Generating and propagation of moored ship screw race jet

GEiïBRATTlïG MHD PROPAGATION OP MOORED S H I P SCREW RAC£ JET V/, R o b a k i e w i o s and V7, S u l i s z

I n s t i t u t e o f H y d r o e n g i n e e r i n g FoXiQh Acadern,/ o f S c i e n c e s , Gdailsk 1 . IITRODUCTION

A r o t a t i n g o h i p ' a p r o p e l l e r i s some k i n d o f a s o u r c e o f '/atcr c t r e a m 'ïrlf>h t h e d e f i n i t e mass f l o w , w h i c h depends on t h e m o t i o n p a r a m e t e r s and g e o m e t r i c a l d i m e n s i o n s o f t h e p r o p e l l e r . The p r o p e l l e r , when r o t a t i n g i n t h e w a t e r e n v i -ronment, g i v e n the d e f i n i t e d i r e c t i o n and v e l o c i t y t o t h e ••vater s t r e a m o f doaraeter e q u a l a p p r o x i m a t e l y t o t h e p r o p e l -•J<:r d i a m e t e r . I f t h e s h i p ' s s t e r n i s s i t u a t e d n e a r t h e w h a r f t h e n tlie p r o p e l l e r g e n e r a t e d w a t e r s t r e a m p r o d u c e s a s t r o n g ••.'ater m o t i o n b e i n g a cause o f t h e b o t t o m e r o s i o n i n t h e s t e r n v i c i n i t : / and a l s o o f t h e ground l o s s e s u n d e r t h e w h a r f s t r u c t u r e and b e h i n d i t . F o r m e r l y t h e s t a b i l i t y o f quay

stï"ncturGs v/as endangered by t h e screw r a c e e f f e c t o n l y i n s h i p b u i l d i n g and rshipre p a i r i n g y a r d s - i n ' t h e ^ - b b l i a r d -'teSt^^ a r e a s . W i t h i n i n c r e a s i n g power o f s h i p e n g i n e s and w i t h i n t h e h a r b o u r operation'téchniqUe'bëing'changëd, t h e b o t t o m erosj.on a p p e a r e d a l s o i n o t h e r h a r b o u r a r e a s .

\-] i; h ; , p /^P'-"}-;?-!,'".'^: Pn p

I t i s the.ref'öti''• mo&oBavf tö'uh'dèrBtMd wéli-^he phenomena o c c u r i n g d u r i n g t h e s h i p * s p r o p e l l e r o p e r a t i o n , t o t r y t o d e t e r m i n e t h o . w a t e r v e l o c i t y f i e l d i n t h e p r o p e l l e r a r e a and t o khoW-hov) t o p r e v e n t t h e a d v e r s e e f f e c t s o f t h e screw r a c e or_how t o make a l l o w a n c e f o r i t d u r i n g t h e d e s i g n w o r k f 02^ • the-ïiëw- b r Pno quays'^aïid wh'a3!'ves.'^'' " •"•* ' • The' problems;^ o f • screw r a c e advei'se ëff e c t s • UpM"^ thö b o t t o m n e a r thé"'harbouï^' quays- a r e b e i n g • investigated'^by'-^.Ift§tj;tute

öf H y d r o e h g i n e e r i n g o f P o l i s l i ' A c a d e m y o f Scièncóö/in 'Gdansk f r b r f l _ t h e - ' b e g i n n i n g - o f • ^ , ' ' ' - .-p-T h i s p'aper ' c o n t a i f t s c o n f irméd • aiid 'próposed • thé'óf-etiöal• So-'

luiiönsp o f '-the söï^eW ra.ce probrëms' 'based öli''the-aü'thór'^ s own'''èxpe^ijiönts-^ ^•'••/^ '" ^ /ro-ir-o IP-,;-: t p -p-.; "

2, DETfimvIIHATJOW OP VELOCITY GENERATED BY SHIP PROPELLER 2 . 1 , Method o f v e l o c i t y d e t e r m i n a t i o n based on t h e p r o p a g a -t i o n o f aubrnerged w a -t e r .je-t

