Study of Algorithms of New Slender Ship

8/14/2019 Study of Algorithms of New Slender Ship

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Journal of Marine Science and Application, Vol. 2, No. 2, D ecember 2003

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

H A N D u a n -f e n g , L I Y u n - B o , a n d H U A N G D e -b oCollege of Shipl)uilding Enginee ring, Ha rbin Engineering University, Hat)in 150001 , China

A bstr ac t : In this paper, Noblesse's New Slender-Ship Wave-Making Th eory was investigated numerically. Detailed expressionsof zeroth and 1st order wav e resistance have been derived and calculation programs have also been com piled. In the single anddouble integral terms of Green function, the kernel function of wave resistance expression, special function exp ansion methodand Chebyshev polynomials approach have been adopted respectively, which greatly simplify the calculation and increase theconvergence speed.K e y word s:new slender-ship wave resistance theory; wave resistance; G reen function; potential; algorithmC L C n u m b e r : U 6 6 1 . 1 D o c u m e n t c o d e: A A r t i cl e I D : 1 6 7 1 - 9 4 3 3 ( 2 0 0 3 ) 0 2 - 0 0 5 3 - 0 8

1 I N T R O D U C T I O NT h e t h e o r e t i c a l s y s t e m o f t h e N e w S l e n d e r

S h i p T h e o r y o f W a v e R e s is t an c e w h i c h w a s p r o p -o s e d b y F . N o b l e s s e i n 1 9 8 3 E'? t h e w e l l - k n o w n f o r -m u l a s o f t h i n s h i p , f l a t s h i p , H o g n e r a n d p ri m a ls l e n d e r s h i p , c o r r e s p o n d t o p a r t i c u l a r c a s e s o f t h ez e r o t h o r d e r a p p r o x i m a t i o n r (~ o f i t. T h e o r e t i c a lp r e d i c t i o n s f o r c l a s s i c a l W i g l e y h u l l b a s e d o n t h ef i r s t o r d e r a p p r o x i m a t i o n r (ly i n d i c a t e s t h e a d v a n -t a g e o f N e w S l e n d e r S h i p T h e o r y o f W a v eR e s i s t a n c e ~31 . I t a p p e a r s t h a t t h i s t h e o r y i s c o n v e -n i e n t t o b e a d a p t e d t o c a l c u l a t e w a v e r e s i s t a n c e o fc o m p l e x h u l l s . I n t h i s p a p e r , a f a s t a n d a c c u r a t ea l g o r i t h m i s p r e s e n t e d . I n s u b s e q u e n t p a p e r s , e x -a m p l e s o f c a l c u l a t i o n o f w a v e r e s i s t a n c e o f s e v e r a lt y p e s o f s h i p w i l l b e g i v e n b a s e d o n t h i s a l g o r i t h m .

F o r r e a d e r ' s c o n v e n i e n c e , a b r i e f i n t ro d u c t i o na b o u t N e w S l e n d e r Sh i p T h e o r y o f W a v e R e si s-t a n c e i s p r e s e n t e d f i r s t l y .2 T H E T H E O R Y S Y S T E M O F T H E N E W

S L E N D E R S H I P T H E O R Y O F W A V E R E -S I S T A N C ET h e p r o b l e m o f p r e d i c t i n g w a v e r e s i s t a n c e e x -

p e r i e n c e d b y a s u r fa c e s h i p w i t h s t i ll - w a t e r l e n g t hL , m o v i n g a t s p e e d U i n s t e a d y r e c t il i n e a r m o t i o ni n c a l m w a t e r o f i n fi n i t e d e p t h a n d u n b o u n d e d h o r i -

z o n t a l e x t e n t i s d i s c u s s e d i n t h i s t h e o r y . W a t e r i sa s s u m e d t o b e h o m o g e n e o u s a n d i n c o m p r e s si b l e f l u -i d w i t h d e n s i t y p . S u r f a c e t e n s i o n , e f f e c ts o f v is -c o s i ty a n d w a v e b r e a k i n g a r e i g n o r e d a n d p o t e n t i a lf l o w i s a l s o a s s u m e d .2 . 1 M a t h e m a t i c a l m o d e l

T h e v a r i ab l e s X = ( X , Y , Z ) a n d 9 a r e m a d en o n - d i m e n s i o n a l w i t h L a n d U a n d w a t e r d e n s i t yp t o g e t ( z , y , z ) = ( X , Y , Z ) / L a n d v e l o c i t y p o -t e n ti a l ~ = ~ / U L .

s a t i s f i e s t h e f o l l o w i n g e q u a t i o n s :V } = 0 i n t h e m e a n f lo w d o m a i n d , ( 1 )

S ~ l S z + F 2S 29 ~/ oq :z -2 F ' 2 q ( ~ ) + 0 ( F 4 ~ 3 ) = 0o n t h e m e a n f r e e s u r f a c e a ( z = 0 ) , ( 2 )

w h e r e F i s t h e F r o u d e n u m b e r F = U / ( g l ) 1 / 2q ( } ) = a [ V]~ leac r - ~ 9 V I ~ ]2/2-

(a~/&r - I ~ la /2)a(a~ /az + F2a2~/asc2)/az,(3)a /a. = n : o n t h e m e a n h u l l s u r f a c e h . ( 4 )

I n a d d i t i o n , t h e u s u al " r a d i a t i o n c o n d i t i o n " ,s p e c i fy i n g t h a t w a v e s a r e n o t p r e s e n t f a r a h e a d o ft h e h u l l , m u s t b e i m p o s e d f o r u n iq u e n e s s o f t h e s o -l u t i o n .

