Bilge keel loads and hull pressures created by bilge keels fitted to a rotating cylinder

Applied Ocean Researcli 53 (2015) 1-14

ELSEVIER

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Applied Ocean Research

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Bilge keel loads and hull pressures created by bilge keels fitted

to a rotating cylinder

R. van't Veer^'*, X.B. Schut^ R.H.M. Huijsmans^

' SBM Offshore, Schiedam, The Netherlands

" Delft University of Technology, Delft, The Netherlands

(D

CrossMark

A R T I C L E I N F O

Article history:

Received 11 December 2014 Received in revised form 2 July 2015 Accepted 6 July 2015

Available online 30 July 2015

Keywords:

Bilge keels normal force Hull pressures Model tests

Drag and inertia coefficients

A B S T R A C T

T h i s p a p e r p r e s e n t s bilge k e e l loads a n d h u l l p r e s s u r e m e a s u r e m e n t s c a r r i e d o u t o n a r o t a t i n g c y l i n d e r in a free s u r f a c e w a t e r b a s i n . A flat plate bilge k e e l a n d o n e m o r e c o m p l e x s h a p e d bilge k e e l w e r e s t u d i e d to investigate t h e g e o m e t r y effect. T h e d r a f t of t h e c y l i n d e r w a s v a r i e d to s t u d y t h e e f f e c t of t h e v i c i n i t y of the free s u r f a c e on the bilge k e e l loads a n d h u l l p r e s s u r e s . T h e rotation axis of t h e c y l i n d e r w a s f i x e d to define a p u r e roll e x p e r i m e n t ( o n e d e g r e e of f r e e d o m ) .

T h e c y l i n d e r w a s s u b j e c t to f o r c e d o s c i l l a t i o n s of v a r y i n g a m p l i t u d e l e a d i n g to a K C r a n g e o f 0 . 3 - 1 6 . U s i n g F o u r i e r a n a l y s i s t h e first t h r e e h a r m o n i c coefficients r e p r e s e n t i n g t h e n o r m a l bilge k e e l load w e r e d e r i v e d . T h e first h a r m o n i c d r a g a n d i n e r t i a coefficients a r e i n good a g r e e m e n t to e x i s t i n g e x p e r i m e n t a l d a t a obtained for w a l l b o u n d e d flat plates fitted i n a U - s h a p e d w a t e r t u n n e l as r e p o r t e d b y S a r p k a y a a n d O'Keefe ( 1 9 9 6 ) . N e w i n s i g h t is g a i n e d by the fact that t h e addition of h i g h e r h a r m o n i c c o n t r i b u t i o n s is e s s e n t i a l to c a p t u r e t h e t i m e v a r y i n g bilge keel n o r m a l force.

T h e p r e s s u r e m e a s u r e m e n t s n e x t to t h e bilge keel a r e c o m p a r e d to m e a s u r e m e n t s r e p o r t e d by I k e d a e t al. ( 1 9 7 9 ) . S i m i l a r f i n d i n g s a r e o b t a i n e d , s h o w i n g that the p r e s s u r e o n t h e h u l l in f r o n t o f t h e m o v i n g bilge k e e l is K C i n d e p e n d e n t w h i l e t h e v o r t e x s y s t e m i n t h e w a k e of the bilge k e e l l e a d s to K C d e p e n d -e n t h u l l p r -e s s u r -e d i s t r i b u t i o n s . T h -e h u l l p r -e s s u r -e j u m p ov-er th-e bilg-e k -e -e l c o r r -e l a t -e s w -e l l to th-e forc-e c o e f f i c i e n t on t h e bilge k e e l . T h e c o m p l e x n a t u r e of t h e v o r t e x i n d u c e d h u l l p r e s s u r e s is m a n i f e s t e d . T h e e m p i r i c a l l y d e r i v e d h u l l p r e s s u r e d i s t r i b u t i o n by I k e d a et al. ( 1 9 7 9 ) f o r the rime i n s t a n t of m a x i -m u -m v e l o c i t y is s h o w n to c o r r e l a t e r e a s o n a b l y w e l l to t h e -m e a s u r e d data w i t h s o -m e c o n s e r v a t i s -m in t h e absolute value.

A l t h o u g h a c y l i n d e r is v e r y d i f f e r e n t f r o m a s h i p - s h a p e d section, the e x p e r i m e n t s p r o v i d e e s s e n t i a l i n s i g h t into t h e p h y s i c s a s s o c i a t e d w i t h r o l l d a m p i n g a n d into t h e factors that s h o u l d b e i n c l u d e d i n a r o l l d a m p i n g p r e d i c t i o n m e t h o d .

© 2 0 1 5 E l s e v i e r Ltd. A l l rights r e s e r v e d .

1. Introduction

To date, b i l g e keels are f i t t e d t o a l m o s t a l l sea g o i n g vessels s i n c e i t is b y f a r t h e m o s t e c o n o m i c a l s o l u t i o n t o r e d u c e vessel r o l l m o t i o n s . D e s p i t e i t s w i d e a p p l i c a t i o n , a n a c c u r a t e n u m e r i c a l p r e -d i c t i o n o f t h e r o l l -d a m p i n g c o n t r i b u t i o n f r o m b i l g e keels r e m a i n s d i f f i c u l t d u e t o c o m p l e x p h y s i c s a s s o c i a t e d w i t h f l o w s e p a r a t i o n at t h e b i l g e k e e l tip. T h e m o s t e x t e n s i v e r e s e a r c h o n r o l l d a m p i n g has b e e n p e r -f o r m e d a b o u t 3 5 y e a r s ago a n d r e p o r t e d i n -f o r e x a m p l e Ikeda e t a l . [ 4 ] , I k e d a e t a l . [ 5 ] a n d H i m e n o [ 2 ] . T h e r e s u l t i n g e m p i r i c a l m e t h o d is w i d e l y u s e d i n t h e i n d u s t r y s i n c e i t is a p r a c t i c a l m e t h o d

* Corresponding author. Tel.:+31 10 2320 000. E-mail addresses: riaan.vantveer®sbmoffshore.com

(R.v. Veer), x a v i e r . s c h u t ® s b m o f f s h o r e . c o m (X.B. Schut), r.h.m.huijsmans@tudelft.nl (R.H.M. Huijsmans).

0141-1187/$ - see front matter © 2015 Elsevier Ltd. All rights reserved. http://dx.doi.Org/10.1016/j.apor.2015.07.001 l e a d i n g t o r e a s o n a b l e r e s u l t s i n m a n y s i t u a t i o n s . I t is r e f e r r e d t o as t h e I T H - m e t h o d i n t h i s paper, a f t e r t h e m a i n a u t h o r s . I n m o r e r e c e n t y e a r s t h e m e t h o d has b e e n a d o p t e d f o r s p e c i a l s h i p t y p e s o r c o n d i t i o n s , see e.g. I k e d a e t a l . [ 3 ] , b u t t h e f u n d a m e n t a l b r e a k -d o w n o f t h e r o l l -d a m p i n g i n -d i f f e r e n t c o n t r i b u t i o n s r e m a i n e -d t h e s a m e . D u e t o i t s success t h e I T H r o l l d a m p i n g m e t h o d has b e c o m e t h e ITTC r e c o m m e n d e d r o l l d a m p i n g p r e d i c t i o n m e t h o d (ITTC [ 7 ] ) . A s h o r t d e s c r i p t i o n o f t h e e s s e n t i a l c o m p o n e n t s a t z e r o s p e e d is p r o v i d e d . T h e p r e s e n t r e s e a r c h is m o t i v a t e d b y t h e f a c t t h a t V e e r e t a l . [ 1 6 ] s h o w s t h a t t h e p r e d i c t i o n o f t h e b i l g e k e e l l o a d s c a n be i m p r o v e d u t i l i z i n g 3 D p o t e n t i a l f l o w w h e n a c c u r a t e l o a d c o e f f i c i e n t s i n c l u d -i n g h -i g h e r h a r m o n -i c s - are a p p l -i e d . H o w e v e r , a d d -i t -i o n a l r e s e a r c h w a s d e e m e d n e c e s s a r y t o b e t t e r q u a n t i f y t h e c o e f f i c i e n t s a n d t o u n d e r s t a n d t h e i n f l u e n c e o f d i f f e r e n t p a r a m e t e r s o n t h e s e c o e f f i c i e n t s . A s s u c h , t h i s p a p e r p r e s e n t s e x p e r i m e n t a l r e s u l t s t h a t d e t a i l t h e i n d u c e d h u l l p r e s s u r e s a n d i n p a r t i c u l a r t h e l o a d c o e f f i c i e n t s

2 R.V. Veer etal. /Applied Ocean Research S3 (2015) 1-14

i n c l u d i n g h i g h e r h a r m o n i c s . These can be u t i l i z e d i n a z e r o speed r o l l d a m p i n g m e t h o d w h i c h is t o be f u r t h e r d e v e l o p e d .

