Applied Ocean Research 39 (2012) 1-10
C o n t e n t s l i s t s a v a i l a b l e at S c i V e r s e S c i e n c e D i r e c t
Applied Ocean Research
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 / I o c a t e / a p o r
Simulation of turning circle by CFD: Analysis of different propeller models and their
effect on manoeuvring prediction
Riccardo Broglia^'", Giulio Dubbioso^ Danilo Du^ante^ Andrea Di Mascio^
'Maritime Researcli Centre, CNR-INSEAN, 00128 Rome. Italy''Istituto per le Applicazioni del Calcolo, CNR-IAC, 00I6I Rome. Italy
A R T I C L E I N F O A B S T R A C T
Article history: Received 27 January 2012
Received in revised form 11 September 2012 Accepted 12 September 2012
Keywords:
Computational methods Marine engineering Twin screw ship Manoeuvring Propeller loads
Propeller modelling in CFD simulations is a key issue for the correct prediction of hull-propeller interactions, manoeuvring characteristics and the flow field in the stern region of a marine vehicle. F r o m this point of v i e w , actuator disk approaches have proved their reliability and computational efficiency; for these reasons, they are c o m m o n l y used for the analysis of propulsive performance of a ship. Nevertheless, these models often neglect peculiar physical p h e n o m e n a w h i c h characterise the operating propeller in off-design condition, namely the in-plane loads that a r e of paramount importance w h e n considering n o n - s t a n d a r d or u n u s u a l p r o p e l l e r / r u d d e r arrangements. In order to e m p h a s i z e the importance of these c o m p o n e n t s (in particular the propeller lateral force) and the need of a detailed propeller model for the correct prediction of the manoeuvring qualities of a ship, the turning circle m a n o e u v r e of a self-propelled fully appended t w i n s c r e w tanker-like ship model w i t h a single rudder is simulated by the unsteady RANS solver xnavis developed at C N R - I N S E A N ; several propeller models able to include the effect of the strong oblique flow c o m p o n e n t encountered d uring a m a n o e u v r e have been considered and compared. It is e m p h a s i z e d that, despite these models account for very complex and f u n d a m e n t a l physical effects, w h i c h w o u l d be lost by a traditional actuator disk approach, the increase in computational resources is a l m o s t negligible. T h e accuracy of these models is assessed by comparison w i t h ex per ime nt al data from free r u n n i n g tests. The m a i n features of the flow field, w i t h particular attention to the vortical structures detached from the hull are presented as w e l l .
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1. Introduction
The n u m e r i c a l p r e d i c t i o n o f the t r a j e c t o r y f o l l o w e d b y a self-p r o self-p e l l e d h u l l i n free m o t i o n e x h i b i t s all t h e c o m self-p u t a t i o n a l issues t h a t can be m e t i n naval h y d r o d y n a m i c s ; t h e m a i n d i f f i c u l t i e s arise i n the accurate evaluation o f the h y d r o d y n a m i c forces and m o m e n t s on the appendages, w h i c h are o f p a r a m o u n t i m p o r t a n c e w h e n assessing the d y n a m i c response of the vessels and its t r a j e c t o r y . W i t h regards to the m a t h e m a t i c a l model, the equations o f r i g i d body m o t i o n m u s t be c o u p l e d w i t h the unsteady Reynolds averaged Navier-Stokes (uRANS) equations and m a n y n u m e r i c a l aspects m u s t be c a r e f u l l y considered. F r o m the discretization p o i n t o f v i e w , at least i n the context o f uRANS solvers f o r structured grids (like the one used here), a d y n a m i c over-l a p p i n g g r i d m e t h o d m u s t be i m p over-l e m e n t e d i n order to over-let the ship m o v e i n a f i x e d b a c k g r o u n d and a l l o w the r u d d e r to m o v e w i t h re-spect to the h u l l . Moreover, due to m e m o r y a n d CPU requirements, i t w o u l d be d i f f i c u l t to o b t a i n results accurate enough f o r practical purposes w i t h a serial code, and t h e r e f o r e p a r a l l e l i z a t i o n m u s t be considered.
H o w e v e r , i n spite of all these d i f f i c u l t i e s , s i m u l a t i o n s p e r f o r m e d
• Corresponding author. Tel.; +39 0650299297; fax: +39 065070519. E-mail address: riccardo.broglia@cnr.it (R. Broglia).
by means o f the n u m e r i c a l s o l u t i o n o f the uRANS equations can be extensively used f o r the analysis o f b o t h static a n d d y n a m i c m a n o e u -vres, as the latest w o r k s h o p s and i n t e r n a t i o n a l conferences (see f o r example [ 15-17,20,21 j ) ha ve d e m o n s t r a t e d .
In this w o r k , the p r e d i c t i o n o f the free t u r n i n g m a n o e u v r e o f a t a n k e r - l i k e ship, already considered i n a recent s t u d y [ 7 ] , has been f u r t h e r analysed to gain f u r t h e r i n s i g h t i n t o some aspects o f p r o p e l l e r m o d e l l i n g w h e n o p e r a t i n g under strong o b l i q u e flows. The test case considered is p a r d c u l a r l y challenging, since the unusual p r o p e l l e r / r u d d e r c o n f i g u r a t i o n (i.e. the t w i n screw vessel w i t h a single r u d d e r ) provides a poor s t a b i l i t y qualities o f the ship. In the previous w o r k , satisfactory results have been obtained i n t e r m s o f t r a j e c t o r y , d r i f t a n d speed d r o p i n the stabilized phase o f the t u r n i n g o n l y w i t h the use o f a "suitable" lateral force acting i n the propeller plane; t h i s approach was necessary because the simple actuator disk m o d e l used (the H o u g h a n d O r d w a y m o d e l [ 9 ] ) i n its o r i g i n a l version provides o n l y t h r u s t a n d t o r q u e .
As a m a t t e r o f fact, detailed measurements o f h y d r o d y n a m i c loads a n d flow features | 1 ) f o r a t w i n - s c r e w f r i g a t e - t y p e vessel d u r i n g a steady t u r n have s h o w n that the side forces g e n e r a t e d b y the p r o -peller can be r e l e v a n t (15-20% o f the t o t a l lateral f o r c e ) and t h e r e f o r e t h e i r c o n t r i b u t i o n to the t o t a l h y d r o d y n a m i c loads a c t i n g o n the h u l l
0141-1187/$ - see front matter ® 2012 Elsevier Ltd. All rights resereed. http://dx.doi.Org/10.1016/j.apor.2012.09.001
R. Brogliaet al./Applied Ocean Research 39 (2012) 1-10
and, consequendy, to its m a n o e u v r i n g behaviour is i m p o r t a n t . I t is ob-vious t h a t the best w a y to p r e d i c t the e x t r e m e l y c o m p l i c a t e d physical phenomena i n the p r o p e l l e r r e g i o n that are responsible f o r the lateral forces (and, i n general, f o r the c o m p l e t e system o f forces and m o -ments acdng o n the p r o p e l l e r disk) w o u l d be achieved by the d i r e c t c o m p u t a d o n o f the r o t a t i n g propellers, b u t the increase i n c o m p u t a -tional resources w o u l d be r a t h e r large. Therefore, w h e n the details of the f l o w are n o t r e q u i r e d , a c o n v e n i e n t choice m i g h t be the i n c l u -sion o f s i m p l i f i e d s e m i - e m p i r i c a l models w h i c h do n o t s i g n i f i c a n t l y increase the o v e r a l l c o m p l e x i t y and resource d e m a n d .
In this w o r k , several p r o p e l l e r models are investigated, w i t h par-ticular a t t e n t i o n t o the side forces they produce a n d t h e i r e f f e c t o n the m a n o e u v r i n g qualities o f the ship. A l l the models are based o n the actuator disk concept; therefore, the m o m e n t u m transferred i n t o the f l u i d b y the p r o p e l l e r is t a k e n i n t o account by means o f a b o d y -force field d i s t r i b u t e d w i t h i n a disk o f finite thickness. T w o p r o p e l l e r models are considered; the first one is based o n the classical H o u g h and O r d w a y m o d e l [ 9 ] ; the second is a m o d e l based o n the blade ele-m e n t t h e o r y (see [10,13,14]). I n the f o r ele-m e r a lateral force is evaluated by means o f t w o s e m i - e m p i r i c a l models, i.e. the m o d e l developed by Ribner [15] and the one already considered i n a previous w o r k [ 7 ] .
d y n a m i c b o u n d a r y conditions read;
p = Xijmnj + ~ + ~
1 W e ^ (2)
T i j n , t i = 0 T y n , t j = 0
w h e r e Ty is the stress tensor, K is the average curvature, W e =
[pUl^L/a)^'^ is the W e b e r n u m b e r ( o b e i n g the surface tension
c o e f f i c i e n t ) , whereas n, t ' and are the surface n o r m a l and t w o t a n g e n t i a l u n i t vectors, respectively.
