Ocean Engineering 64 (2013) 122-134
ELSEVIER
C o n t e n t s l i s t s a v a i l a b l e at S c i V e r s e S c i e n c e D i r e c t
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
Computational analysis of self-pitching propellers performance
in open water
S. Leone, C. Testa, L. Greco *, F. S a l v a t o r e
CNR-INSEAN, The Italian Ship Model Basin, Via di Vallerano 128. 00139 Rome, Italy
A R T I C L E I N F O A B S T R A C T
T h i s p a p e r p r e s e n t s a n u m e r i c a l s t u d y on s e l f - p i t c h i n g p r o p e l l e r s ( S P P ) in o p e n w a t e r a i m e d at the p r e d i c t i o n of t h e p e r f o r m a n c e i n t e r m s of e q u i l i b r i u m p i t c h - s e t t i n g a n d d e l i v e r e d t h r u s t a n d torque. D i f f e r e n t l y f r o m o t h e r t y p e s of propeller, S P P - b l a d e s a r e able to f r e e l y rotate about the pivot axis u p to the s p i n d l e m o m e n t d u e to the c e n t r i f u g a l forces b a l a n c e s that g e n e r a t e d by the h y d r o d y n a m i c loads. In the a t t e m p t to p r o v i d e a r e l i a b l e p r e d i c t i o n o f SPP p e r f o r m a n c e , the e m p h a s i s of t h e w o r k is on p r o p e l l e r h y d r o d y n a m i c s m o d e l l i n g ; in detail, t h e B o u n d a r y E l e m e n t IVIethod ( B E M ) a n d B l a d e E l e m e n t M o m e n t u m T h e o r y ( B E M T ) are h e r e i n p r o p o s e d as fast a n d a c c u r a t e h y d r o d y n a m i c s o l v e r s . B o t h a p p r o a c h e s a r e w i d e l y u s e d in the f r a m e w o r k of r o t a t i n g - b l a d e s p r o p u l s i o n but their a p p l i c a t i o n to s e l f - p i t c h i n g p r o p e l l e r s is n o n s t a n d a r d . H e n c e , SPP d r a w b a c k s a n d p o t e n t i a l i t i e s a r e h i g h l i g h t e d through c o m p a r i s o n s w i t h n u m e r i c a l a n d e x p e r i m e n t a l a v a i l a b l e d a t a . T h e final goal of the p a p e r is to p r o v i d e s o m e g u i d e l i n e s on t h e e f f e c t i v e n e s s a n d r o b u s t e n e s s o f B E M / B E M T - h y d r o d y n a m l c s w h e n a p p l i e d to S P P - b l a d e s ; this m i g h t h a v e p r a c t i c a l i m p l i c a t i o n s for p r e l i m i n a r y d e s i g n a n d o p t i m a l d e s i g n process. © 2 0 1 3 E l s e v i e r Ltd. A l l rights r e s e r v e d . Article history: Received 2 April 2012 Accepted 24 February 2013 Available online 28 March 2013 Keywords:
Self-pitching propellers Hydrodynamics
Boundary Element Method Blade Element Momentum Theory
1. I n t r o d u c t i o n S e l f - p i t c h i n g p r o p e l l e r s (SPP) are u n c o n v e n t i o n a l p r o p e l l e r types w h e r e blades a u t o m a t i c a l l y a d j u s t t h e i r p i t c h t o s u i t t h e a p p l i e d l o a d , a c c o r d i n g t o the p r e v a i l i n g w o r i d n g c o n d i t i o n s . Fig. 1 s h o w s a sketch o f a t h r e e - b l a d e d s e l f - p i t c h i n g p r o p e l l e r . F r o m t h e k i n e m a t i c s t a n d p o i n t , blades f r e e l y t u r n a b o u t a n axis o r t h o g o n a l t o t h e s h a f t ( s p i n d l e axis) so t h a t , w h e n t h e s h a f t rotates, c e n t r i f u g a l f o r c e causes t h e blades t o s w i n g o u t w a r d s w h e r e a s h y d r o d y n a m i c forces t e n d t o p u s h t h e m i n w a r d s ; t h e o p e r a t i n g s e t t i n g is f i n a l l y a c h i e v e d w h e n blades t a k e u p a stable e q u i l i b r i u m p o s i t i o n d u e t o t h e balance b e t w e e n c e n t r i f u g a l a n d h y d r o d y n a m i c loads. T h i s f e a t u r e a l l o w s SPP t o o f f e r g o o d p e r f o r m a n c e i n b o t h ahead a n d reverse m o t i o n . P r e v i o u s research, c a r r i e d o u t i n t h e 1980s a n d e a r i y 1990s, d e m o n s t r a t e s t h e a d v a n t a g e s o f u s i n g f r e e p i t c h i n g p r o p e l l e r s i n t e r m s o f e f f i c i e n c y a n d s h o w s t h a t t h e t o r q u e c h a r a c t e r i s t i c o f t h i s t y p e o f p r o p e l l e r is f l a t t e r t h a n i n case o f a c o n v e n t i o n a l s c r e w w i t h r i g i d p i t c h s e t t i n g (see M i l e s et al., 1 9 9 2 ) w i t h s o m e b e n e f i t s i n t e r m s o f c o u p l i n g b e t w e e n engine a n d p r o p e l l e r s y s t e m s . As s h o w n i n M i l e s et a l . ( 1 9 9 2 ) , t h e e f f i c i e n c y c u r v e o f a s e l f - p i t c h i n g p r o p e l l e r i n o p e n w a t e r f o r m s an e n v e l o p e a r o u n d t h e m a x i m u m
•Corresponding author. Tel.: -F39OS50299313; fax: + 3 9 06 5070619. E-mail address: luca.greco@cnr.it (L. Greco).
0029-8018/$-see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.Org/10.1016/j.oceaneng.2013.02.012
e f f i c i e n c i e s a c h i e v e d w h e n t h e s a m e p r o p e l l e r operates at r i g i d p i t c h settings w i t h i n c r e a s i n g p i t c h r a t i o s . T h i s h y d r o d y n a m i c b e h a v i o u r s t e m s f r o m t h e c a p a b i l i t y o f t h e blades t o a d j u s t t h e m s e l v e s t o a d o p t t h e m o s t e f f i c i e n t p i t c h a n g l e as a f u n c r i o n o f t h e o p e r a t i n g c o n d i t i o n s . For these reasons, SPP have b e c o m e an a p p e a l i n g p r o p u l s i o n u n i t f o r s a i l i n g y a c h t s w h e r e t h e increase o f p r o p e l l e r t h r u s t a n d t h e r e d u c t i o n o f b o t h t r a i l i n g d r a g a n d s h a f t RPM are c h a l l e n g i n g tasks t o be a c h i e v e d , e s p e c i a l l y f o r p r o p e l l e r d i a m e t e r s o f i n c r e a s i n g size. A l t h o u g h SPP are p r i m a r i l y c o n c e i v e d as a u x i l i a r y p r o p u l s i o n systems f o r s a i l i n g yachts, t h e y are o f t e n used also as e n e r g y g e n e r a t i o n devices. Such a v e r s a r i -l i t y makes s e -l f - p i t c h i n g p r o p e -l -l e r s a v e r y a t t r a c t i n g r o t a t i n g - b -l a d e d e v i c e i n t h e range o f l o w - p o w e r engines. L o o k i n g at t h e l i t t l e a v a i l a b l e l i t e r a t u r e o n SPP, i t is easy t o recognise h o w t h e c o m p u t a t i o n a l t o o l s used f o r t h e n u m e r i c a l i n v e s t i g a t i o n are t y p i c a l l y based o n s i m p l i f i e d h y d r o d y n a m i c m o d e l l i n g , y i e l d i n g a r o u g h e s t i m a t i o n o f t h e p e r f o r m a n c e o f t h e p r o p e l l e r ; d e s p i t e t h e h i g h a c c u r a c y a n d r e l i a b i l i t y a c h i e v e d by C o m p u t a t i o n a l F l u i d D y n a m i c s (CFD) i n t h e c o n t e x t o f m a r i n e p r o p e l l e r d e s i g n a n d p e r f o r m a n c e analysis, n o a t t e m p t t o a p p l y m o r e a d v a n c e d h y d r o d y n a m i c a p p r o a c h e s is f o u n d . These c o n s i d e r a t i o n s have i n s p i r e d t h e p r e s e n t w o r k t h a t proposes a n u m e r i c a l c o m p a r i s o n b e t w e e n t h e c a p a b i l i t i e s o f t w o c o m p u t a t i o n a l h y d r o d y n a m i c approaches f o r t h e s t u d y o f t h e h y d r o m e c h a n i c b e h a v i o u r o f SPP i n o p e n w a t e r , n o n - c a v i t a t i n g f l o w c o n d i t i o n s . I n details, t h e Blade E l e m e n t M o m e n t u m T h e o r y
S. Leone et al. / Ocean Engineering 64 (2013) 122-134 123
_ . • '
Fig. 1. Sketch of self-pitching propeller.