J.J. ITH roll damping method

I n t h e I T H r o l l d a m p i n g m e t h o d ( H i m e n o [ 2 ] ) t h e b i l g e keel d a m p i n g a t z e r o f o r w a r d speed is a t t r i b u t e d t o m a i n l y t w o c o n -t r i b u -t i o n s : -t h e d a m p i n g d u e -t o -the n o r m a l f o r c e a c -t i n g o n -t h e b i l g e k e e l plates 5^^,^ a n d t h e d a m p i n g due to t h e h u l l pressures caused b y t h e s e p a r a t i n g b i l g e k e e l v o r t i c e s B;,;, H . For t h e n o r m a l f o r c e , l e a d i n g t o Bb^;,;, t h e M o r i s o n e q u a t i o n is u s e d w i t h a KC d e p e n d e n t d r a g c o e f f i c i e n t . The d r a g c o e f f i c i e n t is d e r i v e d f r o m t h e w o r k o f K e u l e g a n a n d C a r p e n t e r [ 1 0 ] f o r f l a t p l a t e s a n d fitted b y Ikeda t o t h e e q u a t i o n C D = 2 2 . 5 / / C C + 2 . 4 , w h e r e t h e KC n u m b e r is d e f i n e d b y KC=fUATI{2h) w h e r e h is t h e b i l g e k e e l h e i g h t , t h e a m p l i t u d e o f t h e h a r m o n i c m o t i o n , T t h e o s c i l -l a t i o n p e r i o d a n d ƒ a n e m p i r i c a -l v e -l o c i t y c o r r e c t i o n f a c t o r . The b i -l g e k e e l n o r m a l f o r c e o v e r a b i l g e k e e l l e n g t h L is g i v e n b y F(,ft,jv{t) = C D ( l / 2 ) p ( / i L ) y ( f ) | i / ( t ) | i n w h i c h the v e l o c i t y i/(t) i n t h e I T H - m e t h o d is o b t a i n e d f r o m t h e r o l l m o t i o n a n d c o r r e c t e d b y t h e e m p i r i c a l v e l o c i t y f a c t o r / . E q u i v a l e n t l i n e a r i z a t i o n is a p p l i e d t o o b t a i n t h e l i n -e a r i z -e d r o l l d a m p i n g m o m -e n t Bj;; ^ t h a t can b-e us-ed i n t h -e m o t i o n e q u a t i o n . The r o l l d a m p i n g c o m p o n e n t Bj,!;^ d u e t o t h e h u l l surface p r e s s u r e is d e r i v e d i n t h e I T H m e t h o d i n a n a l o g y t o t h e n o r -m a l d r a g f o r c e . T h e pressure c o -m p o n e n t is g i v e n b y p ( t ) = Cp(l/2)pv^{t)\v^{t}\ w h e r e y ^ ( t ) is t h e i n s t a n t a n e o u s r e l a t i v e v e l o c -i t y at t h e b -i l g e r a d -i u s c a l c u l a t e d f r o m t h e r o l l m o t -i o n v e l o c -i t y . The p r e s s u r e c o e f f i c i e n t s a n d its d i s t r i b u t i o n is d e r i v e d f r o m m e a s u r e -m e n t s a n d s o -m e w h a t s i -m p l i f i e d . I n f r o n t o f the -m o v i n g b i l g e keel t h e p o s i t i v e p r e s s u r e c o e f f i c i e n t Cp is l i n e a r l y d e c r e a s i n g t o zero. T h e m a x i m u m p o s i t i v e p r e s s u r e c o e f f i c i e n t is set t o 1.2 a n d f o u n d t o be i n d e p e n d e n t f r o m t h e KC n u m b e r . B e h i n d t h e b i l g e k e e l t h e n e g a t i v e p r e s s u r e c o e f f i c i e n t Cp is set c o n s t a n t o v e r a l e n g t h S/2 t h a t d e p e n d s o n t h e b i l g e keel h e i g h t b e f o r e i t l i n e a r l y decreases t o zero at t h e d i s t a n c e S f r o m t h e b i l g e k e e l . The p r e s s u r e j u m p o v e r t h e b i l g e k e e l d e p e n d s o n t h e KC n u m b e r - since t h e e d d y s h e d d i n g occurs i n t h e w a k e b e h i n d t h e m o v i n g b i l g e k e e l - a n d t h e r e l a t i o n s h i p Co = C + - Cp h o l d s (Ikeda et al. [ 5 ] ) . The l e n g t h o f n e g a t i v e p r e s s u r e r e g i o n is f o r a c y l i n d r i c a l h u l l shape d e f i n e d as S/h = 0 . 4 ( t J 4 r / 2 / i ) + 2.6.

J.2. Further development of an ITH inspired roll damping method

It has b e e n s h o w n i n V e e r et al. [ 1 6 ] t h a t t h e b i l g e k e e l loads o n t h e w e a t h e r a n d l e e w a r d side o f a vessel i n b e a m seas are v e r y d i f f e r e n t . T h i s e f f e c t is o b t a i n e d w h e n t h e r o l l - m o t i o n i n d u c e d v e l o c i t y vit) i n t h e b i l g e k e e l l o a d f o r c e e q u a t i o n is r e p l a c e d b y a local fluid v e l o c i t y f r o m p o t e n t i a l flow c a l c u l a t i o n s w h i c h i n c l u d e t h e c o n t r i b u t i o n s f r o m a l l w a v e c o m p o n e n t s ( i n c i d e n t w a v e , d i f f r a c t e d w a v e , a n d r a d i a t e d w a v e s ) . F u r t h e r i m p r o v e m e n t s are o b t a i n e d w h e n a n i n e r t i a c o n t r i b u t i o n is a d d e d a n d w h e n h i g h e r h a r m o n i c c o m p o n e n t s are i n c l u d e d . I t r e m a i n s o f i n t e r -est t o i n v e s t i g a t e t h e i n f l u e n c e o f f a c t o r s n o t c o n s i d e r e d i n t h e I T H - a p p r o a c h l i k e t h e f r e e s u r f a c e a n d t h e e f f e c t o f t h e b i l g e k e e l g e o m e t r y . I n t h e I T H m e t h o d t h e p r e s s u r e d i s t r i b u t i o n is g i v e n at t h e m o m e n t o f m a x i m u m g l o b a l r o l l m o t i o n v e l o c i t y , a n d n o f u r -t h e r d e -t a i l s are p r e s e n -t e d . V e r i f i c a -t i o n o f -t h e I T H - a p p r o a c h u n d e r d i f f e r e n t c o n d i t i o n s is o f i n t e r e s t a n d hence d e t a i l e d p r e s s u r e m e a -s u r e m e n t -s f o r d i f f e r e n t b i l g e keel g e o m e t r i e -s o -s c i l l a t i n g i n a f r e e s u r f a c e b o u n d e d fluid are e x e c u t e d . A p p l i c a t i o n o f c o m p u t a t i o n a l fluid d y n a m i c s (CFD) to q u a n t i f y t h e pressure d i s t r i b u t i o n a n d it's r e l a t e d r o l l d a m p i n g m o m e n t ( f o r s h i p s h a p e d sections) is as w e l l a p o s s i b i l i t y a n d t o v a l i d a t e CFD a p p l i c a t i o n , d e t a i l e d m e a s u r e m e n t s are r e q u i r e d . The p r e s e n t r e s e a r c h is i n t e n d e d f o r t h a t p u r p o s e as w e l l . 1.3. Present experiments The p r e s e n t e x p e r i m e n t s are c o n d u c t e d w i t h a c y l i n d r i c a l h u l l since s u c h g e o m e t r y w i l l n o t g e n e r a t e h u l l r a d i a t i o n w a v e s u n d e r p u r e r o t a t i o n . For an u n a p p e n d e d c y l i n d e r , t h e local r e l a t i v e v e l o c -i t y at t h e -i n t e n d e d b -i l g e k e e l l o c a t -i o n -is d e t e r m -i n e d s o l e l y b y t h e r o t a t i o n a l v e l o c i t y o f t h e c y l i n d e r a n d hence t h e KC n u m b e r is w e l l d e f i n e d . A test m a t r i x w a s d e f i n e d t o d e t e r m i n e t h e i n f l u e n c e i n t h e l o a d c o e f f i c i e n t s a n d pressure d i s t r i b u t i o n d u e t o t h e presence o f t h e f r e e s u r f a c e , t h e b i l g e k e e l g e o m e t r y a n d the o s c i l l a t i o n f r e -q u e n c y . B o t h i r r e g u l a r a n d h a r m o n i c o s c i l l a t i o n s w e r e a p p l i e d to s t u d y a possible flow m e m o r y e f f e c t , w h i c h w e r e observed i n p r e -v i o u s research r e p o r t e d b y I k e d a e t a l . [ 6 ] f o r flat plates a n d b y V e e r et al. [ 1 5 ] i n r o l l decay e x p e r i m e n t s w i t h a n FPSO h u l l . The c o n t r i b u t i o n o f the h i g h e r h a r m o n i c s i n t h e b i l g e keel n o r m a l f o r c e are p r e s e n t e d i n the same m a n n e r as i n t h e w o r k b y K e u l e g a n a n d C a r p e n t e r [ 10] f o r f r e e plates.

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 s is g i v e n i n Section 2. The data analysis is p r e s e n t e d i n S e c t i o n 3. The n o r m a l b i l g e k e e l f o r c e c o e f f i c i e n t s are discussed i n S e c t i o n 4 a n d t h e pressure r e s u l t s are discussed i n S e c t i o n 6. S e c t i o n 5 discusses the m e a s u r e d loads i n i r r e g u l a r m o t i o n . The c o n c l u s i o n s are d r a w n i n Section 7.

2. Description of experiments

2 . J . Test set-up A n e x p e r i m e n t w a s d e s i g n e d w i t h t h e o b j e c t i v e t o m e a s u r e t h e b i l g e k e e l n o r m a l f o r c e a n d t h e h u l l pressures i n d u c e d b y t h e s e p a r a t i n g v o r t e x s y s t e m f r o m t h e b i l g e keel ( d e n o t e d as h u l l v o r t e x p r e s s u r e ) . For t h i s p u r p o s e a c y l i n d r i c a l h u l l w a s c o n -s t r u c t e d w i t h a n i n t e r c h a n g e a b l e b i l g e k e e l a n d t w o r o w -s o f t e n pressure sensors each.

Forced o s c i l l a t i o n s w e r e a p p l i e d w i t h r e g u l a r ( s i n u s o i d a l ) a n d i r r e g u l a r r o t a t i o n a r o u n d t h e axis o f t h e " 2 D " c y l i n d r i c a l h u l l . The e x p e r i m e n t s w e r e c o n d u c t e d i n t h e N o . 1 T o w i n g T a n k o f t h e L a b o r a t o r y o f S h i p H y d r o m e c h a n i c s at t h e D e l f t U n i v e r s i t y o f T e c h n o l o g y . Fig. 1 s h o w s t h e m o d e l i n t h e f r e e s u r f a c e (a) a n d t h e b i l g e k e e l s e t - u p f r o m u n d e r w a t e r ( b ) .

