Applied Ocean Research 54 (2016) 26-38
E L S E V I E R
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Applied Ocean Research
journal homepage: www.elsevier.com/locate/apor
A P P L I E D O C E A N
RESEARCH
Numerical investigation of the scale effect of hydrodynamic
performance of the hybrid CRP pod propulsion system
Zhan-Zhi Wang*, Ying Xiong, Rui VVang, Chen-liua Zhong
Department of Naval Architecture, Naval University of Engineering, Wulian, ChinaCrossMark
A R T I C L E N F 0 A B S T R A C T
Article history:
Received 2 November 2014
Received in revised form 21 October 2015 Accepted 23 October 2015
Available online 7 December 2015
Keywords:
The hybrid CRP pod propulsion system Scale effect
Hydrodynamic performance RANS
The scale effect of hydrodynamic performance of the hybrid CRP pod propulsion system was investi-gated numerically using the RANS method combined w i t h SST li-a> turbulence model and m o v i n g mesh method. The pod resistance influence factor was introduced to represent the effect of wake field of CRP on the pod resistance. Results showed the pod resistance influence factor to be a function o f t h e Reynolds number and revolution ratio. Representative function expression can be obtained by regression analysis using multiplication of multinomial polynomials and linear function. The standard ITTC 1978 extrapo-lation procedure can be utilized to predict hydrodynamic performance of forward propeller because of the slightness of the influence of the pod unit on the f o r w a r d propeller. The thrust and torque coeffi-cient influence factors of aft propeller were introduced, and they were found to represent the effect of wake field of f o r w a r d propeller and blockage effect of the pod on the hydrodynamic performance o f aft propeller. It shows that thrust and torque coefficient influence factors are independent o f t h e Reynolds number and have a linear relationship w i t h the revolution ratio. On this basis, a method of estimating the hydrodynamic performance was proposed f o r f u l l scale propulsion system.
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1. Introduction
The h y b r i d CRP p o d p r o p u l s i o n s y s t e m is a n e w c o n t r a - r o t a t i n g p r o p u l s o r c o n s i s t i n g o f a f o r w a r d c o n v e n t i o n a l s h a f t p r o p e l l e r a n d a n a f t p o d d e d p r o p u l s o r . I n t h e b a c k g r o u n d o f "Green Ship", t h e h y b r i d CRP p o d p r o p u l s i o n s y s t e m has a b r i g h t f u t u r e d u e t o t h e h i g h p r o p u l s i v e e f f i c i e n c y , g o o d m a n e u v e r a b i l i t y , f l e x i b l e a r r a n g e m e n t , a n d l o w e x h a u s t e m i s s i o n r a t e . The h y b r i d CRP p o d p r o p u l s i o n s y s t e m has b e c o m e t h e f o c u s i n m a r i n e p r o p u l s i o n [ 1 - 8 ] . H o w e v e r , i t is n o t easy t o use i n p r a c t i c a l s i t u a t i o n d u e t o t h e lack o f a n y s a t i s f a c t o r y p r o c e d u r e f o r t h e p r e d i c t i o n o f h y d r o -d y n a m i c p e r f o r m a n c e at f u l l scale. The s t u d y o f t h e scale e f f e c t o f p r o p e l l e r ' s h y d r o d y n a m i c p e r f o r -m a n c e has b e c o -m e -m o r e a n d -m o r e i n t e n s i v e d u e t o t h e c o n t i n u o u s i n v e s t i g a t i o n t h a t has b e e n c o n d u c t e d o v e r t h e past several years, a n d s o m e c o r r e c t i o n p r o c e d u r e s h a v e b e e n p r o p o s e d . The m o s t r e p r e s e n t a t i v e m e t h o d is t h e s t a n d a r d ITTC 1978 e x t r a p o l a t i o n p r o c e d u r e [ 9 ] . I t uses t w o g l o b a l c o r r e c t i o n s , one f o r 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 o t h e r f o r t h e t o r q u e c o e f f i c i e n t . These c o r r e c -t i o n s c o n s i d e r -t h e i n f l u e n c e o f -t h e R e y n o l d s n u m b e r , -t h e p r o f i l e t h i c l m e s s r a t i o , t h e p i t c h r a t i o , t h e blade's n u m b e r , a n d o t h e r f a c -t o r s . I n -t h i s w a y , -t h e s -t a n d a r d ITTC 1 9 7 8 e x -t r a p o l a -t i o n p r o c e d u r e c a n n o t e a s i l y be used t o c o n s i d e r t h e l o c a l flow c o n d i t i o n , so i t is* Corresponding author. Tel.: +86 15807182543; fax: +86 02765461615. E-mail address: wzz200425@126.com (Z.-Z. Wang).
0141-1187/$ - see front matter ® 2015 Elsevier Ltd. All rights reserved. http://dx.doi.0rg/10.1016/j.apor.2015.10.006
n o t a p p r o p r i a t e f o r s o m e Idnds o f special p r o p e l l e r s , l i k e K a p p e l p r o p e l l e r o r CLT p r o p e l l e r .
The r a p i d d e v e l o p m e n t o f CFD has m a d e i t possible f o r t h e p r e d i c t i o n o f p r o p e l l e r s ' h y d r o d y n a m i c p e r f o r m a n c e at f u l l scale, a n d p r o v i d e s a n e w w a y t o s t u d y t h e scale e f f e c t o f p r o p e l l e r s . S t a n i e r [ 1 0 ] , Sanchez [ 1 1 ] , Li e t al. [ 1 2 ] , M u l l e r e t a l . [ 1 3 ] , a n d Krasilnil<ov et a l . [ 1 4 ] h a v e a n a l y z e d t h e scale e f f e c t o f p r o p e l l e r s ' h y d r o d y n a m i c p e r f o r m a n c e u s i n g t h e RANS m e t h o d , a n d t h e y find t h a t t h e viscous f o r c e is d e p e n d e n t o n Reynolds n u m b e r a n d has i n f l u e n c e s o n b o t h t h r u s t a n d t o r q u e . The p r o p e l l e r t h r u s t c o e f f i -c i e n t at f u l l s-cale is g r e a t e r t h a n at m o d e l s-cale, a n d t h i s d i f f e r e n -c e is a f f e c t e d b y p r o p e l l e r g e o m e t r y a n d l o a d i n g . Pressure f o r c e also has a p r o n o u n c e d scale e f f e c t . This c o n t r a d i c t s t h e c o n c l u s i o n d r a w n f r o m t r a d i t i o n a l a i r f o i l t e s t i n g w h i c h suggests t h a t t h e R e y n o l d s n u m b e r has l i t t l e e f f e c t o n t h e l i f t f o r c e c o e f f i c i e n t .
The scale e f f e c t o f p o d resistance is a m a i n p r o b l e m i n h y d r o -d y n a m i c p e r f o r m a n c e c o r r e c t i o n f o r f u l l scale p o -d -d e -d p r o p u l s o r s . The f o r c e a c t i n g o n t h e p o d h o u s i n g has v e r y d i f f e r e n t charac-t e r i s charac-t i c s f r o m charac-t h a charac-t u n d e r u n i f o r m flow c o n d i charac-t i o n due charac-t o charac-t h e flow a c c e l e r a t i o n , p r e s s u r e changes, a n d s w i r i i n g w a k e flow, w h i c h are c a u s e d b y t h e r o t a t i n g p r o p e l l e r . So far, C h i c h e r i n et a l . [ 1 5 ] a n d Park e t al. [ 1 6 ] h a v e i n v e s t i g a t e d t h e scale e f f e c t o f h y d r o d y n a m i c p e r f o r m a n c e o f p o d d e d p r o p u l s o r s n u m e r i c a l l y a n d e x p e r i m e n t a l l y a n d b o t h t e a m s h a v e d r a w n s o m e u s e f u l c o n c l u s i o n s . K r a s i l n i k o v et al. [ 1 7 ] have p r o p o s e d a f u l l scale p r e d i c t i o n m e t h o d f o r p o d -d e -d p r o p u l s o r s . The 2 5 t h ITTC Specialist C o m m i t t e e o n A z i m u t h i n g P o d d e d P r o p u l s i o n [ 1 8 ] has p r o p o s e d a n e x t r a p o l a t i o n m e t h o d f o r
Z.-Z. Wang et al./Applied Ocean Research 54 (2016) 26-38 27 Table 1 Pod parameters. Parameter Value Pod diameter/(mm) 90.9 Pod length/(mm) 278.65 Strut height/(mm) 209.1
Strut chord length/(mm) 145.46
Strut width/(mm) 45.35
Table 2
Parameters of forward and aft propellers.
