD p e r f o r m a n c e , t h e s y s t e m a t i c B E M c a l c u l a t i o n s o f t h e p r e v i o u s p a r a g r a p h have been p e r f o r m e d w i t h t h r e e d i f f e r e n t g r i d densities, a c c o r d i n g to Table 1. For t h e c o m p a r i s o n a m o n g g r i d s , a m e a s u r e f o r t h e m a x i m u m a n d m e a n d e v i a t i o n b e t w e e n t h e m e a n t h r u s t forces p r o d u c e d b y each g r i d , has b e e n d e f i n e d as f o l l o w s ; f o r e a c h c a l c u l a t i o n case a n d each g r i d d e n s i t y , a m e a n v a l u e f o r t h r u s t has b e e n f o u n d . Thus, a table w i t h t h r e e c o l u m n s is o b t a i n e d , i n d i c a t i n g the m e a n t h r u s t f o r each o f t h e t h r e e g r i d s , a n d a n u m b e r o f r o w s e q u a l t o the n u m b e r o f c a l c u l a t i o n cases. For each l i n e o f t h e table, t h r e e m o r e

X

C. Politis, V. Tsarsitalidis / Applied Ocean Research 36 (2012) 36-50 45

Table 1 Grid densities.

Name No. of chordwise No. of circumferential

elements (face + back) elements

G r i d l 20 55

Grid2 26 72

Grids 32 80

c o l u m n s can be a d d e d c o n t a i n i n g the % d i f f e r e n c e s b e t w e e n the m e a n t h r u s t forces o f t h e f i r s t t h r e e c o l u m n s t a k e n i n couples. C o n -s e q u e n t l y , t h r e e m o r e c o l u m n -s o f data are created. For each n e w c o l u m n , the o v e r a l l m e a n and m a x i m u m values are o b t a i n e d (i.e. six m o r e n u m b e r s are o b t a i n e d ) . Those data are s h o w n i n Table 2, w i t h t h e f o l l o w i n g c o n v e n t i o n : above diagonal are o v e r a l l m e a n o f

Z

Fig. 21. Case 3, Wakes for Str=0.29 and theta = 5 (dashed) and 23.54 (solid).

2

Fig. 22. Case 3, Wakes for S f r = 0 . 4 2 and theta = 14.63 (dashed) and 43.53 (solid).

Table 2

Mean (above the diagonal) and m a x i m u m ( b e l o w the diagonal) %deviations in mean thrust calculation.

G r i d l Grid2 Crid3

G r i d l 0 5.1% 5.2%

Grid2 8% 0 0.7%

Grid3 10.1% 3.1% 0

the %differences o f m e a n t h r u s t values a n d b e l o w d i a g o n a l are the m a x i m u m o f t h e %differences o f m e a n t h r u s t values.

It can be seen t h a t G r i d 2 (as w e l l as G r i d S ) can be c o n -s i d e r e d a-s a r e l i a b l e d i -s c r e t i z a t i o n f o r -s y -s t e m a t i c -s i m u l a t i o n -s .

z

Fig. 23. Case 3, Wakes for Str=0.56 and theta = 14,25 (dashed) and 3 2 7 5 (solid).

X

4 6 C. Politis, V. Tsarsitalidis /Applied Ocean Research 36 (2012) 36-50

Fig. 25. Case 4, Wakes for Str=0.29 and theta = 5 (dashed) and 23.54 (solid).

Fig. 26. Case 4, Wakes f o r Str= 0.42 and l:heta = 14.63 (dashed) and 43.53 (solid).

30 40 t h e t a

Fig. 28. Case 1. Mean thrust coefficient, as f u n c t i o n of Ho. Thick lines stand f o r Str and t h i n lines f o r efficiency.

Nevertheless, G r i d l i n s p i t e o f l a c k i n g accuracy, i t is g o o d e n o u g h to s h o w t r e n d s a n d c o u l d be used f o r g e t t i n g q u i c k e x p l o r a t o r y results.