The test b a s i n is 1 4 2 m l o n g , 4.22 m w i d e a n d 2.5 m deep. O n one s i d e t h e r e is a b e a c h t o r e d u c e w a v e r e f l e c t i o n , o n t h e o t h e r side t h e r e is a w a v e m a k e r . The w a v e m a k e r w a s p o s i t i o n e d u n d e r a n angle t o m i n i m i z e w a v e r e f l e c t i o n s since i t is n o t used i n t h e e x p e r i m e n t . Essentially, t h e d u r a t i o n o f t h e e x p e r i m e n t s w a s s u c h t h a t r e f l e c t e d w a v e e n e r g y c o u l d n o t d i s t u r b t h e m e a s u r e m e n t s . A H e x a m o v e w a s used t o g e n e r a t e t h e r o l l m o t i o n s . I t consists o f t w o p l a t f o r m s c o n n e c t e d v i a six h y d r a u l i c d r i v e s . One p l a t f o r m w a s firmly a n c h o r e d t o t h e carriage w h i l e t h e o t h e r w a s c o n n e c t e d t o t h e m o d e l a n d w a s s u b j e c t t o c o n t r o l l e d m o t i o n s . The r o t a -tion axis, b e i n g at t h e o r i g i n o f t h e c i r c l e - c y l i n d r i c a l h u l l shape, w a s p o s i t i o n e d at t h r e e d i f f e r e n t h e i g h t s w i t h r e s p e c t t o t h e c a l m w a t e r l e v e l t h e r e b y s i m u l a t i n g t h r e e d i f f e r e n t d r a f t c o n d i t i o n s o f t h e c y l i n d e r . I n t h e m o s t s h a l l o w c o n d i t i o n t h e b i l g e k e e l operates v e r y close to t h e f r e e surface. The g l o b a l m o t i o n s w e r e r e c o r d e d b y a n i n f r a r e d K r y p t o n m e a s u r i n g s y s t e m w h i c h has a n accuracy o f a b o u t 0.10 m m . The r o l l m o t i o n s w e r e g e n e r a t e d w i t h great a c c u -racy. The s t a n d a r d d e v i a t i o n o f t h e t r a n s l a t i o n s o f t h e c e n t e r o f r o t a t i o n r e c o r d e d d u r i n g t h e e x p e r i m e n t s w a s o n average 0.15 m m a n d at m o s t 0.34 m m f o r t h e largest r o l l angle. The s t a n d a r d d e v i a -t i o n o f -t h e r o -t a -t i o n s o -t h e r -t h a n r o l l w e r e o n average 0.025 d e g a n d at m o s t 0.06 deg.

R.V. Veer et al./Applied Ocean Research 53 (2015) 1-14 3

Table 1

Pressure sensors distance to bilge Iceel.

(a) F r o n t view, m o d e l

^HHHHHHIHHIHHHHHHIHHIHHHHIHIiHIIHHB^BHIHH

( b ) U n d e r w a t e r v i e w , bilge keel

Fig. 1. Experimental set-up.

2.2. Cylindrical hull with single bilge keel

The c y l i n d r i c a l h u l l m e a s u r e d 1218 m m i n l e n g t h a n d h a d a r a d i u s o f 730.5 m m . A cross s e c t i o n ( f r o n t v i e w ) o f t h e m o d e l w i t h t h e s i n g l e b i l g e k e e l a n d p r e s s u r e sensors is g i v e n i n Fig. 2 w h i c h i n d i c a t e s as w e l l t h e t h r e e d i f f e r e n t d r a f t s w h i c h are d e n o t e d b y T364, T 2 7 8 a n d T 1 4 8 . The p r e s s u r e sensors are d e n o t e d b y A t o J a n d t h e distance (S) o f each sensors w i t h respect to t h e b i l g e k e e l l o c a t i o n is s h o w n i n Table 1 . Sensor S ( m m ) S / / ! ( - ) Sensor S ( m m ) S / / l ( - ) A -300.9 - 9 . 6 7 F 25.5 0.82 B - 2 3 2 . 0 - 7 . 4 6 G 94.3 3.03 C -163.2 - 5 . 2 5 H 163.2 5.25 D -94.3 - 3 . 0 3 I 232.0 7.46 E - 2 5 . 5 - 0 . 8 2 J 300.9 9.67 To p r e v e n t 3 D e f f e c t s as m u c h as possible, t w o r i n g s w e r e p l a c e d at each e x t r e m i t y o f t h e m o d e l , so essentially, t h e b i l g e k e e l is f i t t e d b e t w e e n w a l l s . The r i n g s a n d t h e b i l g e k e e l can be seen i n Fig. 3.

T w o d i f f e r e n t i n t e r c h a n g e a b l e b i l g e keels w e r e a p p l i e d w i t h shape a n d d i m e n s i o n s as i n Fig. 4 . Bilge k e e l 1 is a s i m p l e f l a t p l a t e as is o f t e n used i n o f f s h o r e m o d e l t e s t i n g ( a n d f u l l scale). Bilge keel 2 is a m o r e c o m p l e x d e s i g n w i t h a t o p e n d p l a t e a n d a n i n t e r m e -diate plate s t i f f e n e r . B o t h b i l g e keels m e a s u r e 3 1 . 1 m m i n h e i g h t , so t h a t t h e same KC n u m b e r a p p l i e s f o r a g i v e n f o r c e d m o t i o n p a t -t e r n . The -t h i c k n e s s o f -t h e b i l g e k e e l is 1.0 m m . The b i l g e keels w e r e m o u n t e d o n a base p l a t e t h a t fltted i n a g r o o v e i n t h e h u l l w h i c h w a s s l i g h t l y l a r g e r t h a n t h e base p l a t e . The base p l a t e w a s b o l t e d t o t w o f o r c e t r a n s d u c e r s . The f o r c e t r a n s d u c e r s w e r e c a l i b r a t e d b y i n -s i t u t e -s t i n g -s h o w i n g a l i n e a r b e h a v i o r w i t h i n 0.15% accuracy. The mass o f the b i l g e k e e l w a s m e a s u r e d , a n d t h r o u g h o s c i l l a t i o n s i n air t h e d r y i n e r t i a l loads, t h a t need t o be d e d u c e d f r o m t h e m e a s u r e d b i l g e keel f o r c e i n w a t e r , c o u l d be d e t e r m i n e d .

N e x t to t h e b i l g e keels t w è n t y pressure sensors w e r e f i t t e d i n t w o r o w s f l u s h t o t h e h u l l surface, see Fig. 2. T w o t y p e s o f p r e s -sure sensors w e r e a p p l i e d as s h o w n i n Fig. 5. D u e t o t h e v e r y s m a l l p r e s s u r e f l u c t u a t i o n s e x p e c t e d i n the m e a s u r e m e n t a v e r y accurate p r e s s u r e sensor w a s selected ( t y p e I) b u t w i t h a l i m i t e d p r e s s u r e range. For v e r i f i c a t i o n , a d i f f e r e n t p r e s s u r e sensor ( t y p e II) w a s selected w i t h l a r g e r p r e s s u r e range, a l o w e r accuracy, b u t w i t h a s l i g h t l y m o r e f a v o r a b l e c o n s t r u c t i o n o f a s m a l l e r c a v i t y v o l u m e . B o t h p r e s s u r e sensors consist b a s i c a l l y o f a d i a p h r a g m w h i c h d e f o r -m a t i o n is -m e a s u r e d b y s t r a i n gauges. The d i a p h r a g -m is l o c a t e d at t h e e n d o f a s m a l l c h a n n e l ( w i t h d i a m e t e r o f 3 m m ) t h a t c o n n e c t s

I

130 mm

I

5 9 0 mm

Fig. 3. Set-up showing pressure sensors type I and II at 130 mm apart, two force transducers and the complex bilge keel attached to it.

Fig. 2. Cross section view (A) including draft marks (T148, T278 and T364), position of ten pressure sensors (type 1) and the complex bilge Iceel.

( a ) F l a t b i l g e k e e l ( b ) C o m p l e x b i l g e k e e l Fig. 4. Bilge keel configurations.

R.V. Veer et al./Applied Ocean Research 53 (2015) 1-14

n u Casing

O Diaphragm Table 2

KC numbers in regular forced oscillation tests.

J

I

20 mm 25 mm

( a ) T y p e I ( b ) T y p e I I Fig. 5. Pressure sensors.

t h e sensor w i t h t h e e n v i r o n m e n t . For t y p e I t h i s c a v i t y has a d e p t h o f 10 m m , w h i l e i t is o n l y 2 m m deep f o r t y p e 11.

Prior t o t h e e x p e r i m e n t s , t h e p r e s s u r e sensors w e r e s t a t i c a l l y c a l i b r a t e d s h o w i n g a l i n e a r b e h a v i o r o f 0.05% f o r t y p e 1 a n d 0 . 1 % f o r t h e t y p e I I . I n t h i s p a p e r results o f sensor I w i l l be p r e s e n t e d s o l e l y as t h i s sensor p r o v i d e d t h e m o s t accurate p r e s s u r e r e a d i n g s . Sensor I has a l i n e a r range o f 7 0 m b a r a n d a s p e c i f i e d accuracy o f ± 0 . 0 3 5 m b a r .

The d y n a m i c b e h a v i o r o f t h e sensor t y p e I w a s v e r i f i e d as w e l l t h r o u g h a b o v e a n d u n d e r w a t e r o s c i l l a t i o n s . I t w a s c o n c l u d e d t h a t t h e pressure signal w a s n o t a f f e c t e d b y t h e c e n t r i f u g a l forces a n d transverse accelerations a c t i n g o n t h e e n t r a p p e d w a t e r i n t h e c a v i t y i n t h e f r e q u e n c y range o f t h e f o r c e d o s c i l l a t i o n s o f t h e c y l i n d e r .

2.3. Test matrix - harmonic forced oscillations

I n t h e h a r m o n i c f o r c e d o s c i l l a t i o n tests, t h e r o t a t i o n a l m o t i o n o f t h e c y l i n d e r is s p e c i f i e d b y : (^(f) = <^,isin(&)t), w h e r e cpA is the m o t i o n a m p l i t u d e a n d &) t h e o s c i l l a d o n f r e q u e n c y . A l l e x p e r i m e n t s s t a r t e d at t = 0 w i t h a fluid at rest due t o t h e w a i t i n g t i m e i n b e t w e e n d i f f e r e n t tests. Each test w a s e x e c u t e d f o r 45 s ( m o d e l scale) w h i c h r e s u l t s i n a b o u t 1 0 - 2 0 f u l l o s c i l l a t i o n p e r i o d s . The KC n u m b e r i n a r e g u l a r f o r c e d m o t i o n test w i t h a m p l i t u d e (pA is d e f i n e d b y : , ^ ^ ^ ( ^ ^ 2/1 R<p27r 2h OJ R4>ATt h (1) w h e r e h is t h e b i l g e k e e l h e i g h t , R t h e distance f r o m t h e r o t a t i o n axis to t h e base o f t h e b i l g e k e e l ( t h e c y l i n d e r r a d i u s ) , a n d T t h e o s c i l l a t i o n p e r i o d . The KC n u m b e r is f r e q u e n c y i n d e p e n d e n t a n d p r e s c r i b e d b y t h e m o t i o n a m p l i t u d e since a c y l i n d r i c a l h u l l w i l l n o t g e n e r a t e r a d i a t i o n i n d u c e d v e l o c i t i e s a n d the e x p e r i m e n t s are p e r f o r m e d i n c a l m w a t e r . Table 2 d e f i n e s the KC n u m b e r f o r t h e d i f f e r e n t ( t a r g e t ) o s c i l -l a t i o n a m p -l i t u d e s used f o r t h e e x p e r i m e n t s . I n t h e r e s u -l t tab-les s l i g h t l y d i f f e r e n t KC n u m b e r s are seen due to t h e f a c t t h a t t h e m e a s u r e d o s c i l l a t i o n a m p l i t u d e s i n t h e tests are s l i g h t l y d i f f e r e n t .

