2L-1
Ocean Engineering 90 (2014) 119-128
ELSEVIER
Contents lists available at ScienceDirect
Ocean Engineering
journal homepage: www.elsevier.com/locate/oceaneng
Numerical study of a flexible sail plan submitted to pitching: Hysteresis (SN
phenomenon and effect of rig adjustments
B e n o i t A u g i e r ^ F r e d e r i c H a u v i l l e ^ , P a t r i c k B o t ^ ' * N i c o l a s A u b i n ^ M a t h i e u D u r a n d ^ ' ' '
CrossMarl?
= Naval Academy Research Institute - IRENAV CC600, 29240 BREST Cedex 9, France ^ K-Epsilon Company, 1300 route des Cretes - B.P. 255 06905 Sophia Antipolis Cedex, France
A R T I C L E I N F O
Article histoiy:
Received 20 December 2013 Accepted 24 June 2014 Available online 17 July 2014
Keywords:
Fluid structure interaction Dynamic behaviour Yacht sails Pitching Hysteresis Parametric study A B S T R A C T
A n u m e r i c a l i n v e s t i g a t i o n of the d y n a m i c fluid s t r u c t u r e i n t e r a c t i o n ( F S I ) of a y a c h t sail p l a n s u b m i t t e d to h a r m o n i c p i t c h i n g is p r e s e n t e d to analyse the system's d y n a m i c b e h a v i o u r a n d the effects of m o t i o n s i m p l i f i c a t i o n s a n d rigging a d j u s t m e n t s on a e r o d y n a m i c forces. It is s h o w n t h a t the d y n a m i c b e h a v i o u r of a sail p l a n s u b j e c t to y a c h t m o t i o n clearly deviates f r o m the q u a s i - s t e a d y theory. T h e a e r o d y n a m i c forces p r e s e n t e d as a f u n c t i o n of the instantaneous a p p a r e n t w i n d angle s h o w h y s t e r e s i s loops. It is s h o w n that the h y s t e r e s i s p h e n o m e n o n dissipates s o m e e n e r g y a n d t h a t the d i s s i p a t e d e n e r g y i n c r e a s e s strongly w i t h the p i t c h i n g r e d u c e d f r e q u e n c y a n d a m p l i t u d e . T h e effect of r e d u c i n g the real p i t c h i n g m o t i o n to a s i m p l e r s u r g e m o t i o n is investigated. Results s h o w s i g n i f i c a n t d i s c r e p a n c i e s w i t h u n d e r -e s t i m a t -e d a -e r o d y n a m i c forc-es a n d no m o r -e hyst-er-esis w h -e n a surg-e m o t i o n is c o n s i d -e r -e d . H o w -e v -e r , t h -e s u p e r p o s i t i o n a s s u m p t i o n consisting in a d e c o m p o s i t i o n of the surge into t w o t r a n s l a t i o n s n o r m a l a n d c o l l i n e a r to the a p p a r e n t w i n d is v e r i f i e d . T h e n , s i m u l a t i o n s w i t h d i f f e r e n t d o c k tunes a n d b a c k s t a y loads highlight the i m p o r t a n c e of rig a d j u s t m e n t s on the a e r o d y n a m i c forces a n d the d y n a m i c b e h a v i o u r of a sail p l a n . T h e e n e r g y d i s s i p a t e d by the hysteresis is h i g h e r for looser s h r o u d s a n d a tighter backstay.
© 2 0 1 4 E l s e v i e r Ltd. A l l rights r e s e r v e d .
1. Introduction
W h e n a n a l y s i n g t h e b e h a v i o u r o f y a c h t sails, a n i m p o r t a n t d i f f i c u l t y comes f r o m t h e f l u i d s t r u c t u r e i n t e r a c d o n (FSI) o f t h e air f l o w a n d t h e sails a n d r i g ( M a r c h a j , 1996; G a r r e t t , 1 9 9 6 ; Fossad, 2 0 1 0 ) . Yacht sails are s o f t s t r u c t u r e s w h o s e shapes change a c c o r d -i n g t o t h e a e r o d y n a m -i c l o a d -i n g . The r e s u l t -i n g m o d -i f -i e d shape a f f e c t s t h e a i r f l o w a n d t h u s , t h e a e r o d y n a m i c l o a d i n g a p p l i e d t o t h e s t r u c t u r e . This f l u i d s t r u c t u r e i n t e r a c t i o n is s t r o n g a n d n o n - l i n e a r , because sails are s o f t a n d l i g h t m e m b r a n e s w h i c h e x p e r i e n c e l a r g e d i s p l a c e m e n t s a n d accelerations, e v e n f o r s m a l l stresses. As a consequence, t h e actual sail's shape w h i l e s a i l i n g — t h e so-called f l y i n g shape - is d i f f e r e n t f r o m t h e d e s i g n shape d e f i n e d b y t h e sail m a k e r a n d is g e n e r a l l y n o t k n o w n . Recently, s e v e r a l a u t h o r s have f o c u s e d o n t h e fluid s t r u c t u r e i n t e r a c t i o n p r o b l e m t o address t h e issue o f t h e i m p a c t o f t h e s t r u c t u r a l d e f o r m a t i o n o n t h e flow a n d hence t h e a e r o d y n a m i c forces g e n e r a t e d ( C h a p i n a n d H e p p e l , 2010; Renzsh a n d G r a f 2 0 1 0 ) .
A n o t h e r c h a l l e n g i n g task i n m o d e l l i n g r a c i n g y a c h t s is t o c o n s i d e r t h e y a c h t b e h a v i o u r i n a realistic e n v i r o n m e n t ( C h a r v e t e t al., 1996; M a r c h a j , 1996; G a r r e t t , 1996; Fossati, 2010). T r a d i t i o n a l V e l o c i t y P r e d i c t i o n Programs (VPPs) used b y y a c h t designers
'Corresponding author. Tel.: +33 2 98 23 39 86. E-mail address: patrick.bot@ecole-navale.fr (P. Bot).
c o n s i d e r a static e q u i l i b r i u m b e t w e e n h y d r o d y n a m i c a n d aero-d y n a m i c forces. Hence, t h e f o r c e m o aero-d e l s classically useaero-d are e s t i m a t e d i n a steady state. H o w e v e r , i n r e a l i s t i c s a i l i n g c o n d i t i o n s , t h e flow a r o u n d t h e sails is m o s t o f t e n l a r g e l y u n s t e a d y because o f w i n d v a r i a t i o n s , actions o f t h e c r e w a n d m o r e i m p o r t a n t l y because o f y a c h t m o t i o n d u e t o w a v e s . To a c c o u n t f o r t h i s d y n a m i c b e h a v i o u r , several D y n a m i c V e l o c i t y P r e d i c t i o n P r o g r a m s (DVPPs) have b e e n d e v e l o p e d , (e.g. M a s u y a m a et al., 1993; M a s u y a m a a n d Fukasawa, 1997; R i c h a r d t et al., 2 0 0 5 ; K e u n i n g e t al., 2 0 0 5 ) w h i c h n e e d m o d e l s o f d y n a m i c a e r o d y n a m i c a n d h y d r o d y n a m i c forces. W h i l e t h e d y n a m i c e f f e c t s o n h y d r o d y n a m i c f o r c e s have b e e n l a r g e l y s t u d i e d , t h e u n s t e a d y a e r o d y n a m i c b e h a v i o u r o f t h e sails has received m u c h less a t t e n t i o n . Schoop a n d Bessert ( 2 0 0 1 ) first d e v e l o p e d a n u n s t e a d y aeroelastic m o d e l i n p o t e n t i a l flow d e d i cated t o flexible m e m b r a n e s b u t n e g l e c t e d t h e i n e r t i a . I n a q u a s i -static a p p r o a c h , a first step is t o a d d t h e v e l o c i t y i n d u c e d b y t h e yacht's m o t i o n t o t h e steady a p p a r e n t w i n d t o b u i l d a n i n s t a n t a -neous a p p a r e n t w i n d (see R i c h a r d t e t a l , 2 0 0 5 ; K e u n i n g e t a l , 2 0 0 5 ) a n d t o c o n s i d e r t h e a e r o d y n a m i c forces c o r r e s p o n d i n g t o t h i s i n s t a n t a n e o u s a p p a r e n t w i n d u s i n g f o r c e m o d e l s o b t a i n e d i n t h e steady state. I n a r e c e n t s t u d y , G e r h a r d t et a l . ( 2 0 1 1 ) d e v e l o p e d a n a n a l y t i c a l m o d e l t o p r e d i c t t h e u n s t e a d y a e r o d y n a m i c s o f i n t e r a c t i n g y a c h t sails i n 2 D p o t e n t i a l flow a n d p e r f o r m e d 2 D w i n d t u n n e l o s c i l l a t i o n tests w i t h a m o t i o n range t y p i c a l o f a 9 0 -f o o t ( 2 6 m ) r a c i n g y a c h t ( I n t e r n a t i o n a l A m e r i c a ' s Cup Class 3 3 ) . Recently, Fossati a n d M u g g i a s c a ( 2 0 0 9 , 2010, 2 0 1 1 ) s t u d i e d t h e http://dx.doi.org/10.101S/j.oceaneng.2014.06.040 0 0 2 9 - 8 0 1 8 / © 2014 Elsevier Ltd. All rights reserved.
