Ocean Engineering 88 (2014) 607-617
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Ocean Engineering
j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / o c e a n e n g
Scale effects on tip loaded propeller performance using a RANSE solver
( S ) c r o s s M a r kA. Sanchez-Caja^''', J. G o n z a l e z - A d a l i d M . P é r e z - S o b r i n o ^ T. Sipila^
'Ship Hydrodynamics, VTT-Technical Research Center of Finland, Tietotie J A, PL 1000, 02044 Espoo, FinlandSISTEMAR, Spain
A R T I C L E I N F O Article history:
Received 14 February 2014 Accepted 27 A p r i l 2014 Available online 17 May 2014 Keywords;
Scale effects Tip loaded propeller CLT
RANS Extrapolation Circulation
A B S T R A C T
T r a d i t i o n a l l y , t h e q u a l i t y o f a l t e r n a t i v e p r o p e l l e r d e s i g n s at f u l l scale has b e e n assessed b y m e a n s o f m o d e l scale t e s t s . D u e t h e l a r g e d i f f e r e n c e s i n R e y n o l d s n u m b e r s b e t w e e n m o d e l a n d f u l l scales, m o d e l -s c a l e - b a -s e d p e r f o r m a n c e p r e d i c t i o n -s m a y be q u e -s t i o n a b l e i n ca-se-s w h e r e v i -s c o u -s e f f e c t -s are d o m i n a n 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 a b o u t scale e f f e c t s o n p e r f o r m a n c e c o e f f i c i e n t s f o r a CLT p r o p e l l e r w i t h d i f f e r e n t e n d p l a t e g e o m e t r i e s . T w e l v e e n d p l a t e s h a p e v a r i a t i o n s w e r e a n a l y z e d i n m o d e l a n d f u l l scale u s i n g RANS c o d e FINFLO. T h e SST k-co t u r b u l e n c e m o d e l is u s e d as a b a s i c m o d e l f o r t h e s t u d y . A d d i t i o n a l c o m p u t a t i o n s are m a d e w i t h o t h e r t u r b u l e n c e m o d e l s . A s p e c i a l p r o c e d u r e f o r t h e g e n e r a t i o n o f t h e c o m p u t a t i o n a l g r i d s is i m p l e m e n t e d t o m i n i m i z e c o m p u t a t i o n a l e r r o r s i n t h e c o m p a r i s o n o f t h e a l t e r n a t i v e g e o m e t r i e s . T h e s t u d y p r o v i d e s also a R A N S - b a s e d scale e f f e c t o n t h e s h a p e o f r a d i a l c i r c u l a r i o n d i s t r i b u t i o n p r e d i c t e d f o r d i f f e r e n t g e o m e t r y v a r i a t i o n s . T h e r e s e a r c h w o r k g i v e s a n i n s i g h t i n t o w h i c h t y p e o f m o d i f i c a t i o n s a t f u l l scale c o u l d b e a n a l y z e d b y m o d e l scale v i s c o u s f l o w t h e o r y o r m o d e l t e s t s i n r a n k i n g a l t e r n a t i v e d e s i g n s . D i f f e r e n c e s f o u n d b e t w e e n m o d e l a n d f u l l s c a l e n u m e r i c a l r e s u l t s m a k e m o d e l scale a n a l y s i s q u e s t i o n a b l e f o r s o m e s p e c i f i c t y p e o f m o d i f i c a t i o n s w h e n f u l l - s c a l e p e r f o r m a n c e is s o u g h t . © 2 0 1 4 E l s e v i e r L t d . A l l r i g h t s r e s e r v e d .
1. Introduction
M o d e l tests have u s u a l l y been used to p r e d i c t t h e p e r f o r m a n c e o f m a r i n e p r o p e l l e r s i n f u l l scale. Deficiencies i n p r e d i c t i o n s o f p r o p e l l e r p e r f o r m a n c e a n d / o r c a v i t a t i o n b e h a v i o r f o r especial t y p e s o f p r o p e l l e r s have b e e n s o m e t i m e s n o t i c e d i n p a r t i c u l a r a p p l i c a t i o n s l i k e h i g h s k e w e d p r o p e l l e r s , d u c t e d p r o p e l l e r s a n d t i p l o a d e d p r o p e l l e r s . Such deficiencies have l e d t h e 2 7 t h I T I C ( I n t e r n a t i o n a l T o w i n g Tank Conference) t o r e c o m m e n d e x a m i n i n g t h e s c a l i n g p r o c e d u r e s a n d t o encourage e s t a b l i s h i n g research p r o g r a m s i n t e n d e d t o c l a r i f y scaling p r o b l e m s .
F r o m t h e CFD s t a n d p o i n t , l i m i t a t i o n s o f p o t e n t i a l - f l o w - b a s e d m e t h o d s t o p r e d i c t scale effects have l e d researchers t o focus o n RANS approaches. Scale effects o n general p r o p e l l e r s have b e e n c o m p u t e d f o r o p e n p r o p e l l e r s i n Stanier (1998), Funeno ( 2 0 0 2 ) , Li e t a l . ( 2 0 0 6 ) a n d K o u s h a n a n d K r a s i l n i k o v ( 2 0 0 8 ) . For d u c t e d p r o p e l l e r s , scale e f f e c t s have b e e n i n v e s t i g a t e d i n A b d e l - M a k s o u d a n d H e i n k e ( 2 0 0 2 ) , K r a s i l n i k o v et al. ( 2 0 0 7 ) a n d M e r t e s a n d H e i n k e ( 2 0 0 8 ) . Scale effects o n p o d d e d p r o p e l l e r s are p r e s e n t e d i n C h i c h e r i n e t al. ( 2 0 0 4 ) , Lobachev a n d C h i t c h e r i n ( 2 0 0 1 ) a n d Sanchez-Caja et al. ( 2 0 0 3 ) . E f f o r t t o reach a d e e p e r u n d e r s t a n d i n g o n h o w t u r b u l e n c e m o d e l s behave f o r t h e p r e d i c t i o n o f scale
* Corresponding author. Tel.: - f 3 5 8 50 352 4743; fax: 4-358 20 722 7075. E-mail address: antonio.sanche2@vtt.fi (A. Sanchez-Caja).
effects o n p r o p e l l e r s has b e e n m a d e r e c e n t l y . K r a s i l n i k o v e t al. ( 2 0 0 9 ) has s t u d i e d t h e i n f l u e n c e o f blade skew, l o a d i n g a n d area rario o n scaling u s i n g t h e FLUENT solver w i t h t h e SST k-co m o d e l . M M l e r et al. ( 2 0 0 9 ) used t h e ANSYS-CFX solver w i t h t h e SST turib^ulence m o d e l t o n u m e r i c a l l y analyze^a v a r i e t y o f p r o p e l l e r s a t m o d e l a n d f u l l scale i n o p e n w a t e r . K a w a m u r a a n d O m o r i ( 2 0 0 9 ) p r e s e n t e d a c o m p u t a t i o n a l s t u d y o n p r o p e l l e r s c a l i n g o f o p e n w a t e r p e r f o r m a n c e f o r 5 - b l a d e d a n d 4 - b l a d e d p r o p e l l e r s u s i n g FLUENT a n d t h e SST k-co m o d e l w i t h its h i g h a n d l o w Reynolds n u m b e r v e r s i o n . He f o u n d t h a t scale effects e x i s t i n b o t h pressure a n d f r i c t i o n a l forces. B o t h forces a f f e c t t h e scale e f f e c t i n t h r u s t , w h i l e t h e f r i c t i o n a l f o r c e c o n t r i b u t e s m a i n l y t o t h e scale e f f e c t i n t o r q u e . L i m i t a t i o n s o n t u r b u l e n c e m o d e l i n g f o r s i m u l a d n g p r o -p e l l e r flow have b e e n r e -p o r t e d i n Li e t al. ( 2 0 0 6 ) a n d t h e n e e d f o r a g o o d g r i d q u a l i t y has been stressed f r o m t h e s t a n d p o i n t o f w a l l t r e a t m e n t w i t h i n t h e t u r b u l e n c e m o d e l .
Generally, m o d e l scale p r e d i c t i o n s are m o r e c h a l l e n g i n g t h a n f u l l scale ones d u e t o t h e possible s i m u l t a n e o u s presence o f d i f f e r e n t f l o w r e g i m e s a n d t h e d i f f i c u l t y t o a c c u r a t e l y p r e d i c t t r a n s i t i o n i n t h e c o m p u t a t i o n s . S e p a r a t i o n m a y e x i s t at m o d e l scale, w h i c h is n o t p r e s e n t at f u l l scale. E f f o r t s t o i m p r o v e t r a n s i t i o n m o d e l s f r o m l a m i n a r to t u r b u l e n t flow have been m a d e r e c e n t l y i n M e n t e r et al. ( 2 0 0 6 ) .