( B E M T ) a n d B o u n d a r y E l e m e n t M e t h o d ( B E M ) are used, t h r o u g h -o u t t h e paper, as h y d r -o d y n a m i c s-olvers f -o r t h e e v a l u a t i -o n -o f t h e h y d r o l o a d s f o r c i n g t h e e q u a t i o n o f b l a d e m o t i o n . N o t e t h a t t h e BEMT a p p r o a c h h a s j u s t b e e n a p p l i e d i n t h e past t o SPP (see M i l e s et al., 1 9 9 2 ) as w e l l as t o c o n v e n t i o n a l m a r i n e p r o p e l l e r s (see B e n i n i , 2 0 0 4 ) , p r o v i n g t o be a n e f f e c t i v e p r e d i c t i o n t o o l as f a r as c o m p l e x i t y a n d c o m p u t a t i o n a l e f f o r t s are c o n c e r n e d . H o w e v e r , t o o v e r c o m e t h e d r a w b a c k s o f a p p r o a c h e s l i k e BEMT based o n t w o d i m e n s i o n a l f l o w a s s u m p t i o n s a n d p r o v i d e a general r e p r e s e n t a -t i o n o f -t h r e e - d i m e n s i o n a l f l o w e f f e c -t s a r o u n d a r b i -t r a r i l y shaped bodies, B E M h y d r o d y n a m i c s is p r o p o s e d : t o t h e a u t h o r s ' k n o w l -edge, t h e a p p l i c a t i o n o f B E M h y d r o d y n a m i c s is a n o v e l t y i n t h e f r a m e w o r k o f SPP. The assessment o f t h e a f o r e m e n t i o n e d h y d r o d y n a m i c s m o d e l -l i n g is f i r s t c a r r i e d o u t o n a c o n v e n t i o n a -l c o n t r o -l -l a b -l e - p i t c h p r o p e l l e r (CPP) t h r o u g h a d e t a i l e d c o m p a r i s o n b e t w e e n n u m e r i c a l a n d e x p e r i m e n t a l results. T h e n , t w o s e l f p i t c h i n g p r o p e l l e r -t y p e s are n u m e r i c a l l y i n v e s -t i g a -t e d a n d -t h e o b -t a i n e d resul-ts are c o m p a r e d w i t h a v a i l a b l e n u m e r i c a l a n d e x p e r i m e n t a l data. 2 . S e l f - p i t c h i n g b l a d e e q u i l i b r i u m e q u a t i o n A s e l f p i t c h i n g p r o p e l l e r i n u n i f o r m a x i a l onset f l o w is c o n s i d e r e d . The r e s u l t i n g p r o p e l l e r f l o w is a n a l y s e d u n d e r c a v i t a t i o n -f r e e c o n d i t i o n s . The c o o r d i n a t e s y s t e m i n t r o d u c e d t o describe t h e rigid b o d y m o t i o n o f a s e l f - p i t c h i n g blade is t h e r e f e r e n c e f r a m e (0,X/„Y6,Zb) c e n t e r e d at t h e p i v o t ( s p i n d l e ) axis a n d rigidly c o n -n e c t e d t o t h e blade, see Figs. 2 a -n d 3. The e q u a d o -n g o v e r -n i -n g t h e b l a d e m o t i o n a b o u t t h e p i v o t axis is o b t a i n e d f r o m t h e conserva-d o n o f a n g u l a r m o m e n t u m e q u a t i o n w r i t t e n i n t o t h e b l a conserva-d e r e f e r e n c e a n d p r o j e c t e d o n t o t h e s p i n d l e axis Zt. This p r o c e d u r e y i e l d s
J^ë+hyO" cos{2c)+(J^~]yy)^ sm{2e) = m], + m'^ (1) w h e r e JzzJxxJyyJxy are t h e c o m p o n e n t s o f t h e i n e r d a t e n s o r w i t h
respect t o t h e axes o f the blade r e f e r e n c e f r a m e , e is t h e v a r i a t i o n o f a n g u l a r p i t c h s e t d n g due t o t h e f r e e r o t a d o n a b o u t t h e s p i n d l e axis a n d Q is the b l a d e a n g u l a r v e l o c i t y a b o u t t h e s h a f t axis X
Fig. 2. Blade reference: X^Yt plane view. Quantity e/, (blade offset) denotes the distance of blade root with respect to the X(,=0 plane.
0 , Xb Fig. 3. Blade reference: X^Zi, plane view.
(see Fig. 1). A t t h e right h a n d side o f Eq. ( 1 ) , t h e f o r c i n g t e r m c o m e s f r o m h y d r o d y n a m i c a n d b l a d e w e i g h t m o m e n t s a b o u t t h e s p i n d l e axis, a n d m ^ , r e s p e c t i v e l y . N o t e t h a t t h e s p i n d l e m o m e n t is a d m e d e p e n d e n t t e r m because o f t h e c h a n g e o f p o s i t i o n o f blade c e n t r e o f g r a v i t y d u r i n g t h e p r o p e l l e r r e v o l u d o n . A t t h i s stage i t is u s e f u l t o observe t h a t t h e r a t e o f change o f t h e s p i n d l e m o m e n t w i t h respect t o angle c is a n i n d e x o f ' b l a d e p i t c h s t i f f n e s s t h a t s t r o n g l y affects t h e o s c i l l a d o n p e r i o d o f t h e b l a d e a b o u t t h e Zt axis. Previous r e s e a r c h o n t h e s u b j e c t (see M i l e s et al., 1 9 9 2 ) h i g h l i g h t s t h a t n u m e r i c a l l y p r e d i c t e d b l a d e -p i t c h o s c i l l a d o n f r e q u e n c i e s f o r s e l f - -p i t c h i n g -p r o -p e l l e r s i n s t a l l e d o n s m a l l b o a t s are t y p i c a l l y less t h a n o n c e p e r r e v a n d , i n t u r n s , b l a d e p i t c h s e t t i n g can be a s s u m e d t o be stable. T h i s c o n c l u s i o n is f u r t h e r s u p p o r t e d b y r e s u l t s o f c a v i t a t i o n t u n n e l tests (see Fig. 4 ) s h o w i n g a SPP m o d e l d e v e l o p i n g a stable c a v i t a t i n g t i p - v o r t e x i n t h e s a m e f a s h i o n o f a f i x e d p i t c h p r o p e l l e r ; t h u s , t h e a x i s y m -m e t r i c f l o w a s s u -m p t i o n holds, r e q u i r i n g , i n t u r n s , a l l blades t o u n d e r g o t h e same s y n c h r o n o u s p i t c h i n g m o t i o n . A l t h o u g h t h i s b e h a v i o u r d e p e n d s o n blade m a t e r i a l a n d shape, as w e l l as b l a d e s p i n d l e axis bearings, t h e n u m e r i c a l i n v e s d g a d o n c a r r i e d o u t t h r o u g h o u t t h e p a p e r is p e r f o r m e d u n d e r t h e m a i n a s s u m p t i o n
124 S. Leone et al / Ocean Engineering 64 (2013) 122-134
Fig. 4. Self-pitching propeller cavitation tests taken from Miles et al. (1992),
t h a t blade p i t c h o s c i l l a t i o n s f r e q u e n c i e s are v e r y l o w , so t h a t t h e static a p p r o x i m a t i o n m a y be r e a s o n a b l y a d o p t e d .
T h e r e b y , b y n e g l e c t i n g t h e m f j , t e r m , Eq. ( 1 ) y i e l d s t h e f o l l o w -i n g e q u -i l -i b r -i u m e q u a t -i o n f o r e:
J^yQ^ cos(2£e)-hÜ^. -Jyy)^ Sin(2£e) = mfiiCe) (2) w h e r e £e indicates t h e v a r i a t i o n o f blade p i t c h setting at t h e e q u i l i b r i u m p o s i t i o n . Eq. ( 2 ) states t h a t an e q u i l i b r i u m c o n d i t i o n Ce is achieved w h e n t h e spindle m o m e n t s due to c e n t r i f u g a l a n d h y d r o d y n a m i c fields are balanced. For g i v e n SPP geometry, m a t e r i a l a n d o p e r a t i n g conditions, the l e f t - h a n d - s i d e o f Eq. ( 2 ) is c o m p l e t e l y d e f i n e d . The e v a l u a t i o n o f the h y d r o d y n a m i c spindle m o m e n t m f , represents the crucial p o i n t to d e t e r m i n e blade p i t c h e q u i l i b r i u m Ce a n d hence t h e overall p r o p e l l e r p e r f o r m a n c e ; i n v i e w o f this fact, BEM and BEMT h y d r o d y n a m i c approaches are here o u t l i n e d . Details are g i v e n i n A p p e n d i x A f o r completeness.