Nomenclature
A p i t c h i n g o s c i l l a t i o n a m p l i t u d e (deg"")
C sail p l a n c h o r d a t ZCE ( f r o m h e a d - s a i l l e a d i n g edge t o
m a i n s a i l t r a i l i n g edge) ( m ) Cx d r i v i n g f o r c e c o e f f i c i e n t Cy h e e l i n g f o r c e c o e f f i c i e n t ƒ,• f l o w r e d u c e d f r e q u e n c y Fx d r i v i n g f o r c e ( N ) Fy side f o r c e ( N ) Mx h e e l i n g m o m e n t ( N m ) My p i t c h i n g m o m e n t ( N m ) PTOT t o t a l p o w e r o f a e r o d y n a m i c forces ( W ) PLOOP d i s s i p a t e d p o w e r : p o w e r c o n t a i n e d i n t h e hysteresis l o o p ( W ) Pvj5 u s e f u l p o w e r : p o w e r d r i v i n g t h e b o a t f o r w a r d ( W ) S t o t a l sail area ( m ^ ) {0,X,Y,Z) I n e r t i a l f r a m e d e f i n e d f o r a n u p r i g h t b o a t ( o r i g i n 0 at t h e m a s t step, X t h e y a c h t d i r e c t i o n p o i n t i n g f o r w a r d , Y a t h w a r t s h i p s ( u p r i g h t ) p o i n t i n g p o r t s i d e ( l e f t ) , Z v e r t i c a l p o i n t i n g u p w a r d s ) ( m ) ( 0 , x , y , z ) Boat f r a m e d e f i n e d f o r a p i t c h e d a n d h e e l e d b o a t (x y a c h t d i r e c t i o n p o i n t i n g f o r w a r d , y a t h w a r t s h i p s ( h e e l e d ) p o i n t i n g p o r t s i d e ( l e f t ) , z a l o n g m a s t p o i n t i n g u p w a r d s ) ( m ) T p i t c h i n g o s c i l l a t i o n p e r i o d (s) VAW a p p a r e n t w i n d speed ( m s ~ ^ ) VBS b o a t speed ( m s ~ ^ ) VTW ti'ue w i n d speed ( m s - ^ ) Vr flow r e d u c e d v e l o c i t y ZCE i n s t a n t a n e o u s a l t i t u d e o f t h e c e n t r e o f a e r o d y n a m i c forces i n t h e i n e r t i a l f r a m e ( m ) ZCE i n s t a n t a n e o u s z c o o r d i n a t e o f t h e c e n t r e o f a e r o d y -n a m i c forces i -n t h e b o a t f r a m e ( p i t c h e d a -n d h e e l e d ) ( m ) /ï^vv a p p a r e n t w i n d angle (deg"") fieff e f f e c t i v e w i n d a n g l e (deg^) t r u e w i n d angle ( d e g ' ) <p h e e l angle (deg^) 0 t r i m angle (deg'') a h e a d i n g angle ( d e g ' ) p fluid d e n s i t y ( k g m " ^ ) T phase s h i f t (s) i t ( N ) Q[M ( N m ) /i^erodynamic f o r c e m a t r i x : r e s u l t a n t a n d m o m e n t w r i t t e n i n 0 ji^ ( r a d s - ' ) (ms ' ) Boat k i n e m a t i c m a t r i x : r o t a t i o n a n d v e l o c i t y w r i t t e n i n 0 a e r o d y n a m i c s o f m o d e l - s c a l e r i g i d sails i n a w i n d t u n n e l , a n d s h o w e d t h a t a p i t c h i n g m o t i o n has a s t r o n g a n d n o n - t r i v i a l e f f e c t o n a e r o d y n a m i c forces. T h e y s h o w e d t h a t t h e r e l a t i o n s h i p b e t w e e n i n s t a n t a n e o u s forces a n d a p p a r e n t w i n d d e v i a t e s — phase s h i f t s , hysteresis — f r o m t h e e q u i v a l e n t r e l a t i o n s h i p o b t a i n e d i n a s t e a d y state, w h i c h o n e c o u l d have t h o u g h t t o a p p l y i n a q u a s i - s t a t i c a p p r o a c h . T h e y also i n v e s t i g a t e d s o f t sails i n t h e s a m e c o n d i t i o n s t o h i g h l i g h t t h e effects o f t h e s t r u c t u r a l d e f o r m a -t i o n (Fossa-ti a n d Muggiasca, 2 0 1 2 ) .
I n a previous w o r k (Augier et al., 2013), the aero-elastic behaviour o f t h e sail p l a n subjected t o a simple h a r m o n i c p i t c h i n g w a s n u m e r i c a l l y investigated. This study has s h o w n hysteresis p h e n o m e n a b e t w e e n t h e aerodynamic forces and instantaneous apparent w i n d angle. A c o m p a r i s o n b e t w e e n a rigid structure and a realistic s o f t structure s h o w e d t h a t the hysteresis still exists f o r a rigid structure b u t i t is l o w e r t h a n w h e n the structure d e f o r m a t i o n is t a k e n i n t o a c c o u n t However, i n this first w o r k (Augier et al., 2013), t h e question w h e t h e r this hysteresis could be represented b y a simple phase s h i f t b e t w e e n b o t h oscillating signals was n o t clearly elucidated. Moreover, t h e energy exchange associated w i t h the hysteresis p h e n o m e n o n w a s n o t d e t e i T n i n e d . Hence, t h e first a i m o f the present w o r k is to investigate f u r t h e r t h i s hysteresis p h e n o m e n o n , t o q u a n t i f y the phase s h i f t b e t w e e n aerodynamic forces and apparent w i n d angle, and to d e t e r m i n e a n d analyse the associated energy.
M o s t studies o f t h e u n s t e a d y effects d u e t o y a c h t p i t c h i n g h a v e c o n s i d e r e d a 2 D s i m p l i f i e d p r o b l e m a n d t h u s a p p r o x i m a t e d t h e p i t c h i n g m o t i o n b y a t r a n s l a t i o n a l o s c i l l a t i o n a h g n e d w i t h t h e y a c h t c e n t r e l i n e (e.g. F i t t a n d L a t t i m e r , 2 0 0 0 ; G e r h a r d t e t al., 2011). T h e n , t h e u s u a l p r o c e d u r e is t o d e c o m p o s e t h i s surge m o t i o n i n t o o s c i l l a t i o n s p e r p e n d i c u l a r t o a n d a l o n g t h e d i r e c t i o n o f t h e i n c i d e n t flow, w h i c h results i n o s c i l l a t i o n s o f a p p a r e n t w i n d a n g l e a n d speed r e s p e c t i v e l y (Fig. 8 ) . The second a i m o f t h i s w o r k is t o i n v e s t i g a t e t h e e f f e c t s o f s u c h s i m p l i f i c a t i o n s i n t h e y a c h t m o t i o n ,
' In degrees when a value Is mentioned In the text and in radians in all formulae.
t h i s is c o n s i d e r e d b y c o m p a r i n g t h e results o b t a i n e d w i t h t h e sail p l a n s u b j e c t e d t o d i f f e r e n t types o f m o t i o n .
The t h i r d a i m o f t h i s w o r k is t o address t h e e f f e c t o f v a r i o u s r i g a n d sail t r i m s a n d a d j u s t m e n t s c o m m o n l y u s e d b y sailors o n t h e unsteady aero-elastic b e h a v i o u r o f t h e sail p l a n s u b j e c t e d t o p i t c h i n g . T h i s is i n v e s t i g a t e d b y c o m p a r i n g t h e r e s u h s o b t a i n e d w i t h several d o c k t u n e s a n d backstay tensions w h i c h are t y p i c a l l y used w h i l e r a c i n g a 2 8 - f o o t (8 m , J80 class) cruiser-racer.
A n unsteady FSI m o d e l has been developed and validated w i t h experiments i n real sailing conditions (Augier et al„ 2010, 2011, 2012). Calculations are made o n a J80 dass yacht n u m e r i c a l m o d e l w i t h her standard rigging a n d sails designed b y the sail m a k e r DeltaVoiles. The FSI m o d e l is b r i e f l y presented i n Section 2. The m e t h o d o l o g y o f t h e d y n a m i c investigation is g i v e n i n Section 3. I n the c o n t i n u i t y o f a previous w o r k (Augier et al., 2013), Section 4 gives f u r t h e r precisions o n the d y n a m i c behaviour w i t h a particular a t t e n t i o n t o the energy exchange related t o t h e hysteresis p h e n o m e n o n . The analysis o f p i t c h i n g m o t i o n d e c o m p o s i t i o n i n simple translations is given i n Section 5 and the effects o f various dock tunes a n d backstay loads are presented i n Sections 6.1 and 6.2. I n t h e last section, some condusions o f this study are given, w i t h ideas f o r f u m r e w o r k .