The KC range o f 0 . 3 - 1 5 is m o s t r e l e v a n t since the d r a g a n d i n e r t i a c o e f f i c i e n t s v a r y s i g n i f i c a n t l y b e t w e e n these KC n u m b e r s a n d i t is t y p i c a l l y t h e KC range n u m b e r s t h a t can be associated t o a r e a l b i l g e k e e l o n a FPSO i n f a t i g u e sea states. For m u c h h i g h e r KC n u m b e r s the d r a g c o e f f i c i e n t s t e n d t o w a r d s a c o n s t a n t ( m i n i m u m ) v a l u e o f a b o u t

4>A (deg) KC (-) (pA (deg) K C ( - )

0.25 0.32 5.00 6.44 0.50 0.64 6.00 7.73 1.00 1.29 7.00 9.02 2.00 2.58 8.00 10.30 3.00 3.86 10.00 12.88 4.00 5.15 12.00 15.46 T a b l e s

Test matrix of in regular forced oscillation tests.

Series Configuration

A l Plat plate bilge keel

(18 tests) 3 drafts (T364, T278,T148)

6 amplitudes (1,4, 6 , 8 , 1 0 , 1 2 d e g ) 1 frequency (2.62 rad/s)

BI Complex bilge keel

(18 tests) 3 drafts (T364, T278,T148)

6 amplitudes ( 1 , 4 , 6 , 8 , 1 0 , 12deg) 1 frequency (2.62 rad/s)

A2 Plat plate bilge l<eel

(12 tests) 1 draft (T278)

6 amplitudes (1,4, 6, 8 , 1 0 , 1 2 deg) 2 frequencies (1.57,3.64 rad/s)

B2 Complex bilge keel

(50 tests) 1 draft (T278) 10 ampl. ( 0 . 2 5 , 0 . 5 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , i ideg) 5 frequencies (1.57, 2.07, 2.62,3.14, 3.64rad/s) 2.4 as can be seen i n t h e e x p e r i m e n t s b y K e u l e g a n a n d C a r p e n t e r [ 1 0 ] a n d Sarpkaya a n d O'Keefe [ 1 2 ] . I n t h e p r e s e n t e x p e r i m e n t s i t w o u l d r e q u i r e a n u n r e a l i s t i c large m o t i o n a m p l i t u d e o r a m u c h s m a l l e r b i l g e k e e l h e i g h t t o reach s u c h KC n u m b e r s , a n d h e n c e t h i s w a s n o t r e a l i z e d .

The test m a t r i x is s u m m a r i z e d i n Table 3. Series A l a n d A 2 y i e l d r e s u l t s f o r t h e flat p l a t e b i l g e keel t h a t are used t o b e n c h m a r k t h e e x p e r i m e n t s b y Sarpkaya, t o i n v e s t i g a t e t h e h i g h e r h a r m o n i c c o m -p o n e n t s a n d to s t u d y t h e -possible i n f l u e n c e o f t h e f r e e s u r f a c e . Series B I a n d B2 g i v e the s a m e i n s i g h t f o r t h e m o r e c o m p l e x b i l g e keel g e o m e t r y . Series A 2 a n d B2 f u r t h e r i n v e s t i g a t e t h e i n f l u e n c e o f t h e o s c i l l a t i o n f r e q u e n c y . The m e d i a n o s c i l l a t i o n f r e q u e n c y o f 2.62 rad/s c o r r e s p o n d s to a t y p i c a l scaled r o l l n a t u r a l f r e q u e n c y o f a n FPSO.

2.4. Test matrix - irregular forced oscillations

I n r e a l i t y t h e b i l g e k e e l o n a vessel w i l l e x p e r i e n c e i r r e g u l a r m o t i o n s a n d hence v a r y i n g e d d y s h e d d i n g s t r e n g t h a n d p a t t e r n i n t i m e . This i n t r o d u c e s possible t i m e v a r y i n g flow m e m o r y e f f e c t s . To i n v e s t i g a t e r e a l i s t i c s i t u a t i o n s f o u r d i f f e r e n t i r r e g u l a r f o r c e d o s c i l l a t i o n s w e r e p e r f o r m e d . The b i c h r o m a t i c i r r e g u l a r o s c i l l a t i o n s r e p r e s e n t s y n t h e s i z e d m o t i o n p a t t e r n s i n a n i r r e g u l a r sea state. F r o m t h e i r r e g u l a r m o t i o n t i m e trace t h e KC n u m b e r is c a l c u l a t e d f o r each h a l f cycle b e t w e e n t w o z e r o crossings. The m o t i o n a n d KC t i m e trace are p r e s e n t e d i n Fig. 6. The r e s u l t a n t KC r a n g e i n t h e s i m u l a t i o n is l i s t e d i n Table 4.

Table 4

Irregular motion coefficients for Eq. (2).

Bichromatic motion A, (deg) Tl (s) Al(deg) r 2 ( s ) KC range

1 10 2.24 0 _ 0.29-12.88

2 5 2.24 5 2.91 0.14-12.91

3 5 2.24 2 2.76 0.80-14.43

Rv. Veer et al. / Applied Ocean Researcti 53 (2015) 1-14 5 CM 2 , 20 10 0 = - 1 0 - 2 0 • 1 • 1 • • • • 10 2 0 30 T i m e [s] 40 l\/lolion - e - K C

Fig. 6. Irregular motion and KC time traces.

The c o n s t r u c t e d signals are d e f i n e d b y Eq. ( 2 ) w i t h c o e f f i c i e n t s d e f i n e d i n Table 4 . The d u r a t i o n o f t h e s i g n a l is 2 0 o s c i l l a t i o n s o f p e r i o d T i . The r a m p f u n c t i o n Rj m o d i f i e s t h e c o m p l e t e 2 0 p e r i o d s a n d is d e f i n e d b y Eq. ( 3 ) .

i = l

1^1 =sin2(27rt/(20ri)) «2 = 1

R3 = 2f/(2ori) i?4 = 1 + t/(2or,)

2.5. J?eyno;ds number and boundary layer effects

(2) (3) The e x p e r i m e n t a l r e s u l t s f o r a f r e e p l a t e b y K e u l e g a n a n d C a r p e n t e r [ 1 0 ] c o n f i r m t h a t t h e d r a g a n d i n e r t i a c o e f f i c i e n t s r e p r e s e n t i n g t h e n o r m a l f o r c e o n l y d e p e n d o n t h e KC n u m b e r . The R e y n o l d s n u m b e r d e f i n e d b y Re^U^DIv w h e r e D is t h e f r e e plate h e i g h t , v a r i e d b e t w e e n 4 2 0 0 a n d 14,200, b u t no n o t i c e a b l e Reynolds scale e f f e c t s w e r e f o u n d . Reynolds i n - d e p e n d e n c y is o f t e n c o n t r i b u t e d t o t h e f a c t t h a t t h e flow d e t a c h m e n t p o i n t s are w e l l d e f i n e d f o r a s h a r p e d g e d b o d y . W h e n a w a l l is a d d e d , l i k e i n t h e e x p e r i m e n t s o f Sarpkaya a n d O'Keefe [ 12 ] a n d i n t h e p r e s e n t e x p e r i m e n t s , a n o s c i l l a t i n g b o u n d -ary can d e v e l o p o n t h e h u l l n e x t t o t h e b i l g e k e e l w h i c h m i g h t i n f l u e n c e t h e r e s u l t s . T h i s p o i n t w a s addressed b y Sarpkaya a n d t h e Stokes's b o u n d a r y l a y e r t h i c k n e s s <5s w a s r e p o r t e d t o be a b o u t 10 m m l e a d i n g to Ssjh o f 0.10. T h e i n f l u e n c e o f t h e b o u n d a r y l a y e r w a s t h e r e f o r e j u s t i f i e d t o be o f s e c o n d a r y i n f l u e n c e . But since t h e e x p e r i m e n t s w e r e p e r f o r m e d at a single f r e q u e n c y no c o m p a r a -tive i n f o r m a t i o n can be r e t r i e v e d o n t h e i n f l u e n c e o f the o s c i l l a t i n g p e r i o d - a n d hence b o u n d a r y l a y e r - o n t h e results.

I n the p r e s e n t e x p e r i m e n t s t h e Stokes b o u n d a r y layer t h i c k -ness is c a l c u l a t e d t o be 5.2 m m w h e n t h e o s c i l l a t i o n p e r i o d is 4.0 s a n d a b o u t 3.4 m m f o r t h e s h o r t e s t o s c i l l a t i o n p e r i o d o f 1.73 s. The b o u n d a r y layer t h i c k n e s s is c a l c u l a t e d as Ss = 4.6^/2v/(ji), see S c h l i c h t i n g a n d G e r s t e n [13].^ This leads to (5s//i = 0.17 f o r t h e l o n g e s t p e r i o d a n d 0.11 f o r t h e s h o r t e s t p e r i o d .

The Reynolds n u m b e r associated to t h e o s c i l l a t i o n a m p l i t u d e can b e d e f i n e d b y Re = U/iXyi/v w h e r e x^lm] is 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 -t h e bilge keel. U s i n g -t h e a m p l i -t u d e range o f 1 - 1 2 d e g i n the p r e s e n t e x p e r i m e n t s i t leads at t h e h i g h e s t o s c i l l a t i o n p e r i o d t o a Reynolds n u m b e r range o f 2 5 5 to 36.8 x 10^ and f o r t h e s h o r t e s t o s c i l l a t i o n p e r i o d f r o m 6 0 0 t o 85.2 x 10^. These values are r e l a t i v e l y l o w s u g g e s t i n g l a m i n a r flow c o n d i t i o n s . W i t h i n c r e a s i n g Reynolds the b o u n d a r y layer w i l l be t h i n n e r a n d hence this is t h e case w i t h i n c r e a s i n g KC. If f o r o b t a i n e d Re n u m b e r s t h e Stokes b o u n d a r y l a y e r is s t i l l r e p r e s e n t a t i v e a n d i f t h i s r e m a i n s t h e case i n t h e v i c i n i t y o f the b i l g e k e e l t h a t is, at present, y e t u n k n o w n .