H i s t o r i c a l l y , RANS c o m p u t a t i o n s o n CLT p r o p e l l e r s w e r e m a d e w i t h i n EU p r o j e c t L e a d i n g Edge a n d r e p o r t e d i n Sanchez-Caja e t al. ( 2 0 0 6 ) . C o m p a r i s o n o f t h e scale e f f e c t o n flow f e a t u r e s l i k e tip http://dx.doi.org/10.1016/j.oceaneng.2014.04.029
608 h. Sanchez-Caja et al. / Ocean Engineering 88 (2014) 607-617
v o r t i c e s at m o d e l a n d f u l l scaje was presented. H s i n et a l . (2010) a n d Cheng e t al. ( 2 0 1 0 ) s t u d i e d t i p - l o a d e d p r o p e l l e r s o f CLT a n d K a p p e l t y p e . They u s e d b o t h a panel m e t h o d a n d a RANS solver i n t h e analysis. Scale e f f e c t s w e r e f o u n d f o r b o t h p r o p e l l e r types l a r g e r t h a n f o r c o n v e n t i o n a l p r o p e l l e r s . I n a d d i t i o n , scale effects w e r e s t u d i e d i n H a i m o v et al. (2011) f o r CLT p r o p e l l e r s a n d d i f f i c u l t y t o o b t a i n g r i d - i n d e p e n d e n t results u s i n g u n s t r u c t u r e d meshes w a s i l l u s t r a t e d . Gaggero a n d B r i z z o l a r a (2011) have a n a l y z e d a CLT p r o p e l l e r u s i n g t h e RANS code StarCCM 4- a n d a p a n e l code. T h e y p r e s e n t e d n u m e r i c a l analysis u n d e r c a v i t a t i o n c o n d i t i o n s a n d c o m p a r i s o n w i t h e x p e r i m e n t s i n B e r t e t t a e t al. ( 2 0 1 2 ) , b e i n g t h e m a i n f o c u s c o m p a r i s o n o f p a n e l m e t h o d a n d RANS r e s u l t s also f r o m t h e s t a n d p o i n t o f scaling. They p r e s e n t e d r a d i a l d i s t r i b u t i o n s o f c i r c u l a t i o n o b t a i n e d w i t h t h e p a n e l code.
Recently, s y s t e m a t i c v a r i a t i o n s o f t h e e n d p l a t e g e o m e t i y f o r e n d - p l a t e p r o p e l l e r s w e r e m a d e u s i n g RANS code FINFLO i n o r d e r to assess t h e i m p a c t o f t h e blade t i p shape o n p r o p e l l e r p e r f o r -m a n c e at f u l l scale. So-me results w e r e a n t i c i p a t e d i n Sanchez-Caja et a l . ( 2 0 1 2 ) a n d t h e f i n a l analysis w a s r e p o r t e d i n Sanchez-Caja et a l . ( 2 0 1 4 ) . I n t h e p r e s e n t paper, t h e same g e o m e t r y v a r i a t i o n s w i l l be a n a l y z e d f r o m t h e s t a n d p o i n t o f scaling e f f e c t s . Scale effects w i l l be c o m p u t e d u s i n g m a i n l y t h e SST k-co t u r b u l e n c e m o d e l even t h o u g h Chien's k-e and Spalart A l l m a r a s t u r b u l e n c e m o d e l s w i l l be also used f o r a reference g e o m e t r y . A special p r o c e d u r e f o r t h e g e n e r a t i o n o f t h e c o m p u t a t i o n a l g r i d s is i m p l e m e n t e d i n o r d e r t o m i n i m i z e c o m p u t a t i o n a l errors i n t h e c o m p a r i s o n o f t h e a l t e r n a t i v e g e o m e t r i e s . I n t h i s study, w e focus o n n o n -c a v i t a t i n g flow -c o n d i t i o n s i n o r d e r to assess t h e p u r e i m p a -c t o f t h e p l a t e shape o n scaling, l e a v i n g f o r f u t u r e w o r k i n t h e scaling o f c a v i t a t i n g flows. The w o r k presents r a d i a l d i s t r i b u t i o n s o f c i r c u l a -tion o b t a i n e d w i t h t h e RANS solver, w h i c h is a n o v e l t y f o r t h i s t y p e o f p r o p e l l e r s since p r e v i o u s researchers have f o c u s e d m o r e o n c i r c u l a t i o n s based o n p o t e n t i a l flow a n a l y s i s J I h e analysis a l l o w s e v a l u a t i n g t h e scale e f f e c t o n the shspe^f t h e c i r c u l a t i o n curve.
This w o r k has b e e n m a d e w i t h i n t h e EU P r o j e c t TRIPOD w h e r e CLT p r o p e l l e r s w e r e s t u d i e d f r o m t h e s t a n d p o i n t o f e f f i c i e n c y i m p r o v e m e n t .
2. Numerical metliod
T h e RANS e q u a t i o n s are s o l v e d u s i n g the FINFLO code i n i t i a l l y d e v e l o p e d a t t h e L a b o r a t o r y o f A e r o d y n a m i c s a t H e l s i n k i U n i v e r -s i t y o f T e c h n o l o g y ( S i i k o n e n et al., 1990). A d e -s c r i p t i o n o f t h e n u m e r i c a l m e t h o d i n c l u d i n g d i s c r e t i z a t i o n o f t h e g o v e r n i n g equa-tions, s o l u t i o n a l g o r i t h m , etc. can be f o u n d also i n Sanchez-Caja e t a l . ( 1 9 9 9 ) . The c o d e e m p l o y e d i n i t i a l l y t h e a r t i f i c i a l c o m p r e s s i -b i l i t y m e t h o d t o solve t h e RANS e q u a t i o n s a n d has -b e e n r e c e n t l y e x t e n d e d t o i n c o r p o r a t e t h e pressure c o r r e c t i o n m e t h o d . The m o m e n t u m e q u a t i o n s can be w r i t t e n i n t h e f o l l o w i n g f o r m :
p-^+Vp-jlSJ^V =pg +F BE (1) w h e r e is t h e v e l o c i t y vector, p is t h e density, /< is t h e d y n a m i c
viscosity, g" t h e a c c e l e r a t i o n o f g r a v i t y a n d ' f BE are possible b o d y forces. The e q u a t i o n can be expressed i n t e r m s o f a v e c t o r U o f c o n s e r v a t i v e variables ( p , pu, pv, pw, pk, p e f , w h e r e u , v a n d w are t h e absolute v e l o c i t y c o m p o n e n t s , k is t h e t u r b u l e n t k i n e t i c e n e r g y a n d e is t h e d i s s i p a t i o n o f k. For t h e steady-state analysis w i t h a f u l l r e p r e s e n t a t i o n o f t h e propeller, t h e e q u a t i o n s are solved i n a c o - o r d i n a t e s y s t e m t h a t rotates a r o u n d t h e x - a x i s w i t h a n a n g u l a r v e l o c i t y £2. I n t h a t case, t h e RHS o f t h e m o m e n t u m e q u a t i o n has t h e a d d i t i o n a l c o m p o n e n t (0, 0, pQw, -pQv, 0, 0 ) . For time-accurate s i m u l a t i o n s , t h e source t e r m s f o r t h e t u r b u l e n c e e q u a t i o n s are r e t a i n e d , b u t t h e r e are n o source t e r m s i n t h e m o m e n t u m e q u a t i o n s except g r a v i t a t i o n .
FINFLO solves t h e RANS e q u a t i o n s b y a finite v o l u m e m e t h o d u s i n g e i t h e r t h e d e n s i t y - b a s e d p s e u d o - c o m p r e s s i b i l i t y m e t h o d or t h e pressure c o r r e c t i o n m e t h o d . The c o m p u t a t i o n s presented i n t h i s paper have b e e n m a d e w i t h t h e pressure c o r r e c t i o n m e t h o d ( S i i k o n e n , 2011).
I n FINFLO, t h e s o l u t i o n is e x t e n d e d t o t h e w a l l . A n essential f e a t u r e i n the code is t o separate flux c a l c u l a t i o n f r o m the s o l u t i o n . T h e n t h e code m a y use e i t h e r Roe's flux-difference s p l i t t i n g f o r compressible o r a n u p w i n d based scheme f o r i n c o m p r e s s i b l e flows. A m u l t i g r i d m e t h o d m a y be used f o r t h e acceleration o f convergence e v e n t h o u g h w i t h t h e pressure-based s o l u t i o n this k i n d o f m u l t i g r i d acceleration is s t i l l rare ( S i i k o n e n , 2011). Solu-tions i n coarse g r i d levels are used as a s t a r t i n g p o i n t f o r the c a l c u l a t i o n i n o r d e r t o accelerate convergence. Chien's l o w Rey-n o l d s Rey-n u m b e r k-e m o d e l w a s used i Rey-n t h e c o m p u t a t i o Rey-n s .
The b o u n d a r y c o n d i t i o n s f o r t h e cases s t u d i e d are set as usual. The p r o p e l l e r blades are m o d e l e d as r o t a t i n g n o n - s l i p surfaces w i t h t h e v e l o c i t y field m a t c h i n g t h e p r o p e l l e r r o t a t i o n a l speed. A u n i f o r m flow c o n d i t i o n is a p p l i e d t o t h e i n l e t a n d p e r i p h e r a l surfaces. A t t h e o u t i e t , t h e s t r e a m w i s e g r a d i e n t s o f t h e flow variables as w e l l as t h e pressure d i f f e r e n c e are set to zero. For t h e u n i f o r m flow c o m p u t a t i o n s , o n l y t h e p o r t i o n b e t w e e n t w o c o n t i g u o u s blades has been used due t o t h e p e r i o d i c i t y o f the s o l u t i o n .
