Ship Technology Research/Schiffstechnik

Vol. 44 • No. 4 October 1997 T E C ™ « S C H E U N I V E R S r r E r r S c h e e p s b y d r o m e c h a n i c a A r c h i e f M e k e l w e g 2 , 2 6 2 8 C D D e l f t T e l : 0 1 5 - 2 7 8 6 8 7 3 / F a x : 2 7 8 1 8 3 6 N u m e r i c a l S e a k e e p i n g C a l c u l a t i o n s B a s e d o n S l e n d e r S h i p T h e o r y by Masashi K a s h i w a g i O v e r t a k i n g M a n o e u v r e s of S h i p s i n B e a m W i n d by Werner B l e n d e r m a n n S h i p W a v e R e s i s t a n c e B a s e d o n N o b l e s s e ' s S l e n d e r S h i p T h e o r y a n d W a v e - S t e e p n e s s R e s t r i c t i o n by H u a n g De-bo and L i Y u n b o P u b l i s h e d b y S c h i f f a h r t s - V e r l a g „ H A N S A " , H a m b u r g

D a s F a c h b u c h

f ü r S c h i f f b a u ,

S c h i f f s m a s c h i n e n b a u

u n d S c h i f f s t e c h n i i c

Herausgeber Prof. Dr.-lng. H. Keil

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Technologie der Schiffskörperfertigung Dipl.-Ing. H. Wilckens

Binnenschiffe für extrem flaches Wasser -Ergebnisse des VEBIS-Projektes

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SHIP i f H U M y SEARCH

J o u r n a l f o r R e s e a r c h i n S h i p b u i l d i n g a n d R e l a t e d S u b j e c t s

S H I P T E C H N O L O G Y R E S E A R C H / S C H I F F S T E C H N I K was f o u n d e d b y K . W e n d e l i n 1952. I t is edited by H . S ö d i n g and V . B e r t r a m i n c o l l a b o r a t i o n w i t h experts f r o m universities and m o d e l basins i n B e r l i n , D u i s b u r g , H a m b u r g and Potsdam, f r o m Germanischer L l o y d and other research organizations i n Germany.

Papers and discussions proposed for publication should be sent to Prof. H . Söding, Institut für Schiff-bau, Lammersieth 90, 22305 Hamburg, Germany; Fax -f49 40 2984 3199; e-mail soeding@schiffbau. uni-hamburg.de. Rules for authors, newest abstracts, keyword index and editors' software see under http: / / www .schiffbau .uni-hamburg. de

Vol. 44 • No. 4 • October 1997

Masashi K a s h i w a g i

N u m e r i c a l S e a k e e p i n g C a l c u l a t i o n s B a s e d o n S l e n d e r S h i p T h e o r y Ship Technology Research 44 (1997), 167-192

T h e survey of recent developments i n slender ship t h e o r y focusses o n u n i f i e d t h e o r y a n d high-speed slender-body theory. T h e u n i f i e d theory is extended t o include wave d i f f r a c t i o n f r o m the bow p a r t near the free surface. T h e high-speed slender-body t h e o r y is extended t o include a homogeneous component i n the inner s o l u t i o n , accounting f o r the transverse wave system i n a d d i t i o n t o the l o n g i t u d i n a l wave system d o m i n a n t f o r h i g h speeds. Several n u m e r i c a l examples demonstrate the usefulness of these extensions. These theories can be used as a p r a c t i c a l c a l c u l a t i o n m e t h o d , b r i d g i n g the gap between the t r a d i t i o n a l s t r i p theory and the more involved 3-D panel m e t h o d .

K e y w o r d s : slender ship theory, seakeeping, r a d i a t i o n , d i f f r a c t i o n , m o t i o n , added resistance, h i g h speed

Werner B l e n d e r m a n n

O v e r t a k i n g M a n o e u v r e s of S h i p s i n B e a m W^ind Ship Technology Research 44 (1997), 193-197

O v e r t a k i n g manoeuvres of ships are c r i t i c a l ; they m i g h t end i n a collision. T h e risk increases i n beam w i n d . T h e w i n d l o a d i n g of ships i n staggered positions has been measured i n w i n d t u n n e l tests. T h e results were used i n a n u m e r i c a l l y s i m u l a t e d o v e r t a k i n g manoeuvre of t w o ferries. O v e r t a k i n g is less c r i t i c a l o n the leeward t h a n o n the w i n d w a r d side.

H u a n g De-bo, L i Y u n b o

S h i p W a v e R e s i s t a n c e B a s e d o n N o b l e s s e ' s S l e n d e r S h i p T h e o r y a n d W a v e - S t e e p n e s s R e s t r i c t i o n

Ship Technology Research 44 (1997), 198-202

A new m e t h o d f o r wave resistance facilitates the i n t e g r a t i o n of h i g h l y o s c i l l a t o r y f u n c t i o n s a n d results i n a fast and accurate a l g o r i t h m for the K o c h i n f u n c t i o n s i n the f i r s t a p p r o x i m a t i o n of Noblesse's New Slender Ship T h e o r y . B y l i m i t i n g the wave steepness, good agreement w i t h test results f o r W i g l e y a n d Series-60 hulls can be demonstrated, b u t the m e t h o d is n o t suited f o r f l a t ships.

Keywords: wave resistance. Slender Ship Theory, K o c h i n f u n c t i o n

Correction to the article by Hans Meier:

N u m e r i c a l D e s c r i p t i o n o f a Propeller Blade

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N u m e r i c a l S e a k e e p i n g C a l c u l a t i o n s B a s e d o n t h e S l e n d e r S h i p T h e o r y

M a s a s h i K a s h i w e i g i , K y u s h u University^

1. I n t r o d u c t i o n

Because of n u m e r i c a l s i m p h c i t y a n d relatively good agreement w i t h measurements, t h e s t r i p t h e o r y has been used f o r p r e d i c t i n g t h e seakeeping performance of ships. However, t h e s t r i p t h e o r y is deficient i n accounting f o r t h e 3-D effects i m p o r t a n t f o r low frequencies and f o r some forward-speed effects. I n t h e 1960s and 70s, slender-body theories were extensively studied t o overcome t h e defects of s t r i p theory. M a n y slender-ship theories were developed, assuming i n several ways t h e order o f t h e f o r w a r d speed of a ship, U, and t h e frequency of oscillation, uj.

Maruo (1974), Adachi and Ohmatsu (1977), Takagi and Ohkusu (1977) review these theories.

E a r l y slender-ship theories could o n l y vahdate the s t r i p t h e o r y f o r h i g h frequencies. However, since t h e late 1970s, a n u m b e r of u s e f u l theories have been estabhshed, w h i c h account f o r some 3-D and forward-speed effects w h i l e stiU encompassing t h e s t r i p t h e o r y as a special case, e.g.

Yeung and Kim (1985), Maruo (1989). A m o n g these theories, t h i s review focuses on t h e ' u n i f i e d

t h e o r y ' , Newman (1978), and t h e 'high-speed slender-body t h e o r y ' . Chapman (1975, 1976). B o t h theories have been extended and enhanced and are recognized as p r a c t i c a l calculation methods, b r i d g i n g t h e gap between t h e s t r i p t h e o r y and more comphcated 3-D calculation methods. O t h e r related theories, such as t h e ' r a t i o n a l s t r i p t h e o r y ' , Ogilvie and Tuck (1969), are also b r i e f l y reviewed.

2. G e n e r a l D e s c r i p t i o n of S l e n d e r - S h i p T h e o r y

W e consider the hnearized 3-D p r o b l e m , assuming t h e inviscid fluid w i t h i r r o t a t i o n a l m o t i o n . T h e n t h e flow can be described w i t h t h e velocity p o t e n t i a l , w h i c h is expressed as

ncident wave

Fig. 1: Coordinate system

$ = U[-x + ^s{x, y, z)] + U[ 4>{x, y, z) e^* ] (1) { M X , y, z) + (t>r{x, y, z) •

(2)

+iu>Y^^j(j)j{x,y,z)

(po = exp{koz - zA;o(a;cos x + ysm x)} (3) U is t h e constant f o r w a r d speed of a ship, w = CJQ — koUcosx the circular frequency of oscillation, COQ

t h e circular frequency of i n c i d e n t wave, k^ = cJ^/g t h e wavenumber, and x i n c i d e n t wave angle. F i g . 1.

T h e amphtudes of i n c i d e n t wave, a, a n d of t h e j - t h mode of oscillation, { j 6), are aU assumed small. T h e unsteady p o t e n t i a l (p is d i v i d e d i n t o t h e i n c i d e n t wave p o t e n t i a l ^^Oj t h e s c a t t e r i n g p o t e n t i a l 07, and t h e r a d i a t i o n p o t e n t i a l ( f ) j . These potentials are subject t o t h e free-surface c o n d i t i o n of t h e f o r m

/ . Ö \ 2 , d(t> 1.