The method u s i n g t h e o r y and p r i n c i p l e s o f t h e f r e e submerged j e t p r o p a g a t i o n i s now g e n e r a l l y u s e d f o r t h e d e t e r m i n a t i o n o f . t h e s h i p p r o p e l l e r g e n e r a t e d v e l o c i t i e s . Such p r o c e d u r e i s a d m i s s i b l e p r o v i d e d t h a t t h e screw r a c e p r o p a g a t i o n i s s i m i l a r t o t h e f r e e submerged j e t p r o p a g a t i o n . ' L e t u s assume t h a t ; / B u k o w s k i 1970/: 1 , kn a c t u a t o r d i s c i s a c i r c u l a r a r e a /Pp/ o f d i a m e t e r e q u a l t o t h e p r o p e l l o r d i a m e t e r . Bach p a r t i c l e o f t h e f l u i d f l o w i n g t h r o u g h t h i s a r e a i s s u b j e c t e d t o an a b r u p t r i s e o f t h e p r e s s u r e .

2, A s e p a r a t e d column o f t h e f l u i d i n the f o r m o f a body o f r p v o l u t i o n around t h e a c t u a t o r d i s c a x i s appears as a r e -s u l t o f t h e a c t u a t o r d i -s c a c t i o n . The d i a m e t e r o f t h i s

body o f r e v o l u t i o n i s e q u a l t o t h e a c t u a t o r d i s c d i a m e t e r / P l g . i / .

A l l 'tii^''ohBnèeö o f the iöomentum''aüè'to ^thë-actüatór''di!3c a c t i o n a r e , , l i m i t e d , w i t h i n t.he s a i d column o f f l u i d b e i n g

.;..^oa-~beafó:.-:0f..:.5.treamll tti6-i'felatüat.o±i.ia3ÏsöP, .-:™.:lmdi.of^tethëi:^ü^e t h e screw r a c e ,

4^i; The' ^^elo'bit|/Sfa7^ Upstream: fi^otn' thö'^ctua^ör • diSc'^ilè'-'êMal V • t o 'theyufiiformV U h d i s t t i r b b d ' f l o w - s j - e i o b i t ^ ' / ^ f - /i '•^P&F'aó^B-'-^ sti'sate;!'™ the ; disc,•'aë^a 'reMit'•öf U h e a e S ^ l e r a t i a n r

-th:ë"fliiid veïööity y^cii^V" oorov/ -::p pr'-p'ap;o.r-Lnn i.^j 5. The v e l o c i t y d i s t r i b u t i o n in'thé succësive'cross s e c t i o n s

p. • of, the,; screws race.-is uniformv-p/,.

6, The screw r a c e p r e s s i i r e s , f a r . u p s t r e a m ; f r o m - t h e . - a c t u a t o r d i s q a r e e q u a l t o t h e ambient^ p r e s s u r e >/{PQ/«^ " H- -¬

7. The' f l u i d , s t r e a m u p s t r e a m and' downstream f r o m ' thë'-aëtea-tbr„disc i s dèfihed by the f l o w c o n t i n u i t y e q u a t i o n and

by . ^ e r n o u l l i t s Mtheorem.:, - ; ] ,,, , , Making- use oi the'.above assurapti'ons'.we,,5cahVwr0e'."'3erno^

e q u a t i o n , . f o r c r o s s - s e c t i o n 1:,/Pig.1/ and^ iq-r'-ihe - qrosa-^aec- . t i o n d i r e c t l y b e f o r y t h e d i s c : ' " " " " •

For t h e croöa s e c t i o n d i r e c t l y b e h i n c i t h e d i s c and f o r c r o s s s e c t i o n 3 / F i g . 1 / B e r n o u l l i ' s e q u a t i o n g i v e s :

T a k i n g i n t o c o n s i d e r a t i o n t h e f a c t , t h a t t h e p r o p e l l e r g i v e s some amount o f energy t o t h e - j e t , t h e f o l l o w i n g f o r m u l a can bs f o u n d f r o m e q i i a t i o n s and /2.1,2/ /Robakiev;icz

