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ELSEVIER

Marine Structures 9 (1996) 545-575

1996 Elsevier Science Limited

Printed in Great Britain. All rights reserved

0951-8339/96/$15.00

0 9 5 1 - 8 3 3 9 ( 9 5 ) 0 0 0 1 1 - 9

H y d r o d y n a m i c D e s i g n o f M o o r e d F l o a t in g P l a t f o rm s

O guz Yilmaz & Atilla Incecik

Uniw~rsity of G lasgow, Departmen t of Na val Architecture and Ocean Engineering,

Hydrodynamics Laboratory, A cre Road, G lasgow, U K

(Receive d 8 M ay 1994; revised 23 Octo ber 1994;

accepted 1 December 1994)

A B S T R A C T

Sing le po in t mo ored f loa t ing product ion p la t forms prov ide an economical ly

viable opt ion fo r deep water m arginal f ie ld o i l and gas product ion. In the

design o f such systems, non-l inear t ime do main analysis tools are required

to pred ic t the w ave and low f requen cy mo t ions and the moor ing force s due

to non-collinear wave, w ind an d curren t loa d actions. The au thors o f this

paper have deve loped and va l ida ted wi th exper ime n ta l measurem ents

n o n

l i n e a r analysis tools to pred ict the dyna mic m otion response and moo ring

forc es o f a C A L M sys tem due to non-co llinear env ironmenta l forces . In the

f i r s t pa r t o f the paper a b r ie f sum ma ry o f the non-l inear ana lysi s procedure

developed by the authors is g iven, together with som e resul ts obtained fr om

pred ic tions and exper im en ta l measurements . In the second par t o f the paper

the resul ts o f para me tr ic s tudies invest igating the ef fects o f variat ions in

wave, w ind and current mag nitude a nd direct ion, wave and w ind spectral

shapes, the numbe r o f mo oring l ines , hawser length and s t if fness , b uoy s ize

and' thruster cap acity on the steady an d slow ly varying oscillations o f the

C A L M sys tem and on m oor ing and hawser force s wi ll be i llust ra ted.

Key words. s ing le -p o in t m o o re d s y s tem, time d o m a in s imulat ion, s lowly

v a ry in g o s ci ll a tio ns , d y n am ic win d an d cu r r en t f o rce s , p a r am e t r i c s tu d y o f

m o t i o n r e s po n s es an d h aw s e r t en s io n o f C A L M s y stem, m o o r in g fo rce s .

1 I N T R O D U C T I O N

A la rg e, n u m b e r o f S in g le P o i n t M o o r i n g ( S P M ) s y st e m s h a v e b e e n i n s ta l le d

i n v a r io u s p a r t s o f th e w o r l d o v e r t h e p a s t 3 0 y e a rs . A s N o r t h S ea o i l

545

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546 o . Y i lm a z , A . I n c e c i k

p r o d u c t i o n m o v e s t o w a r d s a g r e a te r d e p e n d e n c e o n s m a l le r re s er v o ir s , n o t

o n l y i n c o m p a r a t i v e l y s h a l l o w w a t e r o n t h e c o n t i n e n t a l s h e l f , b u t a l s o i n

d e e p e r w a t e r o f f it , th e r e w i l l b e a c o r r e s p o n d i n g l y g r e a t e r r o l e f o r f l o a t i n g

p r o d u c t i o n f ac il it ie s . A n e x a m p l e o f s u c h a f a ci li ty i s a l a r g e t a n k e r m o o r e d

t o a s i n g l e p o i n t . A s i n g l e p o i n t m o o r e d t a n k e r w e a t h e r v a n e s a c c o r d i n g t o

t h e p r ev a i li n g w e a t h e r co n d i t i o n s , th u s s t ay i n g o n l o c a t io n w i t h a m i n i m u m

o f m o o r i n g l o a d s . S i n g l e p o i n t m o o r i n g s y s t e m s h a v e b e e n i n s t a l l e d i n

v a r i o u s p a rt s o f t h e w o r l d a n d , d e p e n d i n g o n t h e w e a t h e r c o n d i ti o n s , t h e y

v a r y f r o m c h a i n / t u r r e t s y s t e m s t o r i g i d - a r t i c u l a t e d s y s t e m s a n d h y b r i d - t y p e

s t r u c tu r e s . E c o n o m i c v i a b il i ty is o n e r e a s o n f o r t h i s te n d e n c y t o w a r d s S P M

s y s te m s a s t h e y h a v e b e c o m e a l te r n a t i v e s t o f k x e d p l a t f o r m s a n d s u b s e a

p i pe li n es fo r t r a n s p o r t a t i o n o f o il a n d g a s w h i c h b e c o m e s a n i m p o r t a n t p a r t

o f t h e o i l - f i e l d d e v e l o p m e n t a s o f f s h o r e p r o d u c t i o n a c t i v i t i e s m o v e i n t o

d e e p e r w a t e r s . A n o t h e r n o t i c e a b l e d i s t i n c t i o n o f s u c h s y s t e m s i s t h a t t h e y

c a n e n d u r e s e v e r e s e a a n d w e a t h e r c o n d i t i o n s . A s a r e s u l t t h e y e x p e r i e n c e

n u m e r o u s c o m b i n a t i o n s o f w a v e , w i n d a n d c u r r e n t . T h e r e fo r e , d y n a m i c

ana lys i s o f such sys tems i s e s sen t ia l to ensure sa t i s fac to ry ove ra l l pe r fo r -

m a n c e o f t h e s e s y st e m s .