A m a j o r d i f f i c u l t y o f t h e p r e c e d i n g p r o b l e ms t e m s f r o m t h e f re e s u r f a ce c o n d i ti o n ( 2 ) , w h i c h i sn o n l i n e a r. H o w e v e r , t h e n o n l i n e a r t e r m 0 ( F 4 4 3 )i n e q u a t io n ( 2 ) m a y b e n e g l e c t e d a n d t h e n o n l i n e a r

Received date:200 3 05 - 2 6.


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9 54 9 Journal of M arine Science and App lication,Vol. 2, No. 2 , D ecember 2003t e r m F 2 q ( ~ ) i n eq u a t io n ( 2 ) i s o f o r d e r F 2 4 2 ,t h u s c a n b e p r e s u m e d t o b e s m a l l i n c o m p a r i s o nw i t h t h e l i n e ar t e r m a4/a + F 20 2 4 / a x e fo r s l en -d e r s h i p h u l l f o r m s . M o r e g e n e r a l l y , t h i s n o n l i n e a rt e r m F e ( 4 ) w i l l b e i n c o r p o r a t e d i n a n i t e r a t i v em a n n e r b y e x p r e s s i n g t h e f r e e s u r f a c e c o n d i t i o n( 2 ) i n t h e f o r m 8 4 1 8 z + F Z a 2 4 1 8 x 2 = I ~ ' 2 q ( 4 )a n d t re a t i n g t h e r i g h t h a n d s id e a s a n o n h o m o g e -n e o u s t e r m f o r t h e l i n e a r c o n d i t io n a4/a= + F282 4 /a x 2 = 0 , t h i s t e r m m a y t h e n b e n e g l e c t e d in t h ef i r s t a p p r o x i m a t i o n .2 . 2 G r e e n f u n c t i o n

T h e b o u n d a r y - v a l u e p r o b l e m d e f i n e d a b o v ew i l l b e s o l v ed b y f o r m u l a t i n g a n i n t e g r o - d i f f e r e n t i a le q u a t i o n fo r t h e v e l o c i t y p o t e n t ia l 4 b y u s e o f aG r e e n f u n c t i o n s a t i s f y i n g t h e l i n ea r iz e d f r e e s u r f a cec o n d i t i o n . T a k e t h e s o u r c e p o i n t a n d f ie l d p o i n t a sx ( x , y , z ~ 0 ) a n d { ( ~ , r b ~ 0 ) r e s p e c t i v e l y ,t h e n t h e G r e e n f u n c t i o n w il l b e d e n o t e d b y G ( { ,x ; F 2 ) .4 s r G ( { , x ; F e ) = - 1 / r + 1 / / + N ( { , x ; F 2 ) +

W ( { , x ; f 2 ) , ( 5 )w h e r e r , / = [ ( ~ - x ) 2 + ( r ] - y ) 2 + ( ~ g_Z)2] 1 /2 ' I '= (2 / r c f 2 ) I m e x p ( V ) E l ( V ) d t , ( 6 )-1a n d V = [ ( ~ + z ) ( 1 - t 2 ) l l 2 + ( ~ ] - y ) t + i l ~ - x[ ] ( 1 - - t 2 ) l / 2 1 F 2 ,

E l ( V ) = d t .vT h e d o u b l e i n t e g r a l t e r m N r e p r e s e n t s t h e n o n -o s c i l l at o r y n e a r - f i el d d i s t u r b a n c e , I m r e p r e s e n t s t h ei m a g i n a r y p a r t .

4 f ~W = U ( x - ~ ) F 2 I m e x p [ ( [ + z ) ( 1 + t 2 ) ,

F -2 + i { ( ~ - x ) + (7; - y ) t } ( 1 + t2 ) l l2U2]dt -4 S EH ( ~ r - ~:) ~ 2 ~ I m ( { ) E ( x ) d t ( 7 )

T h e s i n g l e i n t e g r a l t e r m W r e p r e s e n t s a w a v y d i s -t u r b a n c e , w h e r e

E = E ( x ) = e x p [ F - a ( 1 + t2 )~ /2 1( 1 + t Z ) l l 2 z -i ( x + t y ) } ] . ( 8 )

E ~ i s t h e c o m p l e x c o n j u g a t e o f E , H ( x - ~ )i s H e a v i s i d e f u n c t i o n .2 . 3 E q u a t i o n s o f p o t e n t i a l a n d w a v e r e s is t a n ce

c o e f f i c i e n tT h e v e l o c i t y p o t e n t i a l e q u a t i o n , w h i c h i s v a l id

b o t h i n n e a r - f ie l d a n d f a r - f i e ld , w i l l b e o b t a i n e d b yu s i n g G r e e n f o r m u l a Ell :

~ ( { ) = [ ( G n , . - 4 a G / a n ) d a + F 2 [ [ G ( n 2 : ~ .d h , 3 c+ G O ~ /O I - n ~ t y a } /O d ) -

- F e f ( 9 )O G / a m ] t y d lw h e r e h i s t h e h u l l s u r f a c e , c i s m e a n w a t e r l i n e ,d i s t h e i n f i n i te p o r t i o n o f t h e p l a n e z = 0 o u t o f c .n i s t h e u n i t o u t w a r d v e c t o r n o r m a l t o h , t i s t h eu n i t v e c t o r t a n g e n t t o c , t h e d i r e c t i o n o f v e c t o r d =n t i s t a n g e n t t o h a n d d o w n w a r d .