The actual p o s i t i o n o f the free surface F(x,y,z,f) = 0 is c o m p u t e d f r o m the k i n e m a t i c c o n d i t i o n ;
m > ^ ^ , (4)
I n i t i a l c o n d i t i o n s ( i n d i c a t e d w i t h a stroke) have to be specified f o r the v e l o c i t y field and f o r the free surface c o n f i g u r a t i o n ;
u , ( x , y , z , 0 ) = W ( x . y . z ) , 1 = 1 , 2 , 3
F ( x , y , z , 0 ) = f ( x , y , z ) '
2. M a t h e m a t i c a l m o d e l
The g o v e r n i n g equations f o r the unsteady m o t i o n o f an i n c o m -pressible viscous fluid can be w r i t t e n i n i n t e g r a l f o r m as
S, U 1105 = 0
- / \idV + é> ( F c - F D ) n d S = 0
o t 7 V J S(V)
w h e r e V i s a c o n t r o l v o l u m e , S( its b o u n d a r y and n is the o u t w a r d u n i t n o r m a l . In the general f o r m u l a t i o n , the equations are w r i t t e n i n an i n e r t i a l f r a m e o f reference, i n o r d e r to take i n t o account the possibility o f g r i d m o t i o n . The equations are made n o n - d i m e n s i o n a l w i t h reference v e l o c i t y Uoc, l e n g t h L and the w a t e r d e n s i t y p.
I n Eq. (1), Fc and FD represent i n v i s c i d ( a d v e c t i o n and pressure) and d i f f u s i v e fluxes, respectively;
Fc = pl + ( U - V ) U
FD = ( j ^ + V t ) ( V U + V U ^ ) (2)
In the previous equation, p = P + z/Fr ^ is the n o n - d i m e n s i o n a l h y d r o d y n a m i c pressure (i.e. t h e d i f f e r e n c e b e t w e e n the t o t a l n o n -d i m e n s i o n a l pressure P an-d the h y -d r o s t a t i c pressure - z / F r ^, Fr =
Uoo/igiy^^ being the Froude n u m b e r and L is the reference l e n g t h ( i n
the present case is the l e n g t h b e t w e e n p e r p e n d i c u l a r ) ; m o r e o v e r g is the acceleration o f g r a v i t y p a r a l l e l to the v e r t i c a l axis z, positive u p w a r d ) , I is the i d e n t i t y tensor, V is the local v e l o c i t y o f the c o n t r o l v o l u m e boundary, Re = U^L/ v is the Reynolds n u m b e r , v is the k i n e m a t i c viscosity, and Vt is t h e n o n - d i m e n s i o n a l t u r b u l e n t viscosity, calculated by means o f a p r o p e r t u r b u l e n c e m o d e l . I n w h a t f o l l o w s , the Cartesian c o m p o n e n t s o f t h e v e l o c i t y vector w i l l be denoted b y u,-w i t h i n d e x n o t a t i o n or by u, v, u,-w .
The p r o b l e m is closed by e n f o r c i n g a p p r o p r i a t e c o n d i t i o n s at p h y s -ical and c o m p u t a t i o n a l boundaries. O n solid walls, the relative veloc-i t y veloc-is set at zero (whereas no c o n d veloc-i t veloc-i o n o n the pressure veloc-is r e q u veloc-i r e d ) ; at the ( f i c t i t i o u s ) i n f l o w b o u n d a r y , v e l o c i t y is set to the u n d i s t u r b e d flow value and pressure is e x t r a p o l a t e d f r o m inside; o n the contrary, pressure is set to zero at the o u t f l o w , w h e r e a s v e l o c i t y is extrapolated f r o m i n n e r points.
A t the free surface, w h o s e l o c a t i o n is one o f the u n k n o w n s o f the p r o b l e m , the d y n a m i c b o u n d a r y c o n d i t i o n requires c o n t i n u i t y o f stresses across the surface; i f t h e presence o f the air is neglected, the
3. N u m e r i c a l method
The n u m e r i c a l s o l u t i o n o f the g o v e r n i n g equations ( 1 ) is c o m -p u t e d b y means o f the solver x^avis, w h i c h is a general -pur-pose s i m u l a t i o n code developed at CNRINSEAN; the code yields the n u -m e r i c a l s o l u t i o n o f the unsteady Reynolds averaged Navier-Stokes (uRANS) equations f o r unsteady h i g h Reynolds n u m b e r ( t u r b u l e n t ) f r e e surface flows a r o u n d c o m p l e x geometries. The m a i n features o f t h e n u m e r i c a l a l g o r i t h m are b r i e f l y s u m m a r i z e d here f o r the sake o f b r e v i t y ; the interested reader is r e f e r r e d to D i Mascio et al. [ 3 - 6 ] , and Broglia et al. [ 2 ] f o r details.
The solver is based o n a cell-centred finite v o l u m e f o r m u l a t i o n . I n the code, the spatial d i s c r e t i z a t i o n can be chosen f r o m a l i b r a r y o f a p p r o x i m a t i o n schemes; i n the present w o r k , a t h i r d o r d e r u p w i n d scheme was used, whereas the d i f f u s i v e terms are discretized w i t h a second order centred scheme. The time i n t e g r a t i o n is made b y a f u l l y i m p l i c i t three points b a c k w a r d second order scheme and the s o l u -tion at each rime step is c o m p u t e d by a p s e u d o - t i m e i t e r a t i o n t h a t e x p l o i t s the Euler i m p l i c i t scheme w i t h a p p r o x i m a t e f a c t o r i z a t i o n , local p s e u d o - t i m e step and m u l t i - g r i d [8] f o r convergence acceler-a t i o n . T u r b u l e n t stresses acceler-are tacceler-aken i n t o acceler-account by the Boussinesq hypothesis; several t u r b u l e n c e models ( b o t h algebraic and d i f f e r e n -t i a l ) are i m p l e m e n -t e d . I n -the presen-t c o m p u -t a -t i o n s -the o n e - e q u a -t i o n m o d e l o r i g i n a l l y i n t r o d u c e d by Spalart and A l l m a r a s [ 1 9 ] has been used. Free surface effects are taken i n t o account by a single-phase level-set a l g o r i t h m [ 4 ] . Complex geometries a n d m u l t i p l e bodies i n relative m o t i o n are h a n d l e d by a d y n a m i c o v e r i a p p i n g g r i d approach [ 6 ] . H i g h p e r f o r m a n c e c o m p u t i n g is achieved b y an e f f i c i e n t shared a n d d i s t r i b u t e d m e m o r y p a r a l l e l i z a t i o n [ 2 ] .
3 . Ï . Propeller models
I n m a n y m a r i n e CFD simulations, the presence o f the p r o p e l l e r is t a k e n i n t o account by models based o n the a c t u a t o r d i s k concept, a c c o r d i n g to w h i c h a field o f body forces are d i s t r i b u t e d w i t h i n a disk o f finite thickness; d i s t r i b u t i o n s o f b o t h axial and t a n g e n t i a l b o d y forces are used i n order to simulate t h e acceleration a n d t h e increase i n s w i r l that the flow undergoes w h e n passing t h r o u g h the p r o p e l l e r . These force fields m i m i c the action o f the blades o n the flow fields a n d are o b t a i n e d b y blade loads averaging i n b o t h time a n d space. T i m e averages are taken over one p e r i o d o f r e v o l u t i o n , w h e r e a s space averages are o b t a i n e d b y d i s t r i b u t i n g blade loads i n c i r c u m f e r e n t i a l d i r e c t i o n over the w h o l e p r o p e l l e r disk.