Fig. 3 6 s h o w s c o m p a r i s o n s o f c a l c u l a t e d t i m e d e p e n d e n t t h r u s t f o r t h r e e d i f f e r e n t g r i d s f o r the case: {Case 1, Str = 0.70, theta = 2 2 . 3 0 } , w h e r e t h e m a x i m u m d i f f e r e n c e o f m e a n values w a s f o u n d . It s h o u l d be n o t e d , t h a t e v e n t h e coarser g r i d , p r o d u c e s results t h a t f o l l o w t h e same t r e n d i n t h e t i m e d o m a i n as w e l l , w h i l e t h e d i f f e r e n c e is l o c a t e d at t h e peak values. I t s h o u l d be also n o t e d t h a t t h e d i f f e r e n c e s i n the c a l c u l a t e d t i m e d e p e n d e n t t h r u s t are r e d u c e d as g r i d d e n s i t y increases, a f a c t w h i c h i n d i c a t e s t h e convergence o f t h e results f o r f i n e r g r i d s .

a.

u

Fig. 27. Case 4, Wakes for Str=0.56 and theta = 14.25 (dashed) and 32.75 (solid).

30 40

theta

Fig. 29. Case 1. Mean p o w e r coefficient, as f u n c t i o n o f Ho. Thick lines stand f o r Str and t h i n forOmax.

C. Politis, V. Tsarsitalidis/Applied Ocean Research 36(2012) 36-50 47

Table 3

Ship particulars.

Ship no: 1 2 3

Type Bulkcarrier Passenger/car ferry Speedboat Propulsor diameter D ( m ) 8.10 4.10 1 4 0 Displacement (tons) 37,288.9- 8917.66 1 6 0 0 0 Number of propulsors 1 2 2

Table 4

Resistance curves of ships.

Ship 1 2 3 Ship V ( m / s ) R{kp) V ( m / s ) R ( k p ) V ( m / s ) R ( k p ) V-R curve 3.05 9877 7.20 29,055 9.25 9204 3.56 14,191 7.71 31,056 10.27 10,471 4.06 18,496 8.23 34,528 11.32 11,519 4.58 23,029 8.75 38,374 11.83 11,962 5.08 27,463 9.26 42,755 12.35 12,367 5.59 34,212 9 7 8 47,292 12.87 12,697 6.10 40,134 10.29 52,301 13.38 13,084 6.61 48,142 10.80 58,153 13.88 13,477 7.12 61,364 11.31 66,426 14.40 13,872 7.63 75,358 11.83 78,009 14.92 14,306 10. A p p l i c a t i o n of a FOD f o r t h e p r o p u l s i o n o f a s h i p -O p t i m u m d e s i g n e x a m p l e s a n d c o m p a r i s o n w i t h c o n v e n t i o n a l p r o p e l l e r s

Three d i f f e r e n t vessels are u s e d i n a f e a s i b i l i t y s t u d y f o r t h e a p p l i c a d o n o f a FOD as an a l t e r n a t i v e p r o p u l s o r to t r a d i t i o n a l p r o -p e l l e r s . The m a i n c h a r a c t e r i s t i c s o f t h e shi-ps are s h o w n i n Table 3 ( D is t h e selected m e a n FOD d i a m e t e r or t h e p r o p e l l e r d i a m e t e r ) . The s h i p bare h u l l resistance d a t a was t a k e n f r o m the database o f t h e NTUA t o w i n g t a n k a n d are s h o w n i n Table 4. W i t h the b a r e h u l l resistance g i v e n , t h e s y s t e m o f algebraic Eqs. ( 1 7 ) - ( 1 9 ) can be s o l v e d f o r a range o f ship speeds V and a n d t h e t o t a l i t y o f design s o l u t i o n s can be p r e s e n t e d i n a d i a g r a m as d i c t a t e d b y Eq. ( 2 0 ) . For t h e n e e d o f the c o m p a r i s o n , w e have a s s u m e d t h a t i n all cases: w = 0.0, t = 0.0, i]R = 1.0. This is a reasonable a s s u m p t i o n o n l y f o r t h e t w o t w i n s c r e w vessels w h i c h have s m a l l p r o p u l s o r h u l l i n t e r -a c t i o n s . For t h e s i n g l e s c r e w s h i p , t h i s s e l e c t i o n is s o u n d solely f o r c o m p a r a t i v e purposes w i t h t h e c o n v e n t i o n a l p r o p e l l e r . The use o f

t h e t a

Fig. 30. Case 2. Mean t h r u s t coefficient, as f u n c t i o n of do- Thick lines stand for Str

and t h i n lines for efficiency.