0 on z 0 (4)

where /it is Rayleigh's a r t i f i c i a l viscosity coefiicieiit ensuring the r a d i a t i o n c o n d i t i o n t o be satis-fied at i n f i n i t y .

T h e u n k n o w n p o t e n t i a l s are (f)j ( j = 1 ~ 7 ) , which can be characterized by the b o d y surface c o n d i t i o n ^ = « , + £ n . , 0 = 1 - 6 ) (5) 0 = 7, ( 6 , where ( m , « 2 , nz) = n ( m i , m2, mz) = -{n • V ) V ( « 4 , ns, ne) = r X n (m4, m s , me) = - ( n • V ) ( r x V ) • (7) r ^ { x , y , z ) V = V[-x + ips{x,y,z)] J

ips{x,y,z) is t h e steady disturbance p o t e n t i a l , w h i c h m a y be c o m p u t e d i n advance, s a t i s f y i n g

the r i g i d - w a l l free-surface c o n d i t i o n on 2: = 0. T h e u n i t n o r m a l vector, n , is positive when p o i n t i n g i n t o the fluid.

I n the slender-ship theory, above equations may be s i m p h f i e d f u r t h e r b y i n t r o d u c i n g the slenderness parameter e as a guide, w h i c h is usually taken asB/LorT/L{B,T,L being ship's b r e a d t h , d r a f t , and l e n g t h , respectively). I n the h m i t of e —> 0, the ship w i U be viewed as a segment i n the x-axis, and t h e n t h e b o d y b o u n d a r y c o n d i t i o n cannot be imposed (which is called the outer problem). T o zoom i n the b o d y surface, the y- and ^-axes m a y be stretched b y the variable t r a n s f o r m a t i o n of y = eY and z = eZ. T h e n , the b o d y b o u n d a r y c o n d i t i o n can be satisfled i n t h e m a g n i f i e d Y-Z plane. O n the other hand, i n this inner problem there is no r a d i a t i o n c o n d i t i o n , because t h a t holds f o r the flow at i n f i n i t y .

Namely, b o t h t h e outer and inner problems can be s i m p h f i e d to some extent, b u t includes unknowns. For a unique s o l u t i o n , the inner and outer problems have t o be matched i n an overlap region.

I n the inner p r o b l e m , besides the variable stretching i n the y- and z-axes, i t is customary t o assume the order of U and oj, because the waves generated b y t h e steady t r a n s l a t i o n and the h a r m o n i c oscillation are d i f f e r e n t i n n a t u r e and those wavelengths are related t o U'^/g and

g/u"^, respectively. These assumptions have produced a number of variations i n t h e slender-ship

theory. Before proceeding t o t h e details of those theories, the outer s o l u t i o n and i t s a s y m p t o t i c and series expansions w i U be summarized flrst. I n w h a t follows, f o r c l a r i t y only the ' l o n g i t u d i n a l p r o b l e m ' , i.e. surge { j = 1), heave ( j = 3), and p i t c h {j = 5) i n the r a d i a t i o n p r o b l e m and the s y m m e t r i c component w i t h respect t o y of the d i f f r a c t i o n p r o b l e m {j = 7 ) , w i l l be discussed. S i m i l a r analyses are possible f o r the ' l a t e r a l p r o b l e m ' (see A p p e n d i x 1).

3. O u t e r S o l u t i o n a n d I t s E x p a n s i o n

I n the outer region f a r f r o m the ship, the ship m a y be viewed as a segment along the x-axis. T h e n the disturbance due t o t h e ship can be described b y a fine d i s t r i b u t i o n of 3-D sources:

<f>'f\x,y,z)= r Qj{OG3D{x-^,y,z)d4 . (8)

J —00

Here Gsd stands f o r t h e 3-D Green f u n c t i o n , equivalent physically to the velocity p o t e n t i a l of t h e source w i t h u n i t s t r e n g t h . Qj is i t s strength, w h i c h is u n k n o w n at this stage, because the b o d y b o u n d a r y c o n d i t i o n is n o t considered.

T h e 3-D Green f u n c t i o n , s a t i s f y i n g the 3-D Laplace equation and (4) together w i t h the r a d i a t i o n c o n d i t i o n , has been extensively studied; its Fourier t r a n s f o r m w i t h respect t o x is

1 roo ^z\/WTrrfl-imy

,i^z-ie^u\y\^l-k^/u^ f o r 1/ > for iz < \k\

+

One i m p o r t a n t i n f o r m a t i o n t o be obtained f r o m the outer s o l u t i o n m a y be the K o c h i n f u n c -t i o n , w h i c h is physically -t h e wave a m p h -t u d e far f r o m -the ship. T h e K o c h i n f u n c -t i o n can be defined by considering t h e a s y m p t o t i c expression of (10) and s u b s t i t u t i n g i t i n t o (8):

Cj{k) = r Qj{x)(^^^dx

J—oo

C{k)^Cr{k) + ' ^ J2 - C j { k )

(12)

(13)

T h e a m p h t u d e o f ship m o t i o n i n the j-ih mode, ^y/a, w i h be given a f t e r solving the m o t i o n equation. F u r t h e r t h e source s t r e n g t h , Qj{x), w i U be determined b y m a t c h i n g (8) w i t h t h e inner s o l u t i o n i n an overlap region.

For the m a t c h i n g procedure t o fohow, expansions of (10) must be sought. For vR » 1, t h e T a y l o r expansion f o r t h e i n t e g r a n d appearing i n (10) gives readily the desired expansion. O n t h e other hand, f o r uR <C 1, we can use a m e t h o d of Kashiwagi and Ohkusu (1989) or Newman's

(1978) analysis extended f r o m Ursell's (1962) analysis f o r = 0. T h e results are: Glj,{k-y,z)=ieke^^'-'^'^\y\) + cos 9

iruR

+0{ (uR)-^, k^/i?, k^y/v) f o r vR » 1

GlD{k-y,z) = -{l + v z ) ( \ n ^ ^ + -i) + -vR{cose + esme)

TT V 2 / TT V i Y V . ^ ^ " ^ ^ ^ ° ^ ' " ' j ^ M ) } V f c v ' ^ - i { - " + ^ ° ^ " ' ( M ) } for yR<^l (14) + - ( l + i/z) TT +0{u^R\k^R^) (15)

where z = —Rcos6, y = RsinO, j is Euler's constant, and the upper and lower expressions i n t h e brackets a p p l y to v > \k\ and v < \k\, respectively.

E q . (14) must be v a h d f o r t h e whole range of U, insofar as the frequency is h i g h enough. However, f u r t h e r s i m p h f i c a t i o n s are possible f o r U = 0 ( 1 ) :

(16)

T h i s is the expression o f the r a t i o n a l s t r i p t h e o r y of Ogilvie and Tuck (1969), under the assumption of w = 0 ( e ~ ^ / ^ ) and U = 0 ( 1 ) .

4. 2 - D G r e e n F u n c t i o n a n d I t s E x p a n s i o n

T h e 2-D Green f u n c t i o n considered here f o r t h e inner p r o b l e m is t h e pseudo 3-D one, satis-f y i n g

(17)

(18)

iiuj-U—'] +g~ + n{iuj-U—]\G^^ = Q on 2 = 0

Lv dxJ dz \ dxJi where 6{x) on t h e r i g h t - h a n d side o f (17) denotes D i r a c ' s d e l t a f u n c t i o n .

T h e e x p h c i t f o r m o f m a y be given by t h e Fourier t r a n s f o r m , e.g. Yeung and Kim (1985):

G « ( x , y, ^; ^, 7?, 0 = In + Gp(x - ^, 12/- 7?!, z + 0 1 r°°

G p ( x , y,z) =

—

E\{u) is the e x p o n e n t i a l i n t e g r a l f u n c t i o n w i t h complex variable.

E q . (20), w h i c h is t h e inverse Fourier t r a n s f o r m of (22), can be w r i t t e n i n t h e f o r m

Gp{x, y, z) = u{-x) é v - ^ r ^ e'' cos{iy) s i n ( ^ / K ^ x ) Jo VÏ u { - x ) e ' v - ^ c ^ ^\ + i\y\ \ ^ ' i where wia) = e - " ' E r f c ( - t a ) , a = - J (23) (24) 2 V N + i M " ^^^^ E r f c ( - 2 ' a ) denotes t h e error f u n c t i o n w i t h complex variable, and u{-x) t h e u n i t step f u n c t i o n ,

equal t o 0 f o r a; > 0 a n d 1 f o r a; < 0.

Gp{x,y,z) represents physically t h e divergent ( l o n g i t u d i n a l ) wave, existing f a r b e h i n d t h e

source p o i n t . T h i s f a c t can be e x p h c i t l y shown either by using t h e expansion o f (24) f o r

y, z =^ 0(s), i n w h i c h exp(—a^) is t h e leading t e r m , or by a p p l y i n g t h e stationary-phase m e t h o d , Faltinsen (1983), t o t h e i n t e g r a l i n (23).