R. Broglia et al./Applied Ocean Research 39 (2012) 1-10 3
Body forces depend o n the actual v e l o c i t y field, w h i c h i n t u r n de-pends on the action o f the blades; therefore any realistic m o d e l s h o u l d take i n t o account t h e i r m u t u a l i n t e r a c t i o n . Moreover, due to t h e n o n -l i n e a r i t y o f this i n t e r a c t i o n , an iterative procedure is r e q u i r e d . T w o m o d e l s are here considered; a m o d e l based o n the s i m p l e H o u g h a n d O r d w a y approach [9] and a m o d e l based o n the blade e l e m e n t m o m e n t u m t h e o r y (BEMT) [ 1 0 ] . The first one, i n its o r i g i n a l v e r s i o n , does n o t take i n t o account the h u l l - p r o p e l l e r i n t e r a c d o n v e l o c i t y ; i.e. i t considers a p r o p e l l e r w o r k i n g i n open w a t e r c o n d i d o n s ; as a consequence, o b l i q u e flow effects are also neglected. In this w o r k t h e o r i g i n a l m o d e l was m o d i f i e d i n order to take i n t o account b o t h the ax-ial f l o w r e d u c t i o n at the p r o p e l l e r disk and the lateral force g e n e r a t e d by an o b l i q u e i n f l o w ; t w o d i f f e r e n t s e m i - e m p i r i c a l m o d e l s f o r t h e side force are tested and b o t h w i l l be denoted as the m o d i f i e d H o u g h and O r d w a y models. The c o u p l i n g a l g o r i t h m b e t w e e n the p r o p e l l e r m o d e l a n d the uRANS solver is sketched in Fig. 3. In the same figure the flowcharts o f b o t h the m o d i f i e d Hough and O r d w a y ( w i t h the i n -c l u s i o n o f the Ribner m o d e l f o r the in-plane load e s t i m a t i o n ) and t h e BEMT models, described i n the f o l l o w i n g sections, are also sketched.
3.1.1. Modified Hough and Ordway models
I n these models,.the p r o p e l l e r l o a d i n g is c o m p u t e d a c c o r d i n g to the idea suggested b y H o u g h and O r d w a y [ 9 ] : g i v e n the advance, t h r u s t and t o r q u e coefficients (ƒ, K 7 , /Cq i n the f o l l o w i n g ) , the a x i a l , r a d i a l and t a n g e n t i a l force d i s t r i b u t i o n s are c o m p u t e d u n d e r the as-s u m p t i o n o f an o p t i m a l d i as-s t r i b u t i o n f o r the c i r c u l a t i o n . The o r i g i n a l m o d e l w a s m o d i f i e d to take i n t o account b o t h the axial flow r e d u c t i o n at the p r o p e l l e r disk and the side force developed b y the p r o p e l l e r . A x i a l flow r e d u c t i o n is accounted f o r by c o m p u t i n g , at each time step, an e s t i m a t i o n o f the advance c o e f f i c i e n t , i n the hypothesis o f c o n s t a n t speed o f t h e r e v o l u t i o n and by u s i n g the instantaneous average axial v e l o c i t y at the propeller d i s k i n f l o w section. Then, n e w values o f Kji^) and K q ( / ) are c o m p u t e d f r o m the propeller characteristic curves.
Lateral forces caused b y n o n a x i a l s y m m e t r y o f the i n f l o w is i n -stead m o d e l l e d by means o f t w o s i m p l i f i e d models; i n b o t h cases the lateral force is considered to be p r o p o r t i o n a l to a proper d r i f t angle. I n the first m o d e l ( r e f e r r e d t o as the m o d i f i e d ''heurxstxC H o u g h a n d O r -d w a y m o -d e l ) , alrea-dy a-dopte-d i n a previous stu-dy [7], the si-de force is e s t i m a t e d to be p r o p o r t i o n a l to the t h r u s t T a n d to the instantaneous angle b e t w e e n the p r o p e l l e r axis e a n d the ship v e l o c i t y Va, i.e.
Yp = Q; r sin ( ^ ) w i t h /3 = arccos (6)
a was chosen equal to 1/3 o n the basis o f e x p e r i m e n t a l obser-vations [ 7 ] . The side force is supposed to lie i n the plane n o r m a l t o e X Va w h e n the t w o vectors are n o t parallel, or i n the h o r i z o n t a l plane o t h e r w i s e (see Fig. 1). It has to be n o t i c e d that, w h e n the l a t e r a l force c o n t r i b u t i o n is e s t i m a t e d b y (6), differences b e t w e e n the lateral forces o f t h e external and the i n t e r n a l propellers are o n l y due to d i f -ferences i n the axial c o m p o n e n t s o f the i n f l o w via the t h r u s t T, t h e lateral c o m p o n e n t b e i n g neglected i n the c o m p u t a t i o n o f the advance c o e f f i c i e n t ] .
I n order to capture the effects o f the i n f l o w lateral c o m p o n e n t and o b t a i n a m o d e l m o r e consistent w i t h respect to the flow physics i n v o l v e d , the s e m i e m p i r i c a l m e t h o d o f Ribner [15] has been c o n -sidered as a plausible a l t e r n a t i v e to the previous one. A l t h o u g h t h e Ribner's m o d e l is v e r y p o p u l a r f o r the e v a l u a t i o n o f s t a b i l i t y q u a l i t i e s i n aeronautics, its a p p l i c a t i o n to m a r i n e propeller is n o t d o c u m e n t e d to the authors' k n o w l e d g e . The m e t h o d was developed o n the basis o f the m a i n flow characteristics a r o u n d the propeller i n o b l i q u e flow, and t h e r e f o r e is s t r o n g l y related to the loads acting i n the p r o p e l l e r plane and provides a reliable e s t i m a t i o n o f the propeller lateral f o r c e on the basis o f theoretical considerations.
Fig. 1 . Modified Hough and Ordway model: side force direction.
W h e n the propeller w o r k s w i t h y a w angle w i t h respect to the i n -c o m i n g flow, i t a-c-celerates the flow b e h i n d the disk, t h e r e f o r e redu-c- reduc-ing the angle o f attack w i t h respect to the shaft. This results i n a lateral m o m e n t u m exerted on the flow by the propeller, and consequently, the propeller experiences a lateral force. This f a c t is accounted f o r i n the Ribner's t h e o r y by means o f an h y b r i d blade e l e m e n t approach (for the e s t i m a t i o n o f the loads a c t i n g o n the p r o p e l l e r blades) and an actuator disk approach ( f o r the e v a l u a t i o n o f the e f f e c t i v e angle o f incidence due t o the p r o p e l l e r i n d u c t i o n e f f e c t ) . In the f o l l o w i n g o n l y the core o f the m o d e l a n d its i n c l u s i o n i n the n u m e r i c a l solver are presented; the interested reader is r e f e r r e d to [10] f o r the details o f its d e r i v a t i o n .
A propeller m o v i n g i n the h o r i z o n t a l plane, at incidence w i t h re-spect to the flow (the t r e a t m e n t is analogous i n the v e r t i c a l plane), experiences a lateral force ( i n the same d i r e c t i o n o f t h e i n p l a n e c o m -p o n e n t o f v e l o c i t y ) d e f i n e d b y the r e l a t i o n :
^PROP = CYPROP^ ^ C YPROP-VPROP (7)
w h e r e CYPROP is the h y d r o d y n a m i c d e r i v a t i v e o f the lateral force for the propeller, fi is the local angle o f attack o f the flow w i t h respect to t h e propeller disk, w h e r e a s VPROP and V are the lateral and the t o t a l speed at t h e p r o p e l l e r disk. It has to be h i g h l i g h t e d t h a t the i n f l o w angle is evaluated b y averaging the v e l o c i t y c o m p o n e n t s o f the fluid over the propeller disk; moreover, the velocities i n the previous r e l a t i o n are r e f e r r e d to the n o m i n a l c o n d i t i o n s , i.e. p r o p e l l e r i n d u c t i o n is n o t considered, its e f f e c t b e i n g i n c l u d e d i n CYPROP by the f o l l o w i n g r e l a t i o n ;
, , 3 aCt ^ F (a)
(8)
w h e r e Z is the n u m b e r o f blades,/4SIDE is the lateral blade p r o j e c t e d area, iJCt/ao; is t h e sectional l i f t c o e f f i c i e n t s , d e r i v e d f r o m t h i n a i r f o i l t h e o r y and F(a) is the p r o p e l l e r load factor, d e f i n e d as
F ( a ) :
(1 + a ) [ ( l -ha) + ( l ^ 2 a f
1 + ( 1 + 2 0 ) (9)
w h e r e a is the i n d u c t i o n factor. I t is to be n o t e d t h a t the lateral force is related to the propeller g e o m e t r y ( l a t e r a l p r o j e c t e d area); f r o m this p o i n t o f v i e w , the p r o p e l l e r can be considered as an a d d i t i o n a l fin w h o s e c o n t r i b u t i o n is analogous to those p r o v i d e d b y a r u d d e r o r a central skeg. Correction factors ka a n d ks are i n t r o d u c e d i n order to account f o r the n o n - u n i f o r m i t y o f the load over the p r o p e l l e r disk i n d u c e d by the s l i p s t r e a m a n d the presence o f the p r o p e l l e r h u b [18], respectively.