I n t h i s p a p e r t h e r e s u l ts o f a s er ie s o f p a r a m e t r i c s t u d i e s a r e p r e s e n t e d t o

i l lu s t r a te t h e e f f ec t s o f e n v i r o n m e n t a l a n d g e o m e t r i c a l c h a r a c t e ri s t ic s o n t h e

d y n a m i c r e s p o n s e a n d m o o r i n g f o r c e s o f t h e t a n k e r - b u o y s y s t e m . T h e

p a r a m e t r i c s t u d i e s w e r e c a r r i e d o u t c o n s i d e r i n g t h e t a n k e r - b u o y s y s t e m

d e s c r i b e d i n F i g . 1 . D u r i n g t h e p a r a m e t r i c s t u d i e s t h e e l a s t i c i t y o f t h e

m o o r i n g l i n e s a n d t h e h a w s e r l i n e , t h e b u o y ' s g e o m e t r y , t h e s e a s t a t e , t h e

w i n d s p e c t r u m , t h e n u m b e r o f m o o r i n g l in e s o f t h e b u o y , t h e h a w s e r l e n g t h

a n d t h e t h r u s t e r c a p a c i t y w e r e v a r i e d t o s t u d y t h e e f f e c t o f v a r i a t i o n s o n

d y n a m i c r e sp o n s e a n d m o o r i n g f o r c es o f th e s y s te m . N u m e r i c a l as p ec ts o f

t h e p r o g r a m , s u c h a s s i m u l a t i o n t im e a n d i n t e g r a t i o n s t e p , a r e d i sc u s se d .

i o n

80mL.J~

Hawser

Buoy

Tanker

~ 7 50v

7

Fig. 1. Co upled tanker-buoy system.

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547

2 B A C K G R O U N D T O T H E P R E D I C T I O N M E T H O D S U S E D IN

T H E S T U D Y

E n v i r o n m e n t a l f o rc e s a c ti n g o n t h e s y s t e m c o n s i s t o f s lo w l y v a r y i n g w a v e

f o r c e s , d y n a m i c w i n d f o r c e s a n d c u r r e n t a n d i d e a l f l u i d f o r c e s . S l o w l y

v a r y i n g w a v e f o r c e s a r e c a l c u l a t e d u s i n g t h e m e a n d r i f t f o r c e s i n r e g u l a r

w a v e s a n d a p p l y i n g a n e x p o n e n t i a l d i s t ri b u t i o n o f t h e w a v e f o rc e re c o r d

i n i r r e g u l a r w a v e s ( N i e n h u i s l ) . M e a n d r i f t f o r c e s w e r e o b t a i n e d u s i n g a 3 -

d i m e n s i o n a l p r o g r a m w r i t t e n b y C h a n . 2

2 . 1 S l c )w l y va ry i n g a n d m ea n w a ve d r i f t f o rce s i n i r reg u la r w a v es

I n i r r e g u l a r s e a s , d r i f t f o r c e s a r e t i m e d e p e n d e n t . T h e s e l o w f r e q u e n c y

d r i f t f o r c e s a r e s m a l l i n m a g n i t u d e b u t m a y c a u s e l a r g e , l o w f r e q u e n c y

o s c i l l a t i o n s o f t h e s i n g l e p o i n t m o o r e d v e s s e l i f t h e v e s s e l ' s n a t u r a l

f r e q u e n c y is e x c i te d .

I n i r r e g u l a r w a v e s , t h e w a v e e l e v a t i o n o n a p o i n t i s w r i t t e n a s

N

~ (t ) ---- E ~i COS ((-D t -~- 8i) (1)

i = l

T h e d r i ft fo r c e is re l a te d t o t h e s q u a r e o f t h e w a v e a m p l i t u d e a n d t h e

s q u a r e o f t h e w a v e e n v e l o p e is

N N 1

~2(t) = Z Z 2

~ i ~ j c s ( ~ i t -[ -

8i) co s( Oy t -]- 8)) (2)

i = 1 j = l

a n d t h e l o w f r e q u e n c y s e c o n d o r d e r w a v e d r i ft f o r c e is w r i tt e n a s f o l lo w s

N N

F (2)

(t) = Z Z ~i~jeijeos{(o~i- o~j)t +

( 8 i - e j ) }

i = 1 j = l

N N 1

+ E Z 2 ~iCjQ ijsin{(~oi-oj)t + ( ~ i - -

gJ )} (3)

i=1 j=l

P a n d Q r e p r e s e n t s y m m e t r i c a n d a s y m m e t r i c m a t r i c e s re s p ec ti v el y :

Pmn = P,~ , Qmn = -Q nm

(4)

I f S~ is t h e w a v e s p e c t r u m t h e n , a c c o r d i n g t o P i n k s t e r , 3 t h e s e c o n d o r d e r

f o r ce s p e c t r u m is

SF(Oj1) ~- 8f0 S ~ ( ( D ) S ~ ( ( D t - ~ - ( D ) [ F ( 2 ) ( ( . D , ( . D - [ - ( . D ) do) (5)

L C ,

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5 4 8

O. Yilmaz, A. Incecik

w he re F(2)(co , co + co ') = v /P /2 j + Q2 j an d the m e an w av e d r i f t fo rce i s

w h e r e F (2 /(co, co) i s th e m e an w av e d r i f t f o r ce in r eg u la r w av es .