Research b y Jensen et al. [ 8 ] suggest t h a t t r a n s i t i o n f r o m l a m i n a r to t u r b u l e n t layers s t a r t a r o u n d Re RilO^ a n d t h a t at Re as l a r g e as 1.6 X 10^ b o u n d a r y l a y e r is n o t f u l l y d e v e l o p e d t o t u r b u l e n t o n e . Clearly, t h e shed eddies w i l l i n t e r f e r e w i t h t h e f l o w b e h i n d t h e o s c i l l a t i n g b i l g e k e e l a n d t h i s w i l l a f f e c t t h e b o u n d a r y layer a n d t h e e x p e c t a t i o n is t h a t t h i s w i l l increase t u r b u l e n c e . A t u r b u l e n t b o u n d a r y layer m i g h t b e t h i c k e r b u t t h e v e l o c i t i e s i n t h e l a y e r w i l l be l a r g e r a n d t h e b o u n d a r y l a y e r p r o f i l e suggests less e f f e c t o n t h e loads f r o m i t . G i v e n t h e b e f o r e m e n t i o n e d , scale e f f e c t s i n the p r e s e n t e x p e r i m e n t s are e x p e c t e d o f s e c o n d a r y o r d e r l i k e i n t h e Sarpkaya a n d O'Keefe [ 1 2 ] e x p e r i m e n t s ; a l t h o u g h t h e Ssjh is s o m e w h a t larger.

3. Data analysis

3.1. Derivation of bilge keel normal force coefficients; C D and C M

T h e m e a s u r e d l o a d i n the l o a d sensor is r e c o r d e d i n a b o d y fixed r e f e r e n c e f r a m e a n d i t c a n be d e c o m p o s e d i n a n m a s s - i n e r t i a l o a d ( e q u a l t o t h e ( s o l i d ) mass o f t h e b i l g e k e e l a n d p a r t o f t h e c o n s t r u c -tion s e t - u p times t h e b o d y fixed a c c e l e r a t i o n ) a n d t h e fluid f o r c e

Fit) a c t i n g o n the b i l g e k e e l . The b o d y - f i x e d a c c e l e r a t i o n is t h e s u m

of a r o l l angle d e p e n d e n t g r a v i t y c o m p o n e n t a n d t h e a c c e l e r a t i o n i n t r o d u c e d b y t h e b o d y m o t i o n . B e f o r e each e x p e r i m e n t t h e l o a d sensor is zeroed so t h a t o n l y t h e v a r i a t i o n o f t h e g r a v i t y c o m p o -n e -n t is m e a s u r e d . W h e -n t h e i -n e r t i a l o a d i -n g is s u b t r a c t e d f r o m t h e m e a s u r e d l o a d t h e r e m a i n i n g f o r c e F(t) is t h e f o r c e o f i n t e r e s t . O s c i l -l a t i o n s i n air w e r e p e r f o r m e d w i t h b o t h b i -l g e kee-ls t o o b t a i n t h e a c t u a l mass o f t h e c o n s t r u c t i o n s e t - u p t h a t needs t o be t a k e n i n t o a c c o u n t . T h e M o r i s o n e q u a t i o n ( M o r i s o n et a l . [ 1 1 ] ) is o f t e n used t o describe t h e fluid forces o n slender o b j e c t s . I t consists o f t w o t e r m s : a n i n e r t i a l o a d i n g g i v e n as a f o r c e p r o p o r t i o n a l t o t h e flow accel-e r a t i o n a n d a d r a g ( d a m p i n g ) f o r c accel-e p r o p o r t i o n a l t o t h accel-e s i g n accel-e d s q u a r e o f t h e i n s t a n t a n e o u s flow v e l o c i t y . W h e n t h e i n e r t i a a n d d r a g c o e f f i c i e n t s are t a k e n as t i m e i n v a r i a n t c o e f f i c i e n t t h e m o t i o n e q u a t i o n s i m p l i f i e s c o n s i d e r a b l y . It leads t o t h e f o l l o w i n g M o r i s o n e q u a t i o n : Fit) = ^pCDApUit)\Uit)\+pVoCMUit) (4)

1 Concerning the Sarpltaya experiments where KC = UTjh and Re = Uhjv and given ^ = Re/KC = /I2/(DT) = 1845, it leads with h = 102 m m to 7= 5.64 s and Ss = 6.2 mm.

6 R.v. Veer et al./Applied Ocean Research 53(2015) 1-14

w h e r e p is t h e f l u i d d e n s i t y , Ap t h e p r o j e c t e d area o f t h e o b j e c t , CQ t h e d r a g c o e f f i c i e n t , VQ is t h e i n e r t i a r e f e r e n c e v o l u m e a n d Cu t h e i n e r t i a c o e f f i c i e n t . For a flat p l a t e b i l g e k e e l t h e p r o j e c t e d area is t h e b i l g e k e e l h e i g h t h times t h e l e n g t h L, Ap = hL. Eq. ( 4 ) c a n be a p p l i e d f o r a fixed o b j e c t s u b j e c t t o an o s c i l l a t o r y fluid, o r f o r a m o v i n g o b j e c t i n a fluid w h i c h is o t h e r w i s e at rest. This is t h e c o n d i t i o n u n d e r w h i c h t h e p r e s e n t e x p e r i m e n t s are c o n d u c t e d .

To be c o n s i s t e n t w i t h t h e analysis a p p r o a c h o n f r e e - p l a t e s b y K e u l e g a n a n d Carpenter [ 1 0 ] , t h e a d d e d mass r e f e r e n c e v o l u m e is t a k e n as h a l f t h e c y l i n d e r w i t h r a d i u s h t h a t c i r c u m f e r e n c e s t h e b i l g e k e e l a n d its m i r r o r i m a g e , t h u s VO = LJTII^I2.

The v e l o c i t y o f t h e b i l g e keel at its r o o t l o c a t i o n is d e f i n e d b y

U{t) = UACOs{cot) = R(0(pACOsicot). w h e r e R is t h e r a d i u s o f t h e c y l i n

-der, (pA t h e m o t i o n a m p l i t u d e a n d &) t h e o s c i l l a t i o n f r e q u e n c y . The f o r c e F(t) i n Eq. ( 4 ) n o w reduces t o a h a r m o n i c f u n c t i o n w h i c h c a n be w r i t t e n as: 2 F ( t ) phWl = CDCOs(wf)|cos(a)t)| C M ^ s i n ( w t ) (5) w h e r e t h e KC n u m b e r is based o n t w i c e t h e b i l g e k e e l h e i g h t as d e f i n e d i n Eq. ( 1 ) .

The M o r i s o n e q u a t i o n is w i d e l y used since i t i n v o l v e s o n l y t w o timeindependent c o e f f i c i e n t s : t h e d r a g CD a n d i n e r t i a C M c o e f f i -c i e n t . H o w e v e r , t h e M o r i s o n e q u a t i o n does n o t a l w a y s r e p r e s e n t t h e p h y s i c a l f o r c e v e r y w e l l . H i g h e r o r d e r h a r m o n i c c o m p o n e n t s m i g h t be r e q u i r e d t o b e t t e r r e p r e s e n t t h e f o r c e time h i s t o r y , as a l r e a d y p o i n t e d o u t b y K e u l e g a n a n d C a r p e n t e r [ 1 0 ] . Since flow r e v e r s a l a f t e r h a l f a p e r i o d w i l l lead to an o p p o s i n g f o r c e o n t h e b i l g e k e e l , t h u s F(t) = - F ( t + r / 2 ) , t h e f o r c e is p e r i o d i c a n d o d d . The l a t t e r i m p l i e s t h a t o n l y t h e o d d h a r m o n i c c o m p o n e n t s w i l l arise a n d h e n c e Eq. ( 5 ) can be e x t e n d e d w i t h h i g h e r h a r m o n i c s t o : = CD c o s ( c ü t ) | cos(d-jf)| -CM^ s i n ( ü ) t ) + B'. cos(3(Wt) phLJJ^ + B 5 cos(5<i)t) + A3 s i n ( 3 ü ) t ) + ^ 5 s i n ( 5 a ) t ) + • • • (6)

The drag, i n e r t i a a n d h i g h e r h a r m o n i c c o e f f i c i e n t s o f Eq. ( 6 ) c a n be o b t a i n e d b y several m e t h o d s o f w h i c h t h e least s q u a r e t e c h n i q u e a n d F o u r i e r analysis are c o n s i d e r e d as t h e m o s t r o b u s t t e c h n i q u e s . I n t h e F o u r i e r analysis t h e m e a s u r e d f o r c e t i m e trace i n t h e analysis c o n c e r n s one o r m u l t i p l e o s c i l l a t i o n p e r i o d s . For r o b u s t n e s s a n d t o a l i g n w i t h K e u l e g a n a n d Carpenter [ 1 0 ] t h e F o u r i e r analysis is s e l e c t e d . This leads t o :

2 F ( t )

•• A-i sinojt+As sm3cüt +As sin Scot+• phWl

+ B\ cos üJt + B3 cos 3cot 4- B5 cos Scot

w i t h F o u r i e r c o e f f i c i e n t s ( n = 1, 3, 5) g i v e n b y : 2 [ 2 F ( t ) s i n ( n ü ) f ) An T phWl '-dt ( 7 )

(8)

2 f 2£(^)cos{m^

C o m p a r i n g Eqs. ( 6 ) a n d ( 7 ) i t is seen t h a t t h e C M d i r e c t l y relates t h e ^ i i n t h e F o u r i e r e x p a n s i o n :

K C ,

(9)

a n d t h a t CQ relates t o B i w i t h s o m e f u r t h e r m a t h e m a t i c a l d e r i v a -tions. This r e l a t i o n s h i p can be f o u n d w h e n t h e cos(a;t)| zos[cot)\ o f