2. Numerical model
To n u m e r i c a l l y i n v e s t i g a t e aero-elastic p r o b l e m s c o m m o n l y f o u n d w i t h sails, t h e c o m p a n y K - E p s i l o n a n d t h e N a v a l A c a d e m y Research I n s t i t u t e have d e v e l o p e d t h e u n s t e a d y f l u i d - s t r u c t u r e m o d e l ARAVANTI m a d e b y c o u p l i n g t h e i n v i s c i d flow s o l v e r A V A N T I w i t h t h e s t r u c t u r a l solver ARA. T h e A R A V A N T I code is able t o m o d e l a c o m p l e t e sail b o a t r i g i n o r d e r t o p r e d i c t forces, t e n s i l e stresses a n d shape o f sails a c c o r d i n g t o t h e l o a d i n g i n d y n a m i c c o n d i t i o n s . For m o r e details, t h e r e a d e r is r e f e r r e d t o Roux e t a l . ( 2 0 0 2 ) f o r t h e fluid s o l v e r A V A N T I a n d t o H a u v i l l e et a l . ( 2 0 0 8 ) a n d Roux e t al. ( 2 0 0 8 ) f o r t h e s t r u c t u r a l s o l v e r A R A a n d t h e FSI c o u p l i n g m e t h o d .
B. Augier et al. / Ocean Engineering SO (2014) 119-128 121
A R A V A N T I i n o d e l has been v a l i d a t e d . N u m e r i c a l a n d e x p e r i -m e n t a l c o -m p a r i s o n s w i t h t h e -m o d e l ARAVANTI are based o n m e a s u r e m e n t s at f u l l scale o n a n i n s t r u m e n t e d 2 8 - f o o t y a c h t (J80 class, 8 m ) . The t i m e - r e s o l v e d sails' flying shape, loads i n t h e rig, yacht's m o t i o n a n d a p p a r e n t w i n d have been m e a s u r e d i n b o t h s a i l i n g c o n d i t i o n s o f flat sea a n d m o d e r a t e head waves a n d c o m p a r e d t o t h e s i m u l a t i o n . The code has s h o w n i t s a b i l i t y t o s i m u l a t e t h e rig's response t o y a c h t m o t i o n f o r c i n g , and t o c o r r e c t l y estimate t h e loads. Thereby, ARAVANTI is a r e l i a b l e t o o l t o s t u d y t h e d y n a m i c b e h a v i o u r o f a sail p l a n s u b j e c t to p i t c h i n g m o t i o n . For a d e t a i l e d d e s c r i p t i o n o f t h e e x p e r i m e n t a l s y s t e m a n d t h e n u m e r i c a l a n d e x p e r i m e n t a l c o m p a r i s o n , see A u g i e r et al. (2010, 2 0 1 1 , 2 0 1 2 ) .
3. Simulation procedure
The y a c h t m o t i o n i n w a v e s i n d u c e s u n s t e a d y effects i n t h e sails' a e r o d y n a m i c s . I n t h i s p a p e r w e w i l l s t u d y separately o n e degree o f f r e e d o m , b y a p p l y i n g s i m p l e h a r m o n i c p i t c h i n g . The reference f r a m e a n d t h e c o o r d i n a t e s y s t e m a t t a c h e d t o t h e y a c h t are i l l u s t r a t e d i n Fig. 1.
3.1. Reference steady case
First, t h e reference steady case is c o m p u t e d w i t h t h e f o l l o w i n g p a r a m e t e r s : t r u e w i n d speed at 10 m h e i g h t VTW = 6.7 m • s " ' (a l o g a r i t h m i c v e r t i c a l w i n d p r o f i l e is i m p o s e d w i t h a roughness l e n g t h o f 0.2 m m ( F l a y 1996)), t r u e w i n d angle / ? j ^ = 4 0 ° , b o a t speed UBs = 2 . 6 m - s - ' , h e e l angle <p = 20° a n d t r i m angle 6=0°. This first c o m p u t a t i o n y i e l d s t h e c o n v e r g e d steady flow, t h e r i g a n d sails' flying shape, a n d enables t h e steady state a e r o d y n a m i c forces a n d centre o f e f f o r t t o be d e t e r m i n e d . The c e n t r e o f e f f o r t is d e f i n e d as t h e i n t e r s e c t i o n o f t h e b o a t s y m m e t r y p l a n e w i t h t h e a e r o d y n a m i c forces m a t r i x c e n t r a l axis, w h i c h is t h e l i n e o f p o i n t s w h e r e t h e m o m e n t o f a e r o d y n a m i c forces is m i n i m a l ( n o t e t h a t t h e r e is n o p o i n t w h e r e t h i s m o m e n t is e x a c t i y zero i n general because t h e sails' shape is n o t d e v e l o p a b l e ) . T h i s c o n v e r g e d steady state is used as t h e i n i t i a l c o n d i t i o n f o r t h e c o m p u t a t i o n s w i t h p i t c h i n g f o r c i n g . The h e i g h t o f t h e c e n t r e o f a e r o d y n a m i c forces ^cEs^,j, = 4.97 m is used t o d e f i n e t h e flow characteristic q u a n t i -t i e s : a p p a r e n -t w i n d speed V / i w = 8 . 3 9 m • s " ' , a p p a r e n -t w i n d angle fi^y^ = 2 8 . 6 4 ° a n d sail p l a n c h o r d C = 4 . 2 5 m d e f i n e d as t h e distance f r o m t h e head-sail l e a d i n g edge t o t h e m a i n sail t r a i l i n g edge a t Zc£s,„„.
Corrections o f t h e a p p a r e n t w i n d angle /J^^y d u e t o c o n s t a n t heel (p ( f i r s t i n t r o d u c e d b y M a r c h a j ( 1 9 9 6 ) ) a n d t r i m 6 are
c o n s i d e r e d t h r o u g h t h e use o f t h e e f f e c t i v e a p p a r e n t w i n d angle fieff (see Jackson, 2 0 0 1 f o r heel e f f e c t , a n d Fossati a n d Muggiasca,
2011 f o r p i t c h e f f e c t ) :
fi^ff
= 2 7 . 1 6 ° i n t h e steady state.3.2. Harmonic pitctiing T h e u n s t e a d y c o m p u t a t i o n s consist o f a 18 s r u n , w i t h f o r c e d h a r m o n i c p i t c h i n g b e i n g i m p o s e d o n t h e r i g , c h a r a c t e r i s e d b y t h e o s c i l l a t i o n a m p l i t u d e A a n d p e r i o d T (Eq. ( 2 ) ) , o t h e r p a r a m e t e r s b e i n g c o n s t a n t a n d e q u a l t o those o f t h e r e f e r e n c e state. 0 = Acos(^ty (2) To a v o i d d i s c o n t i n u i t i e s i n t h e accelerations, t h e b e g i n n i n g o f m o t i o n is g r a d u a l l y i m p o s e d b y a p p l y i n g a r a m p w h i c h increases s m o o t h l y f r o m 0 t o 1 d u r i n g t h e first 3 s o f i m p o s e d m o t i o n (see first p e r i o d i n Fig. 3 ) . The i n v e s t i g a t i o n has b e e n m a d e w i t h variables i n t h e range A = 3 - 6 ° , a n d 1 = 1 . 5 - 6 s, c o r r e s p o n d i n g t o t h e t y p i c a l e n v i r o n m e n t a l c o n d i t i o n s e n c o u n t e r e d , as s h o w n i n t h e e x p e r i m e n t o f A u g i e r et al. ( 2 0 1 2 ) . The u n s t e a d y n a t u r e o f t h e flow is c h a r a c t e r i s e d b y a d i m e n s i o n l e s s p a r a m e t e r d e f i n e d b y t h e r a t i o o f t h e m o t i o n p e r i o d T t o t h e fluid a d v e c t i o n t i m e a l o n g t h e t o t a l sail p l a n c h o r d C. S i m i l a r l y t o t h e closely r e l a t e d l i t e r a t u r e (e.g. Fossati a n d Muggiasca, 2 0 1 2 ; G e r h a r d t e t al., 2011), t h i s p a r a m e t e r is c a l l e d t h e flow r e d u c e d v e l o c i t y Vr ( o r t h e i n v e r s e : t h e r e d u c e d f r e q u e n c y f ) d e f i n e d b y Vr = ^ , f r = ^ . (3) T h e r e d u c e d f r e q u e n c y w a s s h o w n t o be t h e r e l e v a n t p a r a m e t e r t o characterise t h e unsteadiness o f l i f t i n g b o d i e s a e r o d y -n a m i c s (e.g. G l a u e r t , 1926; A b b o t t , 1949; G e r h a r d t e t al., 2011). T h e case Vr <1(fr c o r r e s p o n d s t o quasisteady a e r o d y n a m i c c o n -d i t i o n s . The p i t c h i n g p e r i o -d values i n v e s t i g a t e -d c o r r e s p o n -d t o a r e d u c e d v e l o c i t y Vr f r o m 2.96 t o 11.84 ( r e d u c e d f r e q u e n c y Jr f r o m 0.08 t o 0.34), w h i c h p o s i t i o n s t h i s n u m e r i c a l s t u d y i n a s i m i l a r d y n a m i c range t o t h e e x p e r i m e n t s o f Fossati a n d M u g g i a s c a ( 2 0 1 1 ) w h e r e Vr w a s f r o m 2.3 t o 56 ( r e d u c e d f r e q u e n c y fr f r o m 0.02 t o 0.43) c o r r e s p o n d i n g t o t y p i c a l c o n d i t i o n s e n c o u n t e r e d b y a 4 8 - f o o t y a c h t (14.6 m ) . The c o m p u t e d cases are s u m m a r i s e d i n Table 1.