3. Geometry variations
A f o u r - b l a d e d 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 w o r k i n g u n d e r a m o d e r a t e l y u n l o a d e d o f f - d e s i g n c o n d i t i o n has b e e n chosen as reference g e o m e t r y f o r t h e study. The choice o f t h e o p e r a t i o n a l p o i n t was r e l a t e d t o a p r e v i o u s o p t i m i z a t i o n s t u d y (Sanchez-Caja et al., 2014) i n a c h a l l e n g i n g c o n d i t i o n f o r CLT p r o p e l l e r s since they are e x p e c t e d t o be m o r e b e n e f i c i a l at h i g h loads. I n such c o n d i t i o n , scale effects o n e f f i c i e n c y are e x p e c t e d to be l a r g e r t h a n at the d e s i g n p o i n t . The p r o p e l l e r is a 4.38 m d i a m e t e r w i t h a p i t c h d i a m e t e r r a t i o o f 1.1. T h e h u b d i a m e t e r r a t i o is 0.33. The e x p a n d e d area r a t i o is 0.52 a n d t h e s k e w is m o d e r a t e . The e n d p l a t e is located at t h e pressure side o f t h e blade. The o n s e t flow f o r t h e calcula-t i o n s corresponds calcula-t o a n advance c o e f f i c i e n calcula-t o f 0.90. Full a n d m o d e l scale p e r f o r m a n c e s are s t u d i e d . The scale f a c t o r is 17.962, w h i c h corresponds t o a Reynolds n u m b e r at m o d e l scale o f 10^ a n d at f u l l scale o f 5*10^.
Several t y p e s o f e n d p l a t e m o d i f i c a t i o n s w e r e a n a l y z e d using t h e SST k-O) a n d Chien's l o w Reynolds n u m b e r k-e t u r b u l e n c e m o d e l s . They i n c l u d e v a r i a t i o n s i n p l a t e c o n t r a c t i o n angle, plate sweep, flap angle a n d p l a t e c u t t i n g .
Table 1 s h o w s t h e cross r e l a t i o n a m o n g t h e v a r i a t i o n s o f g e o m e t r y f o r 12 cases a n a l y z e d f o r t h e 4 - b l a d e d p r o p e l l e r relative to t h e baseline. A n i n t e g e r n u m b e r r e p r e s e n t a t i v e o f t h e s t r e n g t h o f t h e m o d i f i c a t i o n is p r e s e n t e d i n each c o l u m n . The larger the n u m b e r , t h e s t r o n g e r t h e m o d i f i c a t i o n a n d v i c e versa. Index n u m b e r 0 m e a n s v a l u e o f t h e baseline p r o p e l l e r ,
3.3. Plate contraction
The c o n t r a c t i o n angle o f t h e p l a t e w a s d e f i n e d as a f u n c t i o n o f t h e t r a i l i n g edge (TE) l e n g t h a n d f o u r v a r i a t i o n s w e r e s t u d i e d (Fig. 1). W i t h t h e g r i d g e n e r a t i o n code, t h e l e n g t h o f the TE line o f t h e p r o p e l l e r was c h a n g e d a n d t h i s w a s t h e p a r a m e t e r used f o r d e f i n i n g t h e r a d i a l i n c l i n a t i o n o f t h e p l a t e . M a x i m u m i n c l i n a t i o n ( c o n t r a c t i o n ) is r e p r e s e n t e d b y t h e s h o r t e s t TE l e n g t h , w h i c h is labeled as t h e 99%, a n d m i n i m u m i n c l i n a t i o n b y the longest TE l e n g t h , labeled as 106% i n t h e sketch. I n o t h e r w o r d s , t h e i n c l i n a -tion varies i n v e r s e l y p r o p o r t i o n a l t o t h e TE l e n g t h . T h e 100% represents i n i t i a l l o c a t i o n o f t h e p l a t e c o r r e s p o n d i n g to an
A. Sanchez-Caja et al. j Ocean Engineering 88 (20U) 607-617 609
Table 1
Cross relation among the variations of geometry f o r 12 cases analyzed f o r the 4-bladed propeller. 0 = b a s e l i n e ; negative number, negative modification; positive number, positive m o d i f i c a t i o n relative to baseline.
endplate outer edge- endplate LE
No. Contract Flap Sweep Cutting
1 0 0 0 0 2 0 1 0 0 3 0 2 0 0 4 - 1 0 0 0 5 1 0 0 0 6 2 0 0 0 7 3 0 0 0 8 0 0 - 1 0 9 0 0 1 0 10 0 0 0 1 11 0 0 0 2 12 0 0 0 3 L E line T E l i n e
Fig. 1. Sketch f o r the d e f i n i t i o n o f plate contraction.
Fig. 2. Sketch f o r the d e f i n i t i o n o f flap angle (s).
e s t i m a t i o n o f t h e e n d p l a t e a l i g n e d w i t h t h e f l o w a t o t h e r w o r k i n g c o n d i t i o n ( t h e d e s i g n p o i n t ) . The TE l e n g t h s w e r e 9 9 , 1 0 2 , 1 0 4 a n d 106 percent. As s h o w n i n Table 1, c o n t r a c t i o n v a r i a t i o n s w e r e l a b e l e d w i t h n u m b e r s 4, 5, 6 a n d 7. 3.2. Flap angle By " f l a p angle", w e m e a n t h e angle o f t h e p l a t e r e l a t i v e t o a n axis passing t h r o u g h t h e c h o r d l i n e at t h e j u n c t u r e o f t h e blade a n d e n d p l a t e (Fig. 2 ) . Z e r o angle m e a n s 'plate at o r d i n a r y p o s i t i o n ' , i.e. a p o s i t i o n such t h a t t h e p l a t e sections m a d e w i t h planes c o n t a i n i n g t h e p r o p e l l e r axis are p a r a l l e l to t h e axis (baseline p r o p e l l e r ) . The angle is p o s i t i v e f o r t h e e n d p l a t e o u t e r edge
backward sweep • - ^ forward sweep
Fig. 3. Sketch f o r the d e f i n i t i o n o f sweep.
blade edge endplate LE •
r
- TE cuttingOuter edge
Fig. 4. Sketch f o r the d e f i n i t i o n o f plate cutting.
r o t a t e d i n w a r d s , t o w a r d s t h e h u b . N e g a t i v e angles ( o u t w a r d r o t a t i o n ) w e r e n o t c o n s i d e r e d as t h e y r e s u l t i n a n increase o f t h e p r o p e l l e r d i a m e t e r , w h i c h w o u l d change t h e basic c r i t e r i o n f o r c o m p a r i s o n (a g i v e n d i a m e t e r ) . Three flap angles w e r e s t u d i e d : 0 ° , 10° a n d 3 0 ° . As s h o w n i n Table 1, f l a p angle v a r i a t i o n s w e r e l a b e l e d w i t h n u m b e r s 1, 2 a n d 3. 3.3. Plate sweep I f w e c o n s i d e r t h e p l a t e as h a l f a w i n g w i t h v e r y l o w aspect ratio, m o d i f i c a t i o n s t h a t r e s e m b l e f o r w a r d o r baclwvard s w e p t w i n g s m a y be i n t r o d u c e d i n t h e e n d p l a t e s (Fig. 3 ) b y m o v i n g t h e o u t e r edge o f t h e e n d p l a t e f o r w a r d o r b a c k w a r d . S w e e p v a r i a t i o n s w e r e l a b e l e d w i t h n u m b e r s 8 (back s w e e p ) a n d 9 ( f o r w a r d s w e e p ) ; see Table 1. 3.4. Plate cutting
The las tsType o f m o d i f i c a t i o n w a s p l a t e c u t t i n g (Fig. 4 ) . T h r e e cuts w e r e m a d e ^ T h e largest c u t r e a c h e d t h e e n d p l a t e b o r d e r a t a l o c a t i o n close t o m i d - c h o r d o f t h e e n d p l a t e r o o t The m o d i f i c a t i o n s d i d n o t a f f e c t t h e l e a d i n g edge (LE). As s h o w n i n Table 1, c u t t i n g s w e r e l a b e l e d w i t h n u m b e r s 10 (large), 11 ( m e d i u m ) a n d 12 ( s m a l l ) .