1966^;/:

ïlv.::: U r i Lhrust . ' d l l ba:

T = F , . . A p = ^ « - V „ ^ ) /2.1.3/

Glnco i;ho u r o p e l j c r thTOGt can a l s o be w r i t t e n i n t h e f o r m :

T=m(V3-Vo)=^,F2.V2(V3-Vo) /2.1.4/ t h e n t h e f o l l o v . i l n g f o r m u l a can be f o u n d f r o m /2.1,3/ and

/ 2 J . 4 / :

/2.1,5/

v/hersas the equati.on of c o n t i n u i t y g i v e s :

/ 2 , U 6 /

where: D,, and are r e s p e c t i v e d i a m e t e r s o f t h e p r o p e l l e r and of tfie j e t i n c r o s s s e c t i o n s 2 and 3«

Assuming t h a t V ^ 0 t h e f o l l o w i n g v a l u e o f t h e f l o w v e l o c i

-t y i n the proneTler g e n e r a t e d s t r e a m can be o b t a i n e d :

where:

rn„ i.s t h e number o f p r o p e l l e r r e v o l u t i o n s p e r second s Krp i s t h e t h r u s t c o e f f i c i e n t , w h i c h can be f o u n d f r o m t h e p r o p . e l l e r c h a r a c t e r i s t i c s v e r s u s t h e p i t c h c o e f f i c i e n t and thcj advance c o e f f i c i e n t 2«g, V e l o c i t . v d e t e r m i n a t i o n w i t h use o f p r o p e l l e r g e n e r a t e d v o r t e x system method T-he s i m p l i f i e d method o f d e t e r m i n a t i o n o f a h i p * s p r o p e l l e r g e n e r a t e d f l u i d v e l o c i t i e s p r e s e n t e d i n t h i s p a r a g r a p h a s -sumes, t h a t a system o f v o r t i c e s g e n e r a t e d by t h e p r o p e l l e r e x i s t s i . e . t h e p r o p e l l e r hub g e n e r a t e s a r e c t l i n e a r v o r t e x whereas each o f t h e b l a d e s g e n e r a t e s a s p i r a l v o r t e x . The c h a r a c t p r i a t i c v a l u e s of t h e v o r t e x system / c i r c u l a t i o n , v o r t e x shape/ depend on t h e p r o p e l l e r geometry and i t s p a r a -n i e t e r s of m o t i o -n . I f 'the r o r t c x system c h a r a c t e r i s t i c s i s Imo-.n t h e n the p r o p e l l e r g e n e r a t e d v e l o c i t i e s i n any p o i n t o f th;: v o N o c i t p f i e l d r e s p e c t i v e I 7 f a r dovmstreajn f r o m t h e propel;: o r c?in be f ound, S p e c i a l a t t e n t i o n w i l l ba p a i d t o th^^ a^/oragp y e l . p o l t i e s due t o t h e o:iraple r e l a t i o n s a l l o w i n g t o f j n d t h o s e v a l u e s .

l o w 'let ua a c c e p t t h e f o l l o w i n g b a s i c a s s u m p t i o n s and d e f i -n i t i o -n s /TuszUowska a-nd K o r o -n o w i c z 1971, S u i i s a 1984/:

1, The f l u i d i s i n c o m p r e s s i b l e and non v i s c o u s .