A n a p p r o x i m a t e m e t h o d is s u g g e st e d b y N e w m a n 4 a n d P i n k s t e r, 3 i n

w h i c h t h e l o w f r e q u e n c y f o r c e s a r e d e r i v e d f r o m m e a n d r i f t f o r c e s i n

r e g u l a r w a v e s . T h i s m e t h o d c a n b e u s e d o n l y w h e n w a v e d i f f r a c t i o n e ff e ct s

a r e d o m i n a n t b e c a u se i t d o e s n o t t a k e a c c o u n t o f t h e fo r c es r el a t ed t o t h e

s e c o n d o r d e r h o r i z o n t a l p r e s s u r e g r a d i e n t . A c c o r d i n g t o t h i s m e t h o d ,

p ( c o r n , c o , ) ~ p ( .c o m c on c orn c o n )

( 7 )

2 ' 2

Q (c o rn c o , ,) , ~ 0

A s p e c t ra l f o r m o f t h i s f o r m u l a h a s b e e n d e v i s e d b y P i n k s t e r ,

o t ] 2

S r ( c o ) = 8 f o S ~ ( c o ) S ~ ( c o + c o ) I F ( 2 ) ( c o + 2 )

de) (8)

(o')

w h er e F (2) co + -~- i s m ea n wa v e d r i f t f o r ce in r eg u la r wav es .

A t i m e h i s t o r y o f sl o w l y v a r y i n g w a v e f o r c e s i n ir r e g u l a r w a v e s c o u l d

b e o b t a i n e d b y u s i n g t h e s u m o f si ne s a p p r o a c h w i t h a r a n d o m p h a s e

d i s t r i b u ti o n b u t t h is a p p r o a c h l e ad s t o a G a u s s i a n d i s t r i b u t i o n o f th e

s l o w l y v a r y i n g f o rc e s. P i n k s t e r 5 a r g u e s t h a t a n e x p o n e n t i a l d i s t r i b u t i o n

o f s l o w l y v a r y i n g f o r c e s i s m o r e r e a l i s t i c a n d h e d e v i s e d a m e t h o d t o

g e n e r a t e a n e x p o n e n t i a l l y d i s t r i b u t e d f o r c e r e c o r d . 1'6 A c c o r d i n g t o t h i s

m e t h o d ,

F(x ) (0 ,, t) = - ~,(2) ~,(2)

x A ( o , ) ( A + 1 ) + ( o , )

F ( y 2 ) ( O , , t ) = - F ( ] ( O , ) ( A +

1 ) + _F(2) (0,1 (9t

F ~2) ( 0 , , t ) = _p (2 )_oA O , ) A

s ig n ( rnd (b) - 0 .5) + ~,~2) (0 ,)

w h e r e A = l n [ r n d ( a )]

r n d (a ) , r n d( b ) = u n i f o r m l y d i s t r i b u t e d n u m b e r b e t w e e n 0 a n d 1

A = l n [r n d (a ) ] h a s a n e x p o n e n t i a l d i s t r i b u t i o n w i t h a v e r a g e - 1 a n d

s t a n d a r d d e v i a t i o n 1. T h e i n c l u si o n o f r n d (b ) i n t h e y a w m o m e n t a s s ur e s

t h a t F ~ 2 )

(Or, t )

h a s a s y m m e t r i c a l d i s t r i b u t i o n , w h i c h i s c o u p l e d t o F ( 2 )

(Or)

a n d F ( 2 ) y

(Or)

i n a m p l i t u d e b u t n o t i n p h a s e .

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H ydrodynamic design of moored loating platforms 549

( 2 ) ~ ( 2 )

F ~ ( ~ r ) , F y A ( 0 ,) a r e d e t e r m i n ed u s i n g t h e d e r i v e d s p ec t r a l d e n s i t y

S~) (0) . T he var i a nce o f ~ .(2) (0 ,) i s g iven b y

a x A

( o , ) ) : = E [ ( F 2 ) ( 8 , ) ) 2 ] - e 2 [ f ( 8 , ) ] =

~ , . L x A

( 0 r ) ) 2 ( 1 0 )

~(2)

A s i m i l a r ex p r e s s i o n can b e d e r i v ed f o r . yA ( 8 ,) . T ak i n g a s am p l i n g

f r e q u e n c y o f e v er y V t , t h e m a x i m u m f r e q u e n c y in t h e w a v e d r i ft f o r c e is

rc / V t . A p p l y i n g a r a n d o m w h i t e no i se p ro c e ss a n d a s s u m i n g th a t S (2) is