Parameter Forward propeller Aft propeller
Diameter/(mm) 240 203.636
Propeller blade number 4 5
PO.7RID 1.1622 1.3027
Hub diameter ratio 0.227 0.216
Section camber and thickness NACA66mod/a = 0.8 NACA66mod/a = 0.8
Rotating direction Left Right
Table 3
Computational parameters of the hybrid CRP pod propulsion system at different scales.
k Hf (r/min) Re y Cell number
27.5 1200 1.146x10^ 150 3454994 16 600 1.693 X 150 5048756 8 420 4.742 X 10^^ 200 8797360 4 300 1.355x10' 250 17691716 2 210 3.793x10' 300 44677288 1 150 1.084 x 10^ 300 55038368 t h i s basis, a m e t h o d o f e s t i m a t i n g h y d r o d y n a m i c p e r f o r m a n c e o f p r o p u l s i o n s y s t e m at f u l l scale is here p r o p o s e d .
2. Numerical model
2.1. Govertiing equations0
Fig. 1. Geometry ofthe hybrid CRP pod propulsion system.
The h y b r i d CRP p o d p r o p u l s i o n s y s t e m w a s a s s u m e d t o o p e r -ate u n d e r u n i f o r m f l o w c o n d i t i o n . The g o v e r n i n g e q u a t i o n s f o r t h e t u r b u l e n t f l o w f i e l d a r o u n d t h e p r o p u l s i o n s y s t e m w e r e t h e i n s t a n t a n e o u s c o n s e r v a t i o n o f mass ( c o n t i n u i t y e q u a t i o n ) a n d m o m e n t u m (Reynolds averaged Navier-Stol<es e q u a t i o n , RANS). These e q u a t i o n s can be expressed as f o l l o w s :
9. N
9 ,
dp 'dX: _9_ 2 dui „9
' 9X; ^ 9Xj (1) (2) p o d d e d p r o p u l s o r . This m e t h o d s p l i t s t h e p o d u n i t i n t o the p o d d e d p r o p e l l e r a n d the p o d a n d t h e n c o r r e c t s b o t h o f t h e m . The s t a n d a r d ITTC 1978 e x t r a p o l a t i o n p r o c e d u r e is used f o r t h e c o r r e c t i o n f o r p o d d e d p r o p e l l e r , w h i l e t h e f o r m f a c t o r a p p r o a c h is used f o r t h e p o d h o u s i n g resistance. This m e t h o d is based o n e m p i r i c a l f o r m u l a a n d needs t o be v e r i f i e d b y m a n y p r o p e l l e r s at f u l l scale.I n t h i s paper, h y d r o d y n a m i c p e r f o r m a n c e s o f a h y b r i d CRP p o d p r o p u l s i o n s y s t e m at d i f f e r e n t scales are c a l c u l a t e d u s i n g t h e RANS m e t h o d c o m b i n e d w i t h SST k - o ) t u r b u l e n c e m o d e l a n d m o v i n g m e s h m e t h o d . V a r i a t i o n s o f t h e l o c a l f l o w f i e l d a n d h y d r o d y n a m i c p e r f o r m a n c e w i t h R e y n o l d s n u m b e r are a n a l y z e d i n d e t a i l . O n
Here, a l l t h e v a r i a b l e s are t i m e - a v e r a g e d , u,, p, p, IXQJU ~pu\u'. are v e l o c i t y , f l u i d d e n s i t y , static pressure, f l u i d v i s c o s i t y , b o d y f o r c e p e r u n i t v o l u m e a n d Reynolds stress, r e s p e c t i v e l y . A n a d d i t i o n a l e q u a t i o n is n e e d e d i n o r d e r t o s o l v e t h e u n k n o w n R e y n o l d s stress. T u r b u l e n c e m o d e l is t h e c l o s u r e e q u a t i o n w h i c h c o m b i n e s t h e f l u c t u a t i n g a n d t i m e average. H e r e , SST iico t u r b u -lence m o d e l is selected. SST fc - w t u r b u l e n c e m o d e l w a s d e v e l o p e d b y M e n t e r [ 1 9 ] t o e f f e c t i v e l y b l e n d t h e r o b u s t a n d a c c u r a t e f o r m u -l a t i o n o f t h e s t a n d a r d fc - o) m o d e -l i n t h e n e a r - w a -l -l r e g i o n w i t h t h e f r e e - s t r e a m i n d e p e n d e n c e o f t h e s t a n d a r d fc-e m o d e l i n t h e f a r f i e l d .
23 Z.-Z. Wang et al./Applied Ocean Research 54 (2016) 26-38
(a) Forwai-d propeller
.-i-ijrr-rt-i-i-fi
(c) The pod
Fig. 3. Comparison of grid cut plane between model scale and full scale.
The t u r b u l e n c e k i n e t i c e n e r g y k a n d t h e s p e c i f i c d i s s i p a t i o n r a t e CO w e r e o b t a i n e d f r o m t h e f o l l o w i n g t r a n s p o r t e q u a t i o n s : 9"j o ,
3
d X j [ U.L + Okll) dxj ( 3 ) an | ( / ) c o ) + | - ( p . . u , ) ft) dUj o l d + 2 ( 1 - F , ) p 1 k "'J dx dk dco ill + o - „ / i ) dXj « f f i u 2 dXj dXj ( 4 ) w h e r e fc = 0 . 4 1 , Q' = F i a , + ( 1 - F i ) q ; 2 , i 3 = F i f t + ( 1 - F , cr;( = F i o - k i + ( l - F i ) c r f c 2 . ffa =FiCTa,!+(1 - F , ) c r f t , 2 , Fi = t a n h ( a r g 4 ) , o r g = m i n [ m a x ( v T < / 0 . 0 9 f t ) y , 500ii/pcoy^), {4k/iaaj2Dt>y'^))]. Dt = maxl[2pi\/a^2(^){dk/dxj){dca/dxj)), 1 0 - ' ° ] . cr^t = 1-176, o - „ i = 2 , ^ 1 = 0 . 0 7 5 , a : i = 0 . 3 1 , ;8* = 0.09, 0:2 = 0.4404, a „ 2 = 1.0, 0-0,2 = 1.168, a n d P2 = 0.0828. 2.2. Geometry T h e h y b r i d CRP p o d p r o p u l s i o n s y s t e m w a s d e s i g n e d f o r a 4 0 0 0 T E U c o n t a i n e r s h i p b y N a v a l U n i v e r s i t y o f E n g i n e e r i n g . I t c o n -sists o f c o n v e n t i o n a l s h a f t p r o p e l l e r a n d a p o d d e d p r o p u l s o r . The p o d i n c l u d e s a p o d h o u s i n g a n d a s t r u t . The m a i n p a r a m e t e r s o f t h e p o d are l i s t e d i n Table 1 ; m a i n p a r a m e t e r s o f t h e f o r w a r d a n d a f t p r o p e l l e r s are l i s t e d i n Table 2, a n d t h e g e o m e t r y o f t h e h y b r i d CRP p o d p r o p u l s i o n s y s t e m is s h o w n i n Fig. 1. T h e a x i a l s p a c i n g d i s t a n c e b e t w e e n f o r w a r d a n d a f t p r o p e l l e r ' s d i s k is 0 . 4 5 4 5 D f , w h e r e Df is t h e d i a m e t e r o f t h e f o r w a r d p r o p e l l e r a n d H/i/np is the d e s i g n e d r e v o l u t i o n r a t i o o f t h e a f t p r o p e l l e r a n d f o r w a r d p r o p e l l e r , w h i c h is here 1.104.2.3. Grid systeirt and boundary condition
The c o m p u t a t i o n a l d o m a i n is s h o w n i n Fig. 2 . I t is a c u b o i d o f l e n g t h 15 Lpod w i t h t h e i n l e t b o u n d a r y l o c a t e d 5 Lpoj u p s t r e a m o f a f t p r o p e l l e r ' s dislc, t h e o u t l e t b o u n d a r y 10 Lpod d o w n s t r e a m o f a f t
Z.-Z, Wangetal./Applied Ocean Research 54 (2016)26-38 29
Fig. 4. Pod dynamometer system, HlOl.