2, The g r a v i t y f o r c e s as b e i n g s m a l l when compared w i t h o t h o r f o r c p s a c t i n g upon the f l u i d , can be n e g l e c t e d , 3, The screw p r o p e l l e r h a v i n g K b l a d e s and t h e r a d i u s e q u a l

t o R.; r o t a t e s w i t h c o n s t a n t a n g u l a r speed i n x m r e s t r i o -t e d f l u i d apace-,

4, I'ho P'arameters of p r o p e l l e r i . e , t h r u s t c o e f f i c i e n t Vi^, t o r q u e c o e f f i c i e n t Kq and p r o p e l l o r p i t c h H are knovmt 5, A s p i r a l v o r t e x i s g e n e r a t e d by each o f t h e b l a d e s . The. v e l o c i t y c i r c u l a t i o n o f each o f t h e s p i r a l v o r t i c e s i s e q n a l t o r . A d d i t i o n a l l y t h e p r o p e l l e r g e n e r a t e s a core-v o r t e x w i t h c i r c u l a t i o n e q u a l t o *r-K i n t h e sense oppo-s i t e t o t h e oppo-s p i r a l v o r t e x v e l o c i t y c i r c u l a t i o n . 6, Par downstream f r o m t h e p r o p e l l e r / p r a c t i c a l l y a l r e a d y i n t h e d i s t a n c e e q u a l t o t h e p r o p e l l e r r a d i u s Rp/ t h e s y s t e m of K t i p v o r t i c e s i s p o s i t i o n e d on t h e a x i a l " c y l i n d r i c a l s i i r f a c e . The c y l i n d e r a x i s i s i d e n t i c a l w i t h t h e p r o p e l l e r a x i s and v a t h the a x i s o f t h e core v o r t e x , t h e cvjLinder r a d i u s i s e q u a l t o R.,. Each o f t h e t i p v o r t i c e s has a f o r m

7« Par u p s t r e a m f r o m t h e p r o p e l l e r t h e v e l o G i t y _ f i e l d /T / H i l l be s t a t i o n a r y , u n i f o r m and r e c t l i n e a r /T\\ coi /. The p r e s s u r e f a r u p s t r e a m and f a r f r o m t h e p r o p e l l e r a x i s i s e q u a l t o p ^ , 8. The f l u i d r e l o c i t y has a p o t e n t i a l e v e r y w h e r e e x c e p t t h e v o r t e x c o r e s . M a k i n g use o f t h e above l i s t e d a s s u m p t i o n s t h e v e l o c i t i e s f o r a p o i n t s i t u a t e d f a r downstream f r o m t h e s h i p ' s p r o p e l -l e r w i -l -l be f o u n d . The v e l o c i t y i n d u c e d by t h e v o r t e x e l e m e n t d l i n P p o i n t / P i g . 3 / w i l l be f o u n d f r o m B i o t - S a v a r t e q u a t i o n /Prosnak 1970/. MP 4ir / 2 . 2 . 1 / R MP I •/ihere P i s t h e v o r t e x c i r c u l a t i o n .

The v e i o c i f c y i n d u c e d by the system o f K r e g u l a r screw l i n e v o r t i c e s as w e l l as by t h e c o r e v o r t e x i n a p o i n t s i t u a t e d f a r downstream f r o m t h e p r o p e l l e r i s f o u n d by t h e i n t e g r a t i o n o f / 2 , 2 . 1 / e q u a t i o n , Por e y l i j n d r i c a l c o o r d i n a t e s / x , ( ^ , 4 ' / y/e; o b t a i n : V3x = 4JrRo

r

M /2.2.2/ /2.2.3/ MP 4arR ^ vjhere 3 k--0 1 P K 1 MP /2.2.4/ 1 r,, = R/R3 R ~ r a d i u s v e c t o r i n r e s p e c t t o t h e a x i s o f r e v o -l u t i o n X, ^„ ~ a n g u l a r c o o r d i n a t e o f v o r t e x element d l on z e r o v o r t e x l i n e /k--0/ % a n g u l a r c o o r d i n a t e o f t h e p i m c t u r e p o i n t o f the z e r o v o r t e x l i n e vi;ith t h e p l a n e p e r p e n d i -c u l a r t o t h e a x i s a t x = 0 .