K o c h i n f u n c t i o n m a y b e e x p r es s e d a sI ~ ( t ) : F - 2 f ( E ~ , . . - + ~ E / ~ n ) d a - I - I [E (T v /2 o ,.h c+ t0 8 r - n = tu S r ) - C a E / O ~ . ]

- f o E q ( 4 ) d a - d y . ( 1 0 )y d lT h e i n t e g r a l o p e r a t o r I i s d e f i n e d a s

I = [ d a ( n . ~ - 4 8 / a n ) + F 2 [ d / [ ( n 2 , +3 h d ct , : a 4 1 a l - n ~ t y a 4 1 8 d ) 4a / a~] , , -F 2 f d x d y q ( 4 ) .

T h e l a s t t e r m i s i g n o r e d t e m p o r a r i l y , d e n o t ee l f d a n ~ + F 2 f d l 2, " 1 1 a . t y ,h c

[ 2 " ~ f h d a 4 3 1 ~ T l - p - F e f < d l ( t ,a 4 / 3 ! -n~tyi )4 /Od 40/O x ) ty ,

t h u s I = Ii + 12 , 4 ( { ) = ( 11 + I 2 ) G , K ( t ) = ~22( I1 + I 2 ) E .B y u s e o f H a v e l o c k f o r m u l a , n o n d i m e n s i o n al w a v er e s i s ta n c e c o e f f i c i e n t m a y b e o b t a i n e d

r = R I p U 2 L 2 - - ( l / 2 s r ) I K ( t ) 12( 1 + t e ) l / 2 d t . (11)

2 . 4 T h e i t e r a ti v e f o r m u l a s o f p o t e n t i a l a n d w a v er e s i s t a n c e

E q u a t i o n ( 9 ) m a y b e e x p r e s s e d i n t h e f o r m4 ( { ) = c ? ({ ) - T ( { , 4 ) , ( 1 2)

wh e re ~0 ( { ) , T ( { , 4 ) a re de f ine d re sp ec t i ve ly by~o(~) I i G f G n ~ d a + F 2 j 2= G n , t y d l ,h c

( 1 2 a )T ( g , 4 ) L ( g , 4 ) + F 2 f G q ( 4 ) d a c d y ,cr

( 1 2 b )


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H a n D u a n - fe n g , e t a l : S t u d y o f a l g o r it h m s o f n e w s l e n de r s h i p t h e o r y o f w a v e r e s is t a n c e 9 5 5 9

w h e r eL ( { ; r = I 2 0

= j h r + F e f c [ r - -G ( G O r - n j g r

( 1 2 c )Th e ve loc ity po te n t ia l r i n e qua t ion ( 12 ) m a y

be so lved u s ing the r e c u r r e nc e r e l a ti on ( 1 3 ) , N or -m a l ly , 9 ( ~ ) m a y be t a ke n a s t he f i rs t o r de r po t e n -t ia l r in the fo l lowing recur re nce re la t ion . I t i sve r i f ied in l i te ra ture>? tha t t he d i f f e re nc e be tw e e nr and ~ is smal l in the l imi ted case tha t F rou denum ber i s c lose to ze ro .

r = ( p ( ~ ) _ _ W(~ ; r ( 1 3 )F or s l e nde r sh ip s , t he i t e r a t i ve app r ox im a t ions a r ede f ine d byl r O ,

r ~ cfl(~) = I 1 O , ( 1 4 )r = I I G + I2 t r ,~ r 1 ) ( ~ ) I z G + I 2 I ~ r O .

Co rrespo ndin g seque nces of K (~ , K ~1) ,K (2) . . . . and t " (~ , r (1) , v (2) . . . . c a n be de f ine d bys im p ly r e p la c ing r by r i n e qua t ion ( 1 0 ) a nd( 1 1 ) .

1~K (~ = ~22I I~=oE = ~ I I E ,1K ( " + l ) ( t ) = F ~ I I r1 6 2 ( 1 5 )1= K ( ~ + ~ I 2 r 1 62

S(~) = (1/2n) I K ( " ) ( t ) 12( 1 + t 2 )~ / Zd t .( 1 6 )

3 A L G O R I T H M F O R Z E R O T H A N D F I R S TO R D E R W A V E R E S I S T A N C E

3 . 1 z e r o t h o r d e r w a v e r e s i s t a n c e r {~W e c a n ob ta in t he f o l low ing e xp r e s s ions f o r

z e r o th o r de r s l e nde r sh ip a pp r ox im a t ion to t heK o c h i n f u n c ti o n f r o m e q u a t i o n ( 1 5 )

1 1K ( ~ 2 ) = ~ I I~ =0 E = ~ I I E =

f E n 2 / s d l + f h E n ~ d a . ( 1 7 )B y subs t it u t i ng it in to e xp r e s s io n ( 16 ) , z e r o th

ord er wav e res is tance r (~ can be obta ine d readi ly .

A n o t h e r m e t h o d i s u s e d t o m a k e n o n d i m e n s i o n -alE4 51

/ /( x ' , y , z ) = ( X / L , Y / B , Z / T ) ,w h e r e L i s sh ip le ng th , B i s sh ip b r e a d th , T i s sh ipd r a ft , ( X , Y , Z ) is d i m e n si o n al a n d ( x ' , y ' , z ' ) isne w nond im e ns iona l c oo r d ina te s .