R. Broglia et al./Applied Ocean Research 39 (2012) 1-10
This m o d e l f o r the side force has been added to the m o d i f i e d H o u g h and O r d w a y m o d e l . In pracdce, the o n l y t e r m to be evaluated at every d m e step is the i n d u c t i o n factor a and the r e s u l t a n t l a t e r a l speed at the p r o p e l l e r disk; i n p a r t i c u l a r :
• the i n d u c t i o n f a c t o r a is easily d e t e r m i n e d f r o m m o m e n t u m c o n -s i d e r a t i o n once the in-stantaneou-s t h r u -s t c o e f f i c i e n t Kj ha-s been d e t e r m i n e d ;
• the r e s u l t a n t lateral speed is evaluated b y averaging the local l a t -eral speed over the disk; moreover, i n o r d e r to take i n t o account the n o m i n a l w a k e , the m o m e n t u m t h e o r y is also considered i n o r d e r to separate the c o n t r i b u t i o n o f s w i r l - i n d u c e d effect.
The a l g o r i t h m is represented by the flowchart d e p i c t e d i n Fig. 3. I t s h o u l d be e m p h a s i z e d that the a d d i t i o n o f the side force m o d e l does n o t increase the c o m p u t a t i o n a l b u r d e n ; this makes the h y b r i d H o u g h and O r d w a y / R i b n e r m o d e l v e r y attractive f o r those p r o b l e m s w h e r e the details o f the flow field a r o u n d the p r o p e l l e r are n o t r e q u i r e d , b u t r a t h e r o n l y the m a i n effects o f the p r o p e l l e r o n the flow field are r e l e v a n t f o r the s i m u l a t i o n . Ship m a n o e u v r i n g is a t y p i c a l f r a m e w o r k , the key issue b e i n g the correct e s t i m a t i o n o f forces and m o m e n t s o n the h u l l , w h o s e m a g n i t u d e has effects o f p a r a m o u n t i m p o r t a n c e f o r t r a j e c t o r y p r e d i c t i o n s .
3.1.2. Blade element momentum theory (BEMT) model
In t h e Blade e l e m e n t m o m e n t u m t h e o r y (BEMT) the p r o p e l l e r is m o d e l l e d as a series o f i n d e p e n d e n t t w o - d i m e n s i o n a l a i r f o i l s ; l i f t and d r a g a c t i n g o n the generic section are easily evaluated once the t w o -d i m e n s i o n a l h y -d r o -d y n a m i c properties o f t h e p r o f i l e are k n o w n ( i n t e r m s o f Q a n d Co) o n the w h o l e range o f incidence angles e x p e r i -enced b y the section d u r i n g a c o m p l e t e blade r o t a t i o n . Usually, w h e n the p r o p e l l e r is o p e r a t i n g d u r i n g a manoeuvre, the local angle o f i n -cidence can be large and stall ( p a r t i c u l a r i y at m o d e l scale) a n d / o r c a v i t a t i o n can occur, a f f e c t i n g the t o t a l load d e v e l o p e d b y the blade. Nevertheless, i f the h y d r o d y n a m i c characteristics o f the section are k n o w n f o r a r e l a t i v e l y broad range o f angles o f incidence, these effects can be reasonably taken i n t o account and m o d e l l e d and this m o d e l is an a t t r a c t i v e a l t e r n a t i v e f o r the purpose o f analysing p r o p e l l e r be-h a v i o u r o p e r a t i n g i n oblique flow.
This m o d e l has already been considered f o r n u m e r i c a l s i m u l a t i o n s of u n m a n n e d u n d e r w a t e r vehicle [13] and o s c i l l a t i n g captive m o d e l test o f a container m o d e l [ 1 4 | , y i e l d i n g p r o m i s i n g results.
In t h e present w o r k , the m o d i f i e d BEMT t h e o r y described i n d e t a i l in [ 1 0 ] has been i m p l e m e n t e d i n the Finite V o l u m e solver x n a W s . The m a i n features o f the t h e o r y are s u m m a r i z e d i n the f o l l o w i n g . The flowchart o f the a l g o r i t h m is r e p o r t e d i n Fig. 3.
A p r o p e l l e r advancing i n o b l i q u e flow at an angle ap is schemat-ically represented i n Fig. 2. Each p r o p e l l e r blade section experiences a flow v a r y i n g p e r i o d i c a l l y b o t h i n m a g n i t u d e and angle o f attack d u r i n g a r e v o l u t i o n ; l i f t and drag on each section can be w r i t t e n as
. 2 d C t ,
dL=0.5pV,'c^ab
dD = 0.5pV^c ( C D O + Com + C D 2 « Ö )
(10)
(11)
w h e r e p is the fluid density, c is the p r o f i l e c h o r d a n d ai, is the e f f e c t i v e sectional angle o f attack (Fig. 2 ) ; h y d r o d y n a m i c l i f t and d r a g c o e f f i c i e n t s are d e t e r m i n e d on the basis o f p r o f i l e g e o m e t r y .
In o r d e r to evaluate ( 1 0 ) and (11), the e f f e c t i v e angle o f attack ( a n d t h e r e f o r e the s e l f - i n d u c e d flow) m u s t be c o m p u t e d ; i n particular, i t can be expressed as (see Fig. 2 ) :
Off, = p - I (12)
Section A - A
Fig. 2. Propeller in oblique flow (from 112]).
w h e r e p is the zero l i f t l i n e angle, <p is the n o m i n a l i n f l o w a n -gle (i.e. w i t h o u t c o n s i d e r i n g i n d u c t i o n velocities) and e,- is the self-i n d u c e d angle o f attack. The e f f e c t self-i v e sectself-ional blade velocself-itself-ies ac-c o u n t f o r the i n d u ac-c t i o n effeac-cts a n d t h e i r expressions are
Vaxial = cos Oïp + Vvi
Vtan =ClrVx sin cZp sin Ö
-(13)
(14)
for the axial and the t a n g e n t i a l c o m p o n e n t s , respectively. In the previous relations, i2 is the angular velocity, r is the r a d i a l p o s i t i o n o f the section, 9 is the c i r c u m f e r e n t i a l p o s i t i o n and finally Vxi and V^i are the axial and t a n g e n t i a l c o m p o n e n t s o f t h e p r o p e l l e r w a k e i n d u c e d flow. These i n d u c e d v e l o c i t y c o m p o n e n t s can be d e t e r m i n e d once the s e l f - i n d u c e d angle o f attack is c o m p u t e d ; i n particular, o w i n g to the Betz c o n d i t i o n ( w h i c h states a r e l a t i o n b e t w e e n sectional c i r c u l a t i o n r and i n d u c e d t a n g e n t i a l speed V^, ):
TZ=Anr¥V0i
w h e r e F i s the Prandtl's tip loss factor:
F = c o s " ' e x p Z ( l 2 sin ((/), ' o r j
(15)
(16)
Z b e i n g the n u m b e r o f p r o p e l l e r blades and (pQj is t h e g e o m e t r i c p i t c h angle o f the blade at tip. F r o m g e o m e t r i c a l relations b e t w e e n sectional velocities r e p o r t e d i n Fig. 2, (15) can be rearranged as
Z c dCj
16r da
( P - c p - S i ) =
F t a n ( e i ) s i n ( 0 + Ê | ) ( 1 7 )Once this n o n l i n e a r e q u a t i o n is solved i n t e r m o f e, ( b y an i t -erative technique), the e f f e c t i v e sectional i n f l o w is d e t e r m i n e d and loads can be evaluated. This procedure is repeated f o r each r a d i a l and c i r c u m f e r e n t i a l positions.