B y s u b s t i t u t i n g i n t o (22) t h e a s y m p t o t i c and power-series expansions of t h e e x p o n e n t i a l i n t e g r a l f u n c t i o n , t h e expansions of G*{k; y, z) f o r i^i? » 1 and vR<^l can be o b t a i n e d i n t h e f o r m

(26)

GUk; y,z) = i efc e K - " . l « l ) + £ ^ + o ( (uR)-^) f o r uR » 1 Gt{k]y,z) = -{l + izz)(]iiiyR-\-^ + mek] +-uRicosÖ + 9 sm0)

^ TT V / TT

+0{iy'^R^) f o r uR<^l . (27)

E q . (26) is i d e n t i c a l t o (14) i n the leading t e r m , i m p l y i n g t h a t t h e pseudo 3-D Green f u n c t i o n is equivalent t o t h e 3-D Green f u n c t i o n f o r aU values of U f o r h i g h frequencies. O n t h e other hand, f o r l o w frequencies, comparison of (15) w i t h (27) gives:

where

T r ^ i " " ' ' ' ^ ' ' ' ' ' ' '

(29)

Since Gp represents o n l y t h e divergent wave, g*{k) given b y (29) can be understood as the correction t e r m associated w i t h the transverse wave w h i c h exists i n the genuine 3-D wave field.

g*{k) reduces t o zero f o r u » \k\, g i v i n g a u t o m a t i c a l l y the r e l a t i o n Gp{x;y,z) « GsDix,y,z).

L e t the zero-speed 2-D Green f u n c t i o n be denoted b y G2D{y,z). Because t h e s u b s t i t u t i o n of J 7 = 0 i n (21) gives t h e same result as A; = 0, the f o l l o w i n g holds

G2Diy,z) = G;iO;y,z) (30)

Therefore, expansions of G2D{y> z) can be r e a d i l y obtained f r o m (26) and (27). T h e r e l a t i o n between G20 and t h e 3-D Green f u n c t i o n is also obtained f r o m (28) and (29), b y s i m p l y p u t t i n g

k = 0, i.e. V = K and = 1.

5. U n i f i e d T h e o r y

5.1 R a d i a t i o n P r o b l e m

I n the inner p r o b l e m , due t o the coordinate stretching, d i f f e r e n t i a t i o n s w i t h respect t o y and z cause the change i n order b y 0{e~^). T h i s slender-body assumption allows us t o use the 2-D Laplace equation as the governing equation. W i t h the same argument, the leading t e r m i n t h e free-surface c o n d i t i o n m a y be the rigid-waU c o n d i t i o n , dcpj/dz = 0. T h i s estabhshes the o r d i n a r y slender-body theory.

However, t o seek a u n i f i e d s o l u t i o n vahd f o r the whole range of frequencies, Newman (1978) considered t h e f o l l o w i n g boundary-value p r o b l e m :

(31)

on z = 0 (32)

( j = l , 3 , 5 ) o n ö ( x ) (33)

where Nj and Mj are slender-body a p p r o x i m a t i o n s of nj and mj defined i n (5), and B{x) denotes t h e contour of t h e transverse section at s t a t i o n x along the ship's l e n g t h .

T h e surge mode { j = 1) is retained i n (33), a l t h o u g h t h e surge mode is of higher order as compared t o heave { j = 3) and p i t c h { j = 5) modes. I n the conventional s t r i p theories, the surge mode has been s i m p l y discarded as higher order. However, w i t h slenderness assumption, the surge mode is of t h e same order as the roU mode w h i c h is c o m m o n l y included i n s t r i p theories. We note t h a t no r a d i a t i o n c o n d i t i o n is specified i n (32). Except f o r t h a t , t h e c o m p u t a t i o n m e t h o d f o r s o l v i n g ( 3 1 ) - ( 3 3 ) can be the same as t h a t used i n s t r i p theories.

T h e u n i f i e d t h e o r y considers a homogeneous solution plus the p a r t i c u l a r s o l u t i o n , i.e. the general inner s o l u t i o n takes t h e f o r m :

(f>f{x; y, z) = ipj{y, z) + ^ f j i v , z) + Cj{x) f n i y , z) ^niy, z) = </33(2/, z) - ip3{y, z) d(p dz -KcP) dN u N i + — M i

where tpj and (pj are the p a r t i c u l a r solutions, corresponding t o the f i r s t and second t e r m s on the r.h.s. of (33) respectively, and the overbar means the complex conjugate. Therefore, ipH{y,z) satisfies the homogeneous b o d y b o u n d a r y c o n d i t i o n .

T h e complex c o n j u g a t e of t h e velocity p o t e n t i a l is physically equivalent t o the s i t u a t i o n i n w h i c h t h e t i m e is reversed, meaning the i n w a r d p r o p a g a t i o n of the wave, w h i c h m u s t be allowed due t o the absence o f t h e r a d i a t i o n c o n d i t i o n . Therefore ipniy, z) represents t h e s t a n d i n g wave and i t s a m p h t u d e , Cj{x), is related t o the m a g n i t u d e of 3-D i n t e r a c t i o n effects among transverse sections; w h i c h is u n k n o w n i n the inner p r o b l e m b u t w i U be d e t e r m i n e d i n the m a t c h i n g procedure w i t h the outer s o l u t i o n .

W i t h (26) and (30) taken i n t o account, t h e outer expansion of (34) can be o b t a i n e d i n the f o r m

(Pf{x; y, z) - ( j j + + Cj{x) {a^ - a^) Gioiv, z) + 2i Cj{x) 0=3 e^^ cos Ky (35) U Kz.

ICO

where aj and denote the 2-D K o c h i n f u n c t i o n s w h i c h can be c o m p u t e d f r o m p a r t i c u l a r solutions of ipj and (pj, respectively.

T h e outer s o l u t i o n is given b y (8) and the 3-D Green f u n c t i o n has i t s inner expansion given as (28). However, i n order t o m a t c h w i t h (35), f u r t h e r approximations may be needed concerning the order of f o r w a r d speed. A c c o r d i n g t o Newman's (1978) analysis, i t takes t h e f o r m

Glnik; y, z) = G2D{y, z ) - - { l + Kz) r { k ) + 0{K^R\ {u - K)R, k^R^) TT where f i k ) + cos - V — (36) (37)

T h e upper and lower expressions i n the brackets apply to v > \k\ and v < \k\, respectively. T h e t e r m (9((^' - K)R) is neglected as a smaU q u a n t i t y . T h i s imphes t h a t t h e u n i f i e d t h e o r y m a y be consistent f o r relatively low f o r w a r d speeds. However, since the genuine 3-D wavenumber, v{k), is retained i n f*{k), forward-speed effects as well as 3-D interactions among transverse sections are t o some extent expected t o be accounted for; these w i U be made clear by comparison w i t h experiments.

S u b s t i t u t i n g (36) i n (8), t h e inner expansion of the outer solution can be expressed as c ^ 1

<Pf\x,y,z)r.Qj{x)G2D{y,z)--{l + Kz) Q j i O f i x - ^ d ^ - (38)

J - 0 0

T h e expression f o r f { x — ^), suitable f o r n u m e r i c a l computations, can be f o u n d i n Newman

and Sclavounos (1980), Sclavounos (1984a), and Sclavounos (1984b).

T h e inner and outer solutions m a y be matched by c o m p a r i n g (35) w i t h (38). T o leading order, the results are o f the f o r m

Qj{x) = ( j j + + Cj{x) {(J3 - 0-3}

TT J — 00

di

(39)

(40)

E h m i n a t i n g Cj{x) f r o m these t w o equations, the i n t e g r a l equation f o r Qj{x) can be obtained i n t h e f o r m

I n n u m e r i c a l c o m p u t a t i o n s , the i n t e g r a t i o n range w i t h respect to ^ m a y be reduced over the ship's l e n g t h . Once (41) is solved, i t is s t r a i g h t f o r w a r d to compute Cj{x) f r o m (39), thereby c o m p l e t i n g t h e inner and outer solutions.

5.2 D i f f r a c t i o n P r o b l e m

T h e b o d y b o u n d a r y c o n d i t i o n (6) can be w r i t t e n as

^ = ko e''°'^"'°y ^ { in2 sin ^ - (ng - m i cos x ) } e'^'^ (42)

where the wavenumber i n the x-axis is denoted by ^ = -A;o cos x, w h i c h w i U be used hereafter. T h e r a p i d l y v a r y i n g p a r t along ship's length is described b y e x p ( i £ x ) . I n beam sea, £ = 0. However, as i n the r a d i a t i o n p r o b l e m , no assumption should be made on the order of i t o o b t a i n a u n i f i e d s o l u t i o n apphcable t o aU the wavelengths and incident-wave angles.