F,

f r e q u e n c y in d e p e n d e n t d u r i n g t h e V t v a r i a n c e c a n b e w r i t t e n a s 7

tp(2)

(aFt2)(0,))2 = Se~2)(o,o,)n/Vt = , - x a (8,)) 2

( b - , ( 2 )

(O'Fy(2) 0,))2 =

SF(y2)(O,O,)r~/vt= , yA

(0r)) 2

(0 V~2) 0,))2 =

SF~2)(o,0,)~/vt

= 2(F~2) (8,)) 2

(11)

2 . 2 D y a a m i c w i nd f o rc e s

I n s i m u l a t i n g t h e d y n a m i c w i n d f o r ce s , u s e w a s m a d e o f t h e d i f f e re n t w i n d

s p e c tr a a n d w i n d v e l o c i ty t i m e h i s to r y w a s c r e a t e d a p p l y i n g a s u m o f s in e s

a p p r o a c h w i t h a r a n d o m p h a s e d i s t r ib u t i o n ( O o r tm e r s se n S ).

C a l c u l a t i o n o f w i n d f o r ce s i s a d i f f icu l t t a sk . M o s t o f t h e t i m e ex p e r i-

m e n t a l d a t a a n d / o r e m p i r i c a l f o r m u l a s h a v e t o b e u s e d . W i n d i s u s u a l l y

t r e a t e d a s a t i m e i n v a r i a n t e n v i r o n m e n t a l e ff ec t. B u t f l u c t u a t io n s o f t h e

w i n d v e l o c i t y ac t i n g o n t h e s u p e r s t ru c t u r e s m ay h av e a l a r g e e f fec t o n t h e

r e s p o n s e o f t h e o f f s h o r e s t r u c t u r e s . Wi n d v e l o c i t y i s ex p r e s s ed b y t h e

f o l lo w i n g fo r m u l a i n w h i c h w i n d s h e a r is c h a r a c t e r is e d b y a p o w e r l a w

ex p r e s s i o n ,9

V t ( z ~ ) - - o ~ ( ~ 0 ) fl ( 1 2 )

- V l h ( 1 0 )

w h e r e

Vt(z)

is w i n d s p e e d a t z a t a n a v e r a g e d t s e c o n d s

Vlh 00 ) i s w i n d s p e ed a t 1 0 m a t a n a v e r ag e d 1 h o u r

is t h e g u s t f a c t o r ( = l )

fl is t h e p o w e r l a w e x p o n e n t ( = 0 . 1 6 s u g g e s te d b y D a v e n p o r t . 10)

D r a g f o rc e d u e t o w i n d l o a d i n g i s e x p r e s se d b y t h e f o l l o w i n g f o r m u l a ;

1

F w(t) : ~ PaCDAp 2 ( t ) (13)

w he re p~, i s a i r d en s i ty

(=O.O012t /m3) , CD

is drag coefficien~t.

Ap is p r o j ec t i o n a r ea , V ( t ) is t im e d e p e n d e n t w i n d v e l o c it y .

B y w r i ti n g V (t) = V + v(t) , m e a n a n d d y n a m i c w i n d f o rc e s a r e o b t a i n e d

as fo l lows ,

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550

O. Yilmaz, A. Incecik

1

FMw(t) = -~

PafDAp'-V 2

(14)

D

F w (t) = PaCD Ap-Vv(t)

(15)

F l u c t u a t i o n s i n w i n d v e l o c i t y co u l d b e m o d e l l ed b y a s p ec t r u m . T h r ee

of the mos t commonly used spec t r a a re as fo l lows :

The H ar r i s Spec t rum 11 i s descr ibed by

SwO)_ 4x)?V~o

f ( 2 -~ J 2) 5/6 (16)

w h e r e ) ? =

1200 f / - f f l o ; f i s

f r equency ; x i s d rag coef f i c ien t ( =0 .005) .

T h e D av e n p o r t s p ec tr a l f o r m u l a t i o n 12 is g iv en b y

S w ( [ ) = 4xfV~ (17)

f ( 2 + ] 2 ) 4 / 3

Ochi and Sh in 13 sugges ted a spec t r a l fo rm ula t io n based on w ind speed

m eas u r em en t s c a r r ied o u t a t s ea . I t h a s t h e f o l lo wi n g f o r m u l a t i o n

583f ,

42 0~ '7

so t , ) = (1 ~ J ~ , '3 5 ) 1 1 5

8 3 8 f ,

(1 + j ~ , ' 3 5 ) 1 1 5

fo r 0 ~< f , ~< 0.003

for 0.003 ~ 0 .1

(18)

whe re f , i s d imen s ion les s f r equency

f , = f z /-Vz

(19)

S0c, )

i s d imen s ion les s spec t r a l den s i ty

S ( f , ) = f S ( f ) / v 2 , (20)

f i s f requ enc y in cps ; z i s heig ht ab ov e sea level in metres ; Vz is m ea n

win d speed a t h e igh t z in m/see ; SO0 i s spec t r a l dens i ty fu nc t ion in mZ/s; v ,

i s shear veloci ty in m/s .