C o n s i d e r i n g p r o p e l l e r h y d r o d y n a m i c s m o d e l l e d b y BEMT, t h e s p i n d l e m o m e n t m f , m a y be e x p r e s s e d as
m f , = / (dTyp-dFQXp) dr (3)
w h e r e rt,oss indicates t h e r a d i u s o f b l a d e r o o t . Coordinates (Xp,yp) d e f i n e t h e l o c a t i o n ( i n t h e XbV;, p l a n e ) o f t h e c e n t r e o f pressure o f a n e l e m e n t a l blade s e c d o n b e t w e e n r a d i a l s t a d o n s r a n d r+dr respect t o t h e b l a d e f r a m e o f r e f e r e n c e , w h e r e a s dT a n d dpQ r e p r e s e n t sectional h y d r o d y n a m i c f o r c e c o m p o n e n t s d e f i n e d i n Fig. 3 5 . Eq. ( 3 ) h i g h l i g h t s h o w t h e p r e d i c t i o n o f t h e s p i n d l e m o m e n t b y BEMT is l a r g e l y a f f e c t e d b y t h e i d e n t i f i c a t i o n o f t h e local c e n t r e o f pressure. Since t h i s p o s i t i o n depends b o t h o n t h e o p e r a d n g c o n d i d o n s a n d h y d r o f o i l c h a r a c t e r i s d c s ( p i t c h i n g m o m e n t a n d l i f t c o e f f i c i e n t s ) , a p r e l i m i n a r y t w o - d i m e n s i o n a l f l o w analysis m u s t be p e r f o r m e d i n case o f blade-shape sections n o t d o c u m e n t e d i n t h e l i t e r a t u r e . D i f f e r e n t l y , p r o p e l l e r m o d e l l i n g b y B E M p r o v i d e s a f u l l y t h r e e -d i m e n s i o n a l f l o w -d e s c r i p t i o n a n -d h y -d r o -d y n a m i c l o a -d i n g r e s u l t s as t h e c o m b i n a d o n o f n o r m a l a n d t a n g e n t i a l stress o n s o l i d surfaces. Blade s p i n d l e m o m e n t m f , is t h e n o b t a i n e d as m f , = ƒ ( - p r X n - i - r r x t) k d S (4) w h e r e Sg represents t h e blade s u r f a c e w h i l e p a n d T i n d i c a t e pressure a n d t a n g e n d a l stress o n SB. S y m b o l r is t h e v e c t o r i d e n d f y i n g t h e p o s i d o n o f a s u r f a c e p o i n t w i t h respect t o t h e o r i g i n o f t h e blade r e f e r e n c e f r a m e , t is t h e u n i t t a n g e n t v e c t o r t o SB a n d a l i g n e d t o local s t r e a m l i n e s , n is t h e u n i t n o r m a l v e c t o r t o Sg w h e r e a s k defines t h e u n i t v e c t o r a l i g n e d t o t h e axis. 3. N u m e r i c a l r e s u l t s I n t h i s s e c d o n BEMT a n d B E M h y d r o d y n a m i c m o d e l s are a p p l i e d t o t h e s t u d y o f SPPperformance i n o p e n w a t e r c o n d i -t i o n s . As s -t a -t e d b e f o r e , -t h e e v a l u a d o n o f -t h e h y d r o d y n a m i c s p i n d l e m o m e n t mf, plays a c r u c i a l r o l e i n t h e d e t e r m i n a d o n o f t h e e q u i l i b r i u m p i t c h s e t t i n g Ce (see Eq. ( 2 ) ) . For t h i s reason, 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 is c a r r i e d - o u t o n a c o n v e n d o n a l c o n t r o l l a b l e - p i t c h p r o p e l l e r (CPP) w h e r e t h e s p i n d l e m o m e n t e s t i m a t i o n is a n issue o f p r i m a r y i m p o r t a n c e f o r t h e s i z i n g o f its p i t c h - c o n t r o l m e c h a n i s m . To t h i s a i m , r e s u l t s o f t h e e x p e r i m e n t a l c h a r a c t e r i s a d o n o f t h e DTRC 4 4 0 2 CPP m o d e l i n B o s w e l l et al. ( 1 9 7 5 ) are c o m p a r e d t o n u m e r i c a l r e s u l t s o b t a i n e d b y t h e p r e s e n t h y d r o d y n a m i c f o r m u l a d o n s . S p e c i f i c a l l y , p r o p e l l e r t h r u s t T, t o r -q u e Q, a n d s p i n d l e m o m e n t m ^ are a n a l y s e d f o r t h r e e b l a d e n o n d i m e n s i o n a l l i n e a r p i t c h s e t t i n g s (P/D). The f o l l o w i n g d e f i n i -t i o n s o f f o r c e a n d m o m e n -t c o e f f i c i e n -t s are u s e d ; T Q prfiD^ (5) w h e r e D, n a n d p d e n o t e p r o p e l l e r d i a m e t e r , r o t a t i o n a l speed a n d w a t e r d e n s i t y , r e s p e c t i v e l y . A t h r e e - d i m e n s i o n a l v i e w o f t h e DTRC 4 4 0 2 p r o p e l l e r is d e p i c t e d i n Fig. 5; m a i n g e o m e t r y p a r a m e t e r s are l i s t e d i n Table 1. A l l results o f c o m p u t a t i o n a l studies b y B E M d e s c r i b e d h e r e a f t e r are o b t a i n e d b y u s i n g p r o b l e m d i s c r e t i s a t i o n settings w h i c h e n s u r e n e g l i g i b l e s e n s i t i v i t y o f r e s u l t s t o f u r t h e r d i s c r e t i s a d o n r e f i n e m e n t s . Results o f g r i d s e n s i t i v i t y studies are n o t d e s c r i b e d here f o r the sake o f conciseness.
Fig. S. Sketch of DTRC-P4402 controllable-pitch propeller.
Table 1
P4402 propeller main geometry parameters.
Blades number, Z 5-CPP
Propeller diameter, D 234.29 mm
Nominal pitch, P/Droy 1.061
Expanded area ratio 0.829
Skew (r/K=0.7) 6.44=
Rake (r/R=0.7) 0.0 mm
Hub/propeller diameter ratio 0.29
S, Leone et al. / Ocean Engineering 64 (2013) 122-134 125
Figs. 6 - 8 s l i o w t h e n u m e r i c a l - e x p e r i m e n t a l c o m p a r i s o n at P/D equal to 0.8, 1.061 and 1.2, respectively, w h e n BEMT-h y d r o d y n a m i c s is used. For tBEMT-hese c o m p u t a t i o n s , sectional aerody-namics properties come f r o m Castagneto a n d M a i o l i ( 1 9 6 8 ) . A t P / D = 0 . 8 , KT and KQ predictions are very g o o d f o r all advance-ratios. However, correlations t e n d t o w o r s e n w h e n blade p i t c h setting P/D increases (see Figs, 7 a n d 8) albeit t h e y r e m a i n acceptable f o r a w i d e range o f o p e r a t i n g conditions. N u m e r i c a l o v e r - p r e d i c t i o n s i n t e r m s o f Kj a n d /Cq arise at advance ratio greater t h a n 0.5 d u e t o the lack o f a realistic h u b m o d e l l i n g i n the BEMT approach w h e r e local f l o w a c c e l e r a t i o n is described i n a s i m p l i f i e d w a y b y t h e G o l d s t e i n -T a c h m i n j i factors (see A p p e n d i x A). -This conclusion is c o n f i r m e d b y an analysis ( n o t s h o w n here f o r conciseness) w h e r e a c o m p a r i s o n b e t w e e n B E M and BEMT predictions o f hubless p r o p e l l e r forces
-0.6 I 1 ^ i > i I
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Advance ratio, J
Fig. 6 . P4402 performance: BEMT vs experiments at P/D=0.8.
Advance ratio, J
Fig. 7. P4402 performance: BEMT vs experiments at P/D=1.061.
0.0 0.2 0.4 0.6 0.8 1.0 1.2 A d v a n c e ratio, J
Fig. 8. P4402 performance: B E M T v s experiments at P/D=1.2.
-0.6 I 1 i ' i i I 0.0 0.2 0.4 0.6 0.8 1.0 1.2
Advance ratio, J
Fig. 9. P4402 performance: BEM vs experiments at P/D=0.8.
shows an excellent a g r e e m e n t at h i g h advance ratios. Conversely, n u m e r i c a l u n d e r - p r e d i c t i o n s f o r Kr a n d KQ at l o w advance r a t i o m a y be p a r t i a l l y e x p l a i n e d b y t h e r o u g h d e s c r i p t i o n o f the w a k e - f i e l d b e h i n d t h e p r o p e l l e r d i s k a n d b l a d e s e c d o n a l d r a g . F i g u r e s a b o v e also s h o w s a t i s f a c t o r y r e s u l t s b e t w e e n B E M T p r e d i c t i o n s a n d e x p e r i m e n t s , as l o n g as Ks c o e f f i c i e n t is c o n c e r n e d ; s p i n d l e m o m e n t c o m p u t a t i o n s s t e m h e r e f r o m a s u i t a b l e t u n i n g process o n t h e f l o w - c u r v a t u r e c o r r e c d o n fi (see Eq. ( 1 5 ) , A p p e n d i x A ) , t a i l o r e d t o f i t d n g e x p e r i m e n t s a t n o m i n a l p i t c h s e t d n g . A v a l u e o f H = 0 . 0 7 5 has b e e n used f o r a l l P/D setdngs. N u m e r i c a l e x p e r i -m e n t a l a g r e e -m e n t i n t e r -m s o f Kj a n d KQ i -m p r o v e s w h e n B E M hydrodynamics is used t o p r e d i c t o p e n - w a t e r propeller performance (see Figs. 9 - 1 1 ) . As s h o w n , the i n c l u s i o n o f the h u b g e o m e t r y and t h e m o d e l l i n g o f t r a i l i n g - w a k e v o r r i c i t y effects yield a better agreement
126 S. Leone et al. / Ocean Engineering 64 (2013) 122-134
a
- 0 . 4
0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1.0 1.2 A d v a n c e r a t i o , J
Fig. 10. P4402 performance: BEM vs experiments at P/D=1.0S1.
0.0 0.2 0.4 0.6 0.8 1.0 1.2 Advance ratio, J
Fig. n . P4402 performance: BEM vs experiments at P/D=1.2.
w i t l i measured data, b o t h at h i g h and l o w advance ratio. Nonetheless, a n enhancement i n the m o d e l l i n g o f viscous f l o w (see A p p e n d i x A ) is expected t o yield better results, especially i n t e r m s o f torque coefficient p r e d i c t i o n at l o w ] values. Above outcomes demonstrate t h a t BEM h y d r o d y n a m i c s predicts CPP-performance slightly bed:er t h a n BEMT approach. Dealing w i t h the spindle m o m e n t . Figs. 9 - 1 1 s h o w h o w BEM-computations are as fair as those c o m p u t e d by t h e B E M T m o d e l l i n g , w i t h o u t i n t r o d u c i n g a n y s o r t o f e m p i r i c a l c o r r e c t i o n . Such a r e s u l t d e m o n s t r a t e s t h e i m p o r t a n t r o l e p l a y e d b y t h r e e - d i m e n s i o n a l e f f e c t s i n t h e e v a l u a t i o n o f Ks. Nevertheless, t h e s i m p l i f i e d BEMT m o d e l l i n g is s t i l l a s u i t a b l e s o l v e r once t h e e m p i r i c a l p a r a m e t e r n is p r o p e r l y t u n e d .