The sym bo l ' i s om i t t e d he r e a f t e r f o r c onve -n i e nc e .T h e n e x p r es s io n ( 8 ) b e c o m e s

+ t 2 ) 1 / 2 F - 2 [ ( 1 + t 2 ) 1 / 2 T zE ( x , t ) e x p { (1- + t y ) ] } .

L e t Y I = F 2 ( 1 + t 2 ) 2 T ,21 = F 2 ( 1 + t 2 ) 1 / 2 / 2 ,

Ba 2 = F -2 ( 1 + t 2 ) 1 / 2 t 2 ~ "E x p a n d i n g E ( x , t ) w i t h L e g e n d r e p o ly n o m i -

al s P . , , P , , , P z , the f i r s t k ind of sphere Besse lfun ctio n j , , ( 21 ) , Jl ( 2 2 ) a nd the f i r s t k ind o f m od i -f ied sp her e Bessel fu nct ion Bessel i . , ( Yl ) .

E ( x , t ) = e x p ( - ) ' , i a 2 ) E E ~ ( 2 m +m n l

1 ) ( 2 n + 1 ) ( 2 l + 1 ) i , , , ( y i ) j , , 9( a l ) J l ( a z ) i U ' + l ) P , , ( 1 + 2 z ) 9P . ( - 2 x ) P , ( 1 - 2 y ) . ( 18 )

Sub st i tu t in g ( 18 ) in to ( 17 ) and des ign a t ingthe r e a l pa r t a nd im a g ina r y pa r t o f z e r o th o r de rK oc h in f unc t ion a s K ~~ ( t ) and K I ~ ( t ) respec-t i ve ly , w e the n ha ve

7r/ ( ( ~ = F e e x p ( - Y l ) C O S [ ~ - ( n + l ) -

0~2~2 E ( 2 m + 1 ) ( 2 n +m n 11 ) ( 2 l + 1 ) i . , ( 7 1 ) j . , ( a 1 ) j , ( a 2 ) 9f ~P m (1 + 2 z ) P , , ( - 2 x ) P , ( 1 -

+ l ) -y ) n . d a + cos[ 7coO~2~E E ( 2 n + 1 ) ( 2 l +,l 1

1 ) j . ( a l ) j ~ ( a g ) f c P , , ( - 2 x ) P z ( 12 y ) 2- n , . t y d l . ( 1 9 2 )

7rK l ~ = F 2 e x p ( - 7 1 ) s i n [ ~ - ( n + l) -


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9 56 9 Journal of Marine Science and Application, Vol. 2, No. 2, December 2003co ~ oo

a 21 ~_ _~ _~ ~ , ( 2 m + 1 ) ( 2 n +m n l1 ) ( 2 / + 1 ) i . , ( ) ' , ) j , , ( c q ) j , ( a 2 ) 9

f , g , , ( 1 + 2 z ) P , , ( 2 x ) P z ( 1 -2 y ) n ~ d a + sin[ 7r + / ) -+ 1)(2 +

1 ) j , , ( a ~ ) j , ( a 2 ) f P n ( 2 x ) P z ( 1- 2 y ) n e t y d l . ( 1 9 b )

I f t h e s h i p h as p o r t an d s t a r b o a r d s y m m e t r y ,ca lcu la t ion cou ld be s impl i f ied somewhat by r ep lac-ing in tegra l domain by hal f hu l l su r f ace and hal fw a t e r l in e ( y > 0 ) .

By us ing th is meth od , funct ions r e la ted toF r o u d e n u m b er F an d t a r e s ep a r a ted f r o m t h a t r e -la ted to hu l l shape . Dur ing the p rocess o f numer ica lca lcu la t ion , in tegra t ion r espect to hu l l shape i s thes am e fo r d i f fe r en t F r o u d e n u m b er F an d t , w h i chsaves g rea t ly the ca lcu la t ion t ime and overcomesthe d i f f icu l ty o f r ap id osc i l l a t ion o f the in tegrand .

Whi le the f i r s t o rder wave r es i s tance are ca lcu-la ted , there i s d i f f er ence wi th ca lcu la t ion o f zero thorder wave r es i s ta nce r (~ , and the near f ie ld po-ten t ia l mu s t be kno wn a p rio r i . The near f ie ld po-ten t ia l can be evaluated numer ica l ly us ing Greenfunct ion wi th ou t d i f f icu l ty in pr inc ip le . In p rac-t i c e , h o w e v e r , t h e r e a r e s i n g u l a ri t y p r o b l em s i n E l( V ) o f ex p r e ss i o n ( 6 ) w h e n V < < I an d d iv e r g en ceprob lem wh en z + ~" 0 , and osc i l la t ion p rob lem inexpress ion (6 ) . Th is r equ i r es tha t comparab le a l-gor i thm be used fo r nume r ica l ly evaluat ion . Beingt h e k e r n e l f u n c t i o n o f N ew S l en d e r S h i p T h eo r y o fW ave Res is tance , Gree n funct ion i s d i f f icu l t toevaluate accura te ly due to the fo rego ing p rob lems .Th ere ar e , some l i t e r a tu res in wh ich a lgor i thms ared iscussed , bu t the ca lcu la t ion accuracy and speedare not s imilar .