p r o p e l l e r ' s disk, a n d t h e t o p , b o t t o m , r i g h t a n d l e f t b o u n d a r i e s are a l l 5 Lpod a w a y f r o m t h e axis o f p o d bossing, a n d Lpod is t h e l e n g t h o f t h e p o d . The t w o r o t a t i n g d o m a i n s e n c l o s i n g t h e f o r w a r d a n d a f t p r o p e l l e r s are t w o c y l i n d e r s w i t h t h e d i a m e t e r o f 1.2Df. The m u l t i -b l o c k s t r u c t u r e d m e s h w a s g e n e r a t e d t o d i s c r e t i z e t h e f l o w d o m a i n . The g r i d s , o f H - 0 t o p o l o g y w e r e r e f i n e d near t h e p r o p e l l e r blade, h u b a n d p o d . G r i d t o p o l o g i e s at d i f f e r e n t scales w e r e f o u n d t o be e x a c t l y t h e same, except i n t h e a b s o l u t e n e a r - w a l l s p a c i n g w h e r e t h e w a l l f u n c t i o n can be used. The g r i d s w e r e also r e f i n e d i n t h r e e d i r e c t i o n s a c c o r d i n g t o t h e increase o f scale, e n s u r i n g t h a t aspect r a t i o s o f n u m e r i c a l cells w o u l d be s m a l l e r t h a n 5 0 o n p r o p e l l e r s a n d t h e p o d . N u m e r i c a l s i m u l a t i o n p a r a m e t e r s i n d i f f e r e n t scales are l i s t e d i n Table 3, w h e r e Re is t h e R e y n o l d s n u m b e r , Re = ( n f 0 ^ ) / ^ , Up is t h e r e v o l u t i o n o f t h e f o r w a r d p r o p e l l e r , D f is t h e d i a m e t e r o f t h e f o r w a r d p r o p e l l e r , v is t h e k i n e m a t i c v i s c o s i t y , y * is t h e n o n d i m e n s i o n a l s p a c i n g near t h e w a l l , a n d X is t h e scale f a c t o r . A c o m p a r i s o n o f t h e g r i d c u t p l a n e at m o d e l scale a n d f u l l scale is s h o w n i n Fig. 3. The m e d i u m w a s set as w a t e r w i t h a d e n s i t y o f 998.2 l < g m - 3 a n d f l u i d v i s c o s i t y o f 0 . 0 0 1 0 0 3 k g m " ' s - ' . T h e i n l e t b o u n d a r y w a s set as t h e v e l o c i t y i n l e t . The t u r b u l e n t i n t e n s i t y w a s 1%, a n d t h e t u r b u l e n t v i s c o s i t y r a t i o is 1. T h e v e l o c i t y v a r i e d a c c o r d i n g t o t h e a d v a n c e v e l o c i t y c o e f f i c i e n t , t h e o u t l e t b o u n d a r y was set as o u t -flow, a n d t h e f a r f i e l d b o u n d a r y w a s set as s y m m e t r y . The p r o p e l l e r b l a d e , h u b , a n d t h e p o d w e r e set as t h e n o - s l i p w a l l . The g o v e r n i n g e q u a t i o n s a n d t u r b u l e n c e m o d e l w e r e d i s c r e t i z e d u s i n g t h e finite v o l u m e m e t h o d w i t h a second o r d e r u p w i n d s p a t i a l d i s c r e t i z a t i o n a n d t h e p r e s s u r e - v e l o c i t y c o u p l i n g w a s d e t e r m i n e d u s i n g SIMPLEC
Fig. 6. Open water test of the hybrid CRP pod propulsion system.
m e t h o d . The t i m e step size is c o r r e s p o n d e d t o t h e r o t a t i o n a l angle 1° f o r t h e f o r w a r d p r o p e l l e r . N u m e r i c a l s i m u l a r i o n s w e r e r e a l i z e d u s i n g ANSYS FLUENT14.0 code, a n d f u l l scale c o m p u t a t i o n w a s p e r -f o r m e d u s i n g p a r a l l e l p r o c e s s i n g i n 64 cores ( I n t e l X e o n E5-2670, 2.6 G H z ) o f D a w n i n g TC4600 h i g h - p e r f o r m a n c e c o m p u t e r .
3. Results and discussion
3.1. Definitions of liydrodynamic performance of tfie liybrid CRP pod propulsion system
D e f i n i t i o n s o f advance c o e f f i c i e n t J, t h r u s t c o e f f i c i e n t o f t h e f o r -w a r d p r o p e l l e r , t o r q u e c o e f f i c i e n t o f t h e f o r -w a r d p r o p e l l e r t h r u s t c o e f f i c i e n t o f t h e a f t p r o p e l l e r KJA , t o r q u e c o e f f i c i e n t o f t h e a f t p r o p e l l e r t h r u s t c o e f f i c i e n t o f t h e p o d u n i t Kju, P o d resistance c o e f f i c i e n t % p o d . t h r u s t c o e f f i c i e n t o f t h e h y b r i d CRP p o d p r o p u l -s i o n -s y -s t e m Kj, t o r q u e c o e f f i c i e n t o f t h e h y b r i d CRP p o d p r o p u l -s i o n s y s t e m a n d o p e n e f f i c i e n c y o f t h e h y b r i d CRP p o d p r o p u l s i o n s y s t e m )/o are as f o l l o w s : !<TA = VA npDf' TA
I<T = Tf + TA+ Rpod
pnjDj QA KTU OF pnjDj TA + Rpod •pod pnpl {TF + TU)VA IniupOF + UAQA) (5)
h e r e p, VA, up, n^, Dp, D^, Qf, Q4, Tf, TA a n d Rpod r e p r e s e n t t h e d e n -s i t y o f t h e fluid, i n f l o w v e l o c i t y , r e v o l u t i o n o f t h e f o r w a r d p r o p e l l e r , r e v o l u t i o n o f t h e a f t p r o p e l l e r , d i a m e t e r o f t h e f o r w a r d p r o p e l l e r , d i a m e t e r o f t h e a f t p r o p e l l e r , t o r q u e o f t h e f o r w a r d p r o p e l l e r , t o r q u e o f t h e a f t p r o p e l l e r , t h r u s t o f t h e f o r w a r d p r o p e l l e r , t h r u s t o f t h e a f t p r o p e l l e r , a n d resistance o f t h e p o d i n p r o p u l s i o n s y s t e m , r e s p e c t i v e l y .
30 Z,-Z. Wang et al. /Applied Ocean Research 54 (2016) 26-38
•• • -«„(RANS) 0.65 r
J
(c) The hybrid CRP pod propulsion system
Fig. 7. Comparisons of hydrodynamic performances of tlie hybrid CRP pod propulsion system between numerical results and experimental data at n/i/nf = 1.104.
Table 4
Calculated results for three grids at n.4/iiF = 1.104.
Grid KTT 1 0 % KTA Km 1 0 % A = 2 7 . 5 Coarse 0 . 1 8 1 5 0 . 3 4 3 7 0 . 2 0 9 9 0 . 1 8 1 6 0 . 4 8 6 4 Medium 0 . 1 8 2 2 0 . 3 4 2 8 0 . 2 1 0 7 0 . 1 8 2 8 0 . 4 8 5 5 Fine 0 . 1 8 2 6 0 . 3 4 2 2 0 . 2 1 1 2 0 . 1 8 3 6 0 . 4 8 4 7 A = 8 Coarse 0 . 1 8 6 8 0 . 3 4 1 4 0 . 2 1 0 8 0.18S5 0 . 4 6 7 7 IWedium 0 . 1 8 7 5 0 . 3 4 0 6 0 . 2 1 1 6 0 . 1 8 9 6 0 . 4 6 6 8 Fine 0 . 1 8 7 8 0 . 3 4 0 1 0 . 2 1 2 0 0 . 1 9 0 4 0 . 4 6 6 1 X = 2 Coarse 0 . 1 9 1 9 0 . 3 4 0 6 0 . 2 1 4 7 0 . 1 9 6 4 0 . 4 6 1 4 Medium 0 . 1 9 2 5 0 . 3 3 9 9 0 . 2 1 5 4 0 . 1 9 7 4 0 . 4 6 0 6 Fine 0 . 1 9 2 8 0 . 3 3 9 4 0 . 2 1 5 8 0 . 1 9 8 1 0 . 4 6 0 0 Table 5
Parametets of the uncertainty verification.