T h e inner s o l u t i o n m a y be sought i n t h e f o r m

ct>?\x;y,z) = {i>s{x;y,z) + i>A{x;y,z)}e''^ (43)

where ips and ipA are s y m m e t r i c and a n t i s y m m e t r i c components w i t h respect t o y = 0, respec-tively, of t h e s l o w l y - v a r y i n g p a r t o f the s o l u t i o n .

For clarity, t h e explanations w i U be made o n l y f o r ips component. S i m i l a r analysis can be done f o r t h e ipA component, A p p e n d i x - 1 .

W i t h the slender-ship assumption, t h e boundary-value p r o b l e m f o r the inner s o l u t i o n can be f o r m u l a t e d as follows:

i>S - = 0 (44) ^ - k o ^ P s = 0 on z = 0 (45)

dip

= koe''°''^^N2smx sin(A;oi/sin x )

-(A^3 - iNi c o s x ) cos(A;oysinx)| on B{x) . (46)

T h e governing equation is n o t the Laplace equation b u t the 2-D m o d i f i e d H e h n h o l t z equation, a n d t h e wavenumber appearing i n the free-surface c o n d i t i o n is n o t K = uP- jg b u t k^ = oJo/g. T h e c o n t r i b u t i o n f r o m the TVi-component (the x-component of the n o r m a l vector) is retained i n (46). I n c o n v e n t i o n a l slender-body theories, the A^i-term has been discarded as higher order b y comparison t o N2 and TVs, i m p l y i n g t h a t the effects of wave d i f f r a c t i o n f r o m the bow p a r t near the water hne cannot be taken i n t o account i n the context of slender-body theory. However, once the value of A^^i is given, no d i f f i c u l t y exists i n solving ( 4 4 ) - ( 4 6 ) w i t h A^i-term kept i n (46). ( T h e o n l y t h i n g t o do i n the p r o g r a m is replacing N3 w i t h Ns — iNi cosx-) I n f a c t , the A^i-term is expected t o be more c r u c i a l t h a n t h e A^2- and 7V3-terms near the ship ends, i n predictions of t h e surge e x c i t i n g force and the added resistance i n head waves.

T h e inner s o l u t i o n can be constructed i n t h e f o r m of the p a r t i c u l a r s o l u t i o n ipg plus a homogeneous s o l u t i o n ipg :

^Psix; y, z) = ip^iy, z) + Crsix) V - f (y, z) (47)

V;|' = -e'=o-cos(/^oysinx) (48)

Here the p a r t i c u l a r s o l u t i o n is taken as the incident wave w i t h opposite sign, and C T S { X )

is t h e u n k n o w n coefficient of homogeneous component. i/'2£)(y! z) denotes a n u m e r i c a l s o l u t i o n f o r ( 4 4 ) - ( 4 6 ) , w h i c h m a y be o b t a i n e d using the integral-equation m e t h o d . I n t h a t m e t h o d , the Green f u n c t i o n s a t i s f y i n g the 2-D m o d i f i e d Hehnholtz equation is needed, for w h i c h Kashiwagi

(1992) adopted the f o l l o w i n g :

Here H2D{£', V, Z ) is the exact expression s a t i s f y i n g the extraneous r a d i a t i o n c o n d i t i o n , w h i c h can be given b y s u b s t i t u t i n g ko and A;o| c o s x l instead of v and |A;|, respectively, i n t o t h e Fourier t r a n s f o r m of the 3-D Green f u n c t i o n given i n ( 9 ) . Therefore, as is clear f r o m (10), t h e i m a g i n a r y p a r t of H2D takes the f o r m

T h i s f u n c t i o n is singular at x = TT (head wave). T h e proper analysis f o r head wave was shown b y Ursell (1968), i n d i c a t i n g t h a t the i m a g i n a r y p a r t is p r o p o r t i o n a l t o \y\ and thus no progressive wave exists i n the y - d i r e c t i o n . T h i s unreahstic p r o p e r t y is inherent i n t h e 2-D head-wave p r o b l e m , and was s u r m o u n t e d b y t h e matched asymptotic-expansion analyses o f Faltinsen

(1971), Maruo and Sasaki (1974), Adachi (1977).

T o seek a u n i f i e d s o l u t i o n w h i c h is vahd n o t only f o r head wave b u t also f o r aU heading angles, Sclavounos (1984a) t o o k o n l y the real p a r t of H2D as the i n n e r - p r o b l e m Green f u n c t i o n ; t h i s is one of the possible choices, because there is no need t o satisfy the r a d i a t i o n c o n d i t i o n i n the inner p r o b l e m . However, n u m e r i c a l computations based o n t h e integral-equation m e t h o d using t h e Green f u n c t i o n of Sclavounos' choice showed the irregular-frequency phenomena at some frequencies. Kashiwagi (1992) resolved these defects b y a d o p t i n g a complex f o r m , (51), as t h e i n n e r - p r o b l e m Green f u n c t i o n , which is another possible choice and i n f a c t regular f o r aU heading angles.

A n efficient c a l c u l a t i o n m e t h o d f o r (52) is less popular t h a n the r a d i a t i o n Green f u n c t i o n ,

G2D{y, z). T h i s seeming c o m p l e x i t y m i g h t be a reason w h y the s t r i p t h e o r y is stiU being used

despite of its shortcomings. A p p e n d i x - 2 provides a calculation m e t h o d , w i t h m u c h a t t e n t i o n p a i d on the c a l c u l a t i o n efficiency.

F r o m (43) and ( 4 7 ) - ( 4 9 ) , t h e outer expansion of the inner s o l u t i o n can be expressed as

4 2 ( x ; y, z) ^ Crsix) ar e''^ H2D{^; y, z) + {Crs{x) - 1} e'^' cos(fco2/sin x ) e^'^ (54) where (J7 denotes t h e 2-D K o c h i n f u n c t i o n t o be computed f r o m ip2Diy, z).

For o b t a i n i n g t h e inner expansion of the outer solution, t h e r e l a t i o n between i?2D(^;2/, ^ ) and Gl]j{k;y,z) m u s t be k n o w n ; w h i c h can be achieved b y n o t i n g t h e s i m i l a r i t y between (9) and (52), i.e. v ^ ko and |A;| —> ko\cosxl- T h e result can be expressed i n t h e f o r m

Gnii; y, z; 77, C) = - ^ { K o i m ) - Koimi)] + H2D{i\ b - r ? U + C) (50)

H2D{^;y,z) = u[n2D{i;y,z)]+ie''°'cos{koysmx) (51) ^[n2D{^;y,z) i c s c x e cos(fco2/sinx) • (53) (f>^"^{x,y,z)r^Q7{x)H2D{i;y,z)--il + koz)Cs{Q7;x) (55) where (56) hsix) = CSC X cosh ^ ( I s e c x l ) - b i ( 2 | s e c x | ) (57)

and the kernel f u n c t i o n f{x — i n (56) is the same as t h a t used i n the r a d i a t i o n problem. T h e m a t c h i n g requirement between (54) and (55) gives t w o equations f o r t w o unknowns,

Crsix) and Qrix). E h m i n a t i n g Crsix) f r o m those t w o equations, we can have the i n t e g r a l

equation f o r Qrix), i n t h e f o r m

Qrix) + -cTr CsiQr, x) = ar é^"" . (58)

TT

A n u m e r i c a l s o l u t i o n of Qrix) determines readily Crsix), c o m p l e t i n g t h e inner and outer solutions. T h e i n t e g r a l equation (58) may be solved w i t h the same scheme as t h a t for t h e corresponding equation, (41), i n the r a d i a t i o n p r o b l e m .

For Fn < 0.15, t h e x-axis may be d i v i d e d i n t o several segments of equal length and on each segment t h e u n k n o w n source s t r e n g t h be assumed t o vary hnearly, w h i c h can set u p a hnear system o f simultaneous equations, g i v i n g stable solutions. However, as t h e Froude number increases, t h a t scheme becomes unstable, p r o b a b l y because the i n t e g r a l equation w i U be of V o r t e r r a t y p e . T o overcome t h i s d i f f i c u l t y , Sclavounos (1984b) proposed a Chebyshev-p o l y n o m i a l reChebyshev-presentation f o r the u n k n o w n source d i s t r i b u t i o n and the use o f Galerkin's m e t h o d t o construct a weU-conditioned m a t r i x . T h i s scheme seems t o give a stable solution especially when the f o r w a r d speed is r e l a t i v e l y high.