W h e n t h e y a c h t is s u b j e c t e d t o p i t c h i n g m o t i o n , t h e a p p a r e n t w i n d is p e r i o d i c a l l y m o d i f i e d as t h e r o t a t i o n adds a n e w c o m p o -n e -n t o f a p p a r e -n t w i -n d w h i c h varies w i t h h e i g h t . F o l l o w i -n g t h e
Fig. 1. Coordinate, angle and motion references for ttie yacht. Z-axis is attached to the earth vertical.
Table 1
Reduced velocity VV, reduced frequency fr, phase shift T determined by cross-correlation between Cx and fieff, phase delay, averaged total power Prar. time-averaged dissipated power Pwop. time-time-averaged useful power Pv^. time-time-averaged driving force Fx, and time-averaged heeling moment Mj" for different pitching amplitudes A and periods T.
T A Vr fr T 2)!r/T PTOT Pv„ Fx Mx Cs) (deg) (s) (rad) (W) (W) ( W ) (N) ( N - m ) 1.5 5 2.96 0.34 0.16 0.670 1454 - 1 0 6 . 4 3 1561 608.8 8290 2 5 3.95 0.25 0.29 0.921 1518 - 5 5 . 5 7 1574 613.3 8274 2.5 5 4.94 0.20 0.50 1.257 1540 - 3 5 . 6 0 1576 614.1 8244 3 5 5.92 0.17 0.76 1.592 1558 - 2 4 . 8 9 1583 616.3 8260 5 5 9.87 0.10 2.70 3.393 1580 - 9 . 9 8 1590 618.5 8266 6 5 11.84 0.08 4.12 4.314 1584 - 7 . 3 7 1592 619.1 8270 5 3 9.87 0.10 2.70 3.393 1588 - 3 . 6 3 1591 619.1 8262 5 5 9.87 0.10 2.70 3.393 1580 - 9 . 9 8 1590 618.5 8266 5 6 9.87 0.10 2.70 3.393 1574 - 1 4 . 4 4 1589 618.0 8268
analysis o f Fossati a n d M u g g i a s c a (2011), t h e a p p a r e n t w i n d a n d p i t c h i n d u c e d v e l o c i t y are c o n s i d e r e d at t h e c e n t r e o f a e r o d y -n a m i c f o r c e a l t i t u d e ZCE- This c e -n t r e o f e f f o r t is a c t u a l l y m o v i -n g d u e t o p i t c h o s c i l l a t i o n , a n d t h e t i m e - d e p e n d e n t c e n t r e o f e f f o r t h e i g h t is considered. T h i s y i e l d s d m e - d e p e n d e n t a p p a r e n t w i n d speed a n d angle, g i v e n b y VAwit)=(iVTw(ZcE) sinflnvf
+ (VTW(ZCE) cos
fijw
+ VBs+ZcEd cos 0 cos(/))^y^^
(4)(5) A „ < O = S , „ - . ( M H ) .
A n d hence t h e t i m e - d e p e n d e n t e f f e c t i v e w i n d angle reads
^ . ^ ( 0 = t a n - ' ( ^ ^ ^ cos cp).
Fig. 2 i l l u s t r a t e s t h e d y n a m i c v e c t o r c o m p o s i t i o n f o r p i t c h i n g v e l o c i t i e s 0 = 0max ( p o i n t b i n Fig. 3 ) , 0 ( p o i n t s a a n d c i n Fig. 3 ) a n d èmin ( p o i n t d i n Fig. 3 ) , a n d Fig. 3 s h o w s t h e r e s u l t i n g d y n a m i c a p p a r e n t w i n d v e l o c i t y a n d angle c o m p u t e d w i t h Eqs. ( 4 ) a n d ( 5 ) . As s h o w n i n Fig. 3, t h e v a r i a t i o n s o f t h e a p p a r e n t w i n d angle are i n phase o p p o s i t i o n w i t h t h e v a r i a t i o n s o f a p p a r e n t w i n d speed.
3.3. Heeling and driving force coefficients
A e r o d y n a m i c forces are c o m p u t e d b y ARAVANTI as t h e r e s u l -t a n -t o f pressures o n -t h e sails a n d -t h e a e r o d y n a m i c f o r c e s m a -t r i x ( r e s u l t a n t a n d m o m e n t ) is w r i t t e n at t h e o r i g i n 0 i n t h e i n e r t i a l f r a m e i l l u s t r a t e d i n Fig. 1.
A t r a n s i t i o n m a t r i x [RT] can be used i n o r d e r to^get f o r c e s i n t h e b o a t Inertialframe T h e [Ra]lReIR^] w i t h ; i 0 u s i n g t h e f o l l o w i n g t r a n s i t i o n m a t r i x e q u a t i o n F Baatframe = [RT] ~ [RT] is d e f i n e d b y [RT] = cos (p sin (p 0 - s i n cos 4 [Rs] = cos 0 0 - s i n 0 sin 0 0 cos 0 lRa] = COS a — s i n a 0 s i n a cos a 0 0 0 1
Fig. 2. Dynamic effect of pitching on the wind triangle (top view). is the wind velocity, VBS is the boat speed, ZCE is the altitude of the aerodynamic centre of effort, Ó is the pitching velocity, p Is the apparent wind angle, subscripts TW and AW stand for true and apparent wind, respectively.
9.4 9.2 9 8.8 8.4 8.2 7.8 7.6 b V^^^ steady - » - V ^ ^ ^ ( t ) pitch V^^^ steady - » - V ^ ^ ^ ( t ) pitch 1: ai Ic i . . . 10 12 14 16 t(s)
Fig. 3. Time dependent apparent wind speed VAW (a); apparent wind angle /?^n, and effective wind angle fieff (b), resulting from pitching oscillation at ZCE with period r = 3 s and a m p l i t u d e / I = 5 ° . We define four reference points to be identified in further figures: bow up for point a 0 = 0,9= -A), horizontal going down (no trim) for point b (fl > 0, ö = 0), bow down for point c (ö = 0, e=A), horizontal going up (no trim) for point d (é < 0, ö = 0).
D r i v i n g a n d h e e l i n g f o r c e c o e f f i c i e n t s i n t h e b o a t f r a m e are o b t a i n e d b y t h e n o r m a l i s a t i o n w i t h t h e p r o d u c t o f t h e i n s t a n t a -neous a p p a r e n t d y n a m i c pressure a n d t h e t o t a l sail area S:
Fx Cx(t)--0.5pVl^(t)S Cy(t) = (6) (7) 0.5pViw(.t)S I n t h e s t e a d y state c a l c u l a t i o n , d r i v i n g f o r c e c o e f f i c i e n t = 0.423 a n d h e e l i n g f o r c e c o e f f i c i e n t Cy = - 1 . 0 8 0 are o b t a i n e d .
4. Dynamic behaviour
Previous studies (Fossati a n d M u g g i a s c a , 2 0 1 1 ; A u g i e r e t al., 2013) h a v e s h o w n t h a t t h e d y n a m i c b e h a v i o u r o f a y a c h t sail p l a n s u b j e c t e d t o p i t c h i n g c l e a r i y d e v i a t e s f r o m t h e q u a s i - s t a t i c a p p r o a c h . Particularly, t h e a e r o d y n a m i c f o r c e s p r e s e n t e d as a f u n c t i o n o f t h e i n s t a n t a n e o u s a p p a r e n t w i n d angle s h o w h y s t e r -esis l o o p s as i l l u s t r a t e d i n Fig. 4 . D i f f e r e n t q u e s t i o n s h a v e b e e n
B. Augier et al. / Ocean Engineering 90 (2014) 119-128 123 -0.7 -0.8 -0.9 -1 -1.1 -1.2 -1.3 -1.4 -1.5 -1.6 — é f - P i t c h A 5 T 1 , 5 • PitchA5T3 — © - PitchA5T5 PitohA5T6 - • - steady 22 24 26 2B 30 32 34 36
Fig. 4. Driving (a) and lieeling (b) force coefficients versus effective wind angle
r a i s e d b y t h i s result, ( i ) Can t h e l o o p s i n t h e Lissajous p l o t s be r e p r e s e n t e d b y a s i m p l e phase s h i f t b e t w e e n t h e signals? This hysteresis p h e n o m e n o n suggests t h a t t h e u n s t e a d y b e h a v i o u r leads t o a e r o d y n a m i c e q u i v a l e n t d a m p i n g a n d s t i f f e n i n g e f f e c t s . T h e area i n c l u d e d i n t h e hysteresis l o o p w a s s h o w n t o increase w i t h t h e m o t i o n r e d u c e d f r e q u e n c y a n d a m p l i t u d e , b u t t h e e x c h a n g e d e n e r g y w a s n o t i n v e s t i g a t e d , ( i i ) Can t h e e n e r g y associated t o t h e hysteresis be d e t e r m i n e d a n d analysed f o r d i f f e r e n t p i t c h i n g f r e q u e n c i e s a n d a m p l i t u d e s ?