(RG) (Pc) 27.5 /CTT 0 . 5 7 1 4 1.5347 0 . 0 0 3 4 IO/CQF 0 . 6 6 6 7 1.1120 0 . 0 0 2 5 I<TA 0 . 6 2 5 0 1.2889 0.0033 Km 0 . 6 6 6 7 1.1120 0 . 0 0 3 4 0 . 7 8 9 5 0.6483 0.0033 S KjT 0 . 5 1 4 7 1 . 8 2 1 4 0.0035 WI<Qf 0 . 6 8 7 5 1.0276 0 . 0 0 1 5 l<TA 0 . 5 7 6 9 1.5085 0 . 0 0 3 8 Kw 0 . 6 9 0 9 1.0140 0 . 0 0 1 9 IOKQA 0 . 7 9 5 5 0 . 6 2 7 6 0 . 0 0 3 2 2 KTF 0 . 5 0 0 0 1.9009 0 . 0 0 3 0 i o ; % 0 . 6 6 2 2 1.1305 0 . 0 0 2 2 KTA 0 . 5 4 9 3 1.6430 0 . 0 0 3 6 Kw 0 . 6 9 6 1 0 . 9 9 3 6 0 . 0 0 1 6 0 . 8 0 7 7 0 . 5 8 5 7 0 . 0 0 3 2 Table 6
Hydrodynamic performance of the hybrid CRP pod propulsion system at n//nf = 1.0546. I Re KTF 1 0 % KTA Km 1 0 % KT-pod 27.5 1.146x10^ 0.1840 0.3457 0.1842 0.1564 0.4466 0.0277 16 1.693x10^ 0.1846 0.3452 0.1844 0.1586 0.4384 0.0258 8 4.742 X 10^ 0.1886 0.3440 0.1852 0.1633 0.4272 0.0219 4 1.355x10' 0.1915 0.3436 0.1868 0.1678 0.4231 0.0190 2 3.793 x 10' 0.1932 0.3433 0.1890 0.1718 0.4208 0.0172 1 1.084 xlO^ 0.1952 0.3431 0.1898 0.1741 0.4190 0.0157 Table 7
Hydrodynamic performance of the hybrid CRP pod propulsion system at n^/np» 1.075. k Re KTF 1 0 % KTA Km 1 0 % 27.5 1.146 X 10" 0.1829 0.3449 0.1951 0.1672 0.4636 0.0280 16 1.693x10" 0.1838 0.3442 0.1954 0.1696 0.4554 0.0259 8 4.742 X 10" 0.1878 0.3430 0.1962 0.1741 0.4445 0.0221 4 1.355x10' 0.1910 0.3425 0.1974 0.1781 0.4389 0.0194 2 3.793x10' 0.1926 0.3423 0.1999 0.1823 0.4375 0.0176 1 1.084 xlO^ 0.1944 0.3420 0.2009 0.1847 0.4368 0.0162 Table 8
Hydrodynamic performance of the hybrid CRP pod propulsion system at n/i/nF = 1.104. A Re KTF 1 0 % KTA Km 1 0 % KT-pod 27.5 1.146 X 10" 0.1815 0.3437 0.2099 0.1816 0.4864 0.0283 16 1.693x10" 0.1827 0.3429 0.2101 0.1841 0.4784 0.0260 8 4.742 X 10" 0.1868 0.3414 0.2108 0.1885 0.4677 0.0223 4 1.355x10' 0.1899 0.3410 0.2125 0.1926 0.4631 0.0199 2 3.793 X 10' 0.1919 0.3406 0.2147 0.1964 0.4614 0.0183 1 1.084 X 10* 0.1934 0.3404 0.2154 0.1985 0.4602 0.0169
Z.-Z. Wang et al./Applied Ocean Researcli 54 (2016) 26-38 31
Table 9
Hydrodynamic performance of the hybrid CRP pod propulsion system at n/i/nf = 1.125. Re 10/% K „ Kw 1 0 % Kj-pod 27.5 1.146x10" 0.1802 0.3429 0.2198 0.1913 0.5021 0.0285 16 1.693x10" 0.1819 0.3420 0.2202 0.1940 0.4946 0.0261 8 4.742x10" 0.1860 0.3404 0.2213 0.1988 0.4836 0.0224 4 1.355x 10' 0.1889 0.3396 0.2231 0.2029 0.4797 0.0202 2 3.793x10' 0.1914 0.3395 0.2250 0.2064 0.4778 0.0187 1 1.084 xlO* 0.1926 0.3393 0.2261 0.2087 0.4763 0.0174 Table 10
Hydrodynamic performance ofthe hybrid CRP pod propulsion system at nA/ni:=1.15.
Re KTT 1 0 % KTA Kw 1 0 % KT-pad 27.5 1.146 xlQi^ 0.1788 0.3418 0.2320 0.2032 0.5198 0.0288 16 1.693x10" 0.1809 0.3411 0.2323 0.2060 0.5125 0.0263 8 4.742x10" 0.1853 0.3395 0.2331 0.2106 0.5018 0.0226 4 1.355x 10' 0.1891 0.3390 0.2348 0.2142 0.4979 0.0206 2 3.793x 10' 0.1908 0.3383 0.2370 0.2178 0.4963 0.0192 1 1.084 xlO» 0.1919 0.3381 0.2380 0.2201 0.4944 0.0179
3.2. Comparisons of numerical results a n d experimental data
A n e x p e r i m e n t a l i n v e s t i g a t i o n w a s p e r f o r m e d at m o d e l scale ( 1 = 2 7 . 5 ) t o v a l i d a t e n u m e r i c a l results. The e x p e r i m e n t w a s c o n d u c t e d i n t h e c a v i t a t i o n t u n n e l at N a v a l U n i v e r s i t y o f E n g i -n e e r i -n g . T h e w o r k i -n g s e c t i o -n w a s a r e c t a -n g u l a r c r o s s - s e c t i o -n o f 0.6 m X 0.6 m X 2.6 m . T h e d i f f e r e n c e i n h e i g h t b e t w e e n t h e c e n t e r l i n e s o f u p p e r a n d l o w e r h o r i z o n t a l s e g m e n t w a s a b o u t 10 m . The d i s t a n c e b e t w e e n t h e c e n t e r l i n e s o f t w o v e r t i c a l seg-m e n t s w a s 18 seg-m . T h e p o d d y n a seg-m o seg-m e t e r s y s t e seg-m , H l O l , s h o w n i n Fig. 4, is a n e w l y d e v e l o p e d a d v a n c e d t e s t e q u i p m e n t s p e c i f i c a l l y d e s i g n e d b y Cussons C o m p a n y f o r h y d r o d y n a m i c p e r f o r m a n c e test o f p o d d e d p r o p u l s o r s . I t c a n m e a s u r e p r o p e l l e r ' s t h r u s t , t o r q u e , s i n g l e - c o m p o n e n t f o r c e o f t h e p o d u n i t u n d e r s t e e r i n g c o n d i t i o n s , and p o d resistance i n t h e flow d i r e c t i o n . T h e l o n g axis p r o p e l l e r d y n a m o m e t e r , s h o w n i n Fig. 5, is a s e l f - d e v e l o p e d test e q u i p m e n t d e s i g n e d b y H u a z h o n g U n i v e r s i t y o f Science a n d T e c h n o l o g y f o r o p e n w a t e r test o f t r a d i t i o n a l p r o p e l l e r s , w h i c h c a n m e a s u r e the p r o p e l l e r ' s t h r u s t a n d t o r q u e .
The f o r w a r d p r o p e l l e r is i n s t a l l e d i n reverse at t h e l o n g axis p r o -p e l l e r d y n a m o m e t e r ; t h e -p o d a n d a f t -p r o -p e l l e r are i n s t a l l e d o n t h e p o d d y n a m o m e t e r s y s t e m . The a r r a n g e m e n t o f o p e n w a t e r test is s h o w n i n Fig. 6. D u r i n g the test, t h e t w o d y n a m o m e t e r s w e r e a d j u s t e d u n t i l the r e v o l u t i o n r a t i o r e a c h e d 1.104. The tests w e r e r e p e a t e d t h r e e t i m e s , a n d t h e m e a n v a l u e s o f t e s t results served as t h e o p e n w a t e r p e r f o r m a n c e o f t h e h y b r i d CRP p o d p r o p u l s i o n . Reynolds n u m b e r s at r/R = 0.7 o f t w o p r o p e l l e r s w e r e o v e r 3 x 10^.
C o m p a r i s o n s o f h y d r o d y n a m i c p e r f o r m a n c e s o f t h e h y b r i d CRP p o d p r o p u l s i o n s y s t e m at n/i/nF= 1.104 b e t w e e n n u m e r i c a l results a n d e x p e r i m e n t a l data are s h o w n i n Fig. 7. I n t h e figure, s o l i d p o i n t s are e x p e r i m e n t a l data, a n d o p e n p o i n t s are n u m e r i c a l r e s u l t s .
It can be seen f r o m Fig. 7 t h a t n u m e r i c a l r e s u l t s w e r e closely c o n s i s t e n t w i t h e x p e r i m e n t a l data a n d t h e o v e r a l l t r e n d w a s consis-t e n consis-t . For f o r w a r d p r o p e l l e r , consis-t h e m a x i m u m e r r o r o f consis-t h r u s consis-t a n d consis-t o r q u e c o e f f i c i e n t s w e r e b e l o w 2.5%. For t h e p o d u n i t , t h e e r r o r s w e r e r e l -a t i v e l y l-arge, especi-ally i n t h e -a d v -a n c e c o e f f i c i e n t s f -a r f r o m d e s i g n c o n d i t i o n , the m a x i m u m e r r o r o f 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 w e r e b e l o w 5%. T h e g r i d t o p o l o g y a n d n u m e r i c a l m e t h o d w e r e f o u n d t o be a p p r o p r i a t e f o r s t u d y o f t h e scale e f f e c t o f h y d r o -d y n a m i c p e r f o r m a n c e s o f t h e h y b r i -d CRP p o -d p r o p u l s i o n s y s t e m .