Next, s e l f - p i t c h i n g propellers are considered. I n order to compare n u m e r i c a l r e s u l t s w i t h a v a i l a b l e e x p e r i m e n t a l d a t a , p r o p e l l e r p e r f o r m a n c e is f i r s t s t u d i e d b y a s s u m i n g t h a t blades are b l o c k e d w i t h respect t o s p i n d l e axis r o t a t i o n s a n d o p e r a t e at a g i v e n p i t c h s e t t i n g . To t h i s a i m m o d e l p r o p e l l e r P2, e x p e r i m e n t a l l y s t u d i e d i n M i l e s et a l . ( 1 9 9 2 ) , is i n v e s t i g a t e d ; a s k e t c h o f t h i s p r o p e l l e r is s h o w n i n Fig. 12. For c o m p l e t e n e s s , s o m e m a i n g e o m e t r y p a r a -m e t e r s are l i s t e d i n Table 2. P e r f o r -m a n c e at f i x e d p i t c h s e t t i n g c o r r e s p o n d i n g t o 0.8 a n d 1.2, respectively, is i n v e s t i g a t e d . For b o t h cases, t w o p r o p e l l e r m o d e l s w i t h t h e s a m e g e o m e t r i c a l characteristics, e x c e p t f o r t h e b l a d e o f f s e t e^, have b e e n t e s t e d i n M i l e s et a l . ( 1 9 9 2 ) . A d e f i n i t i o n o f blade o f f s e t is g i v e n i n Fig. 2 . F r o m a p h y s i c a l s t a n d p o i n t , a d i f f e r e n t p r o p e l l e r o f f s e t i m p l i e s o n l y a s h i f t o f t h e p r o p e l l e r b l a d e a l o n g t h e s h a f t axis a n d h e n c e i t s h o u l d a f f e c t t h e s p i n d l e m o m e n t a n d , s l i g h t l y , t h e t o r q u e . Nevertheless, t h i s is n o t c o n f i r m e d b y e x p e r i m e n t a l results s h o w n i n M i l e s e t a l . ( 1 9 9 2 ) w h e r e t h e same p r o p e l l e r , w i t h 6^ = 0 . 1 8 0 a n d e/i = 0.20D, r e s p e c t i v e l y , e x h i b i t s r e l e v a n t discrepancies i n t e r m s o f m e a s u r e d Kj a n d KQ c o e f f i c i e n t s . The a u t h o r s i n M i l e s et a l . ( 1 9 9 2 ) p o s t u l a t e t h a t such a b e h a v i o u r i n e x p e r i m e n t s is d u e t o a n u n c e r t a i n t y i n t h e p i t c h / d i a m e t e r r a t i o s e t t i n g o f a b o u t 0.05 a r o u n d t h e n o m i n a l P/D at 70% o f blade r a d i u s . Thus, i n t h e f o l l o w i n g i t is a s s u m e d t h a t e x p e r i m e n t a l r e s u l t s are a f f e c t e d b y a n u n c e r t a i n t y o f 0.05 i n t e r m s o f n o m i n a l P/D. Figs. 13 a n d 14 d e p i c t t h r u s t a n d t o r q u e c o e f f i c i e n t s at P / D = 0 . 8 . P e r f o r m a n c e p r e d i c t e d b y B E M T a n d B E M e x h i b i t s discrepancies w i t h r e s p e c t t o e x p e r i m e n t s . S p e c i f i c a l l y , B E M c o m p u t a t i o n s o v e r e s t i m a t e Kj a n d KQ o v e r t h e w h o l e range o f advance r a r i o values, a l b e i t t h e e x p e r i m e n t a l t r e n d is w e l l p r e d i c t e d ; c o n v e r s e l y , BEMT p r e d i c -rions i n t e r s e c t t h e e x p e r i m e n t a l curves so t h a t c o m p u t a t i o n s are s l i g h t l y closer t o e x p e r i m e n t s , e v e n t h o u g h t h e slope o f t h e c u r v e s is n o t w e l l e s r i m a t e d , especially at l o w ] values. For t h i s a n a l y s i s , s e c t i o n a l a e r o d y n a m i c s p r o p e r t i e s i n t h e BEMT m o d e l are t a k e n f r o m A b b o t t a n d v o n D o e n h o f f ( 1 9 5 8 ) . A c o m m o n f e a t u r e o f n u m e r i c a l c o m p u t a r i o n s is t h e t h r u s t o v e r e s t i m a r i o n at h i g h values o f t h e a d v a n c e c o e f f i c i e n t . Such n u m e r i c a l b e h a v i o u r
Fig. 12. Sketch of P2 self-pitching propeller.
Table 2
Self-pitching propeller P2 geometry parameters.
Blades number, Z 3-SPP
Propeller diameter, D 300.0 m m
Design nominal pitch, P/D 1.0 (const.)
Blade area ratio 0.39
Rake 8°
Hub/propeller diameter ratio 0.233
Offset 54.0 mm, 60.0 mm
S. Leone et ai / Ocean Engineering 64 (2013) 122-134 127 0.35 0.30 0.25 0.20 ^ 0.15 0.10 0.05 0.00 -0.05 1 1 I I I I EXP • B E M T *
-1
i BEM ..\ i i 1
-•
1 1
i i i •
xsi
: : : ! ! ! • N S . i i i i i i 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Advance ratio, JF i g . 1 3 . F2 thrust coefficient: numerical vs experiments at P/D=0.8.
0.35 0.30 0.25 0.20 ^ 0.15 0.10 0.05 h 0.00 -0.05
!
l l l l l BEMT - present formulationBEMT-Miles etal (1992) • j - 1
1 i j i
1--
\1 p ^ 5 s j ^ i
-i i i i i i 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Advance ratio, JF i g . 1 5 . P2 thrust coefficient: present BEMT formularion compared to numerical data in Miles et al. (1992) at P/D=0.8.
0.35 0.35
a
-0.05
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Advance ratio, J
F i g . 1 4 . P2 torque coefficient: numerical vs experiments at P/D=0.8.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.1
Advance ratio, J
F i g . 1 6 . P2 torque coefficient: present BEMT formulation compared to numerical data in Miles et al. (1992) at P/D=0.8.
m a y be e x p l a i n e d b y t h e l i m i t s , o f b o t h solvers, i n d e s c r i b i n g h u b e f f e c t s o n p r o p e l l e r loads; i n f a c t , t h e shape o f t h e SPP h u b is d i f f e r e n t f r o m t h e c y l i n d r i c a l g e o m e t r y used i n B E M s i m u l a t i o n s o r m o d e l l e d b y t h e G o l d s t e i n - T a c h m i n j i f a c t o r s i n t h e B E M T solver. A k i n t o t h e analysis p e r f o r m e d o n CPP p r o p e l l e r - t y p e s , a n e n h a n c e d d e s c r i p t i o n o f viscous p h e n o m e n a , r o u g h l y m o d e l l e d i n b o t h BEMT a n d BEM approaches, s h o u l d i m p r o v e n u m e r i c a l p r e d i c t i o n s at l o w j values. For t h e sake o f completeness. Figs. 15 and 16 d e p i c t the n u m e r i c a l c o m p a r i s o n b e t w e e n p e r f o r m a n c e p r e d i c t e d b y t h e p r e s e n t BEMT m o d e l and t h a t i n M i l e s e t a l . ( 1 9 9 2 ) ; as s h o w n , t h e a g r e e m e n t is excellent.
N e x t , P/D = 1.2 p i t c h s e t t i n g is c o n s i d e r e d i n Figs. 17 a n d 1 8 . As expected, BEMT solver s h o w s a l i m i t e d c a p a b i l i t y i n c a p t u r i n g
p r o p e l l e r p e r f o r m a n c e w h e n the sectional angle o f a t t a c k increases, w h e r e a s B E M c o m p u t a t i o n s h i g h l i g h t a v e r y g o o d a g r e e m e n t e s p e c i a l l y . i n t e r m s o f Kj. A n u n d e r p r e d i c t i o n o f t h e t o r q u e c o e f f i c i e n t is o b s e r v e d f o r } values less t h a n 0.65 d u e t o t h e d e t r i m e n t a l e f f e c t o f t h e a p p r o x i m a t e m o d e l l i n g o f v i s c o s i t y d r i v e n p h e n o m e n a . F i n a l l y , e q u i l i b r i u m c o n d i t i o n s ( i n t e r m s o f p i t c h s e t t i n g ) f o r t h e p r o p e l l e r - t y p e h a v i n g a n o f f s e t v a l u e o f 6^ = 0 . 1 8 0 are a n a l y z e d i n Fig. 19, w h e r e p i t c h - d i a m e t e r r a t i o is c o n s i d e r e d at 70% o f b l a d e r a d i u s . E x p e r i m e n t a l results are d e p i c t e d w i t h t h e u n c e r t a i n t y o f 0.05 a r o u n d t h e m e a s u r e d n o m i n a l P/D. C o m p u t a -t i o n s s h o w -t h a -t BEMT p r e d i c -t i o n s y i e l d r e a s o n a b l e a g r e e m e n -t
128 S. Leone et ai / Ocean Engineering 64 (2013) 122-134
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 Advance ratio, J
F i g . 1 7 . P2 thrust coefficient: numerical vs experiments at P/D=1.2.