4.1. Phase shift r
T h e values o f t h e phase s h i f t t b e t w e e n a e r o d y n a m i c forces a n d i n s t a n t a n e o u s w i n d angle have b e e n d e t e r m i n e d f o r each p i t c h i n g p e r i o d a n d a m p l i t u d e b y c r o s s - c o r r e l a t i o n (Table 1). The phase d e l a y 2 ; r T / r i n radians increases ( a l m o s t l i n e a r l y i n t h e i n v e s t i -g a t e d r a n -g e ) w i t h t h e f l o w r e d u c e d v e l o c i t y , m e a n i n -g w i t h t h e m o t i o n p e r i o d , b u t is n o t a f f e c t e d b y t h e o s c i l l a t i o n a m p l i t u d e . W h e n f o r c e c o e f f i c i e n t s Cxj,(t) are p l o t t e d versus t h e t i m e s h i f t e d w i n d a n g l e Peff{'^+-t), t h e l o o p area is s i g n i f i c a n t l y decreased b u t does n o t v a n i s h (see Fig. 5). IVIoreover, as s h o w n i n Figs. 4 a n d 6, t h e l o o p s are n o t p u r e l y e l l i p t i c a l because o f n o n - l i n e a r effects. T h i s s h o w s t h a t t h e hysteresis p h e n o m e n o n c a n n o t be r e d u c e d t o a s i m p l e phase s h i f t b e t w e e n t h e signals. 0.54 0.52 0.5 0.48 PilchA5T5 PitohA5T5, • • = steady
Fig. 5. Driving force coefficient vs. instantaneous apparent wind angle /3,j(t) (blue line with markers), and vs, tlie time shifted instantaneous apparent wind angle Peffit+r) (red line without marker), for a pitching period T = 5 s and amplitude A=5°. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)
4.2. Exchanged energy The hysteresis p h e n o m e n o n o b s e r v e d i n t h e a e r o d y n a m i c s o f t h e p i t c h i n g sail p l a n c o r r e s p o n d s t o a n e x c h a n g e o f e n e r g y b e t w e e n t h e y a c h t m o t i o n a n d t h e aeroelastic s y s t e m . T h e a i m o f t h i s s e c t i o n is t o d e t e r m i n e a n d analyse t h i s e n e r g y f o r d i f f e r e n t values o f t h e m o t i o n p a r a m e t e r s . I n d e e d , t h e e n e r g y p e r u n i t t i m e is c o n s i d e r e d , i.e. t h e e x c h a n g e d p o w e r , w h i c h is m o r e r e l e v a n t to c o m p a r e d i f f e r e n t m o t i o n f r e q u e n c i e s . The area c o n t a i n e d i n t h e hysteresis l o o p o f Fig. 4 does n o t f o r m a l l y c o r r e s p o n d t o a n e n e r g y n o r a p o w e r as fieff is t h e e f f e c t i v e a p p a r e n t w i n d a n g l e a n d i t s r e l a t i o n s h i p t o a d i s p l a c e m e n t o r v e l o c i t y is n o t s t r a i g h t f o r w a r d . T h e d i m e n s i o n a l e n e r g y i n Joules — or d i m e n s i o n a l p o w e r i n W a t t s — is c o n s i d e r e d i n s t e a d o f d i m e n s i o n l e s s q u a n t i t i e s t o a v o i d biased e f f e c t s i n t r o d u c e d b y n o r m a l i z i n g w i t h t h e v a r y i n g d y n a m i c pressure. The i n s t a n t a n e o u s m e c h a n i c a l p o w e r is d e f i n e d b y its g e n e r a l e x p r e s s i o n c o m b i n i n g t h e k i n e m a t i c a n d d y n a m i c m a t r i c e s : PTOT(t) = " ? - + ( 8 ) w h e r e "•" d e n o t e s t h e scalar p r o d u c t b e t w e e n v e c t o r s . For t h e m o t i o n c o n s i d e r e d i n t h i s w o r k ( f o r w a r d t r a n s l a t i o n a n d p i t c h i n g ) , t h i s e x p r e s s i o n reduces t o P r o r ( f ) = f x V B S + l W v ö . (9)
The first t e r m o n t h e right h a n d side FXVBS = f is t h e u s e f u l p o w e r d r i v i n g t h e y a c h t f o r w a r d . T h e second t e r m M y ö = Pioop is t h e p o w e r e x c h a n g e d b y t h e s y s t e m d u e t o t h e hysteresis p h e n o m e n o n . Fig. 6 s h o w s t h e a e r o d y n a m i c f o r c e p i t c h i n g m o m e n t MY as a f u n c t i o n o f t h e t r i m a n g l e 0 f o r d i f f e r e n t values o f t h e o s c i l l a t i o n p e r i o d T f r o m 1.5 u p t o 6 s w i t h a p i t c h i n g a m p l i t u d e A=5°. T h e area c o n t a i n e d i n t h e s e l o o p s is t h e e n e r g y e x c h a n g e d d u r i n g o n e o s c i l l a t i o n p e r i o d b e t w e e n t h e s y s t e m a n d t h e i m p o s e d p i t c h i n g m o t i o n d u e t o t h e hysteresis p h e n o m e n o n . A s s h o w n b y t h e r o t a t i o n d i r e c t i o n i n t h e loops ( F i g . 6 ) a n d t h e c o m p u t e d r e s u l t s (Table 1), t h i s q u a n t i t y is n e g a t i v e w h i c h m e a n s t h a t s o m e e n e r g y is d i s s i p a t e d b y t h e hysteresis p h e n o m e n o n . I n t h e f o l l o w i n g , t h e m e a n p o w e r averaged o v e r o n e o s c i l l a t i o n p e r i o d is c o n s i d e r e d : PTOT d t ( 1 0 )
4500 4000 ^ ^ P i t c h A 5 T 1 . 5 - B — PitchAST2 -e— PitchA5T3 PitchA5T6 E z >: 3000 2500 2000 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.05 0.08 0.1 6 (rad)
Fig. 6. Pitctiing moment My vs. pitcti angle e for pitching periods T=1.5, 2, 3, 6 s. The loop area represents the energy dissipated during the corresponding period (7
times P I O O P ) . -120 -100 1700 iBon 1400 1300
Fig. 7. Dissipated power Pioop, useful power Pv^ and total power PTOT for different reduced frequency fr. - - U ' = \ j PLOOP dt = j j M , Ö dt (11) (12) P^T" is t h e u s e f u l m e a n p o w e r d r i v i n g t h e b o a t f o r w a r d a n d e x t r a c t e d f r o m t h e a i r flow b y t h e sail p l a n . P v j j is p r o p o r t i o n a l t o t h e m e a n d r i v i n g f o r c e Fx as t h e b o a t speed is constant. Pioop is t h e m e a n p o w e r d i s s i p a t e d b y t h e Irysteresis p h e n o m -e n o n f r o m t h -e i m p o s -e d p i t c h i n g m o t i o n a n d c o r r -e s p o n d s t o t h -e l o o p s area i n Fig. 6 d i v i d e d b y t h e p i t c h i n g p e r i o d T. A t first order, t h i s q u a n t i t y is d o m i n a t e d b y JJFXZCEÖ dt.
N o t e t h a t t h e p i t c h i n g m o t i o n i t s e l f i n t r o d u c e s a n a d d e d p o w e r t o t h e s y s t e m c o m p a r e d t o t h e steady case ( n o p i t c h i n g ) .
As s h o w n i n Fig. 7, t h e d i s s i p a t e d average p o w e r absolute v a l u e iPioopI s t r o n g l y increases w i t h t h e m o t i o n r e d u c e d f r e q u e n c y , f r o m z e r o f o r t h e steady case ( v a n i s h i n g f r e q u e n c y ) u p t o 106 W f o r / r = 0 . 3 4 . The n o n - h n e a r i t y o f t h e p h e n o m e n o n is h i g h l i g h t e d b y t h e o b s e r v a d o n t h a t t h e l o o p shape b e c o m e s d i s t o r t e d f o r t h e h i g h e s t values o f t h e r e d u c e d f r e q u e n c y (Fig. 6 ) . The m e a n u s e f u l p o w e r P ^ decreases s l i g h t l y ( a b o u t 2% i n t h e i n v e s t i g a t e d r a n g e ) f o r a n i n c r e a s i n g f r e q u e n c y , s u g g e s t i n g a s m a l l r e d u c t i o n o f a e r o d y n a m i c e f f i c i e n c y f o r a f a s t e r p i t c h i n g m o t i o n .