M a l f j n g unc o f /2.2,2/ - /2.2,4/ r e l a t i o n s we can f i n d t h e i n d u c e d v e l o c i t i e s i n any p o i n t P l o c a t e d r e s p e c t i v e l y f a r downstream f r c m i t h e 3h.ip*s p r o p e l l e r fox* t h e knov.-n v a l u e s o f r , t g ^ 3 and R ,, Since P , tgQ>3 and depend on t h e p r o p e l l e r geometry add. the p a r a m e t e r s o f m o t i o n we w i l l now r e -ro,t8 t h o n s v a l u e s 7;ith t h e b a s i c c h a r a c t e r i s t i c v a l u e s o f the p r o p e l l e r m o t i o n and o f the- p r o p e l l e r i t s e l f /V , co^ , R.-,, K,v,? Iv,s OS o i l l use t o t h i s p u r p o s e t h e r e l a t i o n s f o r t h e p f e o p e l l e r thi.nast and t o r q u e i n 2:'6spect t o t h e a x i s o f r o t a t i o n . The a x i a l t h r u s t /T/ and t h o t o r q u e i n r e s p e c t t o t h e a x i s of r o t a t i o n /Q/ can be f o u n d f r o m t h e f o l l o w i n g f o r m u l a e : T = ?c < K. S '2 2 Q=?c < d; / 2 . 2 . 5 / /2,2.6/ t!U'qu(j c o e f f i c i e n t . Tho t h r u p t and t o r q u e i n r e s p e c t t o t h e p r o p e l l e r a x i s o f r o t a t i o n s h o u l d be e q u a l t o t h e r e s p e c t i v e v a l u e s f o u n d by i n t e g r a t . i - o n of t h e t e r m s d a t e r r a i n i n g t h e t h r u s t and torque-f o r the torque-f l u i d /Prosnak i 9 7 0 / . f t i O " i n t r o d u c t i o n o f t h e cont3;'ol s u r f a c e v;e v / i l l o b t a i n •I r e s ' u l t o f t h e i n t e g r a t e d terms, a n a l y s i s / S u l i s z 1984/: V'3.\ 1 Vo ^3 Vo r p^dS / 2 , 2 . 7 / /2.2.8/ t g ^ 3 i s f o u n d f r o m t h e f o l l o w i n g f o r m u l a / P i g . 4 / : A3 Vo / 2 . 2 . 9 /

w h e r e i a advance c o e f f i c i e n t ' co^H31 whereas S,, i s

a s e c t i o n o f t h o p l a n e p e r p e n d i c u l a r i o t h e p r o p e l l e r - ^ a x i s and s i t u a t e d f a r u p s t r e a m from, t h e p r o p e l l e r e x c l u d i n g t h e t r a n s v e r s e s e c t i o n " ai-'eas o f t h e v o r t e x c o r e s /S^.,/ i n c l u d e d

i n t o t h i s p l a n e , ^ G&t o f e q u a t i o n s /2.2,7/ - /2,2.9/ where t h e v e l o c i t y

com-D o n o n t s a r e g i v e n by r e l a t i o n s /2.2.2/ - /2.2,24/ s h o u l d be s o l v e d bv i t e r a t i o n i n r e s p e c t t o P, t g ^ a a n d A g . H a v i n g f o u n d t h e s e v a l u e s we can f i n d t h e v e l o c i t y f i e l d f a r down-s t r e a m f r o m /2.2,3/ - /2.2,4/ f o r m u l a e . 2 ^ 3 . . C o m p u t a t i o n a l example By u s i n g t h e method d e s c r i b e d u n d e r 2.1 and 2,2 a v e l o c i t y g e n e r a t e d by t h e p r o p e l l e r o f a t y p i c a l c o n t a i n e r c a r r i e r w i l l be c a l c u l a t e d .

L a t u,:; apsunie, t h a t t h e prone H e r a x i s i s 4 ra u n d e r t h e wa-t e r J o v e l and wa-t h a wa-t : K = 4, D.- 5,8 m, H ^ 3,87 m, K^^ =^ 0 , 3 1 , Kq ^ 0,033, Kq /Iv. - 0 / ' - 0,005; mg - i s " " ' .