Eq. ( 6 ) is r e p l a c e d b y its F o u r i e r e x p a n s i o n . This series is o b t a i n e d f r o m t h e i n n e r - p r o d u c t r a t i o : C £ ) C o s ( a ) t ) | c o s ( c ü t ) | = CD 00 { F ( f ) , cos(na)f)) { c o s ( n t ü t ) , cos(nö}f)> f l cos(&)t)| c o s ( ó ) f ) | cos(n&)t)dt n^l / o ' c o s 2 ( n . . t ) d r C D { C I cos(<i}t) + C3 cos(3ait) + C5 c o s ( S c o t ) - f • w i t h : Cn = ( - ! ) '

(11+1

)/2_ n = l , 3 , 5 , . n7r(n2 - 4 ) Hence t h e f o l l o w i n g r e l a t i o n s h i p s are f o u n d :

r

^JTp ( 1 0 ) (11) 53 = « 3 - C D ^ 3' = B 5 + C D - ^ B3 - g B i (12) " " 1 0 5 7 ? - ^ ^ + 1 0 5 ^ ' It can be s h o w n t h a t t h e leastsquare s o l u t i o n a n d a F o u r i e r s o l u -t i o n u s i n g o r -t h o g o n a l basis f u n c -t i o n are e x a c -t l y i d e n -t i c a l ( C o l l i n s [ 1 ] ) . B u t t h i s o n l y h o l d s w h e n all s i g n i f i c a n t t e r m s are i n c l u d e d i n a least-square a p p r o a c h . I f t h e h i g h e r h a r m o n i c c o m p o n e n t s are s i g n i f i c a n t , a least-square fit o n Eq. ( 5 ) w i l l l e a d t o d i f f e r e n t Co a n d CM c o e f f i c i e n t s t h a n w h e n t h e least-square fit is p e r f o r m e d o n Eq. ( 6 ) . The F o u r i e r c o e f f i c i e n t s are. i n d e p e n d e n t f r o m h o w m a n y c o e f f i c i e n t s are i n c l u d e d . Hence t h i s m o r e r o b u s t s o l u t i o n s t r a t e g y is a p p l i e d i n t h e p r e s e n t analysis. A n e x a m p l e o f a n o b t a i n e d r e s u l t is g i v e n i n Fig. 7 w h i c h s h o w s t h e m e a s u r e d a n d c a l c u l a t e d t i m e traces u s i n g t h e F o u r i e r c o e f f i c i e n t s . W h e n o n l y t h e first h a r m o n i c c o e f f i c i e n t s are i n c l u d e d , t h e m e a s u r e d f o r c e is n o t c a p t u r e d w e l l a n d u n d e r p r e d i c t e d . H i g h e r h a r m o n i c c o m p o n e n t s are i n d e e d essential t o find a p r o p e r r e p r e s e n t a t i o n o f t h e m e a s u r e m e n t s . 7.5 2.5Y

o

m -2.5, - 5 1 1 1 J " 1 n \\ : li V IT 1 \ v -jt / ' V ^ . c f f ^ ^ A_/* ftp Ji tl ;V > \ / ' ( . \ . -1' / '

y

26 27 28 T i m e [s] 29 30 — I V I e a s u r e d

— Recalculate using [co, 3co, 5co] — Recalculated using [co]

- o - R e m a i n d e r [3co, 5CD]

R.V. Veer et al./Applied Ocean Research 53 (2015) 1-14 7

Table 5

Flat plat load coefficients, series A l , oscillation period 2.4s.

Draft KC CD _B3 CM A3 As T364 1.38 10.26 - 0 . 7 3 0.52 1.30 - 1 . 5 0 0.11 5.45 5.67 - 1 . 8 5 - 0 . 0 6 2.78 - 1 . 4 8 0.42 8.16 4.35 - 1 . 5 9 0.14 3.19 - 0 . 7 6 0.43 10.86 3.68 - 1 . 3 9 0.22 3.47 - 0 . 3 9 0.43 13.59 3.28 -1.21 0.29 3.68 - 0 . 1 9 0.39 16.31 3.03 - 1 . 0 8 0.37 3.89 - 0 . 0 4 0.33 T278 1.39 10.35 - 0 . 6 7 0.56 1.28 - 1 . 3 4 0.07 5.45 5.71 -1.91 - 0 . 0 8 . 2.81 - 1 . 5 0 0.43 8.14 4.41 - 1 . 6 3 0.11 3.22 - 0 . 7 9 0.45 10.85 3.70 - 1 . 3 9 0.24 3.47 - 0 . 3 7 0.42 13.59 3.33 - 1 . 2 3 0.30 3.68 - 0 . 1 9 0.40 16.32 3.06 - 1 . 0 9 0.36 3.91 - 0 . 0 5 0.34 T148 1.39 9.59 - 0 . 8 4 0.45 1.19 -1.03 0.01 5.46 5.94 - 1 . 6 4 - 0 . 0 4 2.50 - 1 . 3 2 0.40 8.17 4.64 - 1 . 3 9 - 0 . 0 7 2.94 -0.91 0.38 10.87 3.90 - 1 . 3 8 0.11 3.21 - 0 . 4 9 0.41 13.58 3.45 - 1 . 2 4 0.22 3.37 - 0 . 3 Ö 0.38 16.34 3.05 - 0 . 8 4 0.08 3.48 - 0 . 2 9 0.27

3.2. Derivation ofthe hull pressure coefficients; Cp

A n o n - d i m e n s i o n a l p r e s s u r e c o e f f i c i e n t Cp is o b t a i n e d f r o m the pressure v a l u e at the m o m e n t w h e n the r o l l a n g u l a r v e l o c i t y is m a x i m u m ( o r r o l l angle is z e r o ) u s i n g :

p ± p± 2P"ref

( 1 3 )

jP(rco<pA)

w h e r e r, co, cj)f, r e p r e s e n t t h e distance f r o m t h e r o l l axis t o t h e bilge keel, t h e r o l l c i r c u l a r f r e q u e n c y a n d t h e r o l l a m p l i t u d e , r e s p e c t i v e l y . I n t h e e x p e r i m e n t s p e r f o r m e d , t h e m o d e l is o s c i l l a t e d p a r a l l e l t o t h e s u r f a c e . The h y d r o s t a t i c p r e s s u r e v a r i a t i o n is r e m o v e d f r o m t h e m e a s u r e m e n t s b y k n o w i n g t h e d e p t h o f each sensor at every s a m p l i n g , a n d a h u l l - v o r t e x - p r e s s u r e r e m a i n s .

4. Bilge Iceel normal force coefficients

4.J. Flat plate bilge keel

4.1.1. Influence of draft; series Al

A l l l o a d c o e f f i c i e n t s o f Eq. ( 6 ) f o r t h e f l a t p l a t e b i l g e keel i n test series A l are p r e s e n t e d i n T a b l e 5. I t concerns t h e test series f o r t h r e e d i f f e r e n t d r a f t s . A l l e x p e r i m e n t s are c o n d u c t e d at o s c i l l a t i o n p e r i o d 2.4 s. I n Fig. 8 t h e first h a r m o n i c d r a g a n d i n e r t i a c o e f f i c i e n t are s h o w n f o r t h e d e e p e s t d r a f t c o n d i t i o n . The figure i n c l u d e s as w e l l t h e data f o r t h e c o m p l e x b i l g e k e e l t h a t w i l l be discussed i n S e c t i o n 4.2.

As o b s e r v e d i n Fig. 8, the first h a r m o n i c c o e f f i c i e n t s c o m p a r e w e l l t o t h e e x p e r i m e n t a l r e s u l t s o b t a i n e d b y Sarpkaya a n d O'Keefe [ 1 2 ] . B o t h t h e drag a n d i n e r t i a c o e f f i c i e n t s are s l i g h t l y l o w e r t h a n i n t h e Sarpkaya a n d O'Keefe [ 1 2 ] e x p e r i m e n t , b u t t h e d i f f e r e n c e s are s m a l l . The c o e f f i c i e n t s r e p o r t e d b y K e u l e g a n a n d Carpenter [ 10] f o r t h e f r e e plates are g i v e n as w e l l . For a f r e e plate t h e d r a g c o e f f i c i e n t s are f o u n d i n a g r e e m e n t w i t h t h o s e f o r t h e w a l l b o u n d e d plates o v e r t h e c o m p l e t e KC range. B u t t h e i n e r t i a c o e f f i c i e n t s f o r t h e free a n d w a l l b o u n d e d plates are at h i g h e r KC n u m b e r d i s t i n c t l y d i f f e r -ent. The e f f e c t o f t h e w a l l is t h u s m a i n l y f e l t t h r o u g h t h e i n e r t i a c o e f f i c i e n t . The h u l l c u r v a t u r e m i g h t have a n e f f e c t o n t h e i n e r t i a c o e f f i c i e n t s t h r o u g h the i n f l u e n c e o f t h e d e v e l o p i n g v o r t e x sys-t e m a n d isys-ts i n sys-t e r a c sys-t i o n w i sys-t h sys-t h e h u l l , a n d m a r g i n a l l y sys-t h r o u g h sys-the r e f e r e n c e v o l u m e s u r r o u n d i n g t h e b i l g e k e e l .

The i n f l u e n c e o f t h e d r a f t a n d t h u s o f t h e f r e e surface o n t h e load c o e f f i c i e n t s can be s t u d i e d t h r o u g h t h e r e s u l t s i n Table 5. The drag a n d i n e r t i a c o e f f i c i e n t s f o r the t w o deepest d r a f t s ( T 3 6 4 a n d T 2 7 8 ) are w i t h i n 1.5% a n d can t h u s be c o n s i d e r e d to be e q u a l . B u t at the

20

18 16 14

12

10

• S a r p k a y a (1996) o K e u l e g a n (1958) B — Flat bilge keel, T 3 6 4 — A — Complex bilge keel, T 3 6 4

10

KG=U^T/2h [-]

15

₂₀

• S a r p k a y a (1996) o K e u l e g a n (1958) - a — Flat bilge keel, T 3 6 4

A — C o m p l e x bilge keel, T 3 6 4

KG=U^T/2h [-]

Fig. 8. Flat plate and complex bilge keel drag and inertia coefficients (series A l and B I ) . l o w e s t d r a f t ( T 1 4 8 ) t h e r e is a c o n s i s t e n t r e d u c t i o n o f t h e i n e r t i a c o e f f i c i e n t b y a b o u t 9% a n d a n increase i n t h e d r a g c o e f f i c i e n t b y a b o u t 5%. The l o w e r i n e r t i a l o a d m i g h t be r e l a t e d t o t h e f a c t t h a t a b i l g e k e e l i n e r t i a l o a d p e a k occurs w h e n t h e b i l g e k e e l is i n i t s e x t r e m e m o t i o n p o s i d o n ; t h u s i n its closest ( a n d f u r t h e s t ) p o i n t t o t h e f r e e s u r f a c e . A clear p h y s i c a l e x p l a n a t i o n f o r t h e i n c r e a s e d d r a g c o e f f i c i e n t s is n o t y e t f o u n d ; a n d p a r t o f t h e d i f f e r e n c e lies i n t h e m e a s u r e m e n t accuracy.