' 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 Advance ratio, J
F i g . 1 8 . P2 torque coefficient: numerical vs experiments at P/D=1.2.
w i t h m e a s u r e m e n t s f o r / r a n g i n g f r o m 0.4 t o 0.6, w h e r e a s o v e r -e s t i m a t i o n a t } l-ess t h a n 0.5 a n d u n d -e r -e s t i m a t i o n a t J g r -e a t -e r t h a n 0.8 are p r e s e n t . T h e m a j o r drawbacl< i n BEIVIT p r e d i c t i o n s is i n t e r m s o f slope o f t h e P/D c u r v e t h a t is q u i t e f a r f r o m t h e m e a s u r e d one. N o t e t h a t B E M T c o m p u t a t i o n s are o b t a i n e d t h r o u g h a t a i l o r i n g process o n t h e e m p i r i c a l p a r a m e t e r / i (see Eq. ( 1 5 ) , A p p e n d i x A ) , t o m a t c h n u m e r i c a l results a n d e x p e r i m e n t s : i n t h i s case t h e p r o c e d u r e y i e l d s ^ = 0.5. Conversely, B E M c o m p u t a t i o n s h i g h l i g h t a v e r y g o o d agree-m e n t w i t h agree-m e a s u r e agree-m e n t s f o r J r a n g i n g f r o agree-m 0.2 t o 0.9; a l t h o u g h a t h i g h e r ] v a l u e s a n o v e r e s t i m a t i o n o f e q u i l i b r i u m P/D is revealed, t h e g l o b a l t r e n d is g e n e r a l l y w e l l d e s c r i b e d . T h r u s t a n d t o r q u e c o r r e s p o n d i n g t o t h e b l a d e e q u i l i b r i u m p i t c h angle are s h o w n i n Figs. 2 0 a n d 2 1 . I n d e t a i l . Fig. 2 0 s h o w s B E M T a n d B E M p r e d i c t i o n s i n t e r m s o f Kj i n s e l f - p i t c h i n g c o n d i t i o n s : as
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Advance ratio, J
F i g . 1 9 . P2 in self-pitching condition: equilibnum pitch setting.
' 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Advance ratio, J
F i g . 20. P2 in self-pitching condition: thrust coefficient.
e x p e c t e d , BEMT p r o v i d e s s a t i s f a c t o r y p r e d i c t i o n s o n l y i n t h e r a n g e b e t w e e n 0.4 a n d 0.6 w h e r e t h e e q u i l i b r i u m p i t c h s e t t i n g is w e l l c a p t u r e d . O u t o f t h i s range, B E M T is u n a b l e t o d e s c r i b e t h e h y d r o m e c h a n i c b e h a v i o u r o f t h e SPP. D i f f e r e n t l y , B E M c a l c u l a -t i o n s p r o v i d e p e r f o r m a n c e p r e d i c -t i o n s -t h a -t are i n v e r y g o o d a g r e e m e n t w i t h e x p e r i m e n t s , o v e r t h e w h o l e range o f o p e r a t i n g c o n d i t i o n s . T o r q u e analysis is s h o w n i n Fig. 2 1 . E x p e r i m e n t a l r e s u l t s w i t h a n d w i t h o u t t r a n s i t i o n s t r i p s a t t a c h e d t o t h e blades are p r o v i d e d t o h i g h l i g h t scale e f f e c t s o n KQ. M o d e l tests are p e r f o r m e d at d i f f e r e n t p r o p e l l e r r o t a t i o n a l speed a n d i n f l o w v e l o c i t y t o have a d v a n c e r a t i o r a n g i n g b e t w e e n 7 = 0 . 2 a n d J=2.0. C o r r e s p o n d i n g l y , Reynolds n u m b e r at 70% o f b l a d e s p a n
v a r i e d b e t w e e n 1.48 x 10'' at l o w J a n d 0.6 x l O ' ' a t h i g h ] . Same t e s t i n g c o n d i t i o n s are used f o r b o t h u n s t r i p p e d a n d s t r i p p e d blade m o d e l s . S i m i l a r l y t o KT p r e d i c t i o n s , B E M T c a l c u l a t i o n s p r o v i d e s a t i s f a c t o r y results w h e r e t h e e q u i l i b r i u m p i t c h s e t t i n g is w e l l
S. Leone et ai / Ocean Engineering 64 (2013) 122-134 129 0.8 0,7 0.0 EXP Str O EXP no str • B E t ^ T » -BEM J i i L 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Advance ratio, J
Fig. 21. P2 in self-pitcliing condition: torque coefficient.
Table 3
Self-pitching propeller P2fl geometry parameters.
Blades number, Z 3-SPP
Propeller diameter, D 250,0 mm Design nominal pitch, P/D 0.682 (const.)
Blade area ratio 0.41
Rake 8°
Hub/propeller diameter ratio 0.206
Offset 41.75 mm
Blade sections airfoil Refiex
c a p t u r e d ( 0 . 5 < ; < 0 . 7 ) . N e v e r t i i e l e s s , BEMT d a t a are c h a r a c -t e r i s e d b y a d i f f e r e n -t slope w i -t h r e s p e c -t -t o e x p e r i m e n -t s . These results are o b t a i n e d b y a s s u m i n g t w o d i m e n s i o n a l a i r f o i l c h a r -acterisrics o f secrion at 70% o f b l a d e r a d i u s as r e p r e s e n t a t i v e o f a l l secdons s p a n w i s e . Such a w i d e l y used a p p r o a c h i n t h e f r a m e w o r k o f s e c d o n a l a e r o d y n a m i c theories has a d i f f e r e n t i m p a c t o n b l a d e t h r u s t a n d t o r q u e p r e d i c d o n s ; i n f a c t , s m a l l v a r i a d o n s o f s e c d o n a l d r a g m a y d e t e r m i n e large d i f f e r e n c e s i n t h e p r e d i c t e d t o r q u e w h e r e a s a l i m i t e d e f f e c t o n b l a d e t h r u s t is n o d c e d . Conversely, BEM c o m p u t a t i o n s are i n g o o d a g r e e m e n t w i t h r e s p e c t t o n o strips e x p e r i m e n t a l d a t a f o r J < 0.8 w h i l s t e x h i b i t r e l e v a n t d i f f e r -ences w i t h respect t o t h e e x p e r i m e n t a l strip d a t a f o r / < 1.0. T h e v i s c o u s - f l o w correction present i n the BEM m o d e l (see A p p e n d i x A ) is so a p p r o x i m a t e d t h a t i t m a y b e t o o a m b i d o u s t r y i n g t o e s t a b l i s h a r e l a d o n s h i p b e t w e e n n u m e r i c a l r e s u l t s a n d e f f e c t s o n b l a d e loads i n d u c e d b y s d m u l a d n g t u r b u l e n t flow at l e a d i n g edge o f b l a d e m o d e l i n e x p e r i m e n t s .
N e x t , B E M and BEMT m o d e l s are a p p l i e d t o t h e analysis o f a n e w s e l f - p i t c h i n g p r o p e l l e r ( h e r e a f t e r r e f e r r e d t o as t h e P2^) h a v i n g t h e same p l a n f o r m as p r o p e l l e r P2, b u t d i f f e r e n t s e c d o n a l a i r f o i l s . S o m e g e o m e t r i c a l p a r a m e t e r s are d e s c r i b e d i n Table 3. I n Figs. 2 2 - 2 4 BEM a n d BEMT c o m p u t a t i o n s are c o m p a r e d i n t e r m s o f SPP-performance a n d s p i n d l e m o m e n t at i m p o s e d p i t c h s e t t i n g c o r r e s p o n d i n g t o p i t c h / d i a m e t e r r a d o o f 0.682 ( n o m i n a l p i t c h ) . P r o p e l l e r P2^ blades have been d e s i g n e d b y p r o p e l l e r m a n u f a c -t u r e r (see M i l e s e-t al., 1 9 9 2 ) a n d n u m e r i c a l l y i n v e s -t i g a -t e d w i -t h i n t h e f r a m e w o r k o f a n o n g o i n g research p r o j e c t . S e c t i o n a l a e r o -d y n a m i c c h a r a c t e r i s t i c s u s e -d i n B E M T m o -d e l l i n g c o m e f r o m a d e v o t e d a n a l y s i s p e r f o r m e d b y t h e m a n u f a c t u r i n g c o m p a n y
p a r t n e r i n t h e p r o j e c t . N u m e r i c a l results d e m o n s t r a t e a g o o d a g r e e m e n t b e t w e e n BEM a n d BEMT p r e d i c d o n s i n t e r m s o f Kjand KQ c o e f f i c i e n t s . C o n s i d e r i n g t h e s p i n d l e m o m e n t Ks, Fig. 2 4 h i g h -l i g h t s a n e x c e -l -l e n t a g r e e m e n t , e x c e p t f o r s m a -l -l d i f f e r e n c e s at -l o w ] values. N o t e t h a t , d u e t o t h e lack o f e x t e n s i v e e x p e r i m e n t a l k n o w l e d g e o n c o n s i d e r e d SPP m o d e l s , t h e p a r a m e t e r fi i n t h e BEMT m o d e l has been t u n e d t o g e t t h e best a c c o r d a n c e w i t h B E M r e s u l t s a t t h e n o m i n a l p i t c h c o n d i d o n ; a fi v a l u e o f 0.33 has b e e n f o u n d a n d k e p t c o n s t a n t f o r t h e o t h e r p i t c h s e t d n g s discussed i n t h e f o l l o w i n g . A n e x p e c t e d i m p r o v e m e n t o f BEMT t u n i n g process m a y be a c h i e v e d t h r o u g h d e d i c a t e d e x p e r i m e n t a l tests o n SPP m o d e l s . For a P/D v a l u e o f 0.528, b l a d e s e c d o n s o p e r a t e at a n 0.30 0.25 0.20 0.15 0.10 ) ^ 0.05 0.00 -0.05 -0.10 -0.15 -0.20 I I I ! l l l l l BEMT • -BEM
-e-i -e-i -e-i 1 -e-i -e-i -e-i -e-i -e-i 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Advance ratio, J
Fig. 22. P2^ thrust coefficient: numerical BEM vs BEMT predictions at nominal P/D=0.682.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Advance ratio, J
Fig. 23. P2A torque coefficient: numerical BEM vs BEMT predictions at nominal P/D = 0.682.