Finally, i n order to o b t a i n sectional t h r u s t (dT) and t o r q u e (dFp), sectional l i f t ( 1 0 ) a n d d r a g (11) are p r o j e c t e d i n the l o n g i t u d i n a l and c i r c u m f e r e n t i a l d i r e c t i o n 0 ; t o t a l t h r u s t and t o r q u e are calculated b y i n t e g r a t i n g sectional loads along the blade span and a v e r a g i n g over a
R. Brogliaet al./Applied Ocean Researcli 39(2012) 1-10 5
I n i l i j I ComiltiiHis
\
I'lvpJiicriiHHlcI
C.'ojiiputaiiou of iisnJv ^(^r£L•^;. liJJilioli or}.mjn,'o i c n i i ^ in (Itc
linplici! (irDc inri-riidiOiJ K A N ' S cijuuiioii^
OuiljiiiljlKMl dl' liydftHljnütiiic fofi^is uiiJ
l:iul ()l"?,iiii;!l.4lii!ii
Brrcclive wilkc estinialion
Eslrmatioii o f Kj(J) KQ(J)
Estimation o f liuluced factor (fl)
Esliiiiation or)n-j)laiie forces (RIBNER model) Estimation o f induced velocily: V!^=0 Estimation o f nominal wake as V'-Vi" B E M T mode! Iterative computation o f induced angle Compulation o f t h e induced velocity V,"
Open water curves
Propeller geometry: c(r), P(r)/D, 7. Rate o f revolution Blade section parameter:
dCi/da Estimation o f induced velocity: V , ' = W ' Propeller geometry: (•(/ ), P(i )/Ü, Z Rate of revolution Blade section parameters:
dCi/da. Cm. Cm, Cm
Fig. 3. Flowcharts of propeller model/uRANS coupling (left) and of the Hough and Ordway plus Ribner and the BEMT models (top right and bottom right, respectively).
propeller r e v o l u t i o n :
dT = dL cos ( 0 C C + Si) - dD s i n {(f>^ + £?,) dFv = dL sin {(j)oc + £,) + dD cos ( 0 ^ + et)
i-ln
/ / dTdrdO
fR 7 /•27r
In p r i n c i p l e , the i n p u t v e l o c i t y f o r the BEMT m o d e l s h o u l d be the u n d i s t u r b e d i n f l o w v e l o c i t y i n the f a r f i e l d , w i t h o u t the c o n t r i b u d o n o f the p r o p e l l e r acdon (i.e. the n o m i n a l w a k e , f o r a p r o p e l l e r w o r k -i n g b e h -i n d the h u l l ) . As a consequence, the f o l l o w -i n g operadons are ( 1 8 ) carried o u t b e t w e e n t w o successive t i m e steps n a n d n + 1 :
• i n d u c e d v e l o c i t y f i e l d [Vxi and V^-) evaluated at i n s t a n t n obtained by the previous procedure is stored. Propeller loads are evaluated
R. Brogliaet al./Applied Ocean Research 39(2012) 1-10
Fig. 4. Views ofthe model: top, 3D global; bottom, transom region.
and passed to the uRANS solver as a b o d y force d i s t r i b u d o n over the disk a n d the s o l u d o n o f t h e c o m p l e t e f l o w f i e l d is o b t a i n e d ; • at the n e x t d m e step ii + 1, the n e w i n f l o w p r o v i d e d by t h e uRANS
solver accounts f o r t h e p r o p e l l e r s e l f - i n d u c t i o n effects. The BEMT solver evaluates n e w loads and i n d u c t i o n v e l o c i t y c o m p o n e n t .
4. Geometry a n d test conditions
The t a n k e r l i k e m o d e l s h o w n i n Fig. 4 is considered f o r the n u -m e r i c a l s i -m u l a t i o n s . I t can be seen that the -m o d e l considered i n the s i m u l a t i o n has a rather unusual c o n f i g u r a t i o n , b e i n g a t w i n p r o p e l l e r ship w i t h a single r u d d e r ; the m o d e l is f u l l y appended w i t h bilge keels, struts, A-brackets and shafts. The m a i n charactenstics are r e p o r t e d in Table 1 o n l y i n n o n d i m e n s i o n a l f o r m , because o f r e s t r i c t i o n o n d i f -f u s i o n ; -f o r t h e same reason, all the quantities i n the -f o l l o w i n g are r e p o r t e d i n t e r m s o f n o n - d i m e n s i o n a l values (as already said, l e n g t h b e t w e e n perpendiculars tpp and approach v e l o c i t y at m o d e l scale w e r e used as reference q u a n t i t i e s ) . In all the s i m u l a t i o n s the Reynolds n u m b e r is Re = 5 . 0 x 1 0 ^ and the Froude n u m b e r is Fr = 0.217.
The t u r n i n g circle m a n o e u v r e is carried o u t at f i x e d t u r n i n g rate of the propeller. The t u r n i n g rate o f the r u d d e r is 12.23= per time u n i t (at m o d e l scale), and t h e r u d d e r d e f l e c t i o n is 35=. The t u r n i n g circle s i m u l a t i o n is p e r f o r m e d by the f o l l o w i n g steps:
• un-propelled steady state simulation with fixed sinkage and trim:
t h e c o m p u t e d resistance is used to fix the p r o p u l s i o n p o i n t o n the bases o f the o p e n w a t e r characteristics o f the p r o p e l l e r ( t h e p r o p u l s i o n p o i n t is r e p o r t e d i n Table 1);
• acceleration: time resolved 6DoF s i m u l a t i o n i n w h i c h the m o d e l
is accelerated f r o m rest f o r one n o n - d i m e n s i o n a l time u n i t ; an
Table 1
IVlain parameters ofthe model in non-dimensional form.
Dimension Symbol Value
Length between tpp 1
perpendiculars
Speed U„ 1
Displacement A 5.099 X 10-^
Moment of inertia for 1.318 X 10~=
roll
Moment of inertia for h'y 3.012 X 10-''
pitch
Moment of inertia for Izz 3.012 X 10-''
yaw Propeller diameter D 3.261 X 10-2 Number of blades Z 4 Advancement ratio J 0.915 Thrust coefficient KT 0.1914 Torque coefficient KQ 0.03817
Rudder lateral area 0.00115
a d d i t i o n a l fictitious p u s h i n g f o r c e is added to reduce the transient phase;
• stabilization: the fictitious p u s h i n g force is r e m o v e d and a t i m e
accurate 6DoF s i m u l a t i o n is p e r f o r m e d to let the ship achieve the d y n a m i c a l sinkage and t r i m f o r the chosen speed and displace-m e n t c o n d i t i o n s ;
• evolution: once the a t t i t u d e o f t h e ship reached a reasonable stable
c o n d i t i o n , the rudder is r o t a t e d at the prescribed t u r n i n g rate. T i m e resolved 6DoF s i m u l a t i o n is carried out.
5. C o m p u t a t i o n a l m e s h
The physical d o m a i n is discretized by means o f 90 b o d y - f i t t e d patched and overlapped blocks; o v e r i a p p i n g grids' capabilities are e x p l o i t e d to a t t a i n h i g h q u a l i t y meshes, f o r r e f i n e m e n t purposes and to handle the time changes o f the d o m a i n boundary. The w h o l e mesh counts f o r a t o t a l o f 6,166,528 g r i d cells. G r i d d i s t r i b u t i o n is such that the thickness o f the first cell o n t h e w a l l is always b e l o w 1 i n terms o f w a l l units {y* = 0 ( 1 ) , i.e. A/Lpp = O(20/Re), A b e i n g the thickness o f the cell). A n o v e r v i e w o f the c o m p u t a t i o n a l rnesh is s h o w n i n Fig. 5, w h e r e , f o r the sake o f clearness, c h i m e r a cells have been h i d d e n .
Instead o f g e n e r a t i n g a fixed b a c k g r o u n d mesh t h a t covers the w h o l e course o f the h u l l , a r e l a t i v e l y small b a c k g r o u n d mesh a r o u n d the m o d e l has been generated; this b a c k g r o u n d g r i d translates i n the h o r i z o n t a l plane and rotates a r o u n d the v e r t i c a l axis i n order to f o l l o w the m o t i o n o f the h u l l .
6. Results
In this paragraph n u m e r i c a l results are presented. The analysis is focused on the p r o p e l l e r behaviour d u r i n g the manoeuvre, w i t h par-t i c u l a r emphasis par-to par-the lapar-teral force and par-the a s y m m e par-t r i c loads d u r i n g the stabilized phase f o r the d i f f e r e n t p r o p e l l e r models under analy-sis, i.e. the t r a d i t i o n a l H o u g h and O r d w a y actuator disk (i.e. w i t h o u t lateral force correction), its m o d i f i e d versions and the BEMT m o d e l . The e f f e c t o f the lateral force is analysed by c o m p a r i n g the p r e d i c t e d speed drop, d r i f t and y a w rate w i t h e x p e r i m e n t a l m e a s u r e m e n t s o f free r u n n i n g manoeuvres.