M a k i n g use o f t h e method d e s c r i b e d u n d e r 2.1 / t h e o l d m e t h o d / we h a v e :

vv^ 5,15 m/s ; = 4,10 m

whereas f r o m the method d e s c r i b e d u n d e r 2.2 / t h e new m e t h o d / o b t a i n :

V.^ - 5,33 m/s ; = 1 J 5 / r v m/s ,/0,083 < r^^ < 1 / ; V.^ = 0;

D3 " 4,23 JT!

where , V,,^ , V., are r e s p e c t i v e mean c i r c u m f e r e n t i a l v a l u e s t^T the-^axial^'* c i r c u m f e r e n t i a l and r a d i a l v e l o c i t y . The v a l u e s o f t h e a x i a l v e l o c i t y and d i a m e t e r o f t h e j e t

o b t a i n e d by u s i n g t h e method d e s c r i b e d u n d e r 2,1 and 2.2 are a p p r o x i m a t e l y t h e same. The new method g i v e s us t h e v a l u e s o f c i r c u m f e r e n t i a l and r a d i a l comiponants o f t h e v e -l o c i t y g e n e r a t e d by t h e p r o p e -l -l e r . Such are: t h e two main a d v a n t a g e s o f t h e new method.

3. DETERMIMTIOïï OP VELOCITY I N THE SCREW RACE JET

The second stage o f t h e v e l o c i t y f i e l d d e t e r m i n a t i o n assumes a p a t t e r n o f t h e scre-.v r a c e p r o p a g a t i o n - t h e screw r a c e j e t b e i n g d i v e r g e d dovmstreara aa a r e s u l t o f t h e m.om.entum e x c h a n -go bet?;een t h e j e t and t h e e n v i r o n m e n t a l f l u i d .

Duo t o t h e v e r y c o m p l i c a t e d c h a r a c t e r o f t h e phenomena appe-a r i n g i n t h e screw r appe-a c e j e t , t h e t h e o r e t i c appe-a l appe-a n appe-a l y s i s happe-as been g i v e n up and t h e f u r t h e r i n v e s t i g a t i o n s have been made i n t h e h y d r a u l i c l a b o r a t o r y . As f a r as p o s s i b l e t h e r e s u l t s were v e r i f i e d by t h e f u l l s c a l e t e s t s .

T r a w l e r o f B-20 t y p e has been chosen f o r t h e l a b o r a t o r y t e s t s becouse, f o r t h i s t y p e o f s h i p , t h e f u l l s c a l e t e s t r e s u l t s were a v a i l a b l e and a l s o becouse o f t h e a d v a n t a g e o u s parame-t e r s e n a b l i n g parame-the p r o p e r s c a l i n g f o r parame-t h e model parame-t e s parame-t s . The i - e o h n i c a l c h a r a c t e r i s t i c s o f B-20 t r a w l e r a r e as f o l l o w s : S h i p - l e n g t h 55,1 ra - d r a u g h t 5,0 m d e a d w e i g h t 500 t

- main engine r a t e d power 1375 HP / m e t r i c / P r o p e l l e r number o f p r o p e l l e r s 1 " d j . r e c t i o n o f r o t a t i o n c l o c k w i s e - diarnxstar 2,5 m - p i t c h H 1,7 ra - number o f b l a d e s K 4 - hub r a d i u s r 0, 2 7 m r ^ - expanded a r e a S 2,541 m'^ ~ expanded a r e a r a t i o ^KJ^ 0,649 /o " p r o p e l l e r d i s c a r e a / - p i t c h r a t i o H/Dp 0,68 t h r u s t c o e f f i c i s n t 0,3 f o r J = 0 / J - advance r a t i o / - maximum r o t a t i o n a l speed 275 r.p.m.

Two models were made, one t o t h e s c a l e 1t20 and a n o t h e r one t o 1:40, Due t o t h e p r o p c r R e y n o l d s numbers b o t h t h e models met t h e c o n d i t i o n s f o r t h e t u r b u l e n t f l o w . The f o l l o w i n g t e s t , p a r a m e t e r s were a d j u s t a b l e : - p r o p e l l e r r o t a t i o n a l speed /5 s t e p s / , ~ t h r u s t d i r e c t i o n / f o r w a r d - a s t e r n / , - r u d d e r a n g l e , - d r a u g h t , - t r i m .