M ean w i n d s p eed , Vz, and f r i c t ion ve loc i ty , v , , a r e def ined in the

fo l lowing fo rmulas ,

Vz = Vlo + 2-5 v, In (z / lO ) (21)

v, = v/-C~10 V10 (2 2)

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H y d r o d y n a m i c d e si gn o f m o o r e d f l o a t i n g p l a t f o r m s 5 5 1

w h e r e V10 = m e a n w i n d s p e e d a t 1 0 m h e i g h t i n m / s ,

C10 --- su r fa ce d r a g co ef f ic i en t . 13

T i m e d e p e n d e n t w i n d v e l oc i ty is o b t a i n e d b y t h e s u m o f s in es a p p r o a c h

w i t h a r a n d o m p h a s e d i s t r i b u t i o n ,

OG

V ( t ) = V + E V/2 Sw(og .) V w cos (o9 . t + e . ) (23)

n = O

2 .3 Cu rren t a n d i d ea l f l u id fo rces

A s w i t h t h e w i n d f o rc e s , e m p i r i c a l f o r m u l a s h a v e t o b e u s e d i n c a l c u l a t i n g

t h e c u r r e n t f o r c e s. C u r r e n t a n d i d e al f l u i d f o r c e s w e r e e x p r e s s e d u s in g t h e

m e t h o d s o f W i c h e r s 14 a n d M o l i n . 15 I n t h is a p p r o a c h c u r r e n t f o r c e s a n d

m o m e n t s a r e r e p r e s e n t e d as a c o m b i n a t i o n o f t he i d e al f lu i d fo r c es a n d

' re a l ' f o r c es b a s e d o n s e m i - e m p i r i c a l m a t h e m a t i c a l m o d e l s i n c lu d i n g

q u a s i -s t e ad y a n d d y n a m i c c u r r e n t c o m p o n e n t s . I d e a l f l ow f o r c e s a r e g i v en

b y N o r b i n n 16 a s f o l l o w s ,

Fxia = -a xx f t + ayyVO + ayoO 2

Fyia = -a y y f - axxuO - ayoO

(24)

Foid = --aooO -- (ayy -- axx) uv -- ayo ( f + uO)

a n d t h e r e l a ti v e ve l o c i ty c o m p o n e n t s a r e g i v e n a s f o ll o w s

u = ~ - Vc co s (~ - 0)

v = p - Vc s in (a - 0) (25)

T h e r e la t iv e a c c e l e ra t i o n c o m p o n e n t s a r e

= J~ - Vc b si n (ct - O)

(2 6 )

= ~ + E t) cos (~ - 0)

I f e q n s 2 5 a n d 2 6 a r e s u b s t it u t e d i n t o e q n 2 4 , w e o b t a i n

A~xid

:

- axxJC

- (ayy -

axx) V~ si n (zt - O) 0 +

a y y j ; 0 q - a y o 0 2

l~yid :

- - a y y y - - ( a y y - axx) Vc c o s (ct - O) ~) - axx ic 0 - ayo 0 ( 2 7 )

-FOld = - a o o 0 - (ayy - a x x ) u v - aoy x O - aoy Y

A c c o r d i n g t o W i c h e rs , th e M u n k m o m e n t in e q n 27 c a n b e re p l a c ed b y

t h e s te a d y c u r r e n t m o m e n t c o m p o n e n t s a n d t h e e q u a t i o n c a n b e r e w r i t t e n

t o i n c l u d e t h e v i s c o u s f o r c e s a s f o l l o w s ,

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552

O . Y i l m a z , A . I n c e c i k

w h e r e

F xid = --ax x SC + ayy p O + Fxsta 31-F x d y n

F y i d = - - a y y y - - a y 0 0 - - axx Jc O + F y s t a t + F y d yn

FOld =

-aoo 0

- aoy Y + F ostat + F Odyn

(28)

F x s t a =

0 .5

pLs rCxc(~cr) Vc2r

Fys ta t ~ - 0 5 pLs TC y~ (O~cr) V ~r (29 )

Fo~t,,t

= 0 .5

p L 2 TCoc(Ctc,) V2~

F~t,,t

is t h e q u a s i- s te a d y c u r r e n t f o rc e a n d m o m e n t c o m p o n e n t s a c c o r d -

i n g t o t h e r e la t i ve c u r r e n t c o n c e p t ,

w h e r e

C, ,c

i s t h e r e s i s t a n c e c o e f f i c i e n t i n l o n g i t u d i n a l d i r e c t i o n

Cy~ is t h e r e s i s t a n c e c o e f f i c i e n t i n t r a n s v e r s e d i r e c t i o n

Coc

i s t h e r e s i s t a n c e c o e f f i c i e n t i n y a w d i r e c t i o n

c~ = ( u 2 + v 2 ) 5

~ = a r c t a n ( - v / - u )

a n d m o m e n t c o m p o n e n t s a r e e x pr es se d a sy n a m i c c u r r e n t f o r c e

f o l l o w s ,

Fxdyn = --(ayy -- axx) Vc sin (ct - O)/~ + Fxd

Fyayn = -( ay y - axx) Vc co s (~ - O) 0 + Fyd

F O d y n = F o d

(30)