I n t h i s p ap e r , C h eb y s h ev p o l y n o m i a l s ap -p roach m ethod has been adop ted to so lve express ion(6) and the specia l funct ion expans ion method hasbee n adop ted to solve expre ssion (7 ) !2,4,573 . 2 F i r s t o r d e r w a v e r e s i s t a n c e ~/(1)

By us ing express ions ( 5 ) and ( 14 ) , f i r s t o rder

p o t en t ia l m ay b e d en o t ed a s~ ( 1 ) ( r = ~ r ( r + ~(wl)(~;F 2 ) +

~ )! 1) ( r ; f 2 ) , ( 2 0 )w h e r e

4 ~v } r ( { ) = [ ( 1 / r - 1 / r ) n , . d a , ( 2 0 a )d hr~v,~ ( { ; F 2 ) = f I N = N n , d a q- f 2 " ,k

( 2 0 b )} ~ ) ( { ) : I I , w = l l m f L E ~ ( { ) "

1~ I i H ( x - 8 ) E ( x ) d t . ( 2 0 c )The f i r s t and second terms r epresen t nonosci l -

la to ry near f ie ld d i s tu rbance and the a lgor i thmabo ut N wa s p resen ted in l i tera ture ~21 .

The las t t e rm represen ts a wavy po ten t ia l .S ince

2 I, H ( z 8 ) E ( x )F 2 f H ( x 8 ) E ( x ) n , . d a +

J h

; 2( H ( x - ~ ) E ( x ) n ~ t s d l= F 2 fl, E ( x ) n . , . d a + f , . E ( x ) n ~ t . v d [ ,

where h~: and c~ r epresen t the por t ions o f the meanwater l ine and hu l l su r f ace be tween the sh ip bowand the plane x = ~: .

E ( x ) an d E ( { ) c an a im b e s o lv ed u s in g t h especia l funct ion expans ion method l ike tha t o f ex -p r e s s i o n ( 1 8 ) .

The f i r s t o rder wave ampl i tude funct ion cor r e-s p o n d i n g t o ex p r e s s i o n ( 2 0 ) can b e ex p re s s ed a s

/ - ( ( l ) ( t , f 2 ) = f ( ~ 2 ) @ / -( { i] ) ( t , f 2 ) K ~ ) ( t , F 2 ) + K ( I ) ( t , F 2) ( 2 1 )a

wh ere K {~ ( t , F 2 ) i s def ined by express ion (19a) ,( 1 9 b ) ,

( l ) . 1 E ( X ) =K L ( / ~ , f 2 ) = F 2 1 2 r 16 2 N- - S-4(1 + t 2 ) [ E ( X ) ( ~ r q- ~ , ,4 ) [ n z i (n.~. +

d h

t n y ) / ( 1 + t e ) l/ 2 ] d a + F a ( 1 + t2 ) 1/2

f E ( X ) I i ( r a( r+G)9 3 l n ~ ts0 ( r ~ - ~ , N/' ) t 2 ) l/ 2 " ~ j3 d ) / ( 1 + G d l . , ( 2 2 )


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Han Duan-feng, e t a l : S tudy of a lgor ithms of new s lender ship theory of wave res is tance 9 57 9

K ~ ) ( t , F 2 ) = ~ 22I2 r E ( X ) =- F 4 (1 + t e ) f , E ( X ) A v [ n ~ - i ( n ~ + t n y ) / ( 1 +t 2 ) l / 2 ] d a + F 2(1 + t2)1/2 " f e E ( X ) [ iC w + F 2 ( t .3 l n ' J s ( 1 + t 2 ) t /2 t y d l , ( 2 3 )

/ ~ ) ( t , / ; . 2 ) c a n a ls o b e d e f i n e d b y e x p r e s s i o n ( 2 4 )K(1 ) ( t , F 2) = 1 2 I 2 ~ = ~ b E (X )

1 ~ E ( X ) d r7 f _ ~ 1I 2 r 1 6 21 S ~ ( ~ ) ( r , t , F e ) d r ( 24 )

w h e r ek ( l ) ( r t , F 2 ) = 1 2 I 2 r 16 2, E ( X ) =

F 4 ( 1 + t 2 ) f g ( x ) @ [ r / z / ( n r +ht n y ) / ( 1 + t 2 ) l /2 ] d a + F - 2 ( 1 + t2 )~ /2 ~ E ( X ) 9[i@q- 2(t::.~-l.lz,v~31/(l,2)I/21tvd[3 l - 3 d / / - '

( 2 5 3 )w i t h

, d, (g , v , F 2 ) I m E * ( { ) K ~ ~ ( r , F 2 )( 2 5 b )

a n d1 ~ ) E ( x )~ ~ 2 ) = ~ I i H ( : r - =

F 2 J ' , H ( x - ~ ) E ( x ) n . d a + f8 ) E ( x ) n ~ t y d l =

F 2 E ( x ) n j d a + E ( x ) n , t s u d l . ( 2 5 c )h ~ c ~T h e n e x p r e ss i o n ( 2 0 c ) b e c o m e s

( ~ ) = ~ I , wl f 2 I r n E ~ ( ~ ) K ~ ~ ( r , F 2 ) d r1= 7 j ' _ ~ r r , F 2 ) d r .