4. Mesh generation
W e have t r i e d t o reduce r e l a t i v e n u m e r i c a l e r r o r s b e t w e e n g r i d s b y p e r f o r m i n g g e o m e t r i c a l l y s i m i l a r s t r u c t u r e d g r i d s . For a u t o m a t i n g t h e g r i d g e n e r a t i o n process, w e use a n i n - h o u s e d e v e l o p e d p r o g r a m r u n w i t h t h e h e l p o f t e m p l a t e s . By u s i n g t h e same t e m p l a t e s f o r t h e d i f f e r e n t e n d p l a t e shapes, w e g u a r a n t y t h e same n u m b e r a n d d i s t r i b u t i o n o f cells f o r t h e v a r i o u s meshes as w e l l as a s i m i l a r t o p o l o g y . I n t h i s w a y , n u m e r i c a l errors i n t h e c o m p a r i s o n o f p e r f o r m a n c e s f o r d i f f e r e n t e n d p l a t e shapes are r e d u c e d . A n 0 - 0 t o p o l o g y w a s used a r o u n d t h e blades a n d e n d p l a t e s w i t h t h e a i m o f r e d u c i n g t h e n u m b e r o f cells f o r a g i v e n accuracy. This a l l o w e d a c h i e v i n g a g o o d c o n t r o l o f t h e g r i d o r t h o g o n a l i t y o v e r t h e p r o p e l l e r surfaces. Figs. 5 a n d 6 s h o w the d i s t r i b u t i o n o f cells o n t h e blade surfaces o n t h e s u c t i o n a n d pressure sides o f t h e blade, respectively. Fig. 7 shows c o n s t a n t i n d e x cuts o f t h e m e s h o n t h e d i r e c t i o n n o r m a l t o t h e blade.A n u m b e r o f 128 cells a r o u n d t h e blade i n t h e c h o r d w i s e d i r e c t i o n a n d 9 6 i n t h e r a d i a l d i r e c t i o n w e r e used. The f u l l scale grids h a d 16 m o r e cells i n t h e d i r e c t i o n n o r m a l to t h e b l a d e surface. The cell size w a s a d j u s t e d especially at t h e l e a d i n g , t r a i l i n g a n d e n d p l a t e edges i n t h e r e g i o n s o f h i g h c u r v a t u r e . The m e s h consisted o f 19 blocks p e r blade. T h é n u m b e r s o f cells p e r b l o c k i n
610 A. Sanchez-Caja et al. / Ocean Engineering 88 (2014) 607-617
Fig. 5. Grid on t i i e suction side of tlie propeller blade. Endplate located on the pressure side.
eacli o f t l i e t h r e e i n d e x d i r e c t i o n s w e r e m u l t i p l e n u m b e r s o f 16 i n o r d e r t o a p p l y techniques o f m u l t i g r i d a c c e l e r a t i o n . The size o f the fine g r i d was a b o u t 2.1 m i l l i o n cells per b l a d e i n t h e m o d e l scale a n d 2.4 i n f u l l scale. Cyclic b o u n d a r y c o n d i t i o n s w e r e used f o r the c a l c u l a t i o n s i n u n i f o r m flow i n o r d e r t o m o d e l t h e space b e t w e e n blades, a v o i d i n g unnecessary large grids.
5. Convergence
A n u n c e r t a i n t y s t u d y was i n i t i a l l y m a d e . T h r e e m e s h e s w e r e b u i l t f o r a r e f e r e n c e p r o p e l l e r w i t h cell ratios o f 1, 2 a n d square r o o t o f 2 (coarse, fine a n d m e d i u m g r i d ) . The coarse m e s h is o b t a i n e d b y r e m o v i n g e v e r y o t h e r g r i d p o i n t i n t h e t h r e e g r i d i n d e x d i r e c t i o n s f r o m t h e fine one. Tables 2 a n d 3 s h o w t h e p e r f o r m a n c e c o e f f i c i e n t s expressed as percentages o f t h e fine g r i d s o l u t i o n s i n t e r m s o f t h r u s t , t o r q u e a n d e f f i c i e n c y f o r t h e coarse, m e d i u m a n d fine g r i d , respectively. They c o r r e s p o n d to c o m p u t a -t i o n s m a d e w i -t h -t h e SST k-a i n f u l l scale a n d m o d e l scale. Tables 4 a n d 5 i l l u s t r a t e t h e v a r i a t i o n o n p e r f o r m a n c e c o e f f i c i e n t s f o r Chien's k - s m o d e l i n f u l l a n d m o d e l scale. I t is a p p a r e n t t h a t t h e e f f i c i e n c i e s i n Table 5 f o r t h e coarse a n d m e d i u m g r i d are n o t as close t o t h e fine g r i d as i n t h e o t h e r tables. As w i l l be s h o w n i n t h e n e x t section, t h i s is d u e t o t h e p r e d i c t i o n o f l a m i n a r flow d e t a c h m e n t at m o d e l scale w i t h t h e k-e m o d e l i n a s i g n i f i c a n t p o r t i o n o f t h e blade near t h e h u b , w h i c h requires h i g h e r r e s o l u -tion a n d presents h i g h e r u n c e r t a i n t y t h a n u n d e t a c h e d flow.
As t h e fine g r i d s o l u t i o n uses t h e coarse g r i d o n e as a s t a r t i n g p o i n t , t h e results f o r t h e fine a n d coarse g r i d are c o m p u t e d f o r all t h e a l t e r n a t i v e shapes tested.
The c o m p u t a t i o n s w e r e p e r f o r m e d u s i n g W i n d o w s e n v i r o n -m e n t w i t h I n t e l ® X e o n ® 2.67 GHz processors.
Fig. 6. Grid on the pressure side o f the propeller blade.
6. Validation at model scale
I n t h i s section, v a l i d a t i o n d a t a f o r t h e d e s i g n c o n d i t i o n o f an e a r i i e r v e r s i o n o f t h e r e f e r e n c e p r o p e l l e r are p r e s e n t e d . Fig. 8 i l l u s t r a t e s results f r o m p a i n t tests s h o w i n g p o r t i o n s o f l a m i n a r and t u r b u l e n t flows o n s m o o t h a n d LE r o u g h e n e d surfaces. Fig. 9 s h o w s t h e c o r r e s p o n d i n g n u m e r i c a l results o b t a i n e d u s i n g SST k-co a n d Chien's k-e t u r b u l e n c e m o d e l s . The f o r m e r c o m p u t a t i o n is m a d e w i t h f u l l y t u r b u l e n t flow a n d t h e s t r e a m l i n e s are s i m i l a r to t h e r o u g h e n e d surface s t r e a m l i n e s i n t h e tests, e x c e p t f o r the l a m i n a r p a t c h at t h e r o o t o n t h e pressure side. T h e latter c o m p u t a t i o n w a s m a d e t r y i n g t o r e p r o d u c e q u a l i t a t i v e l y a flow p a t t e r n s i m i l a r t o t h a t i n t h e p a i n t tests w i t h s m o o t h surfaces.
Table 2
Performance coefficients expressed as percentages o f the fine g r i d solutions for the coarse, m e d i u m and fine grids. SST k-io turbulence m o d e l in f u l l scale.
Fig. 7. Gnd cuts at constant indexes i n the block surrounding the blade.
I<T « 1 '/ Coarse M e d i u m Fine 101.53 99.85 100.00 101.73 100.25 100.00 99.80 99.50 100.00 Table 3
Performance coefficients expressed as percentages o f the fine g r i d solutions for the coarse, m e d i u m and fine grids. SSI k-co turbulence model i n m o d e l scale.
I<T Coarse M e d i u m Fine 102.33 100.68 100.00 102.55 100.82 100.00 99.782 99.862 100.00
A. Sanchez-Caja et al. / Ocean Engineering 88 (2014) 607-617 611 e n f o r c i n g l o w b a c k g r o u n d t u r b u l e n c e . I n t h i s w a y , an i n d i c a t i o n o f change i n p e r f o r m a n c e c o e f f i c i e n t s w i t h g r o w i n g e x t e n t o f l a m i n a r f l o w is o b t a i n e d . T h e k-8 c o m p u t a t i o n s p r e s e n t l a m i n a r f l o w d e t a c h m e n t at t h e l o w e r r a d i i . Fig. 10 c o m p a r e s n u m e r i c a l p e r f o r m a n c e c o e f f i c i e n t s t o m o d e l tests results. The 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 are w e l l p r e d i c t e d i n t h e c o m p u t a t i o n s as w e l l as t h e e f f i c i e n c y f o r Chien's k-8 m o d e l . The e f f i c i e n c y p r e d i c t i o n f o r t h e SST k-od m o d e l is s o m e w h a t u n d e r p r e d i c t e d . The e f f i c i e n c y is l a r g e r w i t h t h e k-e m o d e l d u e t o t h e l o w e r s k i n f r i c t i o n c o n n e c t e d t o l a m i n a r flow.
The g e o m e t r y v a r i a t i o n s t u d y i n t h e n e x t s e c t i o n w i l l be m a d e w i t h t h e SST k-co m o d e l , a s s u m i n g at m o d e l scale also f u l l y t u r b u l e n t flow i n t h e c o m p a r i s o n s o f t i p l o a d e d p r o p e l l e r s w i t h d i f f e r e n t plate shapes.
Table 4
Performance coefficients expressed as percentages of the fine grid solutions f o r the coarse, m e d i u m and fine grids. Chien's 1<-E turbulence model i n f u l l scale.
I<T Ka '/
Coarse 100.88 101.30 99.59
M e d i u m 100.14 100.36 99.78
Fine 100.00 100.00 100.00
Table 5
Performance coefficients expressed as percentages of the fine g n d solutions f o r the coarse, m e d i u m and fine grids. Chien's k-e turbulence model i n model scale.
KT !<<i '(
Coarse 100.53 102.68 97.908
M e d i u m 99.77 100.87 98.905
Fine 100 100 100
7. Analysis of results
The i n p u t data used f o r t h e n u m e r i c a l analysis are p r e s e n t e d i n Section 3.