For this m o d e l , an extensive f r e e r u n n i n g test p r o g r a m has been carried o u t at t h e lake o f N e m i near Rome ( I t a l y ) [ 1 0 ] and [ 1 1 ] ; in particular, three series o f t u r n i n g circle m a n o e u v r e s w e r e per-f o r m e d ; a m o n g t h e m , all measurements b u t one ( t h a t was thereper-fore neglected) are v e r y close to each other; however, i n order to avoid
R. Broglia et al. /Applied Ocean Research 39(2012)1-10 7
Fig. 5. Computational mesti: top, frontal view; bottom, rear view.
c o n f u s i o n , o n l y one o f t h e m (Serie 111) has been c o n s i i j e r e d as representative o f the t u r n i n g qualities o f the ship. For the sake o f c o m p l e t e -ness, also the p r i n c i p a l features o f the c o m p l e x f l o w f i e l d a r o u n d the h u l l d u r i n g the steady t u r n i n g phase w i l l be described, w i t h focus o n the c o m p l e x v o r t e x structures detached f r o m the h u l l .
I t has to be n o t e d t h a t s i m u l a t i o n s w i t h the H o u g h and O r d w a y models ( w i t h and w i t h o u t lateral force c o r r e c t i o n s ) w e r e p e r f o r m e d on b o t h the finest and the m e d i u m grids i n o r d e r to estimate g r i d uncertainties, whereas o n l y the c o m p u t a t i o n s w i t h the m e d i u m mesh was done f o r the BEMT m o d e l , i n order to save CPU time.
6.J. Manoeuvre analysis
I n t h e f o l l o w i n g analysis, f is the n o n - d i m e n s i o n a l time (i.e. t" =
tUo/Lpp) and f = 0 is the n o n - d i m e n s i o n a l time at w h i c h the r u d d e r
starts its r o t a t i o n ; t h e o r i g i n o f the earth fixed system o f reference is taken as the p o s i t i o n o f the m o d e l at the same instant, w h e r e the x-axis is d i r e c t e d as the a p p r o a c h i n g speed, the z-axis is p o s i t i v e u p w a r d , and finally the y-axis c o m p l e t e a r i g h t - h a n d e d system o f reference. The v e l o c i t y is n o r m a l i s e d w i t h the ship v e l o c i t y at t' = 0 ( n o m i n a l a p p r o a c h i n g speed), i.e. U' = U/UQ. I t has to be n o t e d that, i n the f o l l o w i n g , the y a w rate is made n o n - d i m e n s i o n a l u s i n g the actual v e l o c i t y U instead o f the a p p r o a c h i n g speed UQ, i.e. f = rLpp/U.
N u m e r i c a l s i m u l a t i o n s presented i n [ 7 ] have s h o w n that, i n order to achieve an accurate e s t i m a t i o n o f the t u r n i n g circle m a n o e u v r e , l a t -eral forces developed b y the propeller s h o u l d be p r o p e r l y accounted for. As a m a t t e r o f the fact, w h e n c o m p a r i n g the p r e d i c t i o n o f t r a j e c t o r y , n o n d i m e n s i o n a l m o d e l speed (U'), d r i f t angle ( P ) and n o n -d i m e n s i o n a l y a w rate ( f ) (Fig. 6) w i t h the stan-dar-d an-d the m o -d i f i e -d H o u g h and O r d w a y models, i t is e v i d e n t t h a t the i n c l u s i o n o f the p r o -peller lateral force {"heuristic" c o r r e c t i o n w i t h a; = 1/3 or Ribner's m o d e l ) provides a s t a b i l i s i n g e f f e c t t h a t n o t i c e a b l y i m p r o v e s the e s t i -m a t i o n o f the t r a j e c t o r y and the k i n e -m a t i c para-meters. Note t h a t the s i m u l a t i o n w i t h o u t the lateral force c o r r e c t i o n was s t o p p e d at early time because the discrepancy w i t h the e x p e r i m e n t s (especially f o r the absolute speed and the d r i f t angle) w e r e unacceptably large. On the c o n t r a r y , the s i m u l a t i o n s i n c l u d i n g the lateral force are i n good agree-m e n t w i t h the e x p e r i agree-m e n t s ; i n particular, i n the stabilized phase b o t h
Fig. 6. Trajectoi-y and kinematic parameters: Hough and Ordway model without side force and with different side force models.
Table 2 Trajectories parameters. Exp. Numerical a = 0 a = 1/3 Ribner Advance 2.85 2.99 (4.91%) 2.98(4.56%) 3.02(5.96%) Transfer 1.00 0.98 (2.00%) I.OO(-) 1.02(2.00%) Err. - 3.45% 2.28% 3.98% transient Tactical 2.56 2.34(8.59%) 2.51 (1.95%) 2.48(3.13%) Turning 2.52 NC 2.64(4.76%) 2.60 (2.44%) Err. steady
-
NC 3.36% 2.79%models overestimate the speed loss w i t h an e r r o r a r o u n d 6%, and u n -derestimate the y a w rate w i t h an error o f a r o u n d 7.5%; t h e d r i f t angle is also w e l l p r e d i c t e d ( e r r o r a r o u n d 5%). It is w o r t h to note that, i n t h e first stage o f the t r a n s i e n t phase, b o t h sway and y a w accelerations are u n d e r e s t i m a t e d . This could be caused by several reasons, such as an u n d e r e s t i m a t i o n o f r u d d e r effectiveness, an o v e r e s t i m a t i o n o f i n e r t i a and added mass effects or some deficiencies i n the p r o p e l l e r l a t e r a l force m o d e l , w h i c h does n o t account f o r t r a n s i e n t effects. M o r e o v e r , discrepancies i n the t r a n s i e n t phase m a y be due to some n o n - l i n e a r p h e n o m e n a r e l a t e d to r o l l m o t i o n (caused, f o r e x a m p l e , by an u n -d e r e s t i m a t i o n o f t h e vertical p o s i t i o n o f t h e centre o f g r a v i t y a n -d / o r an over p r e d i c t i o n o f the l o n g i t u d i n a l i n e r t i a ) t h a t are progressively reduced w i t h the t r a n s i e n t phase. H o w e v e r , a deeper analysis o f t h e t r a n s i e n t phase are o u t o f the scope o f the present w o r k and w i l l be addressed i n f u t u r e d e v e l o p m e n t .
Comparisons f o r the t r a j e c t o r y parameters (advance, transfer, tac-tical and t u r n i n g d i a m e t e r s ) are r e p o r t e d i n Table 2; the overall agree-m e n t is c o n f i r agree-m e d to be rather satisfactory f o r the agree-m o d i f i e d agree-models, w h e r e a s large discrepancy is, again, observed w h e n l a t e r a l loads are not taken i n t o account. In the table, i n a d d i t i o n to the t r a j e c t o r y p a -rameters w i t h t h e i r d e v i a t i o n f r o m the e x p e r i m e n t a l values, average values f o r the errors d u r i n g the t r a n s i e n t (advance and t r a n s f e r ) phase and f o r the steady (tactical and t u r n i n g d i a m e t e r s ) are also g i v e n . I n the t r a n s i e n t phase, the o r i g i n a l and the m o d i f i e d m o d e l s p r o v i d e s i m i l a r results, b o t h v e r y close to the e x p e r i m e n t s , w i t h errors less than 4%; moreover, the Ribner's m o d e l p r o v i d e s a s t a b i l i s i n g e f f e c t very close to the m o d i f i e d H o u g h and O r d w a y w i t h O! = 1 / 3 . I t is to be n o t e d t h a t the s i m u l a t i o n w i t h o u t the lateral force ( o ; = 0) provides a rather s m a l l tactical d i a m e t e r i n the steady phase: f r o m the time h i s -tory, i t can be seen t h a t the ship is a d v a n c i n g w i t h a larger d r i f t angle, w h i c h causes a larger speed loss and a s m a l l e r t u r n i n g radius. On t h e contrary, b o t h m o d i f i e d models provide, again, s i m i l a r results i n g o o d agreement w i t h the e x p e r i m e n t s , the average e r r o r b e i n g a r o u n d 3%. If the c o m p a r i s o n is made i n t e r m s o f k i n e m a t i c a l parameters (speed drop, y a w rate and d r i f t angle) at the s t a b i l i z e d phase (see Table 3 ) , s i m i l a r conclusions can be d r a w n . 3
T a b l e s
Kinematic parameters (stabilized phase).