t h e same h u l l - i n t e r a c t i o n c o e f f i c i e n t s f o r t h e FOD a n d the p r o p e l l e r can be c o n s i d e r e d reasonable since i n t e r a c t i o n c o e f f i c i e n t s are ( f o r t h e same s t e r n g e o m e t r y ) m a i n l y f u n c t i o n s o f p r o p u l s o r d i a m e -t e r a n d -t h e d e v e l o p e d -t h r u s -t [ 7 ] . No i n c l i n e d axis c o r r e c -t i o n s w e r e m a d e f o r the c o n v e n t i o n a l p r o p e l l e r s . No c o r r e c t i o n o f t h e b a r e h u l l resistance (Table 4 ) f o r appendages has been m a d e . A s h a f t e f f i -c i e n -c y e q u a l to 1 has b e e n u s e d i n t h e -c a l -c u l a t i o n s . N o -c o r r e -c t i o n s f o r f u l l scale Reynolds n u m b e r have b e e n i n t r o d u c e d f o r e i t h e r t h e FOD or the B-series p r o p e l l e r s .

T h e t o t a l i t y o f design s o l u t i o n s D H P N ( o p t i m u m a n d n o n -o p t i m u m ) f -o r the t h r e e vessels e q u i p p e d w i t h a Case 4-FOD can be f o u n d i n Figs. 37, 39 a n d 4 1 , S i m i l a r l y , t h e t o t a l i t y o f design s o l u t i o n s D H P - N ( o p t i m u m a n d n o n - o p t i m u m ) f o r t h e s a m e ships e q u i p p e d w i t h B5.70 c o n v e n t i o n a l p r o p e l l e r s [ 1 3 ] , can be f o u n d i n Figs, 3 8 , 4 0 a n d 4 2 , On t h e figures, the c o n s t a n t - v e l o c i t y curves a n d t h e c o n s t a n t m a x i m u m p i t c h angle QQ curves (Figs. 3 7 , 3 9 a n d 4 1 ) or the c o n s t a n t P/D curves (Figs. 3 8 , 4 0 a n d 4 2 ) are s h o w n a n d l a b e l e d a c c o r d i n g l y .

14

t h e t a

Fig. 31. Case 2. Mean p o w e r coefficient, as f u n c t i o n of Oo. Thick lines stand for Str

48 C. Politis, V. Tsarsitalidis / Applied Ocean Research 36(2012) 36-50 Table 5

Overall comparison of propulsors.

Ship Speed(kn) Case Revolutions ( r p m ) Power (PS) Overall efficiency {%) Power gain (%) 1 15 FOD (Cr= 0.480) 29.6 9650 79.4 9.81 B-screw 47.0 10,700 71.6 2 23 FOD (Cr= 0.403) 87.5 7550 81.4 3.21 B-screw 145.0 8700 70.7 3 29 FOD (CT=0.401) 305.0 1735 82.0 10.70 B-screw 552.0 1943 73.2 Ttie c o m p a r i s o n s o f t h e o p t i m u m FOD o v e r t h e o p t i m u m B -s c r e w f o r t h e t h r e e -s h i p type-s at c o r r e -s p o n d i n g -selected d e -s i g n speeds can be s u m m a r i z e d i n Table 5. I n atl cases t h e FOD s h o w s e f f i c i e n c i e s o f t h e o r d e r o f 80% a n d the g a i n c o m p a r e d t o t h e B-screw, ranges f r o m 9.81% t o 13.21%. It is also n o t i c e a b l e , t h a t t h e o p t i m u m FOD r e v o l u t i o n s are a l w a y s l o w e r c o m p a r e d t o t h a t o f c o r r e s p o n d i n g c o n v e n t i o n a l o p t i m u m p r o p e l l e r . For i l l u s t r a t i v e

reasons. Fig. 41 i n c l u d e s p r o p u l s i v e e f f i c i e n c y c o n t o u r s s h o w n w i t h dashed lines. I t s h o u l d be s t r e t c h e d t h a t t h e absolute values o f p o w e r a n d o v e r a l l e f f i c i e n c y , c o n t a i n e d i n Table 5, are a p p r o x i m a t e t o t h e e x t e n t o f o u r u n c e r t a i n t y r e g a r d i n g values o f p r o p u l s o r - h u l l i n t e r a c t i o n factors.

30 40 t h e t a

Fig. 32. Case 3. Mean t h r u s t coefficient, as f u n c t i o n of HQ. Thick lines stand for Str

and t h i n lines for efficiency.

30 40 t h e t a

Fig. 34. Case 4. Mean t h r u s t coefficient, as f u n c t i o n o f Ho- Thick lines stand f o r Sfr

and t h i n lines for efficiency.