M o r e o v e r , t h e m e a n d r i v i n g f o r c e Fx is d i f f e r e n t f r o m t h e d r i v i n g f o r c e i n t h e steady case Fx steady as a e r o d y n a m i c forces i n t h e d y n a m i c r e g i m e d o n o t f o l l o w t h e q u a s i - s t a t i c a s s u m p t i o n a n d s o m e p o w e r is e x c h a n g e d d u e t o t h e hysteresis. T h e t o t a l m e a n p o w e r P ^ decreases m o r e ( a b o u t 8% i n the i n v e s t i g a t e d r a n g e ) as t h e d i s s i p a t e d e n e r g y is h i g h e r i n absolute v a l u e . As s h o w n i n Table 1, t h e e f f e c t o f t h e p i t c h i n g a m p h t u d e y i e l d s s i m i l a r t r e n d s t o t h e r e d u c e d f r e q u e n c y , i.e. i n c r e a s i n g I P I O O P I . d e c r e a s i n g Pvss.Fx a n d P ^ f o r a h i g h e r p i t c h i n g a m p l i t u d e . I t s h a l l b e n o t i c e d t h a t Pi^ a n d P ^ are o n e t o t w o o r d e r s o f m a g n i t u d e h i g h e r t h a n \Pl^\, w h i c h m e a n s t h a t t h e u s e f u l p o w e r Pv^s is d o m i n a n t . The a e r o d y n a m i c b e h a v i o u r is n o w c l e a r l y c h a r a c t e r i s e d : a hysteresis p h e n o m e n o n is e v i d e n c e d a n d t h e associated e n e r g y is analysed. T h e n e x t sections address t h e v a r i o u s i n f l u e n c e s o f t h e y a c h t m o t i o n s c o n s i d e r e d a n d o f d i f f e r e n t r i g t r i m s .
5. Pitching decomposition
The real p i t c h i n g m o t i o n is m o d e l l e d i n t h i s w o r k b y a n a n g u l a r o s c i l l a t i o n a r o u n d t h e y a x i s (Fig. 8, p i t c h ) , n o r m a l t o t h e c e n t r e -l i n e w i t h a r o t a t i o n c e n t r e -l o c a t e d a t t h e m a s t step. M o s t o f p r e v i o u s studies o n t h e i n f i u e n c e o f p i t c h i n g h a v e c o n s i d e r e d a 2 D s i m p l i f i e d p r o b l e m a n d t h u s a p p r o x i m a t e d t h e p i t c h i n g m o t i o n b y a t r a n s l a t i o n a l o s c i l l a t i o n a l i g n e d w i t h t h e y a c h t c e n t r e l i n e (Fig. 8, surge). T h e n , t h e u s u a l p r o c e d u r e (see e.g. G e r h a r d t et al., 2011) is t o d e c o m p o s e t h i s m o t i o n i n a n o s c i l l a t i o n p a r a l l e l t o t h e a p p a r e n t w i n d , r e s u l t i n g i n a n o s c i l l a t i o n o f a p p a r e n t w i n d speed, a n d a n o s c i l l a t i o n o r t h o g o n a l t o t h e a p p a r e n t w i n d , r e s u l t i n g m a i n l y i n a n o s c i l l a t i o n o f t h e a p p a r e n t w i n d angle (Fig. 8, d e c o m p o s i t i o n ) . Here, w e w a n t t o test these t w o h y p o t h e s e s b y c o m p a r i n g t h e results o f t h e d y n a m i c s i m u l a t i o n w i t h ARAVANTI o b t a i n e d w i t h d i f f e r e n t i m p o s e d m o t i o n s , a n d i n v e s t i g a t e t h e e f f e c t o n t h e s p e c i f i c d y n a m i c f e a t u r e s h i g h l i g h t e d above. M o t i o n s are based o n t h e r e f e r e n c e p i t c h i n g m o t i o n w i t h a m p l i t u d e A = 5 ° a n d p e r i o d T = 5 s ( A 5 T 5 ) .5.1. Surge
T h e first step is t o c o m p a r e t h e r e s u l t s f o r a real p i t c h i n g m o t i o n ( r o t a t i o n ) t o t h e results f o r a t r a n s l a t i o n a l surge m o t i o n w i t h t h e a m p l i t u d e o f m o t i o n at t h e c e n t r e o f e f f o r t h e i g h t ZCE w h i l e p i t c h i n g . As s h o w n i n Fig. 9 t h e o s c i l l a t i o n o f a e r o d y n a m i c forces is decreased b y 3 0 - 4 0 % a n d phase s h i f t e d ( a r o u n d T / 9 ) w h e n t h e p i t c h i n g is r e d u c e d t o a surge m o t i o n . T h i s r e s u l t gives t h e o r d e r o f t h e e r r o r o n t h e o s c i l l a t i o n a m p l i t u d e o f a e r o d y n a m i c forces i n t r o d u c e d b y c o n s i d e r i n g a surge m o t i o n i n s t e a d o f t h e p i t c h i n g m o t i o n .
Decomposition^/
Fig. 8. Different motions considered: pitching (rotation), surge (translation), surge decomposition into translations collinear to the apparent wind Vc and normal to the apparent wind V„.
B. Augier et al. / Ocean Engineering 90 (2014) 119-128 125 0.55 0.5 0.45 0.4 0.35 1 . . 1 1 i 1 . . . . 1 . . 1 1 1 10 12 14 16 18 25 25.5 26 26.5 27 27.5 28 28.5 29 29.5 30 Pitch - B — Surge - © - O r t h o Coii
Fig. 9. Time series of the driving and heeling force coefficients for FSI simulations of the various motions considered; pitching, surge, translations collinear and perpendicular to the apparent wind (see Fig. 11), corresponding to a pitching amphtude A = 5 ° and period T = 5 s.
-0.8
-1
,=? -1.1
-1.3
25 25.5 26 26.5 27 27.5 28 28.5 29 29.5 30
Fig. 10. Driving and heeling force coefficients versus apparent wind angle for pitch and surge motions. The motion period and amplitude at the centre of effort are identical and correspond to a pitching amplitude A=5° and period T = 5 s.
C o n c e r n i n g t l i e d y n a m i c b e h a v i o u r , i t is i n t e r e s t i n g t o n o t i c e t h a t t h e case o f surge does n o t showr t h e same hysteresis p h e n o m e n o n . I n d e e d , t h e a e r o d y n a m i c b e h a v i o u r i n t h e case o f surge is m u c h closer t o t h e quasi-steady t h e o r y t h a n i n t h e p i t c h i n g case, as c l e a r l y s h o w n i n Fig. 10. The loops o f Cx,y(figff) collapse a n d are s u p e r p o s e d t o t h e steady case l i n e .
5.2. Simple translations decomposition
T h e second step is t o analyse separately t h e effects o f t r a n s l a -t i o n a l oscilla-tions p a r a l l e l Vc (Fig. 11a) a n d o r -t h o g o n a l V„ (Fig. l i b ) t o t h e a p p a r e n t w i n d d i r e c t i o n . I t is observed i n Fig. 9 t h a t t h e m a j o r c o n t r i b u t i o n t o t h e f o r c e o s c i l l a t i o n is d u e t o t h e o r t h o g o n a l o s c i l l a t i o n c o m p o n e n t , w h i c h is associated t o t h e o s c i l l a t i o n o f a p p a r e n t w i n d angle. W h e n t h e v a r i a t i o n s d u e t o b o t h c o m p o -n e -n t s o f m o t i o -n are a d d e d as s h o w -n i -n Fig. 12, t h e r e s u l t is i d e n t i c a l t o w h a t is o b t a i n e d w i t h t h e surge m o t i o n as b o t h curves are s u p e r i m p o s e d , w h i c h j u s t i f i e s t h e l i n e a r s u p e r p o s i t i o n p r i n c i -p l e o f t h i s a -p -p r o a c h . The e f f e c t o f -p a r a l l e l o s c i l l a t i o n - v a r i a t i o n o f Vyiw(t) — is s m a l l , b u t w i t h a m o r e d i s t o r t e d e v o l u t i o n . N o t e t h a t t h e o r t h o g o n a l o s c i l l a t i o n is associated w i t h a n o s c i l l a t i o n of /)fy^{t), a n d t h e e f f e c t o f angle o f attack i n a n a r r o w
a b
X X
Fig. 11. Wind triangle representation for the surge decomposition into 2 transla-tions (a) Vc collinear to V^w and (b) V„ normal to V/iw
range is a l m o s t l i n e a r o n t h e a e r o d y n a m i c l i f t . C o n t r a r i l y , t h e p a r a l l e l o s c i l l a d o n is associated w i t h a n o s c i l l a t i o n o f Vfiv^t), a n d t h e e f f e c t o f w i n d speed is q u a d r a t i c o n a e r o d y n a m i c forces.
0.08 0.06 0.04
i •
u " -0.02 -0.04I A : L I \ ] ] J j
: \ . . L . . / : : : / . J
1 \ \
1 1 1 1 1 1 1 1 1 10 16 t(s) 0.05 - B — Surge - 0 — Orlho+Coli 10 12 14 16 18 l{s)Fig. 12. Comparison of oscillations of aerodynamic force coefficients obtained for a surge motion witli tire sum of oscillations obtained for both translation compo-nents parallel and orthogonal to the apparent wind.
6. Influence of rig adjustments
Before each race, sailors a d j u s t t h e t e n s i o n i n t h e s h r o u d s ( d o c k t u n e ) a c c o r d i n g t o s a i l i n g c o n d i t i o n s , a n d t h e hackstay t e n s i o n is o f t e n a d j u s t e d c o n t i n u o u s l y w h i l e s a i l i n g u p w i n d . I n t h i s s e c t i o n , t h e analysis o f t h e e f f e c t s o f v a r i o u s d o c k tunes a n d backstay loads o n t h e d y n a m i c b e h a v i o u r a n d t h e exchanged e n e r g y is p r e s e n t e d .