Exp. Numerical a = 1/3 Ribner U/Uo 0.54 0.50(7.41%) 0.51 (4.04%) f 46.3 42.74 (7.69%) 43.98> (5.01%) 16.2 17.0 (4.93%) 17.0 (4.93%) Err. steady
-
6.68% 4.66%Fig. 7. Trajectory and kinematic parameters: comparison among different propeller models.
the m o i i i f i e i l ones and the BEMT m o d e l is p e r f o r m e d o n the m e d i u m g r i d . I t can be seen i n Fig. 7 that, f o r the s i m u l a t i o n w i t h the f i n e g r i d , the s i m i l a r i t y o f the p r e d i c t i o n s b e t w e e n the m o d i f i e d H o u g h and Or-d w a y moOr-dels ( b o t h the "heuristic" w i t h a = 1 /3 anOr-d the Ribner's cor-r e c t i o n ) and the good agcor-reement w i t h measucor-rements acor-re c o n f i cor-r m e d ; o n the c o n t r a r y , the BEMT m o d e l provides o v e r - e s d m a t e d values f o r the speed d r o p and the d r i f t angle, and s i m i l a r values f o r the y a w rate. The s i m u l a d o n w i t h o u t any lateral force provides results s i m i l a r to the BEMT m o d e l f o r b o t h the speed d r o p a n d the d r i f t angle, whereas for the y a w rate an o v e r - e s d m a t i o n can be observed.
The parameters o f the t r a j e c t o r y p r e d i c t e d by the various m o d -els are s u m m a r i z e d i n Table 4. To estimate the d i f f e r e n c e a m o n g the models, t h e results o f the m o d i f i e d H o u g h and O r d w a y m o d e l
{"heuristic" a = 1/3) are taken as reference. It can be seen that, also
for these parameters, i t is c o n f i r m e d that the m o d i f i e d H o u g h and Or-d w a y moOr-dels ( b o t h the "heuristic" anOr-d the Ribner c o r r e c Or-d o n ) p r o v i Or-d e s i m i l a r results, the differences b e i n g less t h a n 1 % d u r i n g the t r a n s i e n t phase and a r o u n d 3% d u r i n g the steady phase. On the c o n t r a r y , large d i f f e r e n c e is observed f o r the s i m u l a d o n w i t h o u t the lateral force. For the BEMT m o d e l s i m i l a r results are obtained d u r i n g the t r a n -sient phase, the d i f f e r e n c e b e i n g less t h a n 3%; c o m p a r i n g the values f o r the advance and the transfer, i t is e v i d e n t t h a t the BEMT m o d e l provides a smaller s t a b i l i z i n g e f f e c t than the m o d i f i e d H o u g h and Or-d w a y moOr-dels. This behaviour is c o n f i r m e Or-d i n the steaOr-dy phase, w h e r e b o t h tacdcal a n d t u r n i n g diameters are smaller t h a n those p r e d i c t e d by the m o d i f i e d models. Comparison b e t w e e n k i n e m a d c parameters p r o v i d e d b y the d i f f e r e n t models i n the stabilized phase is g i v e n i n Table 5; i t is c o n f i r m e d t h a t i n c l u d i n g lateral p r o p e l l e r effects (i.e. using either the BEMT or the m o d i f i e d H o u g h a n d O r d w a y m o d e l ) increases the accuracy o f the results.
In Fig. 8 t h e d m e histones o f n o n - d i m e n s i o n a l p r o p e l l e r lateral forces ( l e f t ) a n d lateral f o r c e / t h r u s t rado are reported, w h e r e colours are chosen consistent w i t h those i n Fig. 6, solid and dashed lines refer-ring to i n t e r n a l / l e e w a r d and e x t e r n a l / w i n d w a r d propellers, respec-dvely. Forces are made n o n - d i m e n s i o n a l b y using the w a t e r d e n s i t y
Fig. 8. lateral force: comparison among different models.
( p ) , the speed o f advancement (UQ) and the l e n g t h b e t w e e n p e r p e n -dicular (Lpp), i n p a r t i c u l a r :
y, _ ^prop
Yprop and T being the lateral force and the t h r u s t p r o v i d e d b y the propeller. F r o m these figures, a deeper i n s i g h t about p r o p e l l e r b e h a v i o u r d u r i n g a t i g h t m a n o e u v r e and its c o n t r i b u t i o n to the ship d y n a m i c response can be o b t a i n e d . I t is w o r t h n o t i n g t h a t b o t h the BEMT and Ribner models p r o v i d e lateral forces f o r the l e e w a r d a n d w i n d w a r d propeller i n opposite d i r e c t i o n ; the w i n d w a r d one provides a stabilising effect, whereas the l e e w a r d propeller tends to destabilize the course. This a p p a r e n t l y u n e x p e c t e d behaviour depends o n the stern fineness characteristics, w h i c h s t r o n g l y affects the local fiow field, because the w i n d w a r d p r o p e l l e r experiences a s t r o n g o b l i q u e f l o w f r o m the w i n d to the l e e w a r d side, whereas the flow a r o u n d the l e e w a r d propeller is i n the opposite d i r e c t i o n . The Ribner m o d e l provides higher lateral forces ( i n percent o f the t h r u s t ) w i t h respect to t h e BEMT m o d e l (20% against 5% o n t h e w i n d w a r d shaft, a n d 7% against 2.5% o n the l e e w a r d shaft, respectively); this leads to a m o r e stable vessel, c o h e r e n t l y w i t h the results discussed i n Fig. 7 a n d Table 4. O n the contrary, the m o d i f i e d H o u g h and O r d w a y ( o ; = 1 / 3 ) m o d e l develops lateral forces o f t h e same sign because i t assumes t h a t the i n f l o w angle w i t h respect to the p r o p e l l e r disk coincides w i t h the h y d r o d y n a m i c d r i f t o f the ship. Finally, as already n o t e d , the o r i g i n a l H o u g h and O r d w a y m o d e l ( o ; = 0 ) does n o t develop a n y lateral f o r c e (data are not i n c l u d e d i n the pictures), thus leading to an u n d e r -e s t i m a t i o n o f th-e ship d y n a m i c s t a b i l i t y .
In Fig. 9 time histories o f t h r u s t and t o r q u e developed by the propellers d u r i n g the m a n o e u v r e are r e p o r t e d . It can be observed that, a f t e r the r u d d e r a c t i v a t i o n , the t h r u s t and t o r q u e developed b y b o t h propellers increase, because o f the d r o p o f the advance c o e f f i cients due to speed r e d u c t i o n experienced b y the vessel i n t h e d r i f t -y a w m o t i o n . Moreover, this p h e n o m e n o n is n o t s -y m m e t r i c a l ; i.e. the e x t e r n a l / w i n d w a r d p r o p e l l e r develops higher loads w i t h respect to the l e e w a r d one, because the w a k e , and c o n s e q u e n t l y the i n f l o w i n correspondence o f the propeller plane, is a s y m m e t r i c a l .
The behaviour o f b o t h the o r i g i n a l and the m o d i f i e d H o u g h a n d O r d w a y models is v e r y s i m i l a r to each other (as i t c o u l d be expected), the e x t e r n a l and i n t e r n a l propellers e x p e r i e n c i n g an increase i n t h r u s t and t o r q u e o f about 50% and 40%, respectively. The BEMT m o d e l p r o -vides d i f f e r e n t results: the p r e d i c t e d stabilized t h r u s t is almost s i m i l a r to the previous values f o r the o u t e r propeller, whereas i t is n o t i c e a b l y l o w e r f o r the i n n e r one (20%). Finally, a d i f f e r e n t b e h a v i o u r d u r i n g the t r a n s i e n t phase b e t w e e n BEMT and m o d i f i e d H o u g h and O r d w a y m o d e l s can be also h i g h l i g h t e d : f o r the BEMT m o d e l , the o u t e r p r o -peller is m o r e loaded t h e n the i n n e r one; o n the c o n t r a r y , f o r the H o u g h & O r d w a y m o d e l , the p r o p e l l e r w h i c h experiences a h i g h e r load is the outer one.
R. Broglia et al./Applied Ocean Research 39 (2012) 1-10
Table 4
Trajectories parameters: comparison among different models.
a = 1/3 Q'=0 BEMT Ribner Advance Transfer Err. transient Tactical Turning Err. steady 3.22 1.12 2.78 2.90 3.06 (4.97%) 1.02(8.93%) 6.95% 2.50(10.07%) 2.42(16.55%) 13.31% 3.18(1.24%) 1.07(4.46%) 2.85% 2.58(7.19%) 2.59(10.69%) 8.94% 3.23(0.31%) 1.11 (0.89%) 0.60% 2.67(3.96%) 2.83(2.41%) 3.19% Table 5
Kinematic parameters (stabilized phase): comparison among different models.
a = 1/3 a = 0 BEMT Ribner U/Uo 0.55 0.51 (7.27%) 0.49(10.90%) 0.56(1.82%) f 39.58 46.62 (17.68%) 43.90 (10.91%) 40.64 (2.68%) P 16.24 17.99 (10.76%) 17.67 (8.81%) 16.59 (2.16%) Err. steady
-
11.87% 10.20% 2.22% ^ ^ t ^ ' - —/
V.J . . . . Hit) .1=0 Llinard HM x^O Vtnduard HiO.,= lJlU!lMT<i - - • ;M0ii=Wmaiwo7jRibmr's iTimlel Letward
- ~ • RWiilr's model V/indward - BEI.ITUsmrd ' - - • S S . f f Wiitdward - ^ . ^ ^ . . . ! . . t - — — Hi.0 x=ÖL»wa7d HAO .1=11} Lu *.^Td HA0i'=!/3Viv{^<ifd RiintT'jKifxlrl Lf^v'ii SEKTLieward aEMTWi%du,ard
Fig. 9. Thrust and torque: comparison among different models.