U

30 40 t h e t a

Fig. 33. Case 3. Mean p o w e r coefficient, as f u n c t i o n o f Ho- Thick lines stand for

Str-and t h i n for a,„ax.

a.

u

30 40 t h e t a

Fig. 35. Case 4. Mean p o w e r coefficient, as f u n c t i o n o f Ho- Thick lines stand f o r Str

C. Politis, V. Tsarsitalidis/Applied Ocean Research 36 (2012) 36-50 4 9 \ \ • 1 v - ' V n

\ 1

ê

tn-Fig. 36. Effect of grid density to the calculated instantaneous thrust for the case: {Case 1, Str = 0.70, t h e t a = 2 2 . 3 0 } .

N(rpm)

Fig. 37. Bulk carrier (ship 1) - FOD: totality of design solutions i n the f o r m o f constant-V, constant-Öo grid.

11. C l o s i n g r e m a r k s , c h a l l e n g e s a n d f u r t h e r d e v e l o p m e n t

I n s p i r e d f r o m t h e p r o p u l s i o n m e c h a n i s m s used b y s q u i d s a n d j e l l y f i s h e s , a 3 - D BEM t i m e s t e p p i n g a l g o r i t h m has b e e n a p p l i e d t o i n v e s t i g a t e the o p e n w a t e r p e r f o r m a n c e o f a n e w p r o p u l s o r

N(rpm)

Fig. 38. Bulk carrier (ship 1 )-B5.70 screw; t o t a l i t y of design solutions i n the f o r m of constant-V, constant-P/D grid.

N(rpm)

Fig. 39. Passenger ferry (ship 2)-F0D: totality of design solutions in the f o r m of constant-V, c o n s t a n t - ö o grid. concept, t h e F l e x i b l e O s c i l l a t i n g D u c t . Calculations s h o w t h a t t h e FOD is a v e r y e f f i c i e n t p r o p u l s o r . P r o p u l s i v e c o e f f i c i e n t s o f the o r d e r of 0.82 have b e e n c a l c u l a t e d , w h i c h are h i g h e r t h a n t h a t o b s e r v e d i n c o n v e n t i o n a l p r o p e l l e r s . Systematic c a l c u l a t i o n s w e r e m a d e a n d d e s i g n charts w e r e p r o d u c e d , a l o n g w i t h a d e s i g n m e t h o d o l o g y , w h i c h has been e x p l a i n e d t h o r o u g h l y . Due t o lack o f e x p e r i m e n t a l data, e x t e n s i v e n u m e r i c a l tests w e r e c o n d u c t e d w h i c h has i n d i c a t e the g r i d i n d e p e n d e n c e a n d the e x p e c t e d accuracy o f t h e p r o p o s e d m e t h o d . A f t e r a p p l y i n g t h e design m e t h o d f o r a c t u a l ships, t h e FOD proves to be s u p e r i o r t o c o n v e n t i o n a l p r o p e l l e r s w i t h e f f i c i e n c y gains o f t h e o r d e r o f 9 . 8 1 - 1 3 . 2 1 % . F u r t h e r m o r e at t h e r e g i o n o f o p t i m u m FOD e f f i c i e n c i e s no c a v i t a t i o n issues are e x p e c t e d . The FOD, due t o its shape a n d its n o n - r o t a t i o n a l o p e r a t i o n , seems t o be f r i e n d l i e r t o the e n v i r o n m e n t a n d less c a t a s t r o p h i c f o r a q u a t i c a n i m a l s . O n t h e o t h e r h a n d , t h e p r o b l e m o f c o n s t r u c t i n g s u c h a p r o p u l s o r is a n o p e n c h a l l e n g e w h i c h seems d i f f i c u l t w h e n r e s t r i c t e d t o c o n -v e n t i o n a l m a t e r i a l s a n d m e t h o d s . The usage o f n e w t e c h n o l o g i e s and m a t e r i a l s s u c h as t h e e l e c t r o - a c t i v e p o l y m e r s or p i e z o e l e c t r i c ^ I i I • I I I ' I ! r • I I - - I I I 120 140 160 180 200 220 240 260

N(rpm)

Fig. 40. Passenger ferry (ship 2)-B5.70 screw; totality of design solutions i n the f o r m of constant-V, constant-P/D grid. Results for one of the t w o propulsor u n i t s .