T h e v i s c o us p a r t o f t h e d y n a m i c l o a d c o n t r i b u t i o n r e p re s e n ts t h e e f f ec t s

o f t h e y a w m o t i o n i n t h e r e la t iv e v e lo c i ty fi el d a n d b a s e d o n t h e l o c a l c ro s s

f l o w p r in c i pl e . A c c o r d i n g t o W i c h e r s , t h e v is c o u s p a r t o f t h e d y n a m i c

c u r r e n t l o a d c a n b e a p p r o x i m a t e d a s fo ll ow s ,

Fxd = 0 .5 (ayy -- axx) Vc sin (~ - 0)/~

j

y d = 0.5 p TCyc, (90 ) [(Vc - Or)tVc - O l l - vclvcl]dl

P

Foe

= 0 .5 p T

[Cyc (~

(/)) {(v~ - 0/) 2 - u2} _

Cyc ( ~ ) V~2~] 1 d l

P

(3 1 )

w h e r e

Uc ~ - -U

Vc = - - V

~ c~ (/) = a r c t a n [ ( v , -

Ol)/u~]

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H ydrodynamic design o f m oored loating platforms

553

S u r ge , s w a y a n d y a w m o t i o n s o f t h e t a n k e r a n d s u rg e a n d s w a y m o t i o n s

o f th e b u o y w e r e t a k e n i n t o a c c o u n t i n th e s t u d y . T o f o r m u l a t e t h e m o t i o n

e q u a t i o n s o f th e t a n k e r - b u o y s y st em i n t im e d o m a i n , C u m m i n s ' m e t h o d ]7

w a s u t i l i s e d . C u m m i n s ' m e t h o d u s e s i m p u l s e r e s p o n s e f u n c t i o n s t o d e r i v e

t h e f l u i d r e a c t i o n f o r c e s . I n o r d e r t o s o l v e t h e d i f f e r e n t i a l e q u a t i o n s a n

a l g o r i t h m w r i t t e n b y G e a r ~s i s u s e d . T h e a l g o r i t h m w h i c h is e it h e r a f o r m

o f t h e A d a m s m e t h o d s o r a m e t h o d f o r s ti f f e q u a t i o n s h a s s e v e r al f e a t u re s

s u c h a s t h e a u t o m a t i c s e l e c ti o n o f s t ep s i ze a n d o r d e r f o r t h e m e t h o d u s e d .

I n o r d e r t o a v o i d s h o c k r e s p o n s e o f t h e s y s t e m d u e t o e x t e r n a l fo r c es a n

e x p o n e n t i a l r a m p f u n c t i o n w h i c h e n s u re s t h e g r a d u a l i n c r e a se o f t h e

e x t e r n a l f o r c e f o r a c e r t a i n p e r i o d a t t h e b e g i n n i n g o f t h e s i m u l a t i o n i s

u s e d .

W aw ~- f o r ce s a c t in g o n t h e b u o y m o o r e d t o t h e s e a b o t t o m ( i n d e p e n d e n t

o f t h e t a n k e r ) w e r e ca l c u la t e d u s i n g M o r i s o n ' s e q u a t io n . C a t e n a r y e q u a -

t i o n s w e r e u t i l i s e d t o c a l c u l a t e t h e r e s t o r i n g f o r c e s d u e t o t h e m o o r i n g

l in e s. 19 H y d r o d y n a m i c f o r ce s a c t i n g o n t h e m o o r i n g l in e s w e r e a s s u m e d t o

b e s m a l l a n d t h e r e f o r e t h e s e f o r c e s w e r e n o t i n c o r p o r a t e d i n t h e m o t i o n

e q u a t i o n s .

M o t i o n r e s p o n s e s o f t h e t a n k e r - b u o y s y s t e m a r e c o m p a r e d w i t h h e a d

s e a e x p e r i m e n t a l m e a s u r e m e n t s w h i c h w e r e c a r r i e d o u t a t t h e H y d r o -

d y n a m i c s L a b o r a t o r y o f t h e U n i v e r s i ty o f G l a sg o w . C o m p a r i s o n s s h o w

q u i t e g o o d a g r e e m e n t w i t h t h e e x p e r i m e n t s ( se e F i g s 2 a n d 3 ).

3 P A R A M E T R I C S T U D I E S A N D D I S C U S S I O N O F R E S U L T S

T w o s et s o f p a r a m e t r i c s t u d i e s w e r e c a r r i e d o u t . T h e f i r st o n e i n v e s t i g a te d

t h e e f f e c t s o f d i f f e r e n t w a v e , w i n d a n d c u r r e n t f o r c e m a g n i t u d e s a n d

E

.u.

I - -

Z

I l l

W

W

a

F"

el

0 . 5

0 . 4

0 . 3

0 . 2

0 .1

0 . 0

0 . 4

B M E A S U R E M E N T S

F R E Q U E N C Y D O M A I N S I M U L A T I O N

T I M E D O M A I N S I M U L A T I O N

- [ ]

' ' ' ' ' . ' 6

. 6 0 . 8 1 . 0 1 . 2 1 4 1 .