130 S. Leone et al. / Ocean Engineering 64 (2013) 122-134 o CNJ -0.20 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Advance ratio, J
F i g . 2 4 . P2/1 spindle moment coefficient: numerical BEM vs BEMT predictions at nominal P/D=0.682.
a
-0.20
0.0 0.1 0.2 0.3 0.4 0,5 0.6 0.7 0.8 0,9 1.0 Advance ratio, J
F i g . 2 6 . P2A torque coefficient: numerical BEM vs BEMT predictions at P/D=0.528.
0.30 0.25 0.20 0.15 0.10 0.05 0.00 -0.05 -0.10 -0.15 -0.20 B E M T « -BEM
-e-J I I L 1 I I L 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Advance ratio, J
F i g . 2 5 . P2A thrust coefficient: numerical BEM vs BEMT predictions at P/D=0.528.
angle o f a t t a c k l o w e r t h a n t h e p r e v i o u s c o n d i t i o n : as a conse-quence, v i s c o u s - f l o w a n d t h r e e - d i m e n s i o n a l f l o w e f f e c t s are s u p p o s e d t o have less i m p a c t o n t h r u s t a n d t o r q u e . H e n c e a b e t t e r a g r e e m e n t b e t w e e n B E M T a n d B E M r e s u l t s is e x p e c t e d , see Figs. 25 a n d 2 6 . Nevertheless, Fig. 27 s h o w s a w o r s e n i n g i n t h e n u m e r i c a l a g r e e m e n t i n t e r m s o f s p i n d l e m o m e n t values. Such a b e h a v i o u r is due t o t h e f a c t t h a t t h e t u n i n g process o f t h e e m p i r i c a l p a r a m e t e r /.i ( t o m a t c h B E M r e s u l t s ) has b e e n based o n n o m i n a l p i t c h s e t t i n g . A t p i t c h values l a r g e r t h a n t h e n o m i n a l one, P/D = 0.759, h i g h e r blade l o a d i n g c o n d i t i o n s m a k e t h r e e -d i m e n s i o n a l f l o w e f f e c t s m o r e r e l e v a n t ; t h u s , l a r g e r -discrepancies i n t h e p r e d i c t i o n o f KT, KQ a n d Ks are e x p e c t e d , as c o n f i r m e d b y r e s u l t s s h o w n i n Figs. 2 8 - 3 0 . 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Advance ratio, J
F i g . 2 7 . P2^ spindle moment coefficient: numerical BEM vs BEMT predictions at P/D = 0.528.'
F i n a l l y , n u m e r i c a l s i m u l a t i o n s w h e r e p r o p e l l e r blades are l e f t f r e e t o a d j u s t p i t c h a c c o r d i n g t o b l a d e l o a d i n g at d i f f e r e n t values o f t h e advance c o e f f i c i e n t are c o n s i d e r e d . O b t a i n e d e q u i l i b r i u m p i t c h - s e t t i n g values ( a n d c o r r e s p o n d i n g Kjand KQ) are d e p i c t e d i n Figs. 3 1 - 3 3 , r e s p e c t i v e l y . C o m p u t a t i o n s s h o w t h a t B E M T a n d B E M m o d e l s p r o v i d e p e r f o r m a n c e a n d p i t c h s e t t i n g c u r v e s w i t h d i f -f e r e n t slopes. T h i s r e s u l t c o n -f i r m s a t r e n d a l r e a d y o b s e r v e d -f o r t h e P2 p r o p e l l e r i n Figs. 1 9 2 1 , w h e r e t h e c o m p a r i s o n w i t h e x p e r i -m e n t a l data is d e p i c t e d . Since B E M h y d r o d y n a -m i c s y i e l d s loads p r e d i c t i o n s a c c o u n t i n g d i r e c t l y f o r t h e presence o f t h r e e d i m e n -s i o n a l e f f e c t -s , i t i-s rea-sonable t o a -s -s u m e t h a t t h e m o -s t r e a l i -s t i c r e s u l t s are t h o s e associated t o B E M f o r m u l a t i o n .
S. Leone et ai / Ocean Engmeering 64 (2013) 122-134 131
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Advance ratio, J
F i g . 2 8 . tiirust coefficient: numerical BEM vs BEMT predictions atP/D=0.759.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Advance ratio, J
F i g . 3 0 . P2^ spindle moment coefficient: numerical BEM vs BEMT predictions at P/D=0.759. 0.30 0.25 0.20 0.15 0.10 ^ 0.05 o 0.00 -0.05 -0.10 -0.15 -0.20 1 1 1 BEMT • -BEM
-e-J L J I i L 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Advance ratio, J
F i g . 2 9 . P2/, torque coefficient: numerical BEM vs BEMT predictions atP/D=0.759.
4. C o n c l u s i o n s I n t h i s paper, c a p a b i l i t i e s a n d d r a w b a c k s o f t w o c o m p u t a t i o n a l m e t h o d o l o g i e s a i m e d at t h e analysis o f s e l f - p i t c h i n g p r o p e l l e r s i n u n i f o r m f l o w are i n v e s t i g a t e d b y c o m p a r i s o n s w i t h n u m e r i c a l a n d e x p e r i m e n t a l a v a i l a b l e d a t a . I n t h e a t t e m p t t o p r e d i c t self-p i t c h i n g self-p r o self-p e l l e r s self-p e r f o r m a n c e , t h e Blade E l e m e n t M o m e n t u m T h e o r y a n d B o u n d a r y E l e m e n t M e t h o d are used as h y d r o d y n a m i c solvers t o y i e l d the f o r c i n g t e r m s o f b l a d e m o t i o n e q u a t i o n .
BEMT approach stems f r o m a n o n l i n e a r c o m b i n a t i o n o f Blade Element a n d M o m e n t u m T h e o r y ; i t is based o n sectional steady aerodynamics, enhanced t h r o u g h a s i m p l i f i e d v o r t e x line approach to account f o r t h r e e - d i m e n s i o n a l f l o w effects. O n the o t h e r h a n d ,
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Advance ratio, J
F i g . 3 1 . P2^ in self-pitching condidon: equilibrium pitch setring. BEM vs BEMT numerical simulations. B E M m o d e l is a f u l l y t h r e e - d i m e n s i o n a l , unsteady h y d r o d y n a m i c solver based o n t h e a s s u m p t i o n o f p o t e n t i a l f l o w s . G u i d e l i n e s d e r i v e d f r o m t h e n u m e r i c a l s t u d y h e r e i n p e r f o r m e d c o n f i r m t h e effectiveness o f t h e B E M T b a s e d a p p r o a c h . S p e c i f i c a l l y , t h r u s t a n d t o r q u e c o e f f i c i e n t s p r e d i c t e d a t f i x e d p i t c h -s e t t i n g o f p r o p e l l e r blade-s are r e a -s o n a b l e i n ca-se o f m o d e r a t e l y l o a d e d p r o p e l l e r s w o r k i n g close t o d e s i g n c o n d i t i o n . Conversely, at advance r a t i o causing blade o v e r l o a d i n g o r u n d e r l o a d i n g , t h r e e -d i m e n s i o n a l f l o w e f f e c t s b e c o m e -d o m i n a n t a n -d B E M T p r e -d i c t i o n s can p r o v e t o b e u n r e l i a b l e . C o n c e r n i n g w i t h s p i n d l e m o m e n t p r e d i c t i o n s , BEMT requires a d e v o t e d t a i l o r i n g process a i m e d at d e t e c t i n g f l o w - c u r v a t u r e c o r r e c t i o n e f f e c t s . T h i s f a c t m a k e s BEMT
132 S. Leone et al / Ocean Engmeering 64 (2013) 122-134 0.5 0.4 0.3 0.2 0.1 0.0 n [— i B E M T -BEM • 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Advance ratio, J
Fig. 3 2 . P2A in self-pitcliing condition: thrust coefficient.