The t e a t s v)e3?e made i n t h e c h a n n e l v / i t h s o l i d b o t t o m and w i t h r u b b l e b o t t o m . V e l o c i t y meaf3urements were made and w a t e r s t r e a m p r o p a g a t i o n a f t e r t h e p r o p e l l e r was o b s e r v e d i n t h e s o l i d b o t t o m c h a r u i e l whereas t h e t r o u g h f o r m a t i o n p r o c e s s and t h o s t r e a m p a t t e r n i n t h e t r o u g h were o b s e r v e d i n t h e r u b b l e b o t t o m c h a n n e l . The v e l o c i t y was measured by means o f a h y d r o m o t r i c c u r r e n t m e t e r f o r p r o f i l e s on t h e model e v e r y 5 ora and on t h e each p r o f i l e e v e r y 5 cm. o f t h e d i s t a n c e . F o r a p a r t o f t h e t e s t s t h e m e a s u r i n g p o i n t s were more s e p a r a t e d . The w a t e r j e t s were o b s e r v e d by p h o t o g r a p h i n t h e r o d s p r o v i d e d w i t h t h r e a d s as w e l l as by f i l m i n g t h e " c o l o u r e d w a t e r o r f i l m i n g t h o s p e c i a l m i x t u r e a p p l i e d t o t h e . p r o p e l l e r d i s c . The b o t t o m e r o s i o n changes i n t h e l a b o -r a t o -r y we-re p h o t o g -r a p h e d t h -r o u g h t h e c h a n n e l window d u -r i n g t h e e x p e r i m e n t and sounded a f t e r c o m p l e t i o n o f t h e t e s t s . The :jet p a t t e r n was photog3.-aphed t h r o u g h t h e c h a n n e l window.

3.'^. Iraago o f t h e d i v e r g i n g .jet

For a c h i p i t hout t h e r u d d e r t h o j e t d i v e r g e n c e i s e q u a l i n a l l direcL-:';,ons^,,and has a c o n i c a l shape w i t h apex a n g l e e q u a l t~: a b o u t 35'", The r u d d e r b l a d e p o s i t i o n e d i n t h e j e t a x i s / r u d d e r a n g l e 5= 0 / d i v i d e s t h e j e t i n t o t h e s u r f a c e s t r e a m , •/•hich i s p r o p a g a t e d t o w a r d s t h e f r e e s u r f a c e , and t h e b o t t o m s t r e a m p r o p a g a t e d t o w a r d s t h e b o t t o m and quay / R o b a t i e w i c z 1966-2/, /issuming, t h a t t h e screw r a c e cone apex i s p o s i t i o n e d a f t f r o m t h e p r o p e l l e r d i s c i n t h e d i s -t a n c e o q u a l -t o -t h e p r o p e l l e r radius", R o b a k i e w i c z /1966-2/ f o u n d o x o e r i m e n t a l l y , t h a t t h e screw r a c e d i v e r g e n c e t o -wards t h e b o t t o m I s e q u a l t o 2 5"', 22. - 23° t o w a r d s t h e quay ;ind > 6" t c v a r d s t h e open w a t e r s i d o , Lioreover he has f o u n d , t h a t t h e b o t t o m s t r e a m a x i s i s i n c l i n e d t o w a r d s t h e b o t t o m by 15 and t o w a r d s t h e quay by 1 1 " ,

F i g s 3 t h r o u g h 6 show t h e examples o f screw r a c e j e t v e l o c i t y h o r i z o n t a l compon6.nt d i s t r i b u t i o n s • f o i i n d f r o m t h e l a b o -r a t o -r y t e s t s w i t h t h e model o f 3-20 t -r a w l e -r made t o 1:20 s c a l e ,

