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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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H y d ro d y n a m ic d e sig n o f m o o r e d l o a t in g p l a tfo r m s
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 ,
8/10/2019 513 Attila
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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~]
8/10/2019 513 Attila
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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 =
~
8/10/2019 513 Attila
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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