The h i g h e r h a r m o n i c l o a d c o e f f i c i e n t s (/I3, A 5 , B'j a n d B^) seen i n Table 5 are n o t r e p o r t e d i n t h e U - t u n n e l e x p e r i m e n t s b y Sarpkaya a n d O'Keefe [ 1 2 ] , K e u l e g a n a n d C a r p e n t e r [ 1 0 ] d o p r e s e n t h i g h e r h a r m o n i c c o e f f i c i e n t s f o r t h e f r e e p l a t e b u t these v a l u e s are f o u n d t o be q u i t e d i f f e r e n t f r o m t h o s e o b t a i n e d i n t h e p r e s e n t e x p e r i m e n t . B o t h t h e 3a) d r a g a n d i n e r t i a t e r m s are l a r g e r b y a b o u t a f a c t o r 3 c o m p a r e d t o t h e values r e p o r t e d f o r the f r e e plates b y K e u l e g a n a n d C a r p e n t e r [ 1 0 ] , w h i c h suggests t h a t t h e h i g h e r h a r m o n i c s are m o r e p r o n o u n c e d f o r w a l l b o u n d e d plates t h a n f o r f r e e plates.

The s c a t t e r i n t h e data p o i n t s o f Fig, 9 is due t o t h e f a c t t h a t t h e h i g h e r h a r m o n i c c o e f f i c i e n t s o f a l l e x p e r i m e n t s (series A l t o B2) are i n c l u d e d . D e s p i t e t h i s , t h e scatter r e m a i n s r a t h e r l i m i t e d - f o r m o s t KC values - so t h a t i t c a n be c o n c l u d e d t h a t t h e t r e n d f o r a l l

8 R.V. Veer et al/Applied Ocean Researcli 53 (2015) 1-14 in •3 - 3 • ' • 1 j • • • • 1 • . 1 • 1 , . - n , 1 • • j • 1 • B3' * : : o B5' . © • •

8 H . S . . ^ ^ ' i O

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—i—1 •—1 1 1 •—1 1 1 l _ l 1_1 • • • 1 1 I 1 I 1_1 1 1 1_J 10 KC=U^T/2h I 15 20

i 0

• A3 • A5 • • • 3

-. i -. -. i

0'

I

! .

0,

10 KC=U^T/2h [-] 15 20

Fig. 9. The higher harmonic drag (B3, B'^) and inertia (.43, A5) coefficients (series A l - 2 , B 1 - 2 ) . h i g l i e r h a r m o n i c c o e f f i c i e n t s is s i m i l a r a n d u n i q u e l y d e f i n e d b y t h e KC n u m b e r . F u r t h e r m o r e , t h e results s h o w t h a t the 3 r d h a r m o n i c d r a g c o e f f i c i e n t s are s i g n i f i c a n t l y s m a l l e r t h a n t h e 1st h a r m o n i c c o e f f i c i e n t s , a n d t h a t t h e 5 t h h a r m o n i c c o e f f i c i e n t s are a g a i n s m a l l e r t h a n t h e 3 r d h a r m o n i c values. Hence, t h e i n f l u e n c e o f h i g h e r h a r m o n i c d r a g c o n t r i b u t i o n s is c o n s i s t e n t l y decreasing; l i k e w i s e i t is o b s e r v e d i n t h e f r e e p l a t e e x p e r i m e n t s b y K e u l e g a n a n d C a r p e n t e r [ 1 0 ] . W i t h i n c r e a s i n g KC n u m b e r t h e h i g h e r har-m o n i c d r a g c o har-m p o n e n t s b e c o har-m e s har-m a l l e r i n har-m a g n i t u d e , a n d har-m o s t l i k e l y t h e y t r e n d t o zero. The same is o b s e r v e d f o r t h e h i g h e r h a r m o n i c i n e r t i a c o n t r i b u t i o n s t h a t s h o w a n even clearer t r e n d t o w a r d s z e r o f o r h i g h KC n u m b e r . But, t h e test range is t o o l i m i t e d t o d e f i n i t e l y p r o o f t h e above t h r o u g h m e a s u r e m e n t s .

4.1.2. Influence of oscillation frequency; series A2

In Table 6 t h e results o f test series A 2 are p r e s e n t e d w h i c h c o n -cerns t w o d i f f e r e n t p e r i o d s : 1.72 s a n d 4.0s. A l l these e x p e r i m e n t s are c o n d u c t e d at d r a f t T 2 7 8 . F r o m the results o f series A l o f Table 5,

Table 6

Flat plat load coefficients, series A2, draft T278.

T ( s ) KC CD _^3 CM A3 As 4.0 1.39 10.34 - 0 . 5 9 0.65 1.44 - 0 . 9 8 0.04 5.47 5.53 -1.41 - 0 . 1 9 2.90 - 1 . 5 9 0.20 8.16 4.26 - 1 . 5 6 0.10 3.23 -0.81 0.37 10.87 3.58 - 1 . 4 0 0.22 3.54 - 0 . 4 2 0.42 13.58 3.22 - 1 . 2 3 0.28 3.74 - 0 . 2 6 0.41 16.30 2.97 - 1 . 0 5 0.33 3.87 - 0 . 1 2 0.34 1.73 1.37 10.50 - 0 . 8 8 0.83 1.53 -1.68 0.33 5.40 5.82 - 2 . 1 3 0.11 3.06 -1.44 0.60 8.08 4.46 -1.70 0.27 3.49 - 0 . 6 5 0.46 10.76 3.78 - 1 . 4 2 0.30 3.78 - 0 . 2 8 0.39 13.46 3.40 - 1 . 2 4 0.36 4.03 - 0 . 0 6 0.34 16.15 3.14 -1.11 0.39 4.19 0.07 0.30 a t h i r d o s c i l l a t i o n p e r i o d o f 2.4 s can be i n c l u d e d i n the s e n s i t i v i t y s t u d y . The results can be f o u n d i n Fig. 10.

The r e s u l t s i n d i c a t e t h a t b o t h t h e d r a g a n d i n e r t i a c o e f f i c i e n t s are i n d e p e n d e n t o f t h e o s c i l l a t i o n p e r i o d . The i n e r t i a c o e f f i c i e n t s at Q o KC=U^T/2h [-— 0 [-— Period 1.7s • —B - • Period 2.4s — * — Period 4.0s — 0 — Period 1.7s • —B - • Period 2.4s — * — Period 4.0s

A

. . . . / / /

V / B 10 15 KC=U T/2h [ - ] 20

Fig. 10. Flat plate bilge keel drag and inertia coefficients, oscillation period influ-ence, T278.

R.v. Veer et al./Applied Ocean Research 53 (2015) 1-14 Table 7

Complex bilge l;eel load coefficients, series B I , oscillation period 2.4s.

Table 8

Complex bilge keel load coefficients, series B2, draft T278.

Draft KC CD _{«3 B's} CM A3 As T364 1.38 9.98 0.15 0.17 1.57 - 0 . 4 5 0.15 5.46 5.89 - 1 . 8 7 -0.06 3.03 -1.41 0.61 8.16 4.68 - 1 . 6 4 0.12 3.60 - 0 . 7 9 0.47 10.84 4.11 - 1 . 5 0 0.23 3.94 - 0 . 4 3 0.45 13.61 3.67 - 1 . 2 9 0.27 4.17 - 0 . 2 2 0.46 16.31 3.35 - 1 . 1 6 0.38 4.39 -0.01 0.36 T278 1.39 11.16 - 0 . 9 7 0.53 1.48 -0.98 0.45 5.47 6.09 - 1 . 9 5 -0.07 3.11 - 1 . 5 0 0.54 8.16 4.78 - 1 . 7 4 0.09 3.58 - 0 . 8 7 0.48 10.88 4.15 -1.51 0.24 3.91 - 0 . 4 8 0.45 13.59 3.73 - 1 . 3 3 0.29 4.18 - 0 . 2 4 0.43 16.32 3.38 - 1 . 1 7 0.38 4.41 - 0 . 0 3 0.35 T148 1.39 9.96 - 0 . 6 7 0.27 1.62 - 0 . 6 2 0.29 5.46 6.21 -1.91 - 0 . 0 6 3.06 - 1 . 6 0 0.50 8.17 4.86 - 1 . 7 3 0.12 3.63 - 0 . 8 2 0.53 10.89 4.19 - 1 . 0 9 -0.15 3.92 - 0 . 7 7 0.27 13.58 3.85 - 1 . 4 0 0.29 4.27 - 0 . 2 5 0.44 16.31 3.49 - 1 . 1 3 0.34 4.50 -0.08 0.27 t h e l o w e s t o s c i l l a t i o n p e r i o d are t y p i c a l l y 8% h i g h e r t h a n those at l o n g e r o s c i l l a t i o n periods, b u t i t r e m a i n s d i f f i c u l t t o e x p l a i n t h i s d i f -f e r e n c e a p a r t -f r o m possible m e a s u r e m e n t inaccuracies o r de-fects. The mass o f t h e b i l g e keel is o n l y a b o u t 0.5 k g a n d i t is n o t e d t h a t t h e b o d y - f i x e d r o t a t i o n a l accelerations d o m i n a t e o v e r the g r a v i t a t i o n a l a c c e l e r a t i o n at l o w o s c i l l a t i o n p e r i o d , w h i l e at l o n g e r o s c i l l a t i o n p e r i o d t h e g r a v i t a t i o n a l a c c e l e r a t i o n c o m p o n e n t starts t o b e c o m e m o r e s i g n i f i c a n t . For a l l t h r e e o s c i l l a t i o n p e r i o d s t h e d r a g c o e f f i c i e n t s are w i t h i n 2.7% a n d f o r t h e t w o l o n g e r o s c i l l a t i o n p e r i o d s t h e i n e r t i a c o e f f i c i e n t s are w i t h i n 2.6%; h e n c e these can a l l be c o n s i d e r e d t o be e q u a l a n d t h e c o e f f i c i e n t s are a s s u m e d t o be Reynolds i n d e p e n -d e n t - as e x p e c t e -d f o r a b l u f f b o -d y w i t h a clear flow s e p a r a t i o n edge.