7.1. Influence of grid size on performance prediction
I n general, c o m p u t a t i o n s w i t h t h e coarse meshes s h o w e d the same t r e n d s i n p e r f o r m a n c e p r e d i c t i o n t h a n those w i t h t h e fine ones, w h i c h indicates t h a t e v e n t h e coarse g r i d s c a p t u r e d t h e m a i n features o f the flow a f f e c t i n g g l o b a l forces. For e x a m p l e . Figs. 11-13 s h o w t h e i n f l u e n c e o f t h e g r i d size o n t h e e f f i c i e n c y p r e d i c t i o n f o r t h e g e o m e t r y v a r i a t i o n s s u b j e c t e d t o study. T h e y i l l u s t r a t e t h e cases o f Chien's k-e t u r b u l e n c e m o d e l i n f u l l scale, SST k-co t u r b u l e n c e m o d e l i n f u l l scale a n d SST k - o i t u r b u l e n c e m o d e l i n m o d e l scale. The l i g h t bars r e p r e s e n t c o m p u t a t i o n s o n t h e coarse g r i d , a n d t h e d a r k bars c o m p u t a t i o n s o n t h e fine g r i d . Each n u m b e r i n t h e abscissa represents a v a r i a t i o n o f e n d p l a t e shape (see Table 1). It is a p p a r e n t t h a t t h e t r e n d s are i d e n t i c a l , even t h o u g h the a b s o l u t e l e v e l o f e f f i c i e n c y changes i n a b o u t less t h a n 0.5 p e r c e n t f r o m t h e coarse t o t h e fine g r i d s f o r t h e k-e m o d e l i n f u l l scale. For t h e k-co m o d e l i n f u l l a n d m o d e l scales, t h e d i f f e r e n c e s are e v e n smaller.
S i m i l a r b e h a v i o r w a s f o u n d i n t h e c o m p a r i s o n o f t h r u s t KT a n d KQ c o e f f i c i e n t s .
7.2. Scaling of reference geo metry
Fig. 14a s h o w s t h e pressure d i s t r i b u t i o n s a n d s t r e a m l i n e s p r e d i c t e d at f u l l scale w i t h t h e SST Icco, Chien's l<e a n d S p a l a r t -A l l m a r a s t u r b u l e n c e m o d e l s f o r t h e i n p u t data p r e s e n t e d i n Section 3. There are n o q u a l i t a t i v e d i f f e r e n c e s a m o n g t h e t u r b u -lence m o d e l s . Fig. 14b s h o w s t h e c o r r e s p o n d i n g figures at m o d e l
612 A. Sanchez-Caja et al. / Ocean Engineering 8S (2014) 607-617 Cp 2,00 1.50 1.00 0 50 0.00 -0.50 -1.00 -1 .50 -2,00 ;.o -3 00 c p 2.00 1.50 1.00 0.50 0.00 0.50 1.00 1.50 2.00 2 50 3 00 L
Fig. 9. Streamlines f r o m numerical simulation on the suction ( u p ) and pressure ( d o w n ) sides o f the blade for an earlier, version o f the reference geometry under design condidon. Fully turbulent, SST k-a ( l e f t ) and partially laminar, Chien's k-e (right) flow. The k-e computation was made w i t h l o w background turbulence to obtain a pattern similar to that i n paint tests w i t h smooth surfaces.
1 ^ 0.8 o 0.6 o ^ 0 . 4 ^ 0.2 0 O p e n w a t e r c h a r a c t e r i s t i c s
\
0.2 0.4 0.6 0.1 J 1.2Fig. 10. Comparison of performance coefficients at model scale. IVIodel tests versus numerical results obtained w i t h the SST k-o and Chien's k-e turbulence models.
10 11 12 Fig. 11. Comparison of efficiency predictions for 12 endplate versions using the coarse ( w h i t e ) and fine (black) grids. Chien's k-e turbulence m o d e l i n f u l l scale.
scale. The k03 a n d the S p a l a r t A l l m a r a s t u r b u l e n c e m o d e l s p r e -s e n t a -s m a l l f l o w d e t a c h m e n t area at t h e l o w e r r a d i i o f t h e t r a i l i n g edge m a i n l y o n t h e s u c t i o n side. For t h e k-e m o d e l w i t h l o w b a c k g r o u n d t u r b u l e n c e , t h e d e t a c h m e n t area is m u c h larger a n d t y p i c a l streaiTilines p o i n t i n g s o m e w h a t o u t w a r d s o f l a m i n a r f l o w are a p p a r e n t .
Table 6 s h o w s t h e e f f e c t o f scaling o n t h e p e r f o r m a n c e c o e f f i c i e n t s f o r t h e baseline p r o p e l l e r . The percentages r e p r e s e n t
f u l l - s c a l e values r e l a t i v e t o model-scale ones. I n t h e case o f t h e SST k - w a n d S p a l a r t - A l l m a r a s t u r b u l e n c e m o d e l s , t h e f l o w is t u r b u l e n t at b o t h m o d e l a n d f u l l scale a n d t h e p r e d i c t e d scale e f f e c t o f e f f i c i e n c y f o r t h e l i g h t l y u n l o a d e d c o n d i t i o n is large. The scale e f f e c t o n t h r u s t a n d e f f i c i e n c y is a b o u t 1 p e r c e n t l o w e r f o r the S p a l a r t - A l l m a r a s m o d e l . For t h e k-e t u r b u l e n c e m o d e l , t h e flow is p a r t i a l l y l a m i n a r at m o d e l scale a n d t h e r e l a t e d l o w e r f r i c t i o n m a k e s t h e scale e f f e c t o n e f f i c i e n c y smaller.
A. Sdnchez-Caja et al. / Ocean Engineering 88 (2014) 607-617 613 0.745 0.74 0.7 35 0.75 0.725 0.72 0.715 0.71 0.705 - I 0.7 a E t , i c o d r > e l E U f i n e 1 2 3 4 5 6 7 8 9 10 13 12
Fig. 12. Comparison of efficiency predictions f o r 12 endplate versions using the coarse ( w h i t e ) and fine (blacl<) grids. SST k-e> turbulence model i n f u l l scale.
0 C.75 0.r>7 0.665 H 0.06 0.655 0.65 0.645 0.64 -0.635
1
• L-ta r o a i ' i e • l ; t a f i n t 2 7 f i 10 1 1 12Fig. 13. Comparison o f efficiency predictions for 12 endplate versions using the coarse ( w h i t e ) and fine (black) grids. The baseline is propeller no. 1. SST k-(o turbulence model i n model scale.
t i i a n at f u l l scale e v e n t h o u g h the r a n k i n g o f t h e g e o m e t r i e s f r o m t h e s t a n d p o i n t o f e f f i c i e n c y w o u l d be t h e same ( t h e b e s t g e o m e t r y is n u m b e r 1 ) . The l a t t e r one presents a d i f f e r e n t r a n k i n g : a t m o d e l scale, t h e best g e o m e t r y is n u m b e r 6 and at f u l l scale, n u m b e r 7. For g e o m e t r i e s 8 - 9 ( s w e e p ) , v a r i a t i o n s i n e f f i c i e n c y are s o m e w h a t l a r g e r at f u l l t h a n at m o d e l scale. For g e o m e t r i e s 1 0 - 1 2 , t h e r e l a t i v e changes i n m o d e l a n d f u l l scale are s i m i l a r .
Fig. 18 s h o w s t h e scale e f f e c t o n p e r f o r m a n c e c o e f f i c i e n t s f o r t h e d i f f e r e n t g e o m e t r i e s . The scaling o n t h r u s t is a r o u n d 6 p e r c e n t increase ( s l i g h t l y above) i n average. The scaling o n t o r q u e is a r o u n d m i n u s 3 - 4 percent. The scaling o n e f f i c i e n c y is a r o u n d 10 p e r c e n t increase ( s l i g h t l y above) i n average. Fig. 19 s h o w s t h e scale e f f e c t w h i c h results f r o m e x c l u d i n g t h e forces o n t h e e n d p l a t e . I t is a p p a r e n t t h a t t h e scaling o n t h r u s t is s l i g h t l y b e l o w 6 p e r c e n t a n d t h e scaling o n t o r q u e is a b o u t m i n u s 2 p e r c e n t o n average. T h e e f f i c i e n c y i m p r o v e m e n t is a r o u n d 8 p e r c e n t I n o t h e r w o r d s , t h e e f f e c t o f t h e e n d p l a t e increases t h e scaling o n e f f i c i e n c y i n 2 percent, b e i n g t h e m a i n reason t h e r e d u c t i o n o f t o r q u e .
Fig. 2 0 i l l u s t r a t e s t h e e f f e c t o f scaling o n t h e pressure a n d f r i c t i o n a l c o m p o n e n t o f t h e e n d p l a t e axial d r a g . The d r a g is n o n -d i m e n s i o n a l i z e -d i n the same w a y as t h e t h r u s t c o e f f i c i e n t , i.e. u s i n g t h e d e n s i t y , t h e square o f t h e rps a n d t h e d i a m e t e r t o the f o u r t h p o w e r . T h e f u l l - s c a l e f r i c t i o n a l c o m p o n e n t is r e d u c e d to a b o u t 47 p e r c e n t o f its model-scale m a g n i t u d e . T h e f u l l - s c a l e pressure c o m p o n e n t changes i n + 4 p e r c e n t o f its m o d e l scale v a l u e d e p e n d i n g o n t h e t y p e o f e n d p l a t e m o d i f i c a t i o n : o n l y t h e l o w p l a t e c o n t r a c t i o n cases (nos. 6 a n d 7) a n d t h e p l a t e c u t t i n g cases (nos. 1 0 , 1 1 a n d 12) result i n pressure d r a g decrease. Fig. 21 presents t h e t o t a l d r a g i n t e r m s o f m i n u s Kj.