3.3. Uncertainty analysis and numerical results
U n c e r t a i n t y analysis is u s e d f o r 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 scale f a c t o r X = 27.5, 8, 2. A c c o r d i n g t o f a c t o r s o f s a f e t y m e t h o d f o r
R i c h a r d e x t r a p o l a t i o n , X i n g et al. [ 2 0 ] , t h r e e sets o f c o m p u t a t i o n a l g r i d s (i.e., fine, m e d i u m , a n d coarse g r i d s ) are b u i l t f o r t h e n u m e r i c a l s i m u l a t i o n . The r e f i n e m e n t r a t i o , r, w a s 1.2 i n each d i r e c t i o n o f t h e c o o r d i n a t e . Table 4 s h o w s t h e c a l c u l a t e d 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 o f t h e h y b r i d CRP p o d p r o p u l s i o n s y s t e m f o r t h r e e g r i d s , a n d t h e n u m e r -ical u n c e r t a i n t y analysis p r o c e d u r e is i l l u s t r a t e d b e l o w . The v a l u e s c a l c u l a t e d b y coarse g r i d s , m e d i u m g r i d s a n d fine g r i d s are n o t e d as Sec, SCM. a n d Sep, r e s p e c t i v e l y . T h e g r i d changes e f o r c o a r s e - m e d i u m a n d m e d i u m - f i n e g r i d s are d e f i n e d as f o l l o w s : SGMC = SCM - Sec ecfjw = ScF - SCM {Rc) The g r i d c o n v e r g e r a t i o {RQ) w a s c a l c u l a t e d as f o l l o w s : \£GFM\ ISCMcl (6) (7) (8)
The e s t i m a t e d o r d e r o f accuracy {PC)RE w a s c a l c u l a t e d as f o l -l o w s : {Pc) RE ln(|£cMcl/|gCFMl) I n ( r ) (9) The d i s t a n c e m e t r i c to t h e a s y m p t o t i c r a n g e (Pc> w a s d e f i n e d as f o l l o w s : (PG) {PG)R (Pc)t: ( 1 0 ) h e r e (Pc)t;i w a s t h e t h e o r e t i c a l o r d e r o f accuracy. The g r i d u n c e r t a i n t y (Uc) w a s c a l c u l a t e d as f o l l o w s : m = ( 2 . 4 5 - 0 . 8 5 ( P c ) ) ( 1 6 . 4 ( P G > - 1 4 . 8 ) SCFM r(Pc)RE - 1 Êcfivr rCcliïE _ 1 O < {Pc) < 1 {Pc) > 1 (11) Table 5 s u m m a r i z e s t h e n u m e r i c a l u n c e r t a i n t y assessment r e s u l t s a b o u t t h r u s t and t o r q u e c o e f f i c i e n t s o f t h e h y b r i d CRP p o d p r o p u l s i o n s y s t e m at scale f a c t o r A = 2 7 . 5 , 8 , 2 . O v e r a l l , t h e s o l u t i o n s s h o w e d g o o d m e s h c o n v e r g e n c e b e h a v i o r w i t h v e r y s m a l l e r r o r s . G r i d s t u d y i n d i c a t e s t h a t Kj a n d /Cq v a l u e s are g e n e r a l l y n o t t o o s e n s i t i v e f o r g r i d r e f i n e m e n t , so t h e coarse g r i d w a s u s e d i n t h e f o l -l o w i n g n u m e r i c a -l s i m u -l a t i o n s . H y d r o d y n a m i c p e r f o r m a n c e s o f t h e h y b r i d CRP p o d p r o p u l s i o n s y s t e m i n d i f f e r e n t scales a n d r e v o l u t i o n r a t i o s are l i s t e d f r o m Tables 6 - 1 0 . As s h o w n i n Tables 6 1 0 , v a r i a t i o n o f h y d r o d y n a m i c p e r f o r -m a n c e o f t h e h y b r i d CRP p o d p r o p u l s i o n s y s t e -m w a s s i -m i l a r f o r all r e v o l u t i o n r a t i o s , a n d t h e t h r u s t c o e f f i c i e n t s o f t h e f o r w a r d a n d a f t p r o p e l l e r s i n c r e a s e d as t h e Reynolds n u m b e r i n c r e a s e d , b u t t o r q u e c o e f f i c i e n t s decreased as t h e R e y n o l d s n u m b e r increased, t h e y w e r e c o n s i s t e n t w i t h t r a d i d o n a l s h a f t p r o p e l l e r s . A t t h e same time, t h e p o d resistance c o e f f i c i e n t decreased a n d t h r u s t c o e f f i c i e n t o f t h e p o d u n i t i n c r e a s e d . T h e t h r u s t c o e f f i c i e n t o f t h e p o d u n i t at f u l l scale w a s 9.32% h i g h e r t h a n t h a t a t m o d e l scale, n^ln^ = 1.0546. W h e n r e v o l u t i o n r a t i o is 1.104, c o m p a r i s o n s o f n o n d i m e n -s i o n a l a x i a l v e l o c i t y c o n t o u r -s at xlLpod =-0.2S947, y/Ipod = 0, a n d z/ipod = 0-5 b e t w e e n m o d e l scale (A = 2 7 . 5 ) a n d f u l l scale are s h o w n i n Figs. 8 - 1 0 , r e s p e c t i v e l y .
As s h o w n i n Figs. 8 - 1 0 , t h e b o u n d a r y l a y e r w a s t h i c k e r at m o d e l scale, e s p e c i a l l y i n t h e rear o f t h e s t r u t . The l o c a l flow field a r o u n d t h e p o d at f u l l scale w a s v e r y d i f f e r e n t f r o m t h a t a t m o d e l scale, e s p e c i a l l y i n t h e rear o f t h e s t r u t a n d p o d b o s s i n g , t h e flow v e l o c i t y a r o u n d t h e p o d at f u l l scale w a s r e l a t i v e l y h i g h e r t h a n t h a t at m o d e l scale, a n d t h e area o f h i g h v e l o c i t y w a s l a r g e r .
C o m p a r i s o n s o f pressure c o e f f i c i e n t s o f f o r w a r d a n d a f t p r o p e l l e r s b e t w e e n m o d e l scale a n d f u l l scale are s h o w n i n
32 Z.-Z. Wang et ai/Applied Ocean Research 54 (2016) 26-38
Fig. 8. Nondimensional axial velocity contours on the plane x/Lp„d = -0.2895.
Fig. 9. Nondimensional axial velocity contouts on the plane y/j:,,^ = 0.
Fig. 10. Nondimensional axial velocity contours on the plane z/Lp„d = 0.5.
Figs. 11 a n d 12. O n t h e p r e s s u r e side, t h e m a g n i t u d e a n d area o f pos-i t pos-i v e h pos-i g h pressure w e r e b o t h greater at f u l l scale, a n d o n t h e s u c t pos-i o n side, t h e m a g n i t u d e a n d area o f n e g a t i v e h i g h pressure w e r e also greater at f u l l scale. This p h e n o m e n o n p r o d u c e d t h e h i g h e r t h r u s t c o e f f i c i e n t at f u l l scale.
3.4. Scale effect ofthe pod resistance
The 2 5 t h ITTC Specialist C o m m i t t e e o n A z i m u t h i n g Podded P r o p u l s i o n [ 1 8 ] p o i n t e d o u t t h a t accurate p r e d i c t i o n o f p o d resis-tance at f u l l scale w a s t h e base o f h y d r o d y n a m i c p e r f o r m a n c e
Z-Z. Wang etal. /Applied Ocean Research 54 (2016) 26-38 33 2,S7 1.10 0.64 -0 32 -1.18 •2.04 •£S0 •377 .4.33 -S.49 .6.35 IJ
(a) Pressure side at model scale (b) Pressure side at full scale
Con;ai(t 1 3.13 2.27 1.40 0.54 -0.32 -3.77 -4.S3 -5.49 .6.35
(c) Suction side at model scale (d) Suction side at full scale
Fig. 11. Pressure coefficient distribution of forward propeller. p r e d i c t i o n f o r f u l l scale p o d d e d p r o p u l s o r . H o w e v e r , i t is d i f f i c u l t
t o p r e d i c t p o d resistance a t f u l l scale because o f t h e c o m p l i c a t e d f l o w f i l e d caused b y r o t a t i n g p r o p e l l e r .