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Advance ratio, J
Fig. 3 3 . P2A in self-pitching condition: torque coefficient.
f o r m u l a t i o n q u e s t i o n a b l e f o r t h e n u m e r i c a l analysis o f s e l f -p i t c h i n g -p r o -p e l l e r s -p e r f o r m a n c e . S -p e c i f i c a l l y , i f m o d e l - t u n i n g d a t a are n o t a v a i l a b l e n u m e r i c a l p r e d i c t i o n s m a y c o n s i s t e n t l y d e v i a t e f r o m r e a l i t y . I n c o n t r a s t t o t h i s , B E M h y d r o d y n a m i c s o v e r c o m e s m o s t o f t h e a b o v e m e n t i o n e d l i m i t a t i o n s t h a n k s t o a f u l l d e s c r i p t i o n o f t h r e e -d i m e n s i o n a l f l o w e f f e c t s a r o u n -d p r o p e l l e r bla-des. N u m e r i c a l results d e m o n s t r a t e t h a t f a i r a g r e e m e n t w i t h e x p e r i m e n t a l d a t a is f o u n d f o r b o t h f i x e d a n d s e l f p i t c h i n g blades w i t h o u t i n t r o d u -c i n g e m p i r i -c a l -c o r r e -c t i o n s t h a t r e q u i r e a d - h o -c t u n i n g a n d l i m i t t h e g e n e r a l v a l i d i t y o f results. N u m e r i c a l s t u d i e s d i s c u s s e d i n t h e p r e s e n t w o r k also i d e n t i f y v i s c o u s - f l o w m o d e l l i n g a n d t r a i l i n g v o r t i c i t y d y n a m i c s as areas o f p o s s i b l e i m p r o v e m e n t o f s t a n d a r d B E M m o d e l s f o r t h e analysis o f s e l f - p i t c h i n g p r o p e l l e r s .
The above considerations suggest the use o f BEM h y d r o d y n a m i c s i n v i e w o f applications t o p r e l i m i n a r y design a n d o p t i m i s a t i o n studies. Propeller p e r f o r m a n c e studies b y BEM i n v o l v e a greater e f f o r t as c o m p a r e d to BEMT r e q u i r e m e n t s b u t c o m p u t a t i o n a l costs are s t i l l l i m i t e d to make possible recursive calculations i n the f r a m e -w o r k o f s i m u l a t i o n based design procedures.
F i n a l l y , a great deal o f i n t e r e s t is o n t h e b e h a v i o u r o f SPPs i n t h e presence o f s p a t i a l l y a n d / o r t e m p o r a l l y v a r y i n g f l o w s a r i s i n g d u r i n g o f f - d e s i g n c o n d i t i o n s , as m a n e u v e r s o r r o u g h sea. The i n v e s t i g a t i o n o f s u c h o p e r a t i n g c o n d i t i o n s is c h a l l e n g i n g since i t r e q u i r e s t h e c o m b i n e d analysis o f b o t h blades d y n a m i c s a n d h y d r o d y n a m i c s , i n a h y d r o e l a s t i c f a s h i o n . I n t h i s c o n t e x t , a c r u c i a l p o i n t is t h e e v a l u a t i o n o f blades d y n a m i c e q u i l i b r i u m c o n d i t i o n t o p e r f o r m a f u r t h e r s t a b i l i t y analysis. I n a u t h o r s ' o p i n i o n such a n a c t i v i t y , b e y o n d t h e scope o f t h i s paper, r e p r e s e n t s a n u r g e n t n e e d f o r h y d r o d y n a m i c i s t s t o a d e q u a t e l y i n v e s t i g a t e advantages a n d d i s a d v a n t a g e s o f SPPs against o t h e r s c r e w p r o p e l l e r systems i n o f f - d e s i g n c o n d i t i o n s . A c k n o w l e d g m e n t s Part o f t h e w o r k d e s c r i b e d i n t h i s p a p e r w a s p e r f o r m e d i n t h e f r a m e w o r k o f t h e EUFP7 Research P r o j e c t H y M A R , ' H i g h E f f i -c i e n -c y H y b r i d D r i v e T r a i n s f o r S m a l l a n d M e d i u m Sized M a r i n e C r a f t ' , G r a n t no. 2 6 2 5 5 2 . A p p e n d i x A T h i s s e c t i o n h i g h l i g h t s t h e m a i n aspects o f B E M T a n d B E M m o d e l l i n g p r o p o s e d i n t h i s w o r k . M o r e d e t a i l s m a y be f o u n d i n l i t e r a t u r e papers c i t e d h e r e a f t e r . I n t h e f o l l o w i n g a g i v e n p r o p e l l e r w o r k i n g at a s p e c i f i e d a d v a n c e r a t i o J = V/^/nD is c o n s i d e r e d , w h e r e VA r e p r e s e n t s t h e a d v a n c e v e l o c i t y , D is t h e p r o p e l l e r d i a m e t e r a n d n = Q/2n is t h e r o t a t i o n a l speed.
A.l. Blade Element Momentum Theory
T h i s a p p r o a c h is e s s e n t i a l l y a s t r i p t h e o r y m e t h o d , i n h e r e n t l y steady, t h a t c o m b i n e s basic p r i n c i p l e s f r o m Blade E l e m e n t T h e o r y (BET) a n d M o m e n t u m T h e o r y ( M T ) , i n c l u d i n g s o m e aspects o f v o r t e x t h e o r y . I t a l l o w s a q u i t e r e l i a b l e p r e d i c t i o n o f t h e l o a d d i s t r i b u t i o n a l o n g t h e blade s p a n f o r m o d e r a t e l y l o a d e d p r o p e l -lers w o r k i n g at, or near, t h e i r d e s i g n c o n d i t i o n (see C a r l t o n , 1 9 9 4 ) . T h e n u m e r i c a l a l g o r i t h m a p p l i e d t h r o u g h o u t t h e p a p e r f o l l o w s t h e a p p r o a c h p r e s e n t e d i n M i l e s et a l . ( 1 9 9 2 ) . I n d e t a i l , f r o m the M T t h e i n c r e m e n t a l t h r u s t a n d e f f i c i e n c y o f a r o t o r a n n u l u s o f the p r o p e l l e r d i s k recast (see T o d d a n d C o m s t o c k , 1 9 6 7 ) dT , =4nprViakd1+akp) 11 = Q^r^ a' (6) w h e r e p is the fluid d e n s i t y , r is t h e l o c a l d i s t a n c e o f t h e a n n u l u s f r o m t h e r o t a t i o n a l axis w h e r e a s kfl a n d kc r e p r e s e n t t h e G o l d -s t e i n - T a c h m i n j i c o r r e c t i o n f a c t o r -s (-see C a r i t o n , 1 9 9 4 ) a t t h e p r o p e l l e r d i s k a n d i n t h e u l t i m a t e w a k e , r e s p e c t i v e l y . I n a d d i t i o n , o = Vi/VA is t h e a x i a l i n f l o w f a c t o r , y, i n d i c a t e s t h e i n d u c e d -v e l o c i t y a t t h e p r o p e l l e r d i s k a n d a' is t h e r o t a t i o n a l - i n f l o w f a c t o r (see Fig. 3 4 ) . A k i n t o t h e M T a p p r o a c h , t h r u s t a n d e f f i c i e n c y o f a Blade E l e m e n t o f l e n g t h dr i n s p a n w i s e d i r e c t i o n m a y be e x p r e s s e d t h r o u g h t h e Blade E l e m e n t T h e o r y as (see T o d d and C o m s t o c k , 1 9 6 7 )
S. Leone et aL / Ocean Engineering 64 (2013) 122-134 133
F i g . 3 4 . Blade section velocity diagram.
dT dr n = pcTi/lci+afd . .
COS(0+)')
< cos y 1 - 0 ' tgcj) l + a t g ( ( ^ + y ) (7) w h e r e Z is t h e p r o p e l l e r blades n u m b e r , c is t h e l o c a l c h o r d o f b l a d e s e c t i o n a t r a d i u s r, cl is t h e local l i f t - c o e f f i c i e n t w h i l s t tgy = cd/cl, b e i n g cd t h e s e c t i o n a l d r a g c o e f f i c i e n t ; f o r each b l a d e s e c t i o n , s u c h c o e f f i c i e n t s are a s s u m e d t o be k n o w n f r o m t a b u l a r data. F u r t h e r m o r e , f r o m t h e b l a d e v e l o c i t y d i a g r a m s h o w n i n Fig. 3 4 , 0 a n d i n d i c a t e t h e g e o m e t r i c a l p i t c h a n d l o c a l i n f l o w angles r e s p e c t i v e l y , VR = V^O + a ) / s i n (f> is t h e v e l o c i t y o f t h e fiuid r e l a t i v e t o t h e b l a d e section, a = ( ö - ( / ) ) is t h e l o c a l angle o f a t t a c k ( A O A ) w h e r e a s tg/( = V ^ / f l r .The set o f non-linear equations given b y Eqs. ( 6 ) and ( 7 ) m a y be suitably rearranged b y equating the expressions o f t h r u s t a n d e f f i c i e n c y f r o m M T and BET approaches respectively, and observing t h a t tgcf) = tgP(^'l+d)/{1~a'y. this procedure yields a n o n - l i n e a r equation w r i t t e n i n t e r m s o f the u n l m o w n i n f l o w angle cj). For t h e s o l u t i o n seeldng, the choice o f the starting-value cpg is a crucial p o i n t ; a reasonable initial-guess m a y be assumed f r o m t h e linearised BEMT approach, v a l i d f o r s m a l l values o f cp (see Leishman, 2006), y i e l d i n g
fack VA\\CTCI, r (ud,. \
(8)
w h e r e ?. = (VA + Vi)/QR s (PQT/R, a indicates the r a t i o b e t w e e n blade and disk areas, whereas R and c/^ are blade radius and sectional l i f t slope coefficient, respectively. Once the converged s o l u t i o n f o r the i n f l o w angle cj) is achieved, the operative AOA at each blade section is k n o w n and, i n turns, the angle y is f o u n d ; the axial i n f l o w factor m a y be t h e n d e t e r m i n e d b y
(tg<l>-tgp)
[tg(itg(l>tgi4>+y)+tgp] (9)
whereas t h e f u r t h e r i n t e g r a t i o n o f Eq. (7) radially across t h e blade yields t h e t o t a l t h r u s t delivered by the propeller. Similarly, spanwise i n t e g r a t i o n o f the incremental torque
dr
l p c Z r V ^ ( l + a ) ^ d - i i B M ± Z L
2 sin (p cos y
(10)
provides p r o p e l l e r t o r q u e (see T o d d and Comstock, 1967). By assum-i n g t h a t e l e m e n t a l t h r u s t and t o r q u e act at the centre o f pressure cp o f blade sections (see Fig. 35), spindle m o m e n t due t o t h e h y d r o d y n a m i c
F i g . 3 5 . Forces acting on a blade section: contribution to spindle moment.