T h e t r e a t m e n t h e r e i s to a v o i d t h e i n f lu e n c e o ft r u n c a t i o n o f p o t e n t i a l i n t e g r a t i o n r a n g e o n s u b s e -q u e n t c a l c ul a ti o n o f K o c h i n f u n c t i o n . W e s h o u l dn o t i c e th a t t h e { o f ~b ( { , r , F 2 ) m e a n s t h e p o i n to n e l e m e n t s o f sh i p a n d E ( X ) c a n a ls o b e so l v ed

u s i n g s p e ci a l fu n c t i o n e x p a n s i o n m e t h o d .K (~ ) t , f 2 ) f E ( X ) q ( r ( 2 6 )

~r

w h e r e r ; F 2 ) is f r o m ( 2 0 ) .3 / a n d 3 d m f o r e g o i n g e x p r e s s i o n s c a n b e o b -

t a i n e d b y u s i n g B - s p l i n e t o f it t h e p o t e n t i a l f r o m( 2 0 ) .4 N U M E R I C A L C A L C U L A T I O N M E T H O D4. 1 Hul l sur face in tegra l

T h e h u l l s u r f a c e c a n b e d i s c r e t i z e d w i t h q u a d -r a t e p a n e l s a n d f o u r c o n t r o l p o i n t s c a n b e d e f i n e d i ne a c h p a n e l , t h e c o o r d i n a t e s y s te m t r a n s f o r m a t i o n isa s f o l l o w s

N , = ( 1 - Z , ) ( 1 ) ` 2 ) / 4 ,N z ( 1 + ) ` ~ ) ( 1 ) ` 2 ) / 4 ,N 3 ( 1 + 2 , ) ( 1 + ) , 2 ) / 4 , ( 2 7 )N 4 ( 1 - ) , , ) ( 1 + ) , 2 )/ 4 ,

w h e r e , ( ), 1 , ) ,2 ) i s t h e c o o r d i n a t e i n t h e s t a n d a r dd o m a i n , t h u s t h e c o o r d i n a te o f a n y p o i n t o n a p a n elb e c o m e s

4x - ~ , N ~ x ~ . ( 2 8 )i

3 X 3 XD e n o t e G ( ) , 1 , ) , 2 ) ~T271 3 ) , ~ '3 x : ( 3 ~ i 3 y 3 ~ ) 3 x

where 3 ) , , ' 3 ) , 1 ' 3 ) , 1 ' 3 ) ,2( 3 x 3 y 3 z )

' 3 ) , 2 ' 3 ) , 2 -T h e d i f f e r e n t i a l e l e m e n t i s t h u s

3 X 3 Xd a = ~ ~ d ) , l d ) , 2 = I G ( ) , I , ) , 2 ) I d ) , l d ) , 2 .U n i t n o r m a l v e c t o r s a t is f y i n g r i g h t - h a n d e d s y s t e mi s d e f i n e d a s

n : X G ( ) , I , ), 2 I ; ( 2 9 )( 3 x 3 x )n . d a 3 ) , 1 3) ,2 9 id ) , 1d ) , 2 ,( 3 X 3 X )n s d a = ~ x ~ . j d ) , t d ) , 2 , ( 3 0 )( 3 x d x )n ~ d a = ~ 7 ~ 2 2 9 k d) , l d ) , 2 .

A f t e r t h i s t ra n s f o r m a t i o n , t h e i n te g r a l o v e rh u l l s u rf a c e p a n e l b e c o m e s o v e r t h e s t a n d a r d d o m a -i n , t h e c o o r d i n a t e s y s t e m o f p a n e l o n h u l l s u r f a c e is


6/8

" 5 8 " J o u r n a l o f M a r i n e S c ie n ce a n d A p p l i c a t i o n , V o t . 2 , N o . 2 , D e c e m b e r 2 0 0 3

s h o w n i n F i g 1 .Xa

-1 ~ , 1 )

( _ 1 , _ 1 ) ~ ~ 1 (_ 1,1 ) A~a)

4 ( ) ( ~

)v21Xl

I ( ) ( 2b)F i g . 1 C o o r d i n a te s y s t e m o f q u a d r a n g u l ar p a n e lA ny in teg ra l over a quad r i l a te ra I pane l can be

e x p r e s s e d a s9 d a = d R ld A 2 I G ( ) , i , 2 2 ) I .S 1 - 1 - 1

The hu l l su r face i s d i sc re t i zed wi th m quadr i -l a te ra l pane l s and K l , /~2 Gau ss nodes a re def inedrespe ct ively along 2,~ and 22 direct ion of each p an-e l . Af te r the co o rd ina tes o f Gauss nodes a re in ser t -e d i n to e x p re s s i o n ( 2 8 ) , t h e c o r r e sp o n d i n g h u l lsu r face coo rd ina te xo can be ob ta ined . Le t % bethe weigh t o f co r respond ing Gauss node , each hu l lsu r face in teg ra l can be exp ressed as

f h f ( x ; ~ n .,.d am K I X K 2

~ , ~ , f ( x j ; { ) % C j ( a ~ , a 2 ) . i ,i = l j = lfh,]g'( X ; ~ Hy d a

9 K,~K2 (3 1)X ~, ~ f ( x j ; { ) o o j G j ( 2 , , a 2 ) . j ,i = 1 j = l

f h f ( x ; ~ ) n ~ d a =m K1 K2

~ f ( x j ; { ) o o , g j ( 2 1 , 2 2 ) " k ." = 1 j = l4 . 2 W a t e r l i n e i n te g r a l