I n t h i s paper, t h e d i a m e t e r a n d r e v o l u t i o n o f t h e a f t p r o p e l l e r u n d e r design r e v o l u t i o n r a t i o c o n d i t i o n are used t o n o n d i m e n s i o n -alize t h e resistance c o e f f i c i e n t o f t h e single p o d w i t h o u t p r o p e l l e r u n d e r u n i f o r m i n f l o w c o n d i t i o n . TO-pod Ro (12) h e r e RQ is t h e resistance o f t h e s i n g l e p o d w i t h o u t p r o p e l l e r u n d e r u n i f o r m i n f l o w c o n d i t i o n . I n t r o d u c i n g t h e p o d resistance i n f l u e n c e f a c t o r , i t is d e f i n e d as E q . ( 1 3 ) . T-pod l^TO-pod (13) So t h e p o d resistance i n f l u e n c e f a c t o r , y , represents e f f e c t o f w a k e field o f f o r w a r d a n d a f t p r o p e l l e r s o n t h e p o d drag. The p o d resistance i n f l u e n c e factors, y , at d i f f e r e n t scales a n d r e v o l u t i o n r a t i o s are l i s t e d i n Table 1 1 .
V a r i a t i o n o f t h e p o d r e s i s t a n c e i n f l u e n c e f a c t o r w i t h t h e Reynolds n u m b e r is s h o w n i n Fig. 13, a n d v a r i a t i o n w i t h t h e r e v o -l u t i o n r a t i o is s h o w n i n Fig. 14.
As s h o w n i n Fig. 13, t h e p o d resistance i n f l u e n c e f a c t o r decreased as t h e Reynolds n u m b e r increased. This i n d i c a t e d a n o n l i n e a r r e l a t i o n s h i p a n d t h e curve's slope decreased as w e l l . The p o d resistance i n f l u e n c e f a c t o r increased as the r e v o l u t i o n
r a t i o increased. T h i s i n d i c a t e d a l i n e a r d e p e n d e n c e w i t h r e v o l u t i o n r a t i o . T h i s m e a n t t h a t t h e p o d resistance i n f l u e n c e f a c t o r c h a n g e d c o h e r e n t i y n o t o n l y w i t h t h e Reynolds n u m b e r b u t also w i t h t h e r e v o l u t i o n r a t i o . As s h o w n i n Fig. 14, v a r i a t i o n is s i m i l a r i n d i f f e r e n t Reynolds n u m b e r s . O n t h i s basis, t h e p o d resistance i n f l u e n c e f a c t o r w a s a s s u m e d t o be t h e m u l t i p l i c a t i o n o f Reynolds n u m b e r f u n c t i o n a n d r e v o l u t i o n r a t i o f u n c t i o n , a n d the t w o f u n c t i o n s w e r e i n d e p e n d e n t . The f u n c t i o n is g i v e n b e l o w . (14) I n o r d e r t o regress t h e e x p r e s s i o n o f Reynolds n u m b e r f u n c t i o n , t h e m e a n v a l u e o f t h e p o d resistance i n f l u e n c e f a c t o r s , ymean< m u s t be c a l c u l a t e d at a l l possible r e v o l u t i o n r a t i o s . The v a r i a t i o n o f ymean w i t h Reynolds n u m b e r is s h o w n i n Fig. 15. Here, t h e o p e n p o i n t is c u r v e fit data.
T h e cubic p o l y n o m i a l f u n c t i o n w a s u s e d f o r t h e r e g r e s s i o n o f me).
Kmeon = ao + a i l o g i o ( R e ) - f C 2 ( l o g i o ( R e ) ) ^ - h a 3 ( l o g , o ( R e ) ) ^ (15) here ao, a\, 02, a n d 03 are t h e u n d e t e r m i n e d c o e f f i c i e n t s .
The w e i g h e d least square m e t h o d w a s u s e d t o e s t i m a t e t h e u n d e t e r m i n e d c o e f f i c i e n t , as s h o w n i n Eq. (16):
Ymean = 7 4 . 7 7 0 6 - 27.71771og,o(Re) + 3 . 5 7 0 3 ( l o g i o ( i ? e ) ) '
34 Z.-Z Wang et al I Applied Ocean Researcli 54 (20t6) 26-38 . 1 6.57 4.66 4.14 3 4 3 2.71 2.00 1 28 0 57 •0,15 -0,88 1,53 2.29 ,301 3,72
(a) Pressure side at model scale
(b) Pressure side at full scale
pressure coefficieni Conlour 1 5,57 ,1E-: <! •,! 3,-.3 371 2 C J 1 •<•! Oil ^ ;5 or,6 ,58 - 2 J 9 -3.01 -3,72
(c) Suction side at model scale
(d) Suction side at full scale
Fig. 12. Pressure coefficient distribution of aft propeller.Table 11
The pod resistance influence factor in different scales and revolution ratios.
A = 27.5 X = 16 X = 8 k = 4 A = 2 X = 1 1.0546 3.299 3.066 2.607 2.262 2.044 1.873 1.075 3.331 3.079 2.625 2.303 2.099 1.928 1.104 3.368 3.097 2.654 2.368 2.173 2.009 1.125 3.397 3.111 2.671 2.407 2.221 2.067 1.15 3.428 3.127 2.689 2.448 2.284 2.130 The r e v o l u t i o n r a t i o f u n c t i o n / ( n / i / n F ) c a n be o b t a i n e d b y d i v i d -i n g f u n c t -i o n ( 1 4 ) b y ymean, v a r -i a t -i o n o f yjymean w -i t h r e v o l u t -i o n r a t -i o is s h o w n i n Fig. 16. Here, t h e o p e n p o i n t is c u r v e f i t data.
Linear f u n c t i o n c a n b e used i n t h e r e g r e s s i o n a n d t h e w e i g h e d least square m e t h o d w a s also used t o e s t i m a t e t h e u n d e t e r m i n e d c o e f f i c i e n t . The r e v o l u t i o n r a t i o f u n c t i o n is g i v e n b e l o w : (17) ^ i - = 0 . 7 1 3 8 2 i i - f 0 . 2 1 3 6 Ymean The p o d resistance i n f l u e n c e f a c t o r is t h e m u l t i p l i c a t i o n o f f u n c -t i o n ( 1 6 ) a n d f u n c -t i o n ( 1 7 ) . I-t is g i v e n b e l o w :
+ = ƒ ( « . , • /
( I )
f{Re) = 7 4 . 7 7 0 6 - 27.71771ogio(Re) + 3 . 5 7 0 3 ( l o g i o ( R e ) ) ^ ( i g ) - 0 . 1 5 5 3 ( l o g , o ( i ^ e ) ) ^ ƒ f ü d " ) = 0 . 7 1 3 8 ^ ^ + 0 . 2 1 3 6 3.6 3.4 Ï.2 3.0 2,8 2.6 2.4 2.2 2.0 1.8 - ^ i y n j , = 1 . 0 5 4 6 n^/ny=1.075 - ^ n ^ / i v = 1 . 1 0 4 ~ ^ n / i j , = 1 . 1 2 5 6.0 6.5 7.0 log,„(Re) 7,5 8,0Fi". 13. Variation of the pod resistance influence factor with Reynolds number.
Errors o f t h e p o d resistance i n f l u e n c e f a c t o r 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 regressive data b y f u n c t i o n ( 1 6 ) are l i s t e d i n Table 12. Results s h o w e d t h a t m o s t o f t h e e r r o r s w e r e n o m o r e p r o n o u n c e d t h a n ± 2 % , s h o w i n g t h a t t h i s m e t h o d w a s a p r a c t i c a l a p p r o a c h to t h e f u n c t i o n r e g r e s s i o n o f t h e r e l a t i o n s h i p b e t w e e n R e y n o l d s n u m b e r , r e v o l u t i o n r a t i o , a n d t h e p o d resistance i n f l u e n c e f a c t o r .
Z.-Z Wang et al./Applied Ocean Researcli 54 (2016)26-38 3 5 4.0 3,8 3.6 3.4 3.2 3.0 2.8 2.0 2.4 2,2 2,0 1.8 1.6 l o g , j ( R e ) = 7 . 1 3 2 ( M ) 1.04 1.06 1.08 1.10 1.12 1.14 1.16
Fig. 14. Variation of tlie pod resistance influence factor with revolution ratio.
3,4 3.2 3.0 2.8 2,6 2.4 2,2 2.0 -CFD ouiYe f i t data 6.0 6.5 7.0 logJRe) 7.5
Fig. 15. Variation of mean value of the pod resistance influence factor with Reynolds number. 1.04 1.03 1.02 1.01 1.00 0.99 0.98 0.97 0.96 - C F O
curve
fit daia1.04 1.06 1.08 1.10 1,12 1.14 1.16 Fig. 16. Variation of y/ymeon with revolution ratio.