6.1. Influence of dock tune
T h e i n f l u e n c e o f v a r i o u s d o c k t u n e s o n t h e sail p l a n d y n a m i c b e h a v i o u r is i n v e s t i g a t e d . T h e r e f e r e n c e p i t c h i n g m o t i o n {A=5° a n d T = 1 . 5 s ) is s i m u l a t e d w i t h t h r e e realistic d o c k t u n e s u s e d w h i l e r a c i n g i n d i f f e r e n t w i n d c o n d i t i o n s . D o c k tunes are d e f i n e d as t h e n u m b e r o f s c r e w t u r n s a p p l i e d to t h e s h r o u d s ' t u r n - b u c k l e s . Tune2 is t h e r e f e r e n c e d o c k t u n e used f o r t h e c o n s i d e r e d s a i l i n g c o n d i t i o n s . T h e t h r e e d o c k t u n e s are described b e l o w : • tune-t: - 3 t u r n s o n V I s h r o u d s used i n l i g h t w i n d . • tune2: r e f e r e n c e d o c k t u n e f o r Vnv = 6.7 m . s " ' (13 k n o t s ) . • tunes: +3 t u r n s o n V I s h r o u d s used i n m e d i u m w i n d .
w h e r e V I are t h e o u t e r a n d h i g h e s t lateral s h r o u d s . The o t h e r s h r o u d s are n o t m o d i f i e d .
These t h r e e d o c k t u n e s n o t o n l y m o d i f y t h e rigidity o f t h e f u l l r i g g i n g b u t have a s i g n i f i c a n t i n f l u e n c e o n t h e c a m b e r o f t h e m a s t .
I n c r e a s i n g the V I t e n s i o n makes a s t i f f e r rig, a r e d u c e d f o r e s t a y sag a n d a m o r e b e n t mast, w h i c h results i n f l a t t e r sails. The sails' shape a n d m o r e precisely t h e i r c a m b e r a n d t w i s t are m o d i f i e d by t h e d o c k t u n e . Before t h e p i t c h i n g s i m u l a t i o n , t h e m a i n sail a n d j i b are n u m e r i c a l l y t r i m m e d i n o r d e r t o e n s u r e t h a t t h e c h o r d a t the centre o f e f f o r t h e i g h t has t h e same angle o f a t t a c k f o r t h e d i f f e r e n t tunes t o get a r e l e v a n t c o m p a r i s o n .
Fig. 13 shows t h e energy loops o f p i t c h i n g m o m e n t My versus p i t c h angle f o r t h e three tested d o c k tunes. The loops l o o k similar, however, t h e exchanged energy c o m p u t e d as described i n Section 4 shows variations. Table 2 presents t h e r e l a t i v e e v o l u t i o n o f the m e a n t o t a l p o w e r PTOT, dissipated p o w e r Pïööp a n d j i s e f u l p o w e r Pv-Bs vvhich is equivalent t o the average d r i v e force Fx- Compared to t h e reference d o c k t u n e 2, t h e dissipated p o w e r is increased by 8.5% f o r the loosest rig a n d s i m i l a r f o r t h e tightest r i g . The r e d u c t i o n o f dissipated energy w i t h t h e increase o f r i g t e n s i o n seems t o be due t o a s t i f f e r rig. W i t h m o r e stresses, t h e rig is g e t t i n g closer to a r i g i d s t r u c t u r e a n d c o m p a r i s o n b e t w e e n FSI a n d r i g i d s i m u l a t i o n s has s h o w n t h a t the hysteresis p h e n o m e n o n is s i g n i f i -c a n t l y l o w e r i n t h e rigid -case ( A u g i e r et al., 2013). A n o t h e r fa-ctor m a y be t h a t f l a t t e r sails dissipate less p o w e r .
The u s e f u l p o w e r is s l i g h t l y h i g h e r (1.3%) f o r t h e loosest rig and l o w e r (2%) f o r the tightest rig. This w o u l d suggest t h a t t h e reference d o c k t u n e 2 is n o t o p t i m a l a n d a looser r i g w o u l d be f a s t e r However, i t shall b e recalled t h a t t h i s s i m u l a t i o n is based o n a n i n v i s c i d f l o w w h i c h is k n o w n t o f i n d a h i g h e r d r i v e f o r c e f o r sails w i t h m o r e camber t h a n t h e real o p t i m u m because flow separation is n o t m o d e l l e d . As a looser r i g results i n m o r e c a m b e r e d sails, t h i s m a y be t h e reason w h y t h e m e a n u s e f u l power, o r m e a n d r i v i n g force, is p r e d i c t e d t o be h i g h e r f o r t u n e 1 t h a n f o r t u n e 2. Moreover, a p e r f o r m a n c e analysis s h o u l d also consider t h e side force, a n d the
4500 4000 3500 3000 2500 h 2000 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 e (rad)
Fig. 13. Pitching moment My vs. pitch angle O for different dock tunes, for a pitching amplitude A = 5 ° and period T=1.5 s. The loop area represents the energy dissipated during the corresponding period (T times Pioop).
Table 2
Ivlean total power Pjor. mean dissipated power P I O O P . mean useful power and mean heeling moment for different dock tunes, relative to reference case (tunej), for a pitching amplitude A=5' and period T = l.5s.
Dock tune Prnop P v „
Prorre/ Pioopre/ Mxref
tunei 1.008 1.085 1.013 1.017
tune2 1 1 1 1
B. Augier et al. / Ocean Engineering 90 (2014) 119-128 127
e v o l u t i o n o f t h e m e a n h e e l i n g m o m e n t JWx is also g i v e n i n Table 2 f o r i n f o r m a t i o n .
6.2. Influence of the backstay load
T h e i n f l u e n c e o f a v a r i a t i o n o f t h e backstay t e n s i o n o n t h e d y n a m i c b e h a v i o u r is i n v e s t i g a t e d . The same p i t c h i n g m o t i o n ( A = 5 ° a n d T = 1 . 5 s ) is s i m u l a t e d w i t h f o u r values o f backstay l e n g t h c o r r e s p o n d i n g t o backstay loads o f 1 0 0 0 N , 1500 N , 2 0 0 0 N a n d 2 5 0 0 N i n t h e steady case. T h e case 2 0 0 0 N is t h e reference b a c k s t a y l o a d used f o r t h e p r e v i o u s s i m u l a t i o n s . T h e sail t r i m s are i d e n t i c a l f o r t h e f o u r backstay loads. P r e l i m i n a r y steady s i m u l a -t i o n s w i -t h -t h e f o u r loads have s h o w n -t h e a b i l i -t y o f ARAVANTI m o d e l t o s i m u l a t e t h e e f f e c t o f t h e backstay: t h e m a i n s a i l t w i s t increases, t h e m a i n s a i l c a m b e r decreases a n d moves baclcward w h e n t h e backstay l o a d increases.
Fig. 14 s h o w s t h e e n e r g y loops o f p i t c h i n g m o m e n t M y versus p i t c h angle f o r different_values o f t h e backstay l o a d . As expected, t h e m e a n d r i v i n g f o r c e Fx ( w h i c h is p r o p o r t i o n a l t o P ^ ) a n d t h e m e a n h e e l i n g m o m e n t JWx are g r e a t l y a f f e c t e d b y t h e backstay l o a d , w h i c h changes t h e m a i n s a i l c a m b e r a n d t w i s t (see Table 3 ) .
S i m i l a r l y t o w h a t is s h o w n i n Section 6.1, t h e l o w e s t backstay l o a d l o o k s t o be o p t i m a l i n t e r m s o f m e a n d r i v i n g f o r c e o r u s e f u l p o w e r . Once again, t h e same r e s t r i c t i o n m u s t be m a d e d u e t o t h e i n v i s c i d flow m o d e l w h i c h m a y bias t h e o p t i m i s a t i o n . M o r e o v e r , t h e m e a n h e e l i n g m o m e n t is 20% h i g h e r f o r t h e loosest backstay. T h i s is c o n s i s t e n t w i t h t h e sailors k n o w l e d g e w h o c o m m o n l y tighten t h e backstay t o reduce h e e l .
T h e backstay load also has a g r e a t i n f l u e n c e o n t h e e n e r g y d i s s i p a t e d i n t h e hysteresis l o o p (see Table 3 ) . T h e c o m p u t e d m e a n d i s s i p a t e d p o w e r s t r o n g l y increases w h e n t h e backstay l o a d is
5000
-0.08 -0.06 -0.04 -0.02 0 0.02 0,04 0.06 0.08 8 (rad)
0.1
Fig. 14. Pitching moment My vs. pitch angle 9 for different backstay loads, for a pitching amplitude 4 = 5 ° and period r=1.5 s. The loop area represents the energy dissipated during the corresponding period (T times P^^öF).
Table 3
Mean total power Pror. mean dissipated power Pwop, mean useful power and mean heeling moment iWx for different backstay loads, relative to reference case (2000 N), for a pitching amplitude A=5° and period 7'= 1.5 s.