L e t t h e t w o e n d s o f t h e i t h l i n e s e g m e n t w i t hleng th li be ( :q 2 , yy l ) and (x i 2 , y~2 ) respec t ive lya n d th e d i s t a nc e b e t w e e n a n y p o i n t ( a : , y ) o n t h el in e s e g m e n t a n d o n e e n d ( : q l , y i l ) b e l , a s s h o w n

in F ig . 2 , u s ing coo rd ina te t rans fo rmat ion , one ge t s

(xii , yil )

~ ( X i 2 , yi2 )m X

F i g . 2 S k e t c h o f a t y p ic a l w a t e r l i n e s e g m e n tlil = ~ - ( t + l ) t E ( - 1 , 1 ) , t h u s

lid l = ~ d t ;li = ( ( ,27 i2 - - -Z~il)2 q- ( Y i 2 - - Y i l ) 2 ) 1 / 2 ;l .TEl -- X i 1.7C = ,~C + - - ~i X l

+ 1~ " E ( : < ~ ~ " ) ( t + l ) ' ( 3 2 )l Y = Yil + Yi2 -- Yi l ) ,( lli1= Y il + 2 - ( Y i 2 - Y i l ) ( t + 1 ) .

T h e t w o c o m p o n e n t s o f t a n g e n t v e c t o r t a r et , r - - ( ' T ' i 2 - - " Z 'i l ) / [ i ; t y = (Y,2 - - YiL ) / l i .

( 3 3 )T h e n o r m a l v e c t o r o f w a t e r l in e s e g m e n t i s

chosen as tha t o f the neares t hu l l su r face pane l ase x p r e s s e d i n ( 2 9 ) .

I f the w ate r l ine i s d isc re t ized w i th m l s eg -m e n t s a n d K G a u s s n o d e s a r e d e f in e d o n e a c h p an -e l , then co r respond ing water l ine coo rd ina te x j canb e o b t a i n e d u s in g ( 3 2 ) . L e t % b e t h e w e i g h t o fco rrespond ing Gauss node , any water l ine in teg ra lcan be exp ressed as

f< f ( '=, k 1 ,; { ) d l = ~ ~ f ( x j ; { ) r o j g . ( 3 4 )i = t j = J5 E X A M P L E

The au tho r has ca lcu la ted the zero th , f i r s t o r -der wave res i s tance and po ten t ia l on hu l l su r face o fsubm erged e l lipsoid near the f ree su r face , W ig teyh u ll a n d v e r t ic a l /c a n t e d s t r u t S W A T H s , c o m p a r i-sons wi th w el l - found numer ica l resu l t s have beenma de , wh ich p rove th i s a lgo r ithm in th i s paper isw o r k a b l e . I n v i e w o f t h e l e n g t h o f t h i s p a p e r , o n l ythe numer ica l resu l t s o f wave res i s tance o f s ing le


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Han Duan- feng , et al : Stud y of a lgori thm s of new slender ship theory of wave res is tance 9 59 9

v e r t i c a l s t r u t S W A T H n a me d T - A G O S a r e g iv e n .T h e s k e t c h o f p a n e l a r r a n g e me n t o f S W A T H

T-A GO S hul l i s show n in F ig . 3 . Th e ha l f sur faceof one s t ru t i s d iv ided un i formly in to tw ent y seg-ments in the longi tud ina l d i r ec t ion and ten seg-ments in the ve r t ica l d i r ec t ion , hence the re a re twohundreds pane ls on ha lf surface of one s t ru t . Th eha l f sur face of lower semi-hul l i s d iv ided un i formlyin to twenty segments in the longi tud ina l d i r ec t ionand ten segments in the c i r cumfe ren t ia l d i r ec t ion ,hence the re a re two hundreds pane ls on ha l f sur faceof lower semi-hul l .

Compar isons of the expe r imenta l r e s idua ry r e -s i s tance and ca lcu la ted wave re s i s tance for SW A THT - A G O S a r e s h o w n g r a p h ic a l ly i n F ig . 4 , w h ic hshows tha t the numer ica l r e su l t s o f ze ro th orde rwave re s i s tance a re a lmos t the same wi th numer ica lresults given by Salvesen ES? and Hu an gDingliang ETI , ho we ver , ther e are some phase shif tand va lues d isc repanc ie s be tween ze ro th and 1s t o r -de r wave re s i s tance , and the n umer ica l r e su l t s o f1s t o rde r wave re s i s tance show be t te r agreementw i t h t h e e x p e r ime n ta l d a t a . T h e n o n d ime n s ion a lwave re s i s tance coe f fic ien ts show n in F ig . 4 a re ex-pressed as Czv = 2R /pU 2 S .

Fig . 3 Ske tch o f pane l a r r angemen t o fSWA TH T-AGOS hull

6 CONCLUSIONS

E x p a n d in g E ( x , t ) w i th L e g e n d r e p o l y n o mi -a ls P . .. . Pn , P l , the f i r s t k ind of sphe re bessel func -t ion j , ( a ~ , Jz ( a2 ) and the f irst kind of m odif ie dsphe re Besse l func t ion i,, ( }'1 ) i s show n to be an e l -

g- ,~ Cw 1with line integral,.d Salvensen and Huang Dingliang

~Z Cw 0 with line integral f~ - - - Cw 0 without line integral 7~ Exp.da ta f~ ~ , / /

~)c,iz~ laD

, . . . . . . . . . . . . . . .0.0 5.0 10.0 15.0 20.0 25.0SHIP SPEED/knFig. 4 Wav e res is tance of ver t ical s t ru t