Table 12
Errors of y between numerical results and regressive data.
n/^lnr X = 27.5 A = 1 6 A = 8 X = 4 X = 2 X = l 1.0546 - 1 . 7 0 % - 1 . 9 0 % - 2 . 0 0 % 0.58% 2.64% 3.20% 1.075 - 1 . 1 8 % - 0 . 8 5 % - 1 . 2 1 % 0.26% 1.46% 1.74% 1.104 - 0 . 2 1 % 0.66% - 0 . 2 4 % - 0 . 4 4 % 0.10% - 0 . 2 9 % 1.125 0.42% 1.71% 0.62% - 0 . 5 5 % - 0 . 6 1 % - 1 . 6 4 % 1.15 1.26% 2.96% 1.70% - 0 . 5 2 % - 1 . 6 6 % - 2 . 8 5 % 0.7 - •Ji '^ ^ "
0.6
-
—•— ICjj. (fonvnrd propeller >nly)0,5 S 0.4 0.3 0,2 0,1 10K„,
lOKj^., (fca-\vaid propeller only) - (fonvard propclUr only) • K^Cforwartlpropcllcriiisystcm)
10K.^,(foiwiirt1 propeller in syslrai) • iij (forward propeller in sysicni)
0.5 0.6 0.7 0.9
Fig. 17. Comparison of open water performance between forward propeller only and the hybrid CRP pod propulsion system.
0,8 0.7 0.6 2 0.5 ^ 0.4 0.2 0.1
-K.j.^(pod unit only) K^O'od unit only) - lOK^^Cpod unit only)
iljjCpodunit only) • K.j._^(pod unit in system)
K ^ ( p o d unit iu system) • lOK^j^CpodiuiU in system)
iljjCpod unit in system)
1 0 K „
K T „
0.5 0.6 0 7
J
0.8 0.9
Fig. 18. Comparison of open water performance between pod unit only and the hybrid CRP pod propulsion system.
I n t h i s w a y , f u l l scale p o d resistance c o e f f i c i e n t o f t h e h y b r i d CRP p o d p r o p u l s i o n s y s t e m c a n be p r o d u c e d b a s e d o n f u n c t i o n ( 1 2 ) , f u n c t i o n ( 1 3 ) , a n d f u n c t i o n ( 1 8 ) .
3.5. Scale effect of hydrodynamic performance of forward and aft propellers I n o r d e r t o i n v e s t i g a t e t h e e f f e c t o f t h e p o d p r o p e l l e r i n t e r f e r -ence o n h y d r o d y n a m i c p e r f o r m a n c e o f f o r w a r d a n d a f t p r o p e l l e r s , t h e o p e n w a t e r p e r f o r m a n c e o f single f o r w a r d p r o p e l l e r a n d single p o d u n i t w e r e also c a l c u l a t e d u s i n g t h e RANS m e t h o d . C o m p a r i s o n s o f o p e n w a t e r p e r f o r m a n c e b e t w e e n s i n g l e f o r w a r d p r o p e l l e r , s i n -gle p o d u n i t a n d t h e h y b r i d CRP p o d p r o p u l s i o n s y s t e m are s h o w n i n Figs. 1 7 a n d 18.
Z,-Z. Wang et al./Applied Ocean Research 54 (2016) 26-38 0,94 0,92 0.90 0.S8 0.8IS 0.84 0.82 0,60 0.78 0,76 0.74 nyn,"1.0546 n/nj=1.075 •—»—ii_^/n^= 1.104 " -»-ii/n,=1.125 n/n,=1.15 1.04 1,02 1,00 0,93 0.96 I - , * 0.94 0,92 0.90 o,ss 0,S6 - 1 1 / ^ - 1 , 0 5 4 6 - il/ii,,-1.075 -11/1,-1.104 n / i i , = l . l 2 S n/n,=1.15 7,0 l " E „ ( R e ) loEi.CRc)
{ a ) Thi'ust coefficient influence factor
Fig. 19. Variations of thrust and totque coefficient influence factors of aft propeller with Reynolds number. ( b ) Torque coefficient inlluence factor
- to5|„(R«)-6,059 0^27,5) bsn,(l!i)rö.229(tel6) - l o E , „ ( R t ) - 6 . 6 7 6 ( t e S ) -!o6]oCRi)=7,!32(J.-1) - to8,„tR«)=7.579 lte2) - t o S l o l K ' ) - » » - " » ' - » 1.10 n./n 1.050 1.025 1.000 0.975 , 0.950 0.925 O.90O 0,875 0.850 •»B,5(I!=)-6,I)5'(>-27.S, ,0ïc)^ü.329()."lö) tog,„(n«).6.676()-!) loj,0(K<)=7.13J(W) oil,o(R')"7.579().=2) » S l 0 0 t « ) - S M 5 ( l s l )
( a ) Thrust coefficient influence factor ( b ) Torque coefficient influence factor Fig. 20. Variations of thtust and torque coefficient influence factors of aft propeller with revolution ratio.
open walcr le.sl al model .scale In dirfcrcnl Reynolds numbers and
possible revolulion ralios
(KTF)ni, (KQF)^, {KTA)m, (KQA)™. (KT-pod)m
Ihe Standard ITTC 1978 extrapolation procedure
open water le.sl of single aft propeller at model scale
(KTA0)m^ (KQAo)m
the .standard l i r e 1978 extrapolation procedure
(KTAO).<. (K<3AO)S
resistance test of the pod at mode! scale in uoiform inflow condition
(KTO-pod)m (Kir)... (KQP). = « 1 HA. " f = « j HJL n,, -b. (KTA)S, (KQA)S r = f ( R e ) - f { n J n , ) (KT-pod)s
Z-Z. Wang et al./Applied Ocean Research 54 (2016)26-38 37
As s h o w n i n Fig. 17, t i i e p o d u n i t h a d l i t d e i n f l u e n c e o n the l i y d r o d y n a m i c p e r f o r m a n c e o f t h e f o r w a r d p r o p e l l e r . The d i f f e r -ences i n t h e t h r u s t c o e f f i c i e n t o f t h e s i n g l e f o r w a r d p r o p e l l e r a n d the h y b r i d CRP p o d p r o p u l s i o n s y s t e m w e r e less t h a n 3% u n d e r d e s i g n c o n d i d o n , and t o r q u e c o e f f i c i e n t w a s less t h a n 2%. The m a i n reason f o r t h i s was t h a t t h e s u c t i o n e f f e c t o f a f t p r o p e l l e r a n d b l o c k -age e f f e c t o f the p o d canceled each o t h e r o u t .
As s h o w n i n Fig. 18, t h e w a k e o f f o r w a r d p r o p e l l e r h a d a n i m p o r -t a n -t e f f e c -t o n -the h y d r o d y n a m i c p e r f o r m a n c e o f -the p o d u n i -t . 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 o f t h e h y b r i d CRP p o d p r o p u l s i o n s y s t e m w e r e less p r o n o u n c e d t h a n t h a t o f s i n g l e p o d u n i t i n each advance c o e f f i c i e n t . I n t h i s w a y , the s t a n d a r d ITTC 1978 e x t r a p o l a t i o n p r o c e d u r e w a s p r o p o s e d f o r h y d r o d y n a m i c p e r f o r m a n c e p r e d i c t i o n o f f o r w a r d p r o p e l l e r i n p r o p u l s i o n s y s t e m at f u l l scale. For t h e a f t p r o p e l l e r , t h e 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 i n f l u e n c e factors w e r e i n t r o d u c e d . T h e y are d e f i n e d as f o l l o w s : TA •• KTA !<TAO KQA KQAO ( 1 9 ) ( 2 0 )
here KJAO a n d KQAQ are t h e 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 o f single a f t p r o p e l l e r u n d e r u n i f o r m i n f l o w c o n d i t i o n . So FJA a n d FQA can r e p r e s e n t t h e e f f e c t o f w a k e field o f f o r w a r d p r o p e l l e r a n d blockage e f f e c t o f t h e p o d o n h y d r o d y n a m i c p e r f o r m a n c e o f a f t p r o p e l l e r .
V a r i a t i o n s o f 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 i n f l u e n c e factors o f a f t p r o p e l l e r w i t h Reynolds n u m b e r are s h o w n i n Fig. 19, w i t h r e v o l u d o n r a t i o are s h o w n i n Fig. 2 0 .