6. H y d r o d y n a m i c F o r c e s

S u b s t i t u t i n g t h e completed inner solution i n t o the hnearized B e r n o u f f i equation, i n t e g r a t i n g over t h e mean w e t t e d surface of t h e ship, and using Tuck's theorem, Ogilvie and Tuck (1969), t h e h y d r o d y n a m i c force a c t i n g i n t h e i-ih direction can be summarized as

Fi = -iiw)^ E K + ^ i j/ H^ J ' ( ^ = 1 ' 3 , 5 ) (59)

Aij + Bij/ioj =-pJ^dx j^^^^ (Ni - ^ M j ) ^ipjiv, z) + T^^jiv, z) } ds

- p f dx Cjix) [ (Ni - —Mi) ipHiy, z) ds (60)

where Aij and Bij are t h e added-mass and d a m p i n g coefficients i n the x-th d i r e c t i o n due to the

j-ih mode o f m o t i o n , and J B ( X ) denotes the sectional contour below 2 = 0 at s t a t i o n x . Mi is

t h e slender-body a p p r o x i m a t i o n of the r r > t e r m , defined i n (7). I f t h i s is a p p r o x i m a t e d f u r t h e r b y neglecting t h e steady disturbance p o t e n t i a l (ps, i t foUows t h a t M i = M 3 = 0, M 5 = N3, and N5 = - X A/3. Correspondingly, (pi = ips = 0, (^5 = ips, and (p^ =

-xcps-T h e solutions t o be obtained are cpi and (p^. -xcps-T h e calculation m e t h o d f o r those can be the same as t h a t c o m m o n l y used i n the s t r i p theory, except t h a t the surge mode ( j = 1) is included i n t h e present case.

T h e first hne i n (60) gives i d e n t i c a l results t o t h e s t r i p t h e o r y except f o r surge-related co-efficients, and the second hne i n (60) contains t h e 3-D and forward-speed eff'ects t h r o u g h t h e coefficient of homogeneous s o l u t i o n , C y ( x ) .

I n t h e d i f f r a c t i o n p r o b l e m , n o t o n l y the integrated value of e x c i t i n g forces, b u t also the pres-sure d i s t r i b u t i o n is required t o predict l o c a l wave loads, e.g. Mizoguchi et al. (1992). Neglecting t h e c o n t r i b u t i o n o f t h e steady disturbance p o t e n t i a l and a p p l y i n g the d i f f e r e n t i a t i o n w i t h re-spect t o X o n l y t o t h e r a p i d l y - v a r y i n g t e r m , e x p ( i £ x ) , the s y m m e t r i c p a r t of the d i f f r a c t i o n pressure is

Pa = -pga - ( 1 - - f ) Crsix) (2/, z) UJn V lOJ ox/

^ -pga Cys{x) { ^2D {y,z) + e^°' cos{koy sin x) } e'^^ . (61)

I n t e g r a t i n g (61) over t h e ship huU gives the e x c i t i n g force i n the j - t h d i r e c t i o n :

Ej = pga f dx Crsix) é^"" [ | i v , z) + e''°^ cosikoy sin x ) } n y ds . (62)

J L JB(X)

I n seakeeping, t h e wave-induced steady force and m o m e n t are also i m p o r t a n t . T h e added resistance can be c o m p u t e d b y Maruo's (1960) f o r m u l a , using the K o c h i n f u n c t i o n . M a r u o ' s analysis is based o n the stationary-phase m e t h o d and thus rather comphcated. Kashiwagi

(1991) showed a simpler analysis b y use of Parseval's theorem i n the Fourier t r a n s f o r m , and gave

f o r m u l a e f o r t h e steady l a t e r a l force (Y) and yaw m o m e n t (A'^) as weU; those are summarized as foUows: RAW pga? Y pga^ N pga^ 1 r rf'i r^s 'iirko 1 A-Kko + - s i n x ^[ciko,x) + iSiko,x) - - I + i + / + / \{\Cik)\^ + \Sik)\^} oo Jk2 Jki J ^ rk3 r o o , , + / + / ^2Cik)Sik)\vdk •> Jk2 Jki ^ ^ ' 2 \ 1/(fc - A:oCOSx) fc2 dk (63) (64) Airko 2 ^ Ik ^ Ik \ ^{c'ik)^ik)-S'ik)Cik)^vdk

\ sin X ^[c'iko,x) + iS'iko, X) + ^ ( r + ^ ^ ^ ) {c(A;o, x ) + iSiko, x ) } ] (65)

where

kl k2

(66)

C(fc) is the s y m m e t r i c p a r t of the K o c h i n f u n c t i o n , w h i c h , as shown i n (12) and (13), can be evaluated w i t h t h e source s t r e n g t h Qjix) i n the outer s o l u t i o n . I n the u n i f i e d theory, Qjix) w i U be given as n u m e r i c a l solutions of i n t e g r a l equations (41) and (58). T h e a n t i s y m m e t r i c p a r t o f the K o c h i n f u n c t i o n , Sik), m a y be evaluated i n a similar manner f r o m the doublet d i s t r i b u t i o n along the ship's l e n g t h . A p p e n d i x - 1 . C'ik) and S'ik) i n (65) denote diff'erentiations w i t h respect t o k, and C(A;o, x ) and Siko, x) are the values evaluated at A; = fco cos x and ^/v^ -k'^ = ko sin

x-T h e effects o f wave d i f f r a c t i o n are rationaUy taken i n t o account i n ip2Diy,z) by r e t a i n i n g the A^i-term i n t h e b o d y b o u n d a r y c o n d i t i o n . T h i s means t h a t the bow d i f f r a c t i o n effects are i m p h c i t l y i n c l u d e d i n ar and therefore i n Qrix) as weU, because ar is c o m p u t e d f r o m ip2D and

Qrix) is a s o l u t i o n of (58).

T h e present analysis also includes the surge m o t i o n and its i n d i r e c t effects on the heave and p i t c h m o t i o n s t h r o u g h the cross-couphng terms i n the m o t i o n equations. T h i s is also d i f f e r e n t f r o m the o r i g i n a l u n i f i e d theory; thus the name 'enhanced' u n i f i e d theory, Kashiwagi (1995a), is used.

A few comments should be made on the n u m e r i c a l t r e a t m e n t of the i n f i n i t e integrals i n (63). C o n v e n t i o n a l m e t h o d s based on the s t r i p theory, e.g. Takahashi (1987), usuaUy m u l t i p l y the i n t e g r a n d b y a convergence acceleration factor, hke exp(—i/2;o) w i t h smaU positive value f o r zo, t o ensure t h e convergence at i n f i n i t y . T h i s t r e a t m e n t is apparently inconsistent i n the context of slender-ship theory. Kashiwagi and Ohkusu (1993) showed t h a t no d i f f i c u l t y arises i n the convergence, even i f the sources are placed on z = 0. T h e i r calculation m e t h o d utihzes the Fourier-series representation f o r the hne d i s t r i b u t i o n of sources, and the resultant singular integral, s i m i l a r t o t h a t appearing i n the w i n g theory, is evaluated analytically.

7. H i g h - S p e e d S l e n d e r - B o d y T h e o r y ( H S S B T )

T h e u n i f i e d t h e o r y can account f o r t h e 3-D interactions among transverse sections f o r t h e whole range of frequencies. P a r t i c u l a r l y for zero speed, the u n i f i e d t h e o r y gives satisfactory results. However, f o r r a t h e r h i g h speed, t h e m a t c h i n g procedure shows a h t t l e constraint, because t h e inner s o l u t i o n satisfies the zero-speed free-surface c o n d i t i o n . A s is pointed o u t by Yeung and Kim (1985), t h e 3-D flow is d i c t a t e d b y the inner flow i n t h e u n i f l e d theory. Reversely, t h e 3-D flow should d i c t a t e t h e inner flow.

I n t h a t respect, there is no a m b i g u i t y i n t h e d e r i v a t i o n of (28), the relation between t h e 2-D (pseudo 3-D) Green f u n c t i o n , Gp{x;y,z), and the 3-D Green f u n c t i o n , GzD{x,y,z). Therefore we are t e m p t e d t o consider (18) as t h e free-surface c o n d i t i o n i n the inner p r o b l e m , i m p l y i n g t h e assumption of a; < 0 ( e ~ ^ / 2 ) and U < 0{e~^/'^). T h e n we have

(Pj = 0 (67) ( i u - U ^ ) ' ' c t > j ^ g ^ = Q onz = 0 (68)

^ = N j + ^ M j ( i = 2 ~ 6 ) o n ö ( x ) . (69)

I f t h e above equations are supplemented w i t h the r a d i a t i o n c o n d i t i o n , t h e r e s u l t i n g formular t i o n w i U be the same as t h a t considered b y Chapman (1975, 1976). C h a p m a n solved t h e sway and yaw m o t i o n s of a plate, a p p l y i n g the finite difference scheme t o t h e above boundary-value p r o b l e m .

I n s p i r e d b y excellent agreement of Chapman's results w i t h experiments, a n u m b e r of studies have been made t o extend t o a general ship-hke geometry; f o r l a t e r a l m o t i o n problems, b y

Kashiwagi and Hatta (1984), Kashiwagi (1984), and Yamasaki and Fujino (1983, 1984, 1985).