Fig. 2 2 i l l u s t r a t e s t h e e f f e c t o f scaling o n t h e t o r q u e c o m p o n e n t o f t h e e n d p l a t e . The f u l l - s c a l e t o r q u e is r e d u c e d i n a b o u t
A/Cq=
0 . 0 0 0 3 - 0 . 0 0 0 6 f o r the d i f f e r e n t g e o m e t r i e s . I t is i n t e r e s t i n g to n o t e t h a t i n f u l l scale t h e 1 0 ° flap angle m o d i f i c a t i o n (case n o . 2 ) does n o t p r o d u c e i n c r e m e n t i n t o r q u e , c o n t r a r y t o m o d e l scale. As n o t e d i n Sanchez-Caja et al. (2014), flap angle m o d i f i c a t i o n s r e s u l t i n n e g a t i v e (bacl<ward) e n d p l a t e t h r u s t d u e t o t h e o r i e n t a t i o n o f t h e p l a t e . T h e t o r q u e also decreases since t h e p r o j e c t i o n o f t h e t o t a l e n d p l a t e f o r c e o n t h e c i r c u m f e r e n t i a l d i r e c t i o n is o p p o s i t e to t h e r e l a t i v e i n f l o w d i r e c t i o n due t o t h e plate i n c l i n a t i o n .Fig. 15 illustrates t h e v a r i a t i o n o f scaling w i t h t h e l o a d i n g c o e f f i c i e n t , Cj=8Krl{nf), w h e r e KT is t h e t h r u s t c o e f f i c i e n t a n d ] t h e advance n u m b e r . T h e SST k-o) t u r b u l e n c e m o d e l w a s used. I n general, t h e absolute percentages o f scaling o n e f f i c i e n c y , t h r u s t a n d t o r q u e are larger f o r l o w p r o p e l l e r loadings, b e i n g t h e scaling o n t o r q u e negative. A t t h e design c o n d i t i o n , t h e scaling o n e f f i c i e n c y is a r o u n d 7 - 8 percent, c o r r e s p o n d i n g t o t h e i n n e r p o i n t s i n t h e curves o f t h e figure.
7.3. Influence of geometry on performance scaling
The q u a l i t y o f the g e o m e t r y v a r i a t i o n s can be assessed b y t h e r e l a t i v e changes i n e f f i c i e n c y . The p r e d i c t i o n s o f e f f i c i e n c i e s at f u l l scale f o r t h e SST k-co t u r b u l e n c e m o d e l are s i m i l a r to those f o r Chien's k-e m o d e l . This is i l l u s t r a t e d i n Fig. 16, w h i c h c o m b i n e s t h e fine g r i d e f f i c i e n c i e s o f Figs. 11 a n d 12.
A t m o d e l scale, t h e absolute values o f e f f i c i e n c y are s m a l l e r t h a n at f u l l scale f o r t h e SST k-co m o d e l as e x p e c t e d . T h e r e l a t i v e p e r f o r m a n c e p r e d i c t i o n s at m o d e l scale s o m e w h a t d i f f e r f r o m f u l l scale. This is i l l u s t r a t e d i n Fig. 17, w h i c h c o m b i n e s the fine g r i d efficiencies o f Figs. 12 a n d 13. The d i f f e r e n c e s are v i s i b l e f o r g e o m e t r i e s 1-3 c o r r e s p o n d i n g t o changes i n flap angle a n d f o r g e o m e t r i e s 4 - 7 c o r r e s p o n d i n g t o changes i n p l a t e c o n t r a c t i o n . T l i e f o r m e r one presents l a r g e r v a r i a t i o n s i n e f f i c i e n c y at m o d e l scale
7.4. Circulation scaling
I n Sanchez-Caja e t al. (2014) t h e i n f l u e n c e o f t h e e n d p l a t e shape o n t h e r a d i a l d i s t r i b u t i o n o f c i r c u l a t i o n w a s a n a l y z e d at f u l l scale u s i n g Chien's k-e m o d e l . The shape a n d o r i e n t a t i o n o f t h e e n d p l a t e h a d a clear i n f l u e n c e o n t h e r a d i a l d i s t r i b u t i o n o f c i r c u l a t i o n a t t h e t i p . Here, w e f o c u s o n t h e i n f l u e n c e o f scaling o n t h e r a d i a l d i s t r i b u t i o n o f c i r c u l a t i o n : m o d e l scale a n d f u l l scale g e o m e t r i e s are a n a l y z e d u s i n g t h e SST k-oa m o d e l . T h e m a g n i t u d e o f t h e b o u n d c i r c u l a t i o n is d e r i v e d f r o m t h e field o f i n d u c e d t a n g e n t i a l v e l o c i t i e s b e h i n d t h e p r o p e l l e r as f o l l o w s : w h e r e UT is the^ average i n d u c e d t a n g e n t i a l v e l o c i t y at r a d i u s r b e h i n d t h e p r o p è l l e r a n d Z is t h e n u m b e r o f blades. I n t h e RANS c o m p u t a t i o n s , t h e r a d i a l l o a d i n g can be c a l c u l a t e d a t t h e v e r y t r a i l i n g edge o f t h e blade ( o r v e r y close t o i t ) c o n t r a r y t o m o d e l test m e a s u r e m e n t s . I n this w a y , t h e r a d i a l l o c a t i o n o f c i r c u l a t i o n is n o t a f f e c t e d b y flow c o n t r a c t i o n b e h i n d t h e p r o p e l l e r .
I n fact, t h e c i r c u l a t i o n as d e f i n e d i n Eq. ( 2 ) can be c a l c u l a t e d i n t w o d i f f e r e n t w a y s . I n t h e first w a y , UT is c a l c u l a t e d as t h e averaged t a n g e n t i a l v e l o c i t y over t h e c i r c u l a r arc c o n n e c t i n g t h e t r a i l i n g edges. The second w a y is t h e same b u t e x c l u d i n g t h e w a k e
614 A. Sanchez-Caja et al. / Ocean Engineering 88 (2014) 607-617 FS k-(o FS k-e FS SA MS k-O) MS k-e MS SA
Fig. 14. (a) Comparison of full-scftje pressure distributions and streamlines on the suction ( r i g h t ) and pressure (left) sides o f the blade for the reference geometry w i t h SST k-M, Chien's k-e and Spalart-Allmaras (SA) turbulence models. F S = f u l l scale, (b) Comparison o f m o d e l scale pressure distributions and streamlines on the suction ( r i g h t ) and pressure ( l e f t ) sides o f the blade for the reference geometry w i t h SST k-co, Chien's k-e and Spalart-Allmaras (SA) turbulence models. IV1S= m o d e l scale.
Table 6
Scale Effect on performance coefficients w i t h SST k-co, Chien's k-e and Spalart-Allmaras (SA) turbulence models.
Scale effect
A/C
T
( % ) M<<li%)Ic-oi 6.3 - 3 . 4 10.0 k-e 2.1 - 3 . 0 5.4 SA 4.9 - 3 . 7 8.9
I
12 10 8 64
2
0 -2 4 - - . - - . 1 j 1 4 - - . - - . 1 _ e - K t - A - K q - B - r | | 4 - - . - - . 1 r1 1
: l ; • - | — ^ !1
1
1 0.5 1.5C
T
2.5Fig. 15. Influence o f propeller loading o n scaling i n percentages relative to model scale values. SST k-co turbulence model.
r e g i o n j u s t b e l i i n d t l i e t r a i l i n g edge o f t l i e blade, i.e. t h e v e l o c i t y is averaged o n l y i n the " p o t e n t i a l - f l o w " r e g i o n o f t h e arc. The f o r m e r w a y y i e l d s a t o t a l c i r c u l a t i o n t h a t i n c l u d e s b o t h a " p o t e n t i a l - f l o w " c o m p o n e n t a n d a " v i s c o u s - f l o w " c o m p o n e n t . The second one i n c l u d e s o n l y t h e " p o t e n t i a l - f l o w " c o m p o n e n t
I f w e i n t e n d t o c o m p a r e RANS-based c i r c u l a t i o n s w i t h those o b t a i n e d u s i n g p o t e n t i a l f l o w t h e o r y ( f o r e x a m p l e , l i f t i n g l i n e ) , w e s h o u l d use t h e second w a y o f c a l c u l a t i o n . W i t h t h i s a p p r o a c h , w e are able to assess to w h a t e x t e n t t a r g e t d i s t r i b u t i o n s o f c i r c u l a t i o n i n a p r o p e l l e r design are a c t u a l l y a c c o m p l i s h e d .