FREQUENCY ( H z . )

Fig. 2. Surge motion of the buoy co-linear environmental forces.

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554 O. Yilmaz, A. Incecik

. -

O

E

I-.

-r

ill

1-

u,I

U J

a

I-

. - I

o,,

0.6-

0 . 5 '

0.4'

0.3-

0.2-

0.1

0.0

0.4

[] MEASUREMENTS

FREQUENCY DOMAIN SIMULATION

~

DOMAIN SIMULATION

i i i i i i

0.6 0.8 1.0 1.2 1.4 1.6

FREQUENCY (Hz~

Fig. 3. Surge m otion of the tank er co -linear environm ental forces.

d i r e c ti o n s o n t h e s t e a d y a n d o s c il la t o ry m o t i o n s a n d m o o r i n g f o r c es o f t h e

t a n k e r - b u o y s y st em T h e s e c o n d p a r a m e t r i c s t u d y d e t e r m i n e d th e s e n si -

t iv i ty o f s lo w l y v a ry i n g m o t i o n s a n d h a w s e r f o r ce s to c h a n g e s i n w a v e a n d

w i n d s p ec tr a , t h e n u m b e r o f m o o r i n g l in es o f th e b u o y , h a w s e r l e n g t h a n d

t h r u s t e r c a p a c i t y . T h e f i r s t p a r a m e t r i c s t u d y w a s c a r r i e d o u t i n r e g u l a r

w a v e s w i t h s t e a d y w i n d a n d c u r r e n t p r e s e n t a n d t h e s e c o n d o n e i n i r r e -

g u l a r w a v e s w i t h d y n a m i c w i n d a n d c u r r e n t p r e s e n t.

I n t h e f i r st s et o f p a r a m e t r i c s t u d i e s , si x g r o u p s o f s i m u l a t i o n s t u d i e s

w e r e c a r r i e d o u t u s i n g t h e n o n - l i n e a r t i m e d o m a i n s i m u l a t i o n c o m p u t e r

p r o g r a m b a s e d o n t h e p re d i c t i o n m e t h o d d e s cr ib e d in t h e p r e v io u s

s e ct io n s . A t t h e b e g i n n i n g o f e a c h s i m u l a t i o n t h e t a n k e r w a s p l a c e d a l o n g

t h e x a x is w i t h a z e r o y a w a n g l e a n d t h e h a w s e r w a s u n s t r e t c h e d . R e s u l ts

o f th e p a r a m e t r i c s t u d y a r e t a b u l a t e d b y u s i n g t h e s t e a d y a n d o s c i ll a to r y

m o t i o n r e sp o n s e s o f t h e b u o y a n d t h e t an k e r , w h i c h w e r e o b t a in e d

t h r o u g h a F . F . T . a n a ly s is o f th e t im e d o m a i n s i m u l a ti o n s D u r i n g t h e f i rs t

t h r e e g r o u p s o f s t u d i e s th e e f fe c ts o f d i r e c t i o n a l i t y o f w a v e , w i n d a n d

c u r r e n t f o r c e w e r e i n v e s t i g a t e d a n d t h e r e s u l t s o f t h e s e s i m u l a t i o n s a r e

g i v en in T a b l e s 1 -3 . D u r i n g t h e r e m a i n i n g t h r e e s et s o f s i m u l a t i o n s t h e

e ff ec ts o f v a r i a ti o n s i n w a v e , w i n d a n d c u r r e n t f o r ce m a g n i t u d e s w e r e

i n v e s t i g a t e d a n d t h e r e s u l ts o f t h e s e s tu d i e s a r e g i v e n i n T a b l e s 4 - 6 . T h e

r es u lt s g iv e n in T a b l e 1 i n d i c a te t h a t m a x i m u m s t ea d y a n d o s c i ll a to r y

s w a y a n d y a w m o t i o n s o f t h e ta n k e r o c c u r w h e n w a v e a n d c u r r e n t f or ce s

m a k e a 9 0 a n g l e w i t h t h e w i n d f o rc e s. S i m i l ar ly m a x i m u m s w a y m o t i o n s

o f th e b u o y a n d m a x i m u m h a w s er t e n s i o n o cc u r w h e n w a v e a n d c u r re n t

f o r c e s m a k e a 9 0 a n g l e w i t h t h e w i n d f o r c e s ( F i g . 4 ). T h e r e s u l ts g i v e n i n

T a b l e 2 i n d i c a t e t h a t w i n d d i r e c t i o n d o e s n o t a f f e c t t h e m o t i o n s a n d t h e

h a w s e r t e n s i o n s i g n i f i c a n tl y ( F ig . 5 ). I t c o u l d b e c o n c l u d e d f r o m T a b l e 3

t h a t m e a n s w a y d i s p l a c e m e n t a n d y a w a n g l e i n cr e as e a s th e c u r r e n t

d i r e c ti o n c h a n g e s f r o m 0 t o 9 0 . H o w e v e r , t h e m a x i m u m o s c i ll a to r y s w a y

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yd ro d ynam i c desi g n o f moo red l o a ti n g p l a t f o rm s