For heave a n d p i t c h problems, by Saito and Takagi (1978), Adachi (1980), Yeung and Kim

(1981), and Faltinsen (1983). Recently, Faltinsen and Zhao (1991) and Ohkusu et al.(1991)

apphed t h e same f o r m u l a t i o n t o t h e analysis of a high-speed catamaran.

However, c a l c u l a t i o n m e t h o d s used i n those studies are much more d i f f i c u l t t h a n t h e s t r i p the-o r y the-or t h e u n i f i e d thethe-ory. I n f a c t several sthe-olutithe-on meththe-ods have been develthe-oped; such as Fthe-ourier- Fourier-t r a n s f o r m m e Fourier-t h o d , inFourier-tegral-equaFourier-tion m e Fourier-t h o d using Fourier-t h e Green f u n c Fourier-t i o n of (19), boundary-element m e t h o d w i t h l o g i ï used as t h e Green f u n c t i o n , and so on. Nevertheless, i t seems t h a t a rehab le c a l c u l a t i o n m e t h o d is s t i h n o t estabhshed.

F r o m t h e v i e w p o i n t of t h e m a t c h i n g w i t h t h e outer solution, few studies have been done, except f o r Adachi (1980) and Ohkusu and Faltinsen (1990). T h u s a brief discussion wiU be made below on t h e m a t c h i n g at t h e level of outer and inner Green f u n c t i o n s .

F i r s t l y , i n t h e range of h i g h frequencies assuming w = 0 ( e ~ ^ / 2 ) and U < 0{e~^^'^), as is clear f r o m comparison between (14) and (26), Gp{x] y, z) can be s m o o t h l y matched w i t h G2,D{X, y, z), and thus there is no need t o consider 3-D correction terms (homogeneous solutions). T h i s conclusion is v a h d irrespective of the order of U. Therefore, as suggested i n (16), H S S B T encompasses t h e o r e t i c a l l y t h e r a t i o n a l s t r i p t h e o r y of Ogilvie and Tuck (1969) assuming U = 0 ( 1 ) .

N e x t let us consider t h e case of low frequencies assuming LJ = 0 ( e ) and U < 0{£~^^^). I n this case, as e x p h c i t l y shown i n (28), t h e inner expansion of t h e 3-D Green f u n c t i o n includes 3-D c o r r e c t i o n t e r m s associated w i t h t h e transverse wave i n a d d i t i o n t o t h e divergent wave represented b y Gp{x;y,z). FoUowing the idea of t h e u n i f i e d theory, t h e 3-D correction terms may be m a t c h e d w i t h t h e homogeneous component i n t h e inner solution. However, unhke t h e u n i f i e d theory, c o n s t r u c t i n g t h e homogeneous solution is n o t so easy because of t h e convection t e r m i n t h e free-surface c o n d i t i o n . F i g . 2 sums up the above discussion s h o w i n g where existing

slender-ship theories can be apphed. T h e o r y A p p h c a b l e Region S t r i p T h e o r y R S T H S S B T U T (2) (2) (3) (2) (3) (4) (1) (2)

RST : Rational Strip Theory

HSSBT: High-Speed Slender-Body Theory U T : Unified Theory

0 ( 1 ) h

0 ( 1 )

Fig. 2: Order of parameters valid for various theories

R e t u r n i n g t o t h e homogeneous s o l u t i o n of H S S B T , Kashiwagi (1995b) recently showed an equation f o r t h a t . L e t us denote 0 + f o r t h e s o l u t i o n s a t i s f y i n g ( 6 7 ) - ( 6 9 ) and the real-flow r a d i a t i o n c o n d i t i o n , and hkewise (pJ f o r the reverse-flow and reverse-time s o l u t i o n ( w i t h b o t h signs of U and u> reversed).

E v e n w h e n b o t h U and w are reversed i n sign, (68) and (69) remain unchanged. T h u s (p'^-<f>J gives a possible homogeneous s o l u t i o n . However, this solution is a f u n c t i o n of x, aff'ecting the d o w n s t r e a m sections.

T h e r e f o r e the homogeneous s o l u t i o n should have a f o r m of c o n v o l u t i o n integral. N a m e l y we can w r i t e

.,. roo

cpfix; y, z) = ct>+{x-y,z)+ Cj{0 <t>H{x - y, z) (70)

J—oo

where

<l>H{x]y,z) = (f>j(x;y,z) - (l)j(x;y,z) . (71)

T h e weight f u n c t i o n Cj{i) is u n k n o w n i n the above expression, w h i c h can be m a t c h e d w i t h the outer s o l u t i o n m o r e easily by use of Fourier t r a n s f o r m , Kashiwagi (1995b).

Yeung and Kim (1985) also studied t h e 3-D corrections i n the f r a m e w o r k o f H S S B T . Instead

of (70), t h e y i n t r o d u c e d a "generahzed inner Green f u n c t i o n " , w h i c h is the same as t h e r.h.s. of (28), w i t h {l-\-vz) replaced b y exp(i^z) cos{uy). T h e y proposed a m e t h o d using t h e generahzed inner Green f u n c t i o n i n the 3-D panel m e t h o d , b u t no n u m e r i c a l results were presented.

Lastly, let me refer t o the popular t r a n s f o r m a t i o n of the H S S B T f o r m u l a t i o n i n t o an equiv-alent 2-D i n i t i a l - v a l u e p r o b l e m , w h i c h has been used i n several pubhshed papers.

cj>j{x-y,z) = évi-^/^)^Pj{x-y,z) = e - - * V y ( i * ; y, z) , t* = ( L / 2 - x)/U . (72) W i t h t h i s t r a n s f o r m a t i o n , ( 6 7 ) - ( 6 9 ) can be r e w r i t t e n as V ^ z , ^ y = 0 (73) - ^ + . ^ = 0 o n . = 0 (74) ^ = ( ^ ^ - + Ü = 2 - 6 ) onB{x). (75)

I f t* is viewed as t h e t i m e variable, the above is a 2-D t i m e - d o m a i n p r o b l e m and thus various e x i s t i n g s o l u t i o n methods m a y be used.

E q u i v a l e n t i n i t i a l c o n d i t i o n can be given b y

V'i = 0 , ^ = 0 a t r = 0 , (76)

w h i c h means physicaUy no disturbances at the bow.

Kashiwagi (1995b) showed n u m e r i c a l results based on this initial-value f o r m u l a t i o n , using the

q u a d r a t i c isoparametric elements i n the boundary-element m e t h o d and a n u m e r i c a l absorbing beach t o s a t i s f y t h e r a d i a t i o n c o n d i t i o n .

8. C o m p a r i s o n o f N u m e r i c a l R e s u l t s w i t h E x p e r i m e n t s

F i g . 3 shows t h e zero-speed results for the r a d i a t i o n p r o b l e m of a catamaran w i t h half-immersed spheroid oi L/B = 8 used as a demihuU, Kashiwagi (1993). I n this analysis, the inner region is defined as flow field near the r i g h t (or l e f t ) demihuU, and the i n t e r a c t i o n effects f r o m the other demihuU are taken i n t o account t h r o u g h the m a t c h i n g w i t h t h e outer solu-t i o n . T h e resulsolu-ts f o r a monohuU are also shown and compared w i solu-t h independensolu-t resulsolu-ts b y a more rigorous 3-D panel m e t h o d . T h e agreement is very good for the whole range of frequencies.

K B / 2 K B / 2

Fig. 3: Heave added-mass and damping coefficients of twin half-immersed spheroids w i t h

L/B = 8 and D/B = 2a.tU = 0{D being the separation distance between twin hulls)

I t has been said t h a t the surge mode and the d i f f r a c t i o n i n the x - d i r e c t i o n near the ship ends cannot be c o m p u t e d w i t h the slender-ship theory. However, once t h e x-component of the n o r m a l vector is given, there are no f u n d a m e n t a l difiiculties i n c o m p u t a t i o n s , as was shown by

Kashiwagi (1995a) w i t h various n u m e r i c a l examples. F i g . 4 shows t h e wave-exciting surge force

o n a rather b l u n t hah-immersed spheroid of L/B = 5. T h e results agree w e l l w i t h the 3-D panel m e t h o d , despite t h e s m a l l a m p h t u d e of the force. T h e d o t t e d hne is t h e result b y the F o u d e - K r y l o v force only, w h i c h has been used i n the s t r i p t h e o r y b u t is obviously n o t enough.