Results s h o w e d t h a t t h e 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 i n f l u -ence f a c t o r s o f a f t p r o p e l l e r r e m a i n e d a l m o s t c o n s t a n t as t h e Reynolds n u m b e r increased, b u t t h e y increased as t h e r e v o l u d o n r a t i o increased, s h o w i n g a n e a r l y l i n e a r d e p e n d e n c e . So t h e 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 i n f l u e n c e f a c t o r s w e r e a s s u m e d to o n l y be t h e r e v o l u t i o n r a d o f u n c t i o n :
TA ( 2 1 )
'•--MS
The l i n e a r f u n c t i o n c a n be used f o r t h e regression a n d t h e w e i g h e d least square m e t h o d can also b e used t o e s t i m a t e the u n d e -t e r m i n e d c o e f f i c i e n -t . The r e v o l u -t i o n r a -t i o f u n c -t i o n s are as b e l o w : = 2 . 0 0 4 9 — - 1 . 3 6 8 4 tlF • 0 . 8 1 0 8 ( 2 3 ) ( 2 4 )
3.6. Algorithm for extrapolation to full scale
A n e x t r a p o l a t i o n m e t h o d o f h y d r o d y n a m i c p e r f o r m a n c e f o r t h e h y b r i d CRP p o d p r o p u l s i o n s y s t e m at f u l l scale w a s p r o p o s e d . The e x t r a p o l a t i o n m e t h o d w a s expressed as f o l l o w s :
Step 3: A n o p e n w a t e r test o f t h e h y b r i d CRP p o d p r o p u l s i o n sys-t e m asys-t m o d e l scale w a s c o n d u c sys-t e d i n d i f f e r e n sys-t possible r e v o l u sys-t i o n r a t i o s ( o r u s i n g CFD m e t h o d ) , w h i c h c o v e r e d several service speeds i n o r d e r t o achieve a large r a n g e o f R e y n o l d s n u m b e r s . The t h r u s t c o e f f i c i e n t s , t o r q u e c o e f f i c i e n t s o f f o r w a r d a n d a f t p r o p e l l e r s , a n d t h e p o d resistance c o e f f i c i e n t w e r e c a l c u l a t e d i n o r d e r t o c o n d u c t t h e r e g r e s s i o n s t u d y .
Step 2: A n o p e n w a t e r test o f s i n g l e a f t p r o p e l l e r at m o d e l scale w a s p e r f o r m e d (or use CFD, p a n e l s u r f a c e m e t h o d ) , KJAQ a n d KQAO w e r e c a l c u l a t e d .
Step 3: A resistance t e s t o f t h e p o d w i t h o u t p r o p e l l e r s at m o d e l scale w a s c o n d u c t e d ( o r use CFD m e t h o d ) , a n d KjQ.pod w a s c a l c u -l a t e d .
Step 4: The s t a n d a r d ITTC 1978 e x t r a p o l a t i o n p r o c e d u r e w a s used to p r e d i c t h y d r o d y n a m i c p e r f o r m a n c e o f f o r w a r d p r o p e l l e r i n p r o p u l s i o n s y s t e m at f u l l scale. Step 5: 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 i n f i u e n c e f a c t o r s o f t h e a f t p r o p e l l e r w e r e c a l c u l a t e d , a n d the l i n e a r f u n c t i o n s i m i l a r to f u n c -t i o n ( 2 3 ) a n d f u n c -t i o n ( 2 4 ) w e r e used -to c o n d u c -t -t h e regression analysis o n 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 i n f l u e n c e f a c t o r s i n d i f -f e r e n t r e v o l u t i o n r a t i o s , a n d the w e i g h t e d least s q u a r e m e t h o d w a s used t o e s t i m a t e t h e u n d e t e r m i n e d c o e f f i c i e n t .
Step 6: The s t a n d a r d ITTC 1978 e x t r a p o l a t i o n p r o c e d u r e w a s u s e d to p r e d i c t h y d r o d y n a m i c p e r f o r m a n c e o f the single a f t p r o p e l l e r at f u l l scale i n u n i f o r m i n f l o w c o n d i t i o n , t h e n t h e 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 i n f i u e n c e f a c t o r s o b t a i n e d i n step 5 w e r e m u l t i p l i e d to d e t e r m i n e t h e h y d r o d y n a m i c p e r f o r m a n c e o f a f t p r o p e l l e r i n p r o p u l s i o n s y s t e m at f u l l scale. Step 7: A f u n c t i o n s i m i l a r to f u n c t i o n ( 1 8 ) w a s u s e d f o r r e g r e s -s i o n analy-si-s o f t h e p o d re-si-stance i n f l u e n c e f a c t o r i n d i f f e r e n t R e y n o l d s n u m b e r s a n d r e v o l u t i o n ratios, a n d t h e w e i g h t e d least square m e t h o d was used t o e s t i m a t e t h e u n d e t e r m i n e d c o e f f i c i e n t . Step 8: The p o d resistance c o e f f i c i e n t i n u n i f o r m i n f l o w c o n d i t i o n o b t a i n e d i n step 3 w a s m u l t i p l i e d b y t h e p o d resistance i n f l u -ence f a c t o r o b t a i n e d i n step 7 to d e t e r m i n e t h e p o d resistance c o e f f i c i e n t i n p r o p u l s i o n s y s t e m at f u l l scale.
The flow c h a r t is s h o w n i n Fig. 2 1 . To e v a l u a t e t h e r e l i a b i l i t y o f t h e p r o c e d u r e above, m o r e h y b r i d CRP p o d p r o p u l s i o n s y s t e m s s h o u l d be c o l l e c t e d a n d v e r i f i e d .
4. Conclusions
I n t h i s paper, t h e h y d r o d y n a m i c p e r f o r m a n c e o f a h y b r i d CRP p o d p r o p u l s i o n s y s t e m at d i f f e r e n t scales w a s c a l c u l a t e d u s i n g t h e RANS m e t h o d c o m b i n e d w i t h SST w t u r b u l e n c e m o d e l a n d m o v i n g m e s h m e t h o d . V a r i a t i o n s i n t h e local flow field a n d h y d r o -d y n a m i c p e r f o r m a n c e w i t h the Reynol-d's n u m b e r w e r e a n a l y z e -d i n d e t a i l , a n d a m e t h o d o f e s t i m a t i n g t h e h y d r o d y n a m i c p e r f o r m a n c e o f t h e h y b r i d CRP p o d p r o p u l s i o n s y s t e m at f u l l scale w a s p r o p o s e d . This w o r k s h o w e d t h e f o l l o w i n g : (1) T h r u s t c o e f f i c i e n t s o f f o r w a r d a n d a f t p r o p e l l e r s increase as t h e R e y n o l d s n u m b e r increases, b u t t h e t o r q u e c o e f f i c i e n t decreases. This is c o n s i s t e n t w i t h results o b s e r v e d o n t r a d i -t i o n a l s h a f -t p r o p e l l e r s . A -t -t h e s a m e -time, -t h e p o d resis-tance c o e f f i c i e n t also decreases a n d t h e t h r u s t c o e f f i c i e n t o f t h e p o d u n i t increases.(2) The B o u n d a r y layer is thicl<er at m o d e l scale, e s p e c i a l l y i n t h e r e a r o f t h e s t r u t . T h e local flow field a r o u n d t h e p o d at f u l l scale a n d a t m o d e l scale is v e r y d i f f e r e n t , especially i n t h e rear o f t h e s t r u t a n d p o d b o s s i n g area. The flow v e l o c i t y a r o u n d t h e p o d at f u l l scale is h i g h e r t h a n t h a t at m o d e l scale, a n d t h e area o f h i g h v e l o c i t y is larger.
(3) T h e p o d u n i t has l i t t l e i n f l u e n c e o n h y d r o d y n a m i c p e r f o r m a n c e o f t h e f o r w a r d p r o p e l l e r , a n d t h e w a k e o f f o r w a r d p r o p e l l e r has a n i m p o r t a n t e f f e c t o n t h e h y d r o d y n a m i c p e r f o r m a n c e o f t h e p o d u n i t .
(4) The p o d resistance i n f l u e n c e f a c t o r changes a l o n g w i t h t h e R e y n o l d s n u m b e r a n d r e v o l u t i o n r a t i o o f t h e f o r w a r d a n d a f t p r o p e l l e r s . ( 5 ) 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 i n f l u e n c e f a c t o r s o f a f t p r o p e l l e r are i n d e p e n d e n t o f t h e Reynolds n u m b e r , b u t t h e y i n d i c a t e a n e a r l y l i n e a r d e p e n d e n c y w i t h r e v o l u t i o n r a t i o o f t h e f o r w a r d a n d a f t p r o p e l l e r s .
38 Z-Z. Wang et al./Applied Ocean Research 54 (2016) 26-38
Acknowledgements
The p r e s e n t w o r k w a s s u p p o r t e d b y the N a t i o n a l N a t u r a l Science F o u n d a t i o n o f China ( G r a n t N o . 5 1 4 7 9 2 0 7 ) a n d t h e H i g h T e c h n o l -ogy M a r i n e S c i e n t i f i c Research P r o j e c t o f M i n i s t r y o f I n d u s t r y a n d I n f o r m a t i o n T e c h n o l o g y o f China ( G r a n t N o . [ 2 0 1 2 ] 5 3 4 ) . W e w o u l d l i k e t o express o u r deep a p p r e c i a t i o n t o t h e M i n i s t r y o f I n d u s t r y a n d I n f o r m a t i o n T e c h n o l o g y o f China. W e t h a n k LetPub ( w w w . l e t p u b . c o m ) f o r its l i n g u i s t i c assistance d u r i n g t h e p r e p a r a t i o n o f t h i s m a n u s c r i p t .
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