SWA TH named T-A(X)Sfec t ive me th od by whic h the coord ina te s o f sourcepoin t can be sepa ra ted f rom those of f ie ld po in ts ,r e la ted va r iab le s to F roude nu mb er F and t , andwavy poten t ia l and Kochin func t ion can be ca lcu la t -ed . Dur ing the process of numer ica l ca lcu la t ion , in -tegra t ion o ve r the hu l l sur face i s r equi red to be ca l -cu la ted on ly once and for al l F roude n um ber F andt , which makes the ca lcu la t ion more e f f ic ien t andovercomes the d i f f icu l ty of the r ap id osc i lla t ion ofthe in tegrand . Th e prec is ion of double in tegra lte rm in Green func t ion may be of f ive to s ix dec i -ma ls , us ing Chebyshe ve po lynomia ls approach a ssugge sted by Ne wm an Ez? .

T h e n e w s imp le h ig h - a c c u r a t e a l g o r i t h m f o rze ro th and f i r s t o rde r wave re s i s tance based on NewSlende r Ship Th eor y i s p re sen ted in th is pape r .T h i s r e s e a r c h ma y p r o mo te t h e N e w S l e n d e r S h ipT h e o r y o f W a v e Re s i s t a n c e , b e c au s e th e N e w S l e n-d e r S h ip T h e o r y o f W a v e Re s i s t a n c e c o v e r s ma n yothe r wave re s i s tance theo ry and th is a lgor i thm isconvenien t fo r adapt ing to com plex hu l l s . Th e nu-me r i ca l r e s u lt s f o r v e r t ic a l s tr u t S W A T H s h o w th a tthe re a re some phase sh i f t and va lues d isc repanc ie sbe tween ze ro th and 1s t o rde r wave re s i s tance , thenumer ica l r e su l t s o f ze ro th orde r wave re s i s tanceare c lose to n ume rical results given by Satvense Es?and Hu ang Dinglian g I77 , and the num erical resultsof 1s t o rde r wave re s i s tance show be t te r agreementwi th t he expe r im enta l da ta , which prove th is a lgo-r i thm in th is pape r i s workable .REFERENCES

[ 1 ] NOB LES SE F. A s l ende r sh ip theo ry o f wave r e s i st ance


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9 60 9 Journal of Marine Science and Application,Vol. 2, No. 2, December 2003

[ J ]. J SR , 2 7 ( 1 ) : 1 3 - 3 3 , 19 83 .[ 2] NEW MA N J N . Evaluation of the wave Green func-

tion :Part I - The double integral l][ [J ] . JSR, 1987,1 31( 2 ) : 7 9 - 9 0 .

[3] NOBLE SSE F, Triantafyllou G. Explicit approxima-tions for calculating potential f low about a bo dy [J ] .JSR, 1983,27(1 ) :1 - 12.

[4] HU AN G De-bo, LI Yunbo. Ship wave resistance basedon Noblesse' s slender ship theory and w ave-syeepnessrestric tion [Rl. Schiffstechnik Bd. 44-1997.

[5 ] SALVEN SEN. Hydro-numeric design of SWAT H Ships[ J ] . T r a n s o f S N A M E , 1 9 8 5 ( 9 3 ) : 3 2 5 3 46 .

[ 6 ] H AN D uanfen. Research on the metho d and calculationof the n ew slender-ship wave resistance Theo ry [D ].Harbin: Harbin Engineering Un iversity, 2002.

[ 7 ] H UA NG Dingliang. Performance principle of SW AT H[ M ]. BeiJing: National Defense Industry PublishingCompany , 1993 ( in Chinese).

HAN Duanfeng Ph. D. Associate pro-fessor, Fac ulty of Ship Building Engi-neering, Harbin Engineering Universi-ty. Graduated from Dept. of Naval Ar-chitecture and Ocean Engineering,Harbin E ngineering U niversity ( 1988 :Bachelor Degree of Ship Design; 1994:

Master Degree of Hydrodynam ics; 2002: Doctor Degree ofHydrodynamics). He has worked in a ship yard for fouryears and paid half years academic visit in HIT AC HI ZosenShip Y ard in 2001. His interests are in the fields of Ship I)e-

sign and Ship Hydrodynamics. He published more than 10papers or books.

L1 Yunbo Ph. D. Associate professor,Faculty of Ship Building Engineering,Harbin Engineering University. Gradu-ated from Dept. of Naval Architectureand Ocean Engineering, Harbin Engi-neering University (198 6: BachelorDegree of Ship Design; 1989: Master

Degree of Hydrodynamics; 1999: Doctor Degree of Hydro-dyna mics). His principal research interests concern the resis-tance and wav e-making characterist ics of high speed craft .He published more than 10 papers or books.

H U A N G De-Ix) Professor, Fac ulty ofShip Building Engineering, Harbin En-gineering University. Graduated fromDept. of Ma th& M echanics, TsinghuaUniversity (1967 ) ; 1979 - 1981 : fur-ther studied for two y ears in Universityof California at Berkeley, U. S . as a

visiting scholar; 1987 1988: paid one year in ShipbuildingInsti tute, Hamburg University, ( ;emlany, as an visi tedscholar. His interest is in the field of Ship Hydrod ynam ics,especially ship wave resistance and its applications. He pub-lished around 40 papers or books on ship wave, theory ofwave resistance and ship-fore1 optimization etc.

Documents

Study of Algorithms of New Slender Ship