4.2. Complex bilge keel

4.2.1. Influence of draft; series BI

As i n t h e p r e v i o u s s e c t i o n f o r t h e flat b i l g e k e e l , a l l l o a d c o e f f i c i e n t s o f Eq. ( 6 ) f o r t h e c o m p l e x b i l g e k e e l u n d e r t h r e e d i f -f e r e n t d r a -f t s are p r e s e n t e d i n Table 7.

I n Fig. 8 t h e first h a r m o n i c d r a g a n d i n e r t i a c o e f f i c i e n t are s h o w n f o r t h e deepest d r a f t c o n d i t i o n . The h i g h e r h a r m o n i c c o e f f i c i e n t s can be f o u n d i n Fig. 9. I t is o b s e r v e d t h a t t h e b i l g e k e e l g e o m e t r y has n o s i g n i f i c a n t e f f e c t o n t h e Co c o e f f i c i e n t s b u t t h e CM v a l u e s o f t h e c o m p l e x b i l g e keel are c o n s i s t e n t i y a b o u t 13% h i g h e r t h a n t h o s e f o r t h e flat p l a t e b i l g e keel. This d i f f e r e n c e can be e x p l a i n e d b y t h e c h o i c e o f t h e i n e r t i a r e f e r e n c e area ( v o l u m e ) i n Eq. 4. For t h e flat p l a t e the s u r r o u n d i n g h a l f - c i r c l e w a s t a k e n , b u t this w i l l i n t e r s e c t t h e t o p - p l a t e o f t h e c o m p l e x b i l g e k e e l . W h e n t h e i n e r t i a r e f e r e n c e area f o r t h e c o m p l e x b i l g e k e e l is t a k e n as t h e h a l f - c i r c l e p l u s t h e r e c t a n g l e w i t h d i m e n s i o n s t o p - p l a t e w i d t h times b i l g e keel h e i g h t t h e r a t i o o f t h e t w o i n e r t i a r e f e r e n c e areas ( v o l u m e s ) is f o u n d t o be ^fiat/^compiex = 1.114 a n d hence the i n e r t i a c o e f f i c i e n t s w o u l d be w i t h i n 2% f o r b o t h b i l g e keels.

S i m i l a r as f o r t h e flat p l a t e keel, t h e d r a f t i n f l u e n c e o n the d r a g c o e f f i c i e n t s f o r t h e c o m p l e x b i l g e k e e l is n e g l i g i b l e . This h o l d s as w e l l f o r t h e i n e r t i a c o e f f i c i e n t s w h i l e t h e flat p l a t e b i l g e keel i n e r -t i a c o e f f i c i e n -t s r e d u c e d b y a b o u -t 9%. W i -t h d e c r e a s i n g d r a f -t b o -t h t h e d r a g a n d i n e r t i a c o e f f i c i e n t s o f t h e c o m p l e x keel r e m a i n w i t h i n a b o u t 3% o f t h e deep d r a f t values. O n l y at t h e l o w e s t KC n u m b e r the d i f f e r e n c e s are larger, b u t t h i s m i g h t be a t t r i b u t e d to m e a s u r e m e n t inaccuracies associated w i t h t h e l o w f o r c e l e v e l . r ( s ) KC CD _^3 CM A3 As 4.0 0.36 13.72 2.46 -0.18 1.41 3.60 2.89 0.70 11.05 0.12 1.04 1.48 0.64 0.97 1.39 10.64 - 0 . 7 7 0.74 1.70 - 0 . 8 0 0.37 2.75 9.18 -1.51 0.05 2.18 - 1 . 7 0 - 0 . 1 0 4.12 7.52 - 1 . 5 7 - 0 . 2 9 2.68 - 2 . 2 9 0.20 5.46 5.93 - 1 . 8 8 0.02 3.10 - 1 . 3 4 0.47 6.82 5.24 - 1 . 6 9 0.01 3.54 - 1 . 1 0 0.45 9.52 4.42 - 1 . 6 4 0.17 3.82 - 0 . 6 4 0.49 9.52 4.42 - 1 . 6 4 0.17 3.82 - 0 . 6 4 0.49 10.85 3.90 -1.51 0.18 4.05 - 0 . 4 5 0.51 3.0 0.36 12.92 0.66 0.44 1.37 3.54 1.61 0.70 10.81 - 0 . 0 4 0.82 1.45 0.34 0.82 1.39 10.01 - 1 . 0 4 0.61 1.60 -1.11 0.13 2.75 9.12 - 1 . 4 5 0.19 2.12 - 1 . 6 0 0.08 4.11 7.24 - 1 . 6 6 - 0 . 3 5 2.71 - 2 . 2 2 0.22 5.46 5.92 -2.61 0.63 3.17 0.26 - 0 . 2 4 6.81 5.16 - 1 . 7 4 0.08 3.52 -1.01 0.46 9.54 4.46 - 1 . 6 4 0.22 3.81 - 0 . 5 9 0.47 9.54 4.46 - 1 . 6 4 0.22 3.81 - 0 . 5 9 0.47 10.90 4.00 - 1 . 5 0 0.17 3.93 - 0 . 5 0 0.46 2.4 0.36 9.86 0.53 - 0 . 1 3 1.37 3.45 0.43 0.70 10.55 - 0 . 2 7 0.71 1.42 0.34 0.96 1.39 11.16 - 0 . 9 7 0.53 1.48 - 0 . 9 8 0.45 2.74 9.08 - 1 . 3 2 - 0 . 0 0 2.05 -1.71 0.13 4.11 7.28 - 1 . 7 5 - 0 . 3 4 2.71 -2.21 0.21 5.47 6.09 - 1 . 9 5 - 0 . 0 7 3.11 - 1 . 5 0 0.54 6.81 5.21 -1.71 0.08 3.40 -1.01 0.44 8.16 4.78 - 1 . 7 4 0.09 3.58 - 0 . 8 7 0.48 9.52 4.51 - 1 . 6 4 0.18 3.82 - 0 . 6 5 0.49 10.88 4.15 -1.51 0.24 3.91 - 0 . 4 8 0.45 2.0 0.35 10.79 0.20 - 0 . 4 0 1.33 3.61 1.31 0.69 10.32 - 0 . 1 9 1.09 1.40 0.25 1.29 1.38 9.48 -0.98 0.38 1.54 - 1 . 4 4 0.29 2.73 8.77 - 1 . 4 8 0.01 2.09 - 1 . 6 6 0.25 4.10 7.29 - 1 . 5 0 - 0 . 2 9 2.71 - 2 . 3 7 0.07 5.43 6.08 - 1 . 8 2 - 0 . 0 4 3.09 - 1 . 5 2 0.47 6.79 5.30 - 1 . 6 5 0.08 3.37 - 1 . 0 6 0.46 8.13 4.81 - 1 . 6 2 0.10 3.60 - 0 . 8 7 0.45 9.49 4.41 - 1 . 5 6 0.12 3.77 - 0 . 7 0 0.45 10.85 4.16 - 1 . 4 7 0.18 3.92 -0.51 0.45 1.73 0.35 9.19 - 0 . 2 0 - 1 . 1 2 1.33 3.00 1.59 0.68 9.75 0.14 0.63 1.39 0.25 1.38 1.37 9.40 -0.71 0.24 1.54 - 1 . 3 0 0.42 2.71 8.89 - 1 . 5 8 0.07 2.05 - 1 . 7 4 0.25 4.07 7.49 - 1 . 5 5 - 0 . 2 4 2.69 - 2 . 3 7 0.15 5.41 6.16 - 1 . 7 6 0.01 3.06 - 1 . 4 5 0.53 6.74 5.36 - 1 . 7 0 0.14 3.35 - 1 . 0 2 0.48 8.11 4.86 -1.51 0.03 3.57 - 0 . 9 0 0.41 9.45 4.48 - 1 . 5 5 0.17 3.76 -0.61 0.44 10.77 4.19 - 1 . 5 0 0.22 3.94 - 0 . 4 0 0.43

4.2.2. Influence of oscillation frequency; series B2

The l o a d c o e f f i c i e n t s f o r t h e c o m p l e x b i l g e k e e l f o r d i f f e r -e n t o s c i l l a t i o n p -e r i o d s at d r a f t T 2 7 8 ar-e s u m m a r i z -e d i n Tabl-e 8. The ( f i r s t h a r m o n i c ) d r a g a n d i n e r t i a c o e f f i c i e n t s are p r e s e n t e d i n Fig. 1 1 . The s a m e c o n c l u s i o n s h o l d f o r t h e c o m p l e x b i l g e keel as f o r t h e flat p l a t e : t h e first h a r m o n i c d r a g a n d i n e r t i a c o e f f i c i e n t s are n o t e f f e c t e d w h i l e the i n e r t i a c o e f f i c i e n t c o n s i s t e n t l y decreases w i t h i n c r e a s i n g o s c i l l a t i o n p e r i o d . The h i g h e r h a r m o n i c c o e f f i c i e n t s ( d r a g a n d i n e r t i a ) are v e r y s i m i l a r f o r a l l o s c i l l a t i o n p e r i o d s as w e l l .

5. Bilge keel loads in irregular motion

The l o a d c o e f f i c i e n t s as o b t a i n e d f r o m t h e h a r m o n i c o s c i l l a t i o n tests are u t i l i z e d t o c a l c u l a t e the b i l g e keel n o r m a l f o r c e i n i r r e g u l a r m o t i o n t i m e t r a c e 1-4 (see Table 4 ) . F r o m t h e m e a s u r e d r o l l s i g n a l i n t h e i r r e g u l a r m o t i o n t h e v e l o c i t y a n d t h e a c c e l e r a t i o n at t h e l o c a -tion o f t h e b i l g e keel r o o t p o i n t is c a l c u l a t e d . For e a c h q u a r t e r o f a n o s c i l l a t i o n p e r i o d ( t i m e b e t w e e n a zero c r o s s i n g a n d a p e a k v a l u e )