P r o j e c t e d A r e a
F i g . 3 6 . Propeller geometrical features and local force due to elemental torque.
loads is g i v e n by
m f , = / (dTyp-dF^Xp) dr (11)
w h e r e Vboss indicates the radius o f the propeller hub, (Xpj/p,Zp) d e f i n e the location o f the centre o f pressure o f the elemental section w i t h respect to t h e blade f r a m e o f reference, dpQ = (dQ,/r)cos £, comes f r o m the elemental torque, w i t h £, representing the angle, measured o n the projected plane, b e t w e e n t h e Z/, axis a n d vector ( c p - 0 ) (see Fig. 36). Distances Xp and yp d e p e n d o n blade o p e r a t i n g c o n d i t i o n s ; as s h o w n i n T o d d and Comstock (1967), t h e y m a y be obtained b y yp = r sin ^
Xp = -eh + ng6-^ngO
( 1 2 )w h e r e e;, a n d è represent the blade offset f r o m the p i v o t axis (see Fig. 2 ) a n d t h e rake angle, respectively, w h i l s t t h e ^ angle m a y be c o m p u t e d by
[r2-h(rtgO)2]V2 (13)
being dhd a n d d^p the distance o f the blade generator line f r o m the leading edge o f the section and the l o c a t i o n o f t h e centre o f pressure f r o m the leading edge, respectively (see Fig. 3 7 ) . Neglecting
134 S. Leone et aL / Ocean Engineering 64 (2013) 122-134
v e l o c i t y cp can be d e t e r m i n e d f r o m t h e Laplace e q u a t i o n t h r o u g h t h e f o l l o w i n g b o u n d a r y i n t e g r a l r e p r e s e n t a t i o n :
E x p a n d e d A r e a
Fig. 3 7 . Propeller geometrical features: expanded area.
the drag c o n t r i b u t i o n , d^p m a y be evaluated by
OTla
d cos « + 0.25C (14)
once t h e local l i f t coefficient and the p i t c h i n g m o m e n t c o e f f i c i e n t w i t h respect to the aerodynamic centre {cmac) are l<nown. The p i t c h i n g m o m e n t c o e f f i c i e n t m a y be corrected f o r the effects i n d u c e d by t h e v a r i a t i o n o f the i n f l o w factors a and a' across the blade w i d t h t h a t cause a curved f l o w experienced by propeller sections; this f l o w cui-vature is equivalent to a change i n the e f f e c t i v e camber o f h y d r o f o i l s , b o t h as a m o u n t o f camber and its c h o r d w i s e d i s t r i b u t i o n . F o l l o w i n g the scheme proposed b y L u d w i e g a n d Ginzel (see Miles et al., 1992), one obtains
cm'^^ = cmac+l.i{k]k2-1)d (15) w h e r e /x is a n e m p i r i c a l parameter d e p e n d i n g o n the m e a n - l i n e shape
o f the h y d r o f o i l and operating conditions, k^ a n d k j are the camber c o r r e c t i o n coefficients (see Eckhardt and M o r g a n , 1955) accounting for c u r v a t u r e effects whereas cl is the l i f t coefficient.
A.2. Boundary Element Method
S t a r r i n g f r o m mass a n d m o m e n t u m c o n s e r v a t i o n e q u a t i o n s f o r a n i n c o m p r e s s i b l e f l o w , a g e n e r a l f o r m u l a t i o n t o d e s c r i b e a t h r e e -d i m e n s i o n a l b o -d y a r b i t r a r i l y m o v i n g w i t h r e s p e c t t o a f l u i -d is d e r i v e d . D e t a i l s o f t h e p r e s e n t f o r m u l a t i o n c a n be f o u n d i n Greco e t a l . ( 2 0 0 4 ) . A s s u m i n g t h a t t h e i n c o m i n g f l o w is i n v i s c i d a n d t h e v e l o c i t y p e r t u r b a t i o n i n d u c e d b y t h e b o d y is i r r o t a r i o n a l e x c e p t f o r a t h i n l a y e r o f v o r t i c i t y shed a t p r o p e l l e r b l a d e s t r a i l i n g edges, i t is p o s s i b l e t o i n t r o d u c e a scalar p o t e n t i a l cp a n d r e p r e s e n t t h e p e r t u r b a d o n v e l o c i t y i n g r a d i e n t f o r m as v = V t p . Mass c o n s e r v a -t i o n e q u a -t i o n y i e l d s -t h a -t -t h e v e l o c i -t y p o -t e n -t i a l is g o v e r n e d b y -t h e Laplace e q u a d o n V^<p = 0, w h e r e a s t h e m o m e n t u m e q u a t i o n m a y be m a n i p u l a t e d t o o b t a i n B e r n o u l l i ' s e q u a t i o n dtp 1 . P 1 (16) I n Eq. ( 1 6 ) , w r i t t e n i n t h e r o t a t i n g f r a m e o f r e f e r e n c e (0,Xj,,y)„Zb), V, d e n o t e s t h e v e l o c i t y o f f l o w i n c o m i n g t o t h e p r o p e l l e r disk, q = V(p+Vi t h e t o t a l v e l o c i t y a n d gZo is t h e h y d r o s t a t i c h e a d . A classical d e r i v a t i o n (see M o r i n o , 1 9 9 3 ) y i e l d s t h a t the p e r t u r b a t i o n
(17)
w h e r e Sp denotes t h e b o d y surface {i.e.,, the p r o p e l l e r ) , Sw is t h e t r a i l i n g v o r d c a l surface ( t h e w a k e ) , a n d n is the u n i t n o r m a l t o these surfaces. The s y m b o l A denotes d i s c o n t i n u i t y o f q) across t h e w a k e surface, a n d G, 8C/8n are r e s p e c t i v e l y u n i t source a n d d i p o l e s i n g u l a r i d e s i n t h e u n b o u n d e d t h r e e - d i m e n s i o n a l space. Eq. ( 1 7 ) is used t o e v a l u a t e cp a n d h e n c e t h e v e l o c i t y f i e l d o n t h e b o d y surface a n d i n t h e f l u i d d o m a i n once q u a n t i t y 8(p/8n is d e t e r m i n e d b y i m p o s i n g the i m p e r m e a b i l i t y c o n d i t i o n o n Sp a n d A<p is d e t e r -m i n e d b y u s i n g a K u t t a - t y p e c o n d i t i o n . Once pressure o n t h e s o l i d surface is d e t e r m i n e d by Eq. ( 1 6 ) , a n d d e n o t i n g by T t h e t a n g e n t i a l (viscous) stress o n Sp, h y d r o d y n a m i c forces f a n d m o m e n t s m o a c t i n g o n t h e p r o p e l l e r are c o m p u t e d as f = <p ( - p n - h T t ) d S m o = 9 ( - p r X n - j - t r x t) dS ./Sp (18) w h e r e r = x - X o , t is t h e u n i t t a n g e n t t o Sp a l i g n e d t o local s t r e a m l i n e s . P r o p e l l e r t h r u s t T a n d t o r q u e Q easily f o l l o w b y t a k i n g a x i a l c o m p o n e n t s o f f a n d mo respectively. N e x t , s p i n d l e m o m e n t mff is e v a l u a t e d as t h e c o m p o n e n t o f m o a l o n g b l a d e s p i n d l e axis i n t e g r a t e d o n a single blade. R e c a l l i n g B E M is b a s e d o n i n v i s c i d f l o w a s s u m p t i o n s , a d d i -t i o n a l m o d e l l i n g is r e q u i r e d -t o p r e d i c -t -t h e -t a n g e n -t i a l s-tress x a n d h e n c e t o e s t i m a t e v i s c o s i t y c o n t r i b u t i o n s t o p r o p e l l e r loads i n Eq. ( 1 8 ) ; i n t h e p r e s e n t w o r k , a s e m i - e m p i r i c a l a p p r o a c h is u s e d i n w h i c h l o c a l d i s t r i b u t i o n o f x o n b l a d e surface is d e r i v e d f r o m classical l a w s f o r a t t a c h e d l a m i n a r a n d t u r b u l e n t b o u n d a r y l a y e r o n a f l a t p l a t e (see C a r i t o n , 1 9 9 4 ) . I f l o c a l e f f e c t i v e a n g l e o f a t t a c k is h i g h e r t h a n a p r e s c r i b e d t h r e s h o l d , v i s c o s i t y e f f e c t s i n Eq. ( 1 8 ) have t o be f u r t h e r c o r r e c t e d t o a p p r o x i m a t e l y a c c o u n t f o r t h e a d d i d o n a l d r a g i n d u c e d b y b o u n d a r y l a y e r f l o w s e p a r a t i o n . R e f e r e n c e s
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