0.74
0.72
0.7
-0.68
0.66
-0,64
0.62
0.6
• EtaKOFS • Eta KEFS -1 1— 7-1 1 1 1— 'r
I4 5 6 7 8 9 ao 11 12
Fig. 16. Comparison o f efficiency predictions for 12 endplate versions at f u l l scale using the SST k-co ( w h i t e ) and Chien's l o w Reynolds n u m b e r k-e (black) turbulence model. The baseline is propeller no. 1.Figs. 2 3 - 2 6 i l l u s t r a t e t h e i m p a c t o f scaling o n the radial d i s t r i b u t i o n s o f c i r c u l a t i o n f o r t h e cases o f t h e baseline propeller, t h e p r o p e l l e r w i t h m i n i m u m c o n t r a c t i o n , t h a t w i t h m a x i m u m plate c u t t i n g , a n d t h a t w i t h 3 0 ° flap angle, respectively. The c i r c u l a t i o n is iTiade n o n - d i m e n s i o n a l w i t h 2nRV, R b e i n g the p r o p e l l e r radius a n d V t h e i n f l o w v e l o c i t y . B o t h t h e t o t a l and the p o t e n t i a l - f l o w (PF) d i s t r i b u t i o n s o f c i r c u l a t i o n are p r e s e n t e d at m o d e l ( M S ) a n d f u l l (FS) scale.
I n general, t h e total c i r c u l a t i o n curves are v e r y s i m i l a r at the i n n e r r a d i a l stations, i n d e p e n d e n t l y o f t h e scale. I n contrast, a t the o u t e r r a d i a l stations, t h e y are h i g h e r at m o d e l scale t h a n at f u l l scale, w i t h p o s i t i v e slopes o f c i r c u l a t i o n l a r g e r at m o d e l scale. Viscous d r a g a f f e c t i n g t h e v e l o c i t y field i n t h e w a k e o f t h e blades is t h e m a i n responsible f o r t h e d i f f e r e n c e s i n t o t a l c i r c u l a t i o n .
The potential flow c i r c u l a t i o n curves at m o d e l scale are l o w e r t h a n a t f u l l scale over m o s t o f t h e s p a n w i t h larger average slopes.
fl. Sanchez-Caja et al / Ocean Engmeering 88 (2014) 607-617 615
10 1 1 12 Fig. 17. Comparison of efficiency predictions f o r 12 endplate versions using the SST k-a turbulence m o d e l . M o d e l ( w h i t e ) and f u l l (black) scales. The baseline is propeller no. 1. 11.0 9.0 - I 7.0 5.0 - I i.O 1.0 -1.0 -3.0 • 5.0 -B l a d e Including e n d p l a t e • « E t a 1 8 9 10 1 1 12
Fig. 18. Scale effect o n performance coefficients f o r 12 CLT propellers w i t h d i f f e r e n t endplates using t h e SST k-a turbulence model. Full scale relative to m o d e l scale values i n percentages. The baseline is propeller no. 1.
11.0 9.0 - I 7.0 5.0 3,0 1,0 -1.0 -3.0 •] -5.0 B l a d e m i n u s e n d p l a t e • % K t P % K i | • 'VTt.i 1 5 10 11 12
Fig. 19. Scale effect i n percentages o n performance coefficients f o r 12 CLT propellers w i t h d i f f e r e n t endplates using the SST k-co turbulence model. Forces o n the endplate are n o t considered. The baseline is propeller no. 1.
0.0180 0.0160 0.0140 H 0.0120 0.0100 H 0.0080 0.0060 H 0.0040 0.0020 0.0000 • PresMirea.-vial tIrag-ivIS • ['re-:.stM"e axial drag-FS • F r i c t i o n a l a,>:ifll (Irag-MS i F r i c t i o n a l axial <lrag-FS 3 10 11 12 0.0040 0.0020 0.0000
Ql
D m i n u s K t e n t l p l a t e MS • n i i i u i s K t e n d p l a t e FS Fig. 20. M o d e l versus f u l l scale f r i c d o n a l and pressure components o f endplate axial drag f o r 12 shape variations using the SST k-co turbulence m o d e l . Forces are expressed i n terms o f minus Kr. The baseline is propeller no. 1.0.0200 0.0180 0.0160 0.0140 0.0120 0.0100 0.0080 -0.0060 10 11 12 Fig. 21. Scale effect o n endplate axial drag f o r 12 shape variations using the SST k-co turbulence model. Drag is expressed i n terms o f minus 7Cr. The baseline is propeller no. 1.
10 11 Fig. 22. Scale effect o n endplate torque f o r 12 shape variations u s i n g the SST l<-co turbulence model. The baseline is propeller no. 1.
L a r g e r slopes i n potential-flow c i r c u l a t i o n d i s t r i b u t i o n s are i n d i c a t i v e o f l a r g e r i n d u c e d d r a g . C o n s e q u e n t l y , t h e p r o p e l l e r i n d u c e d d r a g a t m o d e l scale is s o m e w h a t l a r g e r t h a n t h a t a t f u l l scale i n a l l cases. T h e s e o b s e r v a t i o n s a r e i n l i n e w i t h t h e f i n d i n g s i n S a n c h e z - C a j a e t a l . ( 2 0 1 4 ) t h a t p r o p e l l e r s n u m b e r 7 ( F i g . 2 4 ) a n d n u m b e r 10 (Fig. 2 5 ) h a v e b e t t e r e f f i c i e n c y at f u l l scale t h a n p r o p e l l e r s n u m b e r 1 ( F i g . 2 3 ) a n d n u m b e r 3 ( F i g . 2 6 ) , b e i n g t h e s m a l l e r d r a g o f t h e f o r m e r o n e s ( s m a l l e r c i r c u l a t i o n s l o p e s ) t h e r e a s o n f o r b e t t e r e f f i c i e n c y . T h e f i g u r e s p r e s e n t s o m e d i s t u r b a n c e s i n t h e c i r c u l a t i o n a t t h e p r o p e l l e r t i p because s o m e p o i n t s u s e d f o r t h e e v a l u a t i o n o f c i r c u l a t i o n f a l l i n s i d e t h è w a k e b e h i n d t h e e n d p l a t e s .
616 A. Sanchez-Caja et al. / Ocean Engineering 88 (2014) 607-617
Case 1
Case 3
0.040 0.030 0.020 4 0.010 0.000 • F S Circulation MS Circulation ' • F S P F Circulation • M S P F Circulation 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Fig. 23. Radial distribution o f bound circulation f o r the baseline propeller (1).
0.040
0.020
Fig. 26. Radial distribution o f bound circuladon f o r the propeller w i t h largest flap angle (3). 0.040 0.030 + 0.020
Case 7
0.010 0.000 c 1 T i -F S Circulation MS Circulation - - - F S P F Circulation — — MS P F Circulation i -F S Circulation MS Circulation - - - F S P F Circulation — — MS P F Circulation 1 . •'• . . .i
Jt'^ 1 1S/^l.-S" i 1
\ \ 0.3 0.4 0.5 0.6 0.7 0.8 0.9Fig. 24. Radial distribution of bound circulation f o r the propeller w i t h lowest contraction (7).
Case 12
0.040 0.030 0.020 0.010 0.000 — F S Circulation MS Circulation F S P F Circulation MS P F CirculationFig. 25. Radial distribution of bound circulation f o r the propeller w i t h largest c u t d n g (12). The b l a d e g e o m e t r y r e m a i n e d u n t o u c h e d i n t h e d i f f e r e n t p r o p e l l e r v e r s i o n s . N o t e t h a t c o n t r a r y to t h e p o t e n t i a l f l o w c i r c u l a t i o n curves, t h e t o t a l c i r c u l a t i o n curves c a n n o t be d i r e c t l y r e l a t e d t o t h e a m o u n t o f t h r u s t d e v e l o p e d b y t h e p r o p e l l e r . This is d u e t o t h e s p u r i o u s viscous c o m p o n e n t o f c i r c u l a t i o n p r e s e n t i n t h e t o t a l c i r c u l a t i o n .
8. Discussion
Several v a r i a t i o n s o f e n d p l a t e g e o m e t r y have been t e s t e d f r o m t h e s t a n d p o i n t o f scaling u s i n g code FINFLO w i t h t h e pressure c o r r e c t i o n m e t h o d . The s t u d y has been m a d e m a i n l y w i t h t h e SST k-co t u r b u l e n c e m o d e l , e v e n t h o u g h some c o m p u t a t i o n s w e r e m a d e also w i t h Chien's k-e t u r b u l e n c e m o d e l h k e the analysis o f all f u l l scale cases a n d o f t h e m o d e l scale case f o r t h e baseline g e o m e t r y , a n d w i t h t h e S p a l a r t - A l l m a r a s t u r b u l e n c e m o d e l f o r the baseline g e o m e t r y . The Reynolds n u m b e r at m o d e l scale is 10*^ and at f u l l scale 5*10''. The c o m p u t a t i o n s w i t h t h e SST k-co m o d e l w e r e all f u l l y t u r b u l e n t a n d presented v e r y l i m i t e d t u r b u l e n t separation at t l i e t r a i l i n g edge close t o t h e h u b a t m o d e l scale. C o m p u t a t i o n s w i t h Chien's k-e m o d e l a t m o d e l scale w e r e m a d e w i t h l o w b a c k g r o u n d t u r b u l e n c e i n o r d e r to have a n u n d e r s t a n d i n g of trends i n s c a l i n g effects w i t h p a r t i a l l a m i n a r f l o w . The c o m p u t a -t i o n s p r e s e n -t e d a-t -t h e l o w e r r a d i i p o r -t i o n s o f l a m i n a r f l o w w i -t h s t r e a m l i n e s p o i n t i n g s o m e w h a t o u t w a r d f r o m t h e axis f o l l o w e d by f l o w d e t a c h m e n t . Due t o t h e l o w e r f r i c t i o n a l d r a g c o n n e c t e d to l a m i n a r f l o w , t h e m o d e l scale p e r f o r m a n c e is closer t o t h e f u l l -scale one, p r o v i d e d t h a t f l o w s e p a r a t i o n is n o t extensive. Extensive f l o w s e p a r a t i o n at m o d e l scale w o u l d r e s u l t i n d e t e r i o r a t i o n o f p e r f o r m a n c e , v i s i b l e as a r e d u c t i o n o f t h r u s t a n d t o r q u e c o e f f i -cients. A t f u l l scale, b o t h t u r b u l e n t m o d e l s y i e l d s i m i l a r results.