6.2. Flow field
For the sake o f completeness, the l o n g i t u t j i n a l v e l o c i t y (i.e. the x c o m p o n e n t o f the f l o w v e l o c i t y i n the ship fixeij f r a m e o f reference) d u r i n g the stabilized phase at d i f f e r e n t cross-secdons a l o n g the h u l l is presented i n Fig. 10, w h e r e the c o m p l e x i t y o f the f l o w f i e l d a r o u n d the vessel clearly appears. A t section x = 0.3 ( b o w region) the genera-t i o n o f a clockwise v o r genera-t e x (seeing f r o m genera-the b o w ) can be observed: genera-this v o r t e x is due to the cross f l o w a r o u n d the bulbous b o w , t h e d i r e c d o n o f this transversal f l o w b e i n g f r o m the i n n e r t o the o u t e r side w i t h respect to the centre o f the t r a j e c t o r y (the lateral v e l o c i t y due to the y a w rate overtakes the d r i f t m o d o n , and t h e r e f o r e the n e t f l u x at the b o w is f r o m the i n n e r to the o u t e r side); once this v o r t e x is gener-ated, i t is d r i v e n by the i n c o m i n g f l o w , and t h e r e f o r e i t is convected t o w a r d s the p o r t side. A t section x = - 0 . 2 t w o counter-clockwise vortices, detached f r o m the l e e w a r d and w i n d w a r d bilges, can be o b -served. These vordces are convected d o w n s t r e a m as w e l l : the one on the p o r t / l e e w a r d side is stronger a n d i t is clearly observable up to the last section x = - 0 . 4 8 6 i n the t o p right region o f the f l o w field. The v o r t e x on the s t a r b o a r d / w i n d w a r d side is convected t o w a r d s the p o r t side and i t merges w i t h the intense keel v o r t e x (see section at
\
...
X __ J • ^Fig. 10. Axial velocity at several cross-sections.
X = - 0 . 3 8 2 ) ; moreover, i t can be observed t h a t the w i n d w a r d shaft bossing generates an a d d i t i o n a l v o r t e x , w h i c h is convected d o w n -stream. A t section X = - 0 . 4 5 3 the s w i r l and t h e acceleration caused by t h e propellers, together w i t h the i n t e r a c t i o n o f t h e wakes o f the ap-pendages w i t h t h e propeller are evident.'The f l o w a r o u n d the r u d d e r and i n its w a k e is r e p o r t e d i n the last t w o sections, w h e r e i t is s h o w n that the r u d d e r is p a r t i a l l y i n the s l i p s t r e a m o f the o u t e r propeller. A t sections x = - 0 . 4 7 0 and x = - 0 . 4 8 6 , i t can be also observed t h a t the r u d d e r is i n the w a k e o f the skeg, w h i c h is the probable cause o f the s t r o n g i n e f f i c i e n c y o f the r u d d e r i t s e l f In the last cross-section, flow separation i n t h e suction side o f the rudder, as w e l l as a t i p v o r t e x generated b y t h e cross flow, is evident.
7. C o n c l u s i o n s
In present w o r k the i n f l u e n c e o f the p r o p e l l e r m o d e l l i n g o n the p r e d i c t i o n o f the d y n a m i c properties o f a m a n o e u v r i n g vessel has been i n v e s t i g a t e d ; to this a i m , the t u r n i n g circle m a n o e u v r e o f a t w i n screw ship w i t h a single r u d d e r have been e x t e n s i v e l y analysed b y means o f the uRANS solver / n o v i s c o u p l e d w i t h d i f f e r e n t p r o p e l l e r models. I n particular, m o d i f i e d H o u g h and O r d w a y and BEMT m o d -els have been considered and p r o p e r i y i n c l u d e d i n the CFD solver, i n o r d e r to account f o r the inplane loads w h i c h arise w h e n the p r o -peller w o r k s i n o b l i q u e flow, a typical s i t u a t i o n experienced d u r i n g t i g h t manoeuvres. It has been emphasized t h a t accurate n u m e r i c a l p r e d i c t i o n s carried o u t w i t h proper p r o p e l l e r models (able to p r o -v i d e a lateral f o r c e ) can be obtained, b o t h i n t e r m s o f t r a j e c t o r y a n d k i n e m a t i c response (speed d r o p and y a w rate); o n the contrary, unsat-i s f a c t o r y predunsat-ictunsat-ions w e r e obtaunsat-ined w unsat-i t h models t h a t do n o t account
R. Broglia et al/Applied Ocean Researclj 39 (2012) 1-10
f o r the side force developed by.the propellers, f o r unusual p r o p e l l e r -r u d d e -r c o n f i g u -r a t i o n s , like the one conside-red i n the p-resent pape-r. It has to be h i g h l i g h t e d that, using these models, the increase o f the c o m p u t a t i o n a l resources is negligible.
It has been d e m o n s t r a t e d that, w h e n u s i n g m o d i f i e d H o u g h and O r d w a y models, either the "heuristic" or the Ribner's correction, ac-curate p r e d i c t i o n o f the t u r n i n g circle m a n o e u v r e ( i n terms o f b o t h t r a j e c t o r y and k i n e m a t i c parameters) is achieved. I n particular, the c o m p a r i s o n w i t h free r u n n i n g test has s h o w n that b o t h the transient and the stabilized phases are w e l l captured, the averaged comparison errors b e i n g less t h a n 4%. Needless to say t h a t the advantages to use the Ribner's m o d e l , w i t h respect to the "heuristic" one, are related to t h e larger physical i n f o r m a t i o n contained i n the m o d e l itself, and the smaller n u m b e r o f parameters to tune.
The c o m p a r i s o n b e t w e e n the d i f f e r e n t models considered, i.e. the o r i g i n a l H o u g h and O r d w a y m o d e l , the m o d i f i e d ones and the BEMT, has been p e r f o r m e d o n the m e d i u m mesh. It has been c o n f i r m e d t h a t the "heuristic" and the Ribner's models p r o v i d e s i m i l a r results w i t h good agreement w i t h experiments, whereas differences can be seen w h e n c o n s i d e r i n g m o d e l s unable to y i e l d side forces. The BEMT m o d e l provides results s i m i l a r to the corrected H o u g h and O r d w a y models in the t r a n s i e n t phase, the d i f f e r e n c e w i t h the "heuristic" m o d e l being less t h a n 3%; s l i g h t l y larger differences (less t h a n 9%) w e r e observed in the stabilized phase.
It has to be p o i n t e d o u t t h a t propeller models considered i n this w o r k are based on a quasi-steady hypothesis o f the flow field, and t h e r e f o r e all those e f f e c t due to s t r o n g unsteadiness (such as t r a i l i n g a n d s h e d d i n g w a k e structures o f variable strength, added mass and d y n a m i c i n f l o w effects) are n o t e x p l i c i t l y considered; therefore, the r e l i a b i l i t y o f this models s h o u l d be also v e r i f i e d a n d v a l i d a t e d f o r unsteady manoeuvres, like zig-zag or P M M o s c i l l a t o r y m o d o n s ; this w i l l be addressed i n f u t u r e research w o r k s .
A c k n o w l e d g e m e n t s
The w o r k has been done i n the f r a m e w o r k o f t h e EDA p r o j e c t
"Submarine Coupled 6DoF Motions including Boundary Effects"
finan-cially s u p p o r t e d by the Italian Navy. The authors w i s h to thanks Dr. Salvatore M a u r o f o r p r o v i d i n g e x p e r i m e n t a l data. N u m e r i c a l c o m p u -tations presented here have been p e r f o r m e d o n the parallel machines o f CASPUR S u p e r c o m p u d n g Center (Rome); t h e i r s u p p o r t is g r a t e f u l l y a c k n o w l e d g e d .
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