W i t h surge-related r a d i a t i o n forces and wave-exciting force, the surge m o t i o n was c o m p u t e d ( F i g . 5). T h e sohd hne was obtained f r o m the coupled m o t i o n equation between surge and p i t c h , and the dashed hne is the s o l u t i o n as t h e single mode of surge. T h e noticeable discrepancy

between t h e t w o and the good agreement between the sohd hne and the 3-D panel m e t h o d imphes t h a t the c o u p h n g effects between surge and p i t c h must be taken i n t o account.

| E j / p f f a A ^ 180 90 -90 -180 . 1 . 1 ' ' ' ' : • O üniflmd Thmoxy

; \

Fig. 4: Exciting surge force on a half immersed spheroid of L/B = bin head waves at [/ = 0 1.. J i ^ 90 i-0 t -90 -180 w i t h Fitch Coupliaff SurffB Slnffl» HodB o 3-D P a n a l M9thod

Fig. 5: Surge motion of a half i m -mersed spheroid of L/B = 5 in head waves at C/ = 0 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0 -0.2 D i f f r a c t i o n

Unified Theory (with nl-term) Do. (trithout nl-teim)

O 3-D Panel Method

. . , .

3 X / L 4

Fig. 6: D r i f t force on a fixed spheroid of L / B = 5 i n head waves at f7 = 0 (diffraction only) 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0 -0.2

R^^/paa-CB'/L) _{notion Free}

Unified Theory (with nl-term) Do. (without nl-term)

O 3-D P a n e l Hethod

. t i

.\v'...o.\..i..i^..-. .\v'...o.\..i..i^..-. .\v'...o.\..i..i^..-. , i .\v'...o.\..i..i^..-. , , < 1 1 1 1 , . . .

1 3 X / L 4

Fig. 7: D r i f t force on a freely oscillat-ing spheroid of L / B = 5 i n head waves at i7 = 0

T h e effects o f wave d i f f r a c t i o n near the bow are expected t o be pronounced i n the d r i f t force i n head waves. F i g . 6 shows the results exerted b y the wave d i f f r a c t i o n only, and F i g . 7 the results i n c l u d i n g aU effects of ship motions, d e m o n s t r a t i n g the i m p o r t a n c e of the A'^i-term i n the b o d y - b o u n d a r y c o n d i t i o n .

A33/PV Vniflad Theory (Tn'0.1) O Eitveriment (rn'0.1) unified Theory (Tn'0.2) • Experiment (rn'0.2) S t r i p Method 0.2 (Fn-<rn. " um fled Theory C Experiment (Fn-<rn. '0 1) ₁₎ (Tn^O 2) • Experiment 0 2) S t r i p Method 0.15 0.10 0.05 B33/P V / f f / l 1 1 1 1 1 1 1 1 1 1 1 ' 0 . . . . 1 . . . . 1 . 0 1 2 3 Cü' 4 fl55/pVL/&r 1 1 1 • 1 1 . • I 1 • 1 • ' ' • 1 • • ' • 1 ' ' ' • 1 ' • • • ( ' ' ' ' 1 ' \ _t IQ \ m-0.1 / • / ° o • a-g* . . . . 1 . . . . 1 . . . . 1 . . . . 1 . . . . 1 . . . . 1 . . . . 1 . . . . 1 .

Fig. 8: Added-mass and damping coefRcients in iieave and p i t d i of a mathematical ship model at F„ = 0.1 and 0.2. Experiments are reproduced from Matsunaga and Maruo

(1981)

ship m o d e l {L/B = 8) w i t h transverse sections represented by the Lewis f o r m t o experiments of Matsunaga and Maruo (1981). Except for the p i t c h d a m p i n g coefficient at F „ = 0.2, the results of the u n i f i e d t h e o r y agree weU, i n c l u d i n g the r a p i d change near the c r i t i c a l frequency at T = 1/4. T h e u n i f i e d t h e o r y is apparently superior t o the i n t e r p o l a t i o n t h e o r y developed

Fig. 9: Surge added-mass and damping coefRcients of a half-immersed ellipsoid {L/B = 4,

by Matsunaga and Maruo (1981, 1982) and comparable t o the 3-D Green f u n c t i o n m e t h o d of

Inoue and Makino (1989).

E x p e r i m e n t a l results f o r the surge added-mass and d a m p i n g coefficients are pubhshed i n very few papers. Comparisons are made here w i t h experiments done b y Kobayashi (1981), using a half-immersed effipsoid w i t h length r a t i o L/B = 4 and B/2T = 1.25. L/B = 4 is b l i m t considering the assumption of slender-ship theory.

E x p e r i m e n t a l values f o r F „ = 0 scatter due t o the t a n k - w a U interference, F i g . 9. F u r t h e r m o r e , the measurement of damp-i n g coeffdamp-icdamp-ient at Fn = 0.3 was n o t accu-rate, Kobayashi (1981) . W i t h these taken i n t o account, t h e 'enhanced' u n i f i e d the-o r y accthe-ounts weU f the-o r t h e fthe-orward-speed ef-fects.

Figs. 10, 1 1 , and 12 compare the wave-e x c i t i n g surgwave-e forcwave-e, hwave-eavwave-e forcwave-e, and p i t c h m o m e n t , respectively, at F „ = 0.3 w i t h experiments of Kobayashi (1981) and 3-D panel m e t h o d results o f Lin et al. (1993). T h e unified t h e o r y underestimates t h e surge exciting force, b u t captures the ten-dency. Predictions i n heave and p i t c h agree weU w i t h experiments. I n the 'en-hanced' u n i f i e d theory, t h e effects of t h e bow d i f f r a c t i o n are taken i n t o account i n t h e pressure level, a n d thus t h e heave force and p i t c h m o m e n t must, t o some extent, d i f f e r f r o m the results of o r i g i n a l u n i f i e d t h e o r y by Sclavounos (1984a).

T h e c o n t r i b u t i o n o f t h e A'^i-term w i U be understood clearer b y F i g . 13, w h i c h shows h y d r o d y n a m i c pressure d i s t r i b u -tions i n the head-sea d i f f r a c t i o n prob-l e m of A / L = 1.0, measured at Fn = 0.1 and 0.3 and at three d i f f e r e n t transverse sections (x/{L/2) = ^ = 0.793, 0.131, - 0 . 7 9 3 ) . T h e section at ^ = 0.793 is near the b o w and 6 = 90° i n the abscissa is t h e center of the section.

T h e s t r i p t h e o r y does n o t contain the 3-D effects of wave a t t e n u a t i o n along the ship, r e s u l t i n g i n t h e same pressure at = 0.793 and - 0 . 7 9 3 . I n contrast, the u n i f i e d t h e o r y accounts f o r t h e 3-D effects, and the agreement w i t h measured values is remarkable. P a r t i c u l a r l y at Fn ~ 0 . 1 , the results w i t h t h e A^i-term show a siz-able i m p r o v e m e n t over t h e results neglect-i n g the A^neglect-i-term. | E j / p g a A ^ _Fn=0.3 0.2 0.1 T Exp. by Kobayashi • Vnitiaa Theory • 3-D Panel Method 0 2 4 X / L

Fig. 10: Exciting surge force of a half inuner-sed ellipsoid in head waves at F „ = 0.3

1.5 1.0 0.5 | E j / p g a A ^ _Fii=0.3 a Exp. by Kobayaabl Unified Theory 3-D Panel Method

Fig. 11: Exciting heave force of a half immer-sed ellipsoid i n head waves at F^ = 0.3

0.2 I M / P ^ ^ \ ^ Fn=0.3 0.1 D Exp, by Kobayashi Unified Theory — 3-D Panel Method 0 2 4 X / L

Fig. 12: Exciting pitch moment of a half im-mersed ellipsoid in head waves at F „ = 0.3

Unified Theory (with nl-term) Strip Method Do. (without nl-term) O • Exp. by Kobayashi

1.4 1.2 1.0 cn Q. 0.8 CM 0.6 0.4 0.2 0 S=0.793 -60 e 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0 1=0.131 j ( Fn—0.2 1

-

--

Fig. 13: Hydrodynamic pressure distributions on a half-immersed ellipsoid {L/B = 4,

B/2T = 1.25) i n head waves of A / L = 1.0 (diffraction problem)

R^^/pga'(BVL) Fn=0.2 R A W / O Q A ' C B V L )

• Exp. at Hlroabima Univ. Valfled Theory (with al-tarmfi

Do. fwltiout: nl-term) • CBIEM Method 3-D Penal Method / [ / [ _{: • n . ^ ° ' ° ° 0} ( . . \ ^^\.= . , . , i , , . , 1

Fig. 14: Added resistance on a half-immersed spheroid of L / i J = 5 i n head waves at F „ = 0.2 (motions restrained) Exp. Basin O MHI O Osaka Univ. • RIAM A IHI V NKK Strip Method Unillad Theory

Fig. 15: Added resistance of SR108 container ship in head waves at F „ = 0.2 (free to surge, heave and pitch)