Even t h o u g h t h e r a n k i n g o f t h e v a r i o u s g e o m e t r i e s f r o m t h e s t a n d p o i n t o f e f f i c i e n c y is s i m i l a r f o r m o s t o f t h e a l t e r n a t i v e e n d p l a t e shapes w h e n u s i n g e i t h e r m o d e l - or f u l l - s c a l e results, d i f f e r e n t trends are f o u n d i n some cases. I n p a r t i c u l a r , f o r plate c o n t r a c t i o n , m o d e l scale results r a n k g e o m e t r y n u m b e r 6 as the best o n e w h e r e a s f u l l - s c a l e results r a n k n u m b e r 7. For s w e e p and f l a p angle v a r i a t i o n s , t h e r e l a t i v e i n c r e m e n t s i n e f f i c i e n c y also vary f r o m m o d e l t o f u l l scale. For plate c u t t i n g , r e l a t i v e i n c r e m e n t s i n e f f i c i e n c y are s i m i l a r .
For t h e m o d e r a t e l y u n l o a d e d o f f d e s i g n c o n d i t i o n , t h e SST k-co t u r b u l e n c e m o d e l p r e d i c t s a scaling o f a b o u t 6 p e r c e n t o n t h r u s t a n d o f a b o u t m i n u s 4 p e r c e n t o n t o r q u e , w h i c h a m o u n t s t o a 10 p e r c e n t i n e f f i c i e n c y . The S p a l a r t - A l l m a r a s m o d e l y i e l d s s i m i l a r results a l t h o u g h t h e s c a l i n g i n t h r u s t was 1 p e r c e n t less, w h i c h r e s u l t e d i n 9 p e r c e n t scaling o n e f f i c i e n c y . The k-e m o d e l w i t h p a r t i a l l a m i n a r f l o w predicts a scale e f f e c t o f 5 p e r c e n t i n e f f i c i e n c y . The s c a l i n g o n e f f i c i e n c y is s t r o n g l y d e p e n d e n t o n the t y p e o f f l o w r e g i m e at m o d e l scale.
W h e n forces o n t h e blade alone w i t h o u t e n d p l a t e are c o n s i d -ered, t h e scale e f f e c t o n e f f i c i e n c y w i t h t h e SST k-CD m o d e l d r o p s f r o m 10 to 8 p e r c e n t The scaling o f d r a g o n t h e e n d p l a t e is responsible f o r t h e d i f f e r e n c e . C o m p a r i n g to ordinai-y p r o p e l l e r s .
A. S&nchez-Caja et al. / Ocean Engineering 8S (2014) 607-617 617
t h e CLT propeller s c a l i n g o n e f f i c i e n c y is e x p e c t e d t o be f u r t h e r l a r g e r t h a n t h a t 2 p e r c e n t (10 m i n u s 8 ) f o r t h i s p a r t i c u l a r w o r k i n g p o i n t T h i s is because c o n v e n t i o n a l p r o p e l l e r s are less l o a d e d a t t h e t i p , w h i c h w o u l d m a k e m o s t p r o b a b l y t h e i r o w n s c a l i n g e f f e c t u n d e r s i m i l a r w o r k i n g c o n d i t i o n s l o w e r t h a n 8 p e r c e n t .
T h e scaling o n the f r i c t i o n a l c o m p o n e n t o f a x i a l d r a g is s i m i l a r f o r t h e various endplates, a r o u n d 4 5 - 4 9 p e r c e n t r e d u c t i o n . O n t h e o t h e r h a n d , t h e pressure c o m p o n e n t o f d r a g changes i n + 4 p e r c e n t o f its m o d e l scale value d e p e n d i n g o n t h e t y p e o f e n d p l a t e m o d i f i c a t i o n . The l a t t e r is usually larger i n absolute value t h a n t h e f o r m e r , a n d t h e r e f o r e , i t is the m a i n c o n t r i b u t o r t o changes i n e f f i c i e n c y a m o n g t h e d i f f e r e n t endplates. I n o t h e r w o r d s , pressure forces o n t h e e n d p l a t e s h o u l d be c o n t r o l l e d i n t h e shape o p t i m i z a t i o n process. T h e e f f e c t o f s c a l i n g o n t h e r a d i a l d i s t r i b u t i o n o f c i r c u l a t i o n o v e r t h e b l a d e span is i l l u s t r a t e d f o r s o m e e n d p l a t e s . I n g e n e r a l , a l l t h e curves p r e s e n t l a r g e r p o s i t i v e slopes o f c i r c u l a t i o n at m o d e l scale t h a n a t f u l l scale. C o n s e q u e n t l y , t h e p r o p e l l e r v i s c o u s a n d i n d u c e d d r a g c o m p o n e n t s a t m o d e l scale are l a r g e r t h a n a t f u l l scale.
9. Conclusions
T h e i n c o m p r e s s i b l e v i s c o u s f l o w a r o u n d CLT p r o p e l l e r s w i t h d i f f e r e n t t y p e s o f e n d p l a t e s has b e e n s i m u l a t e d b y s o l v i n g t h e RANS e q u a t i o n s w i t h v a r i o u s t u r b u l e n c e m o d e l s u s i n g t h e p r e s -s u r e c o r r e c t i o n m e t h o d . T h e FINFLO code w a -s u -s e d f o r t h e c a l c u l a t i o n s . A g r i d s t u d y w a s m a d e u s i n g 3 g r i d s w i t h r a t i o s 1, 1.41 a n d 2 f o r t h e c o m p u t a t i o n s m a d e a t ' m o d e l a n d f u l l scales. T h e finest g r i d s c o n t a i n e d a b o u t 2.4 m i l l i o n cells. T r e n d s i n p e r f o r -m a n c e c o e f f i c i e n t s s c a l i n g w e r e s h o w n f o r several t y p e s o f shape v a r i a t i o n s . I m p o r t a n t f e a t u r e s o f t h e f l o w a f f e c t i n g s c a l i n g w e r e i d e n t i f i e d o n a basis o f p r e s s u r e d i s t r i b u t i o n s a n d o v e r a l l forces o n t h e blades a n d e n d p l a t e s . T h e s t u d y reveals also h o w t h e r a d i a l d i s t r i b u t i o n o f c i r c u l a t i o n is a f f e c t e d b y scaling. T h e d i f f e r e n c e s b e t w e e n t h e c o m p u t a t i o n a l r e s u l t s o b t a i n e d u s i n g e i t h e r f u l l y t u r b u l e n t o r p a r t i a l l y l a m i n a r f l o w at m o d e l scale a n d t h e i r c o r r e s p o n d i n g s c a l i n g are i n d i c a t i v e o f t h e i m p o r t a n c e o f t a k i n g i n t o a c c o u n t t h e right t y p e o f flow r e g i m e i n m o d e l t e s t e x t r a p o l a t i o n p r o c e d u r e s . A w a y o f e x t r a c t i n g t h e " p o t e n t i a l flow" c o m p o n e n t o f t h e r a d i a l d i s t r i b u t i o n o f c i r c u l a t i o n is p r e s e n t e d , w h i c h a l l o w s asses-s i n g t o w h a t e x t e n t t a r g e t d i asses-s t r i b u t i o n asses-s o f c i r c u l a t i o n i n a p r o p e l l e r d e s i g n are a c t u a l l y a c c o m p l i s h e d i n a viscous flow c o n t e x t (RANS), D i f f e r e n c e s f o u n d b e t w e e n m o d e l a n d f u l l scales i n r a n k i n g a l t e r n a t i v e designs m a k e m o d e l scale analysis q u e s t i o n a b l e f o r s o m e t y p e o f m o d i f i c a t i o n s w h e n f u l l - s c a l e p e r f o r m a n c e is s o u g h tAclatowledgments
T h i s w o r k has b e e n m a d e w i t h i n t h e E u r o p e a n U n i o n TRIPOD p r o j e c t u n d e r t h e 7 t h f r a m e w o r k p r o g r a m ( G r a n t # 2 6 5 8 0 9 ) . T h e a u t h o r s w i s h t o t h a n k t h e p a r t n e r s i n t h e TRIPOD c o n s o r t i u m f o r t h e i r s u p p o r t a n d o b s e r v a t i o n s . P a r t i c u l a r l y t h a n k s are g i v e n t o Rasmus Folso a n d M a a r t e n N i j l a n d f r o m A.P. M o l l e r - M a e r s k , T o m i V e i k o n h e i m o f r o m A B B , R a m o n Q u e r e d a a n d J a i m e M a s i p f r o m CEHIPAR a n d A i t o r A u r i a r t e f r o m C i n t r a N a v a l - D e f C a r .References
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