Load
1000 N 1.174 0.686 1.140 1,198
1500 N 1.088 1.072 1.052 1.098
2 0 0 0 N 1 1 1 1
2 5 0 0 N 0.895 1.211 0.916 0.890
increased (|Pioop| a l m o s t d o u b l e s w h e n t h e backstay is tighten f r o m 1000 N u p t o 2 5 0 0 N ) . I t is w o r t h n o t i c i n g t h a t t h i s t r e n d is opposite t o t h e o n e o b s e r v e d f o r a t i g h t e r d o c k t u n e b e i n g closer t o a r i g i d s t r u c t u r e as s h o w n i n Section 6 . 1 . I n t h e p r e s e n t case, m o r e t e n s i o n o n t h e backstay results i n flatter sails, b u t t h e m a i n sail leech is also looser. T h i s m a y r e s u l t i n m o r e flapping o f t h e m a i n sail w h i l e p i t c h i n g w h i c h can dissipate m o r e p o w e r .
7. Conclusions
The u n s t e a d y fluid s t r u c t u r e i n t e r a c t i o n o f t h e sails a n d r i g o f a 2 8 f o o t (8 m ) y a c h t u n d e r h a r m o n i c p i t c h i n g has b e e n i n v e s t i -gated i n o r d e r t o h i g h l i g h t t h e c o n t r i b u t i o n s o f t h e r i g a d j u s t m e n t s a n d t h e c o n s i d e r a t i o n o f a realistic p i t c h i n g m o t i o n i n t h e d y n a m i c b e h a v i o u r o f a sail p l a n . The ARAVANTI m o d e l is based o n a n i m p l i c i t u n s t e a d y c o u p l i n g b e t w e e n a v o r t e x l a t t i c e fluid m o d e l a n d a finite e l e m e n t s t r u c t u r e m o d e l , a n d has b e e n p r e v i o u s l y v a l i d a t e d w i t h f u l l scale e x p e r i m e n t s i n u p w i n d r e a l c o n d i t i o n s ( A u g i e r et al., 2 0 1 2 ) . Previous studies (Fossati a n d M u g g i a s c a , 2012; A u g i e r et al., 2 0 1 3 ) have s h o w n t h a t t h e a e r o d y n a m i c c o e f f i c i e n t s p l o t t e d against t h e i n s t a n t a n e o u s a p p a r e n t w i n d a n g l e e x h i b i t a n hysteresis l o o p . T h e p r e s e n t results c o n f i r m t h a t t h e d y n a m i c b e h a v i o u r o f a sail p l a n s u b j e c t t o y a c h t m o t i o n deviates f r o m t h e quasi-steady t h e o r y a n d a n a e r o d y n a m i c e q u i v a l e n t d a m p i n g e f f e c t is h i g h l i g h t e d . Oscillations o f t h e a e r o d y n a m i c forces e x h i b i t a n hysteresis p h e n o m e n o n w h i c h increases w i t h t h e m o t i o n r e d u c e d f r e q u e n c y a n d a m p l i t u d e .
I n t h i s paper, i t is s h o w n t h a t t h e hysteresis l o o p area is n o t o n l y d u e t o a phase s h i f t b e t w e e n t h e signals. A f t e r s h i f t i n g b y t h e phase delay T , t h e hysteresis l o o p o f Cx,y=f(fieffit+T)) does n o t collapse i n t o a s i n g l e l i n e . The p o w e r o f a e r o d y n a m i c forces is i n v e s t i g a t e d a n d a n a l y s e d i n t e r m s o f u s e f u l p o w e r a n d p o w e r e x c h a n g e d b e t w e e n t h e s y s t e m a n d m o t i o n t h r o u g h t h e hysteresis p h e n o m e n o n . I t is s h o w n t h a t s o m e e n e r g y is d i s s i p a t e d b y t h e aeroelastic s y s t e m f r o m t h e e n e r g y i n p u t b y t h e m o t i o n . This d i s s i p a t e d e n e r g y increases w i t h t h e m o t i o n r e d u c e d f r e q u e n c y a n d a m p l i t u d e . T h e u s e f u l e n e r g y associated t o t h e d r i v i n g f o r c e is l o w e r f o r a f a s t e r a n d h i g h e r a m p l i t u d e p i t c h i n g m o t i o n . T h e m o t i o n c o n s i d e r e d i n t h i s w o r k is a c o n s t a n t b o a t speed a n d f o r c e d p u r e h a r m o n i c p i t c h i n g only, a n d a l l o t h e r degrees o f f r e e d o m are k e p t c o n s t a n t . I n reality, w h e n t h e a e r o d y n a m i c forces oscillate, t h e heel a n g l e v a r y a c c o r d i n g l y , a n d t o a s m a l l e r e x t e n t t h e b o a t speed a n d leeway, so o t h e r t e r m s m u s t be c o n s i d e r e d i n t h e e x p r e s s i o n o f p o w e r . F u r t h e r w o r k is n e e d e d t o i n v e s t i g a t e t h e e f f e c t o f o t h e r types o f m o t i o n o n t h e e x c h a n g e d energy. I t w o u l d be i n t e r e s t i n g t o t r y a n d find s o m e f a v o u r a b l e m o t i o n r e s u l t i n g i n a h i g h e r u s e f u l p o w e r a n d m e a n d r i v i n g f o r c e t h a n t h e steady case. F r o m sailors experience w h o s o m e t i m e s f o r c e a r o l l i n g m o t i o n , c a l l e d r o c k i n g , w e expect t h a t t h i s m a y be o b t a i n e d f o r a p r o p e r t y chosen r o l l m o t i o n o f t h e r i g . T h i s i n t e r e s t i n g b e h a v i o u r w o u l d r e s e m b l e a flapping w i n g p r o d u c i n g t h r u s t . Pure h a r m o n i c s u r g e m o t i o n is c o m p a r e d to p i t c h i n g m o t i o n i n o r d e r t o h i g h l i g h t t h e i m p o r t a n c e o f a r e a l i s t i c 3 D m o t i o n . Oscillations o f t h e a e r o d y n a m i c c o e f f i c i e n t s decrease b y 3 0 - 4 0 % i n t h e case o f a n e q u i v a l e n t surge m o t i o n c o m p a r e d t o t h e p i t c h i n g m o t i o n case. M o r e o v e r , i n t h e case o f t h e surge m o t i o n , t h e hysteresis p h e n o m e n o n is a l m o s t cancelled, so t h a t t h e d y n a m i c b e h a v i o u r is s i m i l a r t o t h e q u a s i - s t e a d y t h e o r y . W h e n t h e s u r g e m o t i o n is d e c o m p o s e d i n t o t w o c o m p o n e n t s , p e r p e n d i c u l a r t o a n d a l o n g t h e a p p a r e n t w i n d d i r e c t i o n , i t is s h o w n t h a t t h e m a j o r c o n t r i b u t i o n t o f o r c e o s c i l l a t i o n s is d u e t o t h e o r t h o g o n a l o s c i l l a -tion c o m p o n e n t , w h i c h is associated t o t h e o s c i l l a t i o n o f a p p a r e n t w i n d angle.
F i n a l l y , a p i t c h i n g m o t i o n o f t h e s t r u c t u r e w i t h v a r i o u s s h r o u d s ' d o c k t u n e s a n d b a c k s t a y t e n s i o n loads is s i m u l a t e d i n o r d e r t o s t u d y t h e i n f l u e n c e o f t h e r i g g i n g stresses o n t h e d y n a m i c b e h a v i o u r . T i g h t e r s h r o u d s r e s u l t i n g i n flatter sails a n d a m o r e r i g i d s t r u c t u r e t e n d t o decrease t h e e n e r g y d i s s i p a t e d b y t h e s y s t e m . C o n t r a r i l y , m o r e l o a d o n t h e b a c k s t a y r e s u l t s i n a h i g h e r e n e r g y d i s s i p a t i o n w h i c h m i g h t be e x p l a i n e d b y m o r e flapping o f t h e sails because o f a l o o s e r l e e c h , d e s p i t e t h e i r r e d u c e d c a m b e r . I n b o t h cases, t h e u s e f u l p o w e r p r e d i c t e d b y t h e s i m u l a t i o n is h i g h e r f o r a l o o s e r r i g , c o r r e s p o n d i n g t o m o r e c a m b e r e d sails. D i r e c t a p p l i c a -tion o f t h i s c o n c l u s i o n ( l o o s e r r i g / f u l l e r sails r e s u l t i n g i n a h i g h e r d r i v i n g f o r c e ) t o t h e r e a l case m u s t be m o d e r a t e d b y t h e a s s u m p -t i o n o f i n v i s c i d flow u s e d i n -t h i s w o r k w h i c h is k n o w n -t o l e a d -t o a n o p t i m a l s a i l s h a p e w i t h m o r e c a m b e r t h a n t h e a c t u a l o p t i m a l because flow s e p a r a t i o n is n o t m o d e l l e d . M o r e o v e r , t h e side f o r c e a n d h e e l i n g m o m e n t m u s t b e c o n s i d e r e d as w e l l t o o p t i m i s e t h e sails t r i m as t h e y a f f e c t t h e p e r f o r m a n c e d u e t o l e e w a y a n d h e e l . A f u l l V e l o c i t y P r e d i c t i o n P r o g r a m i n c l u d i n g h y d r o d y n a m i c f o r c e s m u s t b e u s e d f o r a r e a l i s t i c o p t i m i s a t i o n .
Aclmowledgments
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