5 5 5

0

6 6 6 6 6 6 6

6 6 6 6 6 6 6

666666

o

H

0

~

H

0

E

H

0

6

II

~ o

8/10/2019 513 Attila

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556

O . Y i l m a z A . I n c ec i k

0

, , , ~

~d

e~

.~ ~

II

6 6 6 6 6 6 6

~ o o o ~ II

~ ~ ' ~

~ ~

I I

0

6 6 6 6 6 6 6 N

6

I I

~ o

e ~

~ H

6 6 6 I I =

~

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yd ro d ynam i c desi g n o f moo red l o a ti n g p l a t f o rm s

5 5 7

F

~

6 6 6 6 6 6 6

6 6 6 6 6 6 6

6 6 6 6 6 6 6

6 6 ~ ~

~ ~

o

I I

=

0

I I

ID

I=

f l

o

6

I I

6 6 ~ 6 6 6 ~

II =

8/10/2019 513 Attila

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558 O. Yilmaz A. I ncecik

0

~d

6 ~ 6 6

66o6

6

I I

e ~

6 6 6 & I 1 ~

$

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yd ro d ynam i c desi g n o f moo re d l o a ti n g p l a tf o rm s

5 5 9

0

I ~ ~

. ~ ~

~ . . . . _

II

0

~ o o o ~

II

. . . . 0

6 6 6 6

~ 6 II

0

6 6 6 6

6

II

~o

e ~

6 6 6

I 1 ~

~ . ~

" ~

8/10/2019 513 Attila

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5 6 0

O. Yil maz A. ln cecik

o

0

~ -~

~ ~ 0 ~ o ~

6 o 6 6 I I

o

o o o o

II

0 0 0 ~

o o o o

E

II

c,;

0

0

6 ~

I I I I

0

~ ' ~

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H ydrodynamic design o f moored loating platforms 561

I Cunent

Wind

W a v e

20000-

[] MAXTENSION / ~

/

15000- MEANTENSION I

/

5000

0

25 50 75

W a v e Heading IDeg)

Fig. 4. Effect of wave direction.

l

ioo

~4~ ~Current

i

~

-='Wave

W ir~d

~5000 1 [ ]

O

lOOGO

5ooo

0 1 ~ , ,

0 25 50 75

Wind Heading Deg)

Fig. 5. Effect of wind direction.

MAXIMUMTENSION

MEANTENSION

I

1oo

motion of the buoy occurs when wave force direction makes a 0 and wind

and current directions make a 45 angle with the hor izontal axis. Maxi-

mum steady and oscillatory surge motions of the buoy and ship occur

when wave, wind and current forces act co-linearly. The maximum hawser

tension stays relatively low, below 10 000 kN (Fig. 6). Tables 4 and 5 show

that the mean mooring line forces are generally not very sensitive to the

changes in current and wind loading since the dominant load on the

system is due to wave induced oscillatory and steady forces (Figs 7 and 8).

Table 6 shows that there is no linear relationship between the wave height

and the motion response or the mooring force values of the CALM system

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562 O. Yilmaz, A. Incecik

r W i n d

W a v e

90*

Current

1 0 0 0 0

7500 -

5000

2500 2

0

: 5 + o 7 ;

Current Heading i Deg)

F i g . . E f f e c t f c u r r e n t i r ec t io n .

I"1 MAXIMUM TENSION

O MEAN TENSION

|

1 0 0

F i g . 9 ). T h i s i n d i c a t e s t h a t s u c h s y s t e m s m u s t b c a n a l y s e d i n t h c t i m e

d o m a i n u s i n g n o n l i n c a r a n a l y s i s t oo ls .

A s c c o n d s et o f p a r a m e t r i c s t u d i e s w a s c a r r i e d o u t t o d c t c r r n i n c t h c

s cn si ti vi ty f s l o w l y v a r y i n g m o t i o n s a n d h a w s e r f o r c c s t o c h a n g e s i n t h c

c n v i r o n m c n t , t h c n u m b e r o f m o o r i n g l i ne s o f t h e b u o y , h a w s e r l e n g t h a n d

t h r u s t e r c a p a c i t y f o r t h e C A L M s y s t c m i l l us t ra t e d n F i g . I. I n t h e s i m u -

l a t i o n s t h c t a n k e r w a s g i v c n a n i ni ti al - 5 y a w a n g l c w i t h r c s p c c t t o t h c

c u r r e n t a n g l c , a n d t h c b o w h a w s c r w a s u n s t r c t c h e d a n d p a r a l l c l t o t h e

c u r r e n t . D u r i n g t h e s e s i m u l a t i o n s f ir st r d e r w a v e f o r c c s w c r c n e g l e c t e d .

1 0 0 0 0

9 0 = ~ 7500-

W i r ~ 4 + 5

Current ~ 5 ~ -

~ 0 o

w a v e 25oo

t' l MAXIMUM TENSION

0 MEAN TENSION

0

; 1

i~

Current V e l o c i t y

m l s e c )

F i g . 7. E f f e c t o f c u r r e n t v e l o c i t y .

5 ;

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ydrodynamic design of moored loa t in g p la t fo rm s

563

10000-

9

W in d I / c S u * r r e n t

W a v e