For precise p r e d i c t i o n s of t h e added resistance i n short wavelengths i t has been argued t h a t the wave d i f f r a c t i o n i n x - d i r e c t i o n near t h e bow should be t a k e n i n t o account b u t the slender-ship t h e o r y cannot, e.g. Takahashi (1987). W i t h t h i s reasoning, Fujii and Takahashi

(1975) proposed a s e m i - e m p i r i c a l f o r m u l a , and other t h e o r e t i c a l studies have been also made, Faltinsen et al.(1980), Nakamura et al.(1980), Sakamoto and Baba (1986), Ohkusu (1986). As

the wave d i f f r a c t i o n near the bow. One example of t h a t is shown i n F i g . 14, w h i c h gives the added resistance on a half-immersed spheroid of L / B = 5 at F „ = 0.2. E x p e r i m e n t a l d a t a were obtained at H i r o s h i m a University, using the unsteady wave-pattern analysis proposed b y

Ohkusu (1980). O t h e r c a l c u l a t i o n b y Lin et al.(1993), using C B I E M ( C o m b i n e d B o u n d a r y

I n t e g r a l E q u a t i o n M e t h o d ) and the forward-speed version of the 3-D Green f u n c t i o n m e t h o d w i t h flat-panel a p p r o x i m a t i o n , are also reproduced. T h e m a g n i t u d e of wave d i f f r a c t i o n effect i n the x - d i r e c t i o n is shown b y the difference between the sohd and d o t t e d hnes p r e d i c t e d b y the u n i f i e d theory. T h i s difference is essential i n the p r e d i c t i o n of the added resistance i n head waves.

A n o t h e r example of the added resistance is shown i n F i g . 15 for the SR108 container ship free t o surge, heave and p i t c h , r u n n i n g at F „ = 0.2 i n head waves. Due possibly to wave breaking or related phenomena, t h e p r e d i c t i o n i n short wavelengths is s t i f i smaUer t h a n measured forces, b u t the agreement becomes better as the wavelength increases. I n head waves, the encounter frequency is r e l a t i v e l y h i g h at F „ = 0.2, w h i c h may be a good circumstance for the s t r i p t h e o r y shown b y the dashed hne. However, the u n i f i e d t h e o r y stands o u t i n t h a t i t gives stable and favorable results f o r aU heading angles, Kashiwagi and Ohkusu (1993).

T h e results demonstrated above are a f i related t o ' l o n g i t u d i n a l ' ship motions. A s summarized i n A p p e n d i x 1, t h e u n i f i e d t h e o r y can be apphed t o the l a t e r a l s h i p - m o t i o n problems t o o , t a k i n g account of the 3-D and forward-speed effects. However, since the s t r i p t h e o r y is vahd at the h m i t i n g cases w —> 0 and w —» oo, no correction terms are needed i n t h e u n i f i e d t h e o r y at those h m i t i n g cases. I n f a c t , unhke ' l o n g i t u d i n a l ' motions, 3-D effects on the l a t e r a l m o t i o n s are n o t so large, and forward-speed effects may be more p r o m i n e n t , Kashiwagi (1985).

0 5 10 KL 15 0 5 10 KL 15 Fig. 16: Added-moment of inertia and damping coefRcient i n yaw of a half-immersed spheroid of L / B = 5 at C/ = 0

F i g . 16 shows t h e 3-D effects on the added moment of i n e r t i a and d a m p i n g coefficient i n yaw mode at [ / = 0 f o r a h a f f - i m m e r s e d spheroid of L / B = 5. I n the h m i t w ^ 0, the theory shown i n A p p e n d i x 1 includes no 3-D corrections (the t e r m \k\^y hi{\k\R) appearing i n the first hne on the r.h.s. of ( A . 3 ) is i g n o r e d ) . T h e r e f o r e the u n i f i e d t h e o r y is i d e n t i c a l t o the s t r i p t h e o r y at w —> 0, which is d i f f e r e n t f r o m t h e result of the 3-D panel m e t h o d . Except f o r t h a t p o i n t , t h e u n i f i e d t h e o r y accounts for t h e 3-D effects over the w i d e range of frequencies. I t is n o t e w o r t h y t h a t t h e 3-D effects on the sway mode are smafier t h a n i n F i g . 16, b u t q u a h t a t i v e l y the same. I t is stiU d i f f i c u l t t o make a d e f i n i t i v e j u d g m e n t on the Froude number range i n w h i c h the u n i f i e d t h e o r y is expected t o give relatively good results. J u d g i n g f r o m comparisons w i t h exper-iments i n the past pubhshed papers, i t seems t h a t F „ = 0.25 0.3 is a h m i t i n g value. For the Froude n u m b e r higher t h a n t h a t , pseudo 3-D methods hke H S S B T m a y be recommended. For comparison. F i g . 17 shows results of the u n i f i e d theory, c o m p u t e d by Newman and Sclavounos

(1980), and F i g . 18 corresponding results of the H S S B T , c o m p u t e d by Yeung and Kim (1981);

b o t h are compared w i t h the same experiments at F „ = 0.35. A p p a r e n t l y cross-couphng t e r m s between heave and p i t c h are w e l l predicted b y H S S B T , b u t the degree of agreement i n the heave added-mass and d a m p i n g coefficients is more or less the same.

T h e forward-speed t e r m i n the free-surface c o n d i t i o n influences the cross couphng coeffi-cients as a correction of 0{U) t o t h e s t r i p theory, whereas no corrections are necessary on t h e heave d i a g o n a l t e r m s and 0{U'^) corrections on the p i t c h diagonal terms, Ogilvie and Tuck

(1969). These t h e o r e t i c a l results are c o n f i r m e d numericaUy b y Faltinsen (1974). RecaUing t h a t

H S S B T encompasses the r a t i o n a l s t r i p theory, we can expect t h a t the cross-couphng coefficients predicted b y H S S B T agree weU w i t h experiments.

C o m p u t a t i o n s f o r sway and yaw motions using H S S B T were i n i t i a t e d b y Chapman (1975,

1976), and foUowed b y Kashiwagi and Hatta (1984), Kashiwagi (1984), and Yamasaki and Fujino (1983, 1984, 1985). I n p a r t i c u l a r , Y a m a s a k i and F u j i n o conducted extensive comparisons

w i t h experiments f o r a flat plate, W i g l e y ship. Series 60, and SR108 container ship models, showing encouraging agreement.

A c c o r d i n g t o t h e comparison w i t h experiments f o r the SR108 container ship by Takaki and

Tasai (1973), H S S B T accounts weU f o r the forward-speed effects compared t o the s t r i p theory.

However, at F „ = 0.15, the degree of agreement seems n o t enough i n low frequencies. I n t h a t range, 3-D i n t e r a c t i o n effects among transverse sections become i m p o r t a n t , w h i c h are n o t taken i n t o account i n H S S B T .

Troesch (1981) extended the r a t i o n a l s t r i p t h e o r y t o the l a t e r a l m o t i o n p r o b l e m and

Fig. 18: Radiation force coefRcients of a frigate hull {Cb = 0.55) due to heaving at F „ = 0.35 (computed by Yeung and Kim (1981) using high-speed slender-body theory)

Troesch's results are a p p a r e n t l y superior t o the s t r i p t h e o r y b u t n o t so good as compared t o the results of H S S B T .

9. C o n c l u d i n g R e m a r k s

T h e u n i f i e d t h e o r y encompasses the s t r i p theory and includes t h e 3-D and forward-speed effects. However, f o r higher Fb-oude numbers, we had better rely on a pseudo 3-D m e t h o d hke H S S B T , a l t h o u g h its n u m e r i c a l calculation scheme is n o t fuUy vahdated. F u r t h e r m o r e , H S S B T m u s t become faster t o serve as a p r a c t i c a l t o o l hke s t r i p theory.

I t has been said t h a t the slender-ship t h e o r y cannot account for the wave d i f f r a c t i o n i n x-d i r e c t i o n near t h e ship enx-ds anx-d t h a t t h e surge mox-de shoulx-d be treatex-d uncouplex-d using only the F r o u d e - K r y l o v force. However, these defects are resolved b y the 'enhanced' u n i f i e d t h e o r y by r e t a i n i n g t h e x-component of t h e n o r m a l vector i n the b o d y b o u n d a r y c o n d i t i o n . N u m e r i c a l examples showed t h a t t h i s t h e o r y can predict reasonably surge-related h y d r o d y n a m i c forces and also r e m a r k a b l y improves t h e pressure d i s t r i b u t i o n and the added resistance i n head waves.

R e f e r e n c e s

A D A C H I , H . (1977), On the interaction between head sea waves and slender ship (part 1 and 2), Rep. of Ship Res. Inst. 14/4 and 14/6 (in Japanese)

A D A C H I , H.; O H M A T S U , S. (1977), Ship motion analysis by slender-ship theory (part 1), Rep. of Ship Res. Inst. 14/5 (in Japanese)