3,2, V e l o c i t y changes i.n t h e screw r a c e j e t

The l a b o r a t o r y t e s t s w i t h t h e model o f B-20 t r a w l e r have shown, t h a t t h e screw r a c e j e t v e l o c i t y d i s t r i b u t i o n i s a s o p h i s t i c a t e d f u n c t i o n o f t h e measured p o i n t l o c a t i o n . T h i s d i s t r i b u t i o n i s even more c o m p l i c a t e d f o r t h e s h i p w i t h X'udder and t h i s case i s j u s t t h e most i n t e r e s t i n g f r o m t h e p r a c t i c a l p o i n t o f v i e w . T h e r e f o r e and a l s o becouse t h e maximum v e l o c i t i e s a p p e a r i n g i n t h e screw r a c e a x i s a r e t h e most d e t r i m e n t a l , t h i s paper w i l l d e a l o n l y w i t h t h e s e v e l o c i t i e s .

An e m p i r i c a l f o r m u l a f o r c a l c u l a t i o n t h e screw r a c e j e t a x i s v e l o c i t y has been f o u n d w i t h use o f v e l o c i t y d i s t r i -b u t i o n s o -b t a i n e d f o r s e p a r a t e secti.ons f r o m t h e l a -b o r a t o r y t e s t s o f t r a w l e r 1:20 s c a l e m o d e l , Por t h e s h i p w i t h r u d d e r /5==0/, Robalriewicz has f o u n d , by means o f t h e g r a -p h i c method, t h e f o l l o v . d n g f o r m u l a f o r t h e v e l o c i t y i n t h e screw r a c e a x i s : V f l o w v e l o c i t y i n t h e p o i n t s i t u a t e d one p r o p e l l e r r a -d i u s a f t from, t h e p r o p e l l e r -d i s c , V. - maximum v e l o c i t i e s a i o n g t h e a x i s o f a gj.ven j e t , ^ ••- s c a l e c o s f f i c i e n t / s c a l e 1 : o C / , X ™ d i s t a n c e a l o n g t h e s h i p a x i s f r o m t h e measured p o i n t t c t h o p o i n t s i t u a t e d one p r o p e l l e r r a d i u s a f t f r o m th(: p r c o e l l e r d i s c . F o r m u l a / 3 . 1 / t h o u g h d e v e l o p e d f o r v e l o c i t i e s i n the. s u r -f a c e s t r e a m a x i s , i s a l s o v a l i d -f o r t h e b o t t o m a x i s . T h i s f a c t has been confirm.ed hy c a l c u l a t i o n s f o r t h e measured b o t t o m a x i s v e l o c i t i e s . P i g s 7 and 8 show t h e screw r a c e

j e t a x i s v e l o c i t i e s o b t a i n e d f r o m t h e c a l c u l a t i o n s and t h o s e o b t a i n e d f r o m t h e l a b o r a t o r y experim.ents. I t s h o u l d b a d d e d p t h a t due t o ver.y s t r o n g t u r b u l e n c e i n t h e p r o p e l l e r v i c i n i t y , t h i s f o r m u l a can be p r a c t i c a l l y used f o r d i s t a n c e s g r e a t e r than one p r o p e l l e r d i a m e t e r f r o m t h e p r o ¬ p e 11 e r 1 o c a t i on,

F o r m u l a / 3 . 1 / was i n t h e p a s t many t i m e s v e r i f i e d by mo-d e l t e s t r e s u l t s / e x p e r i m e n t s -in P o l i s h , Swemo-dish anmo-d D u t c h l a b o r a t o r i e s / as w e l l as by f u l l s c a l e t e s t s . These v e r i -f i c a t i o n s have c o n -f i r m e d t h e g r e a t u s e -f u l n e s s o -f t h i s em-p i r i c a l f o r m i u l a . The f o l l o w i n g examem-ple shows t h e r e s u l t s o f t h e r e c e n t measurements f o r a t y p i c a l c o n t a i n e r c a r r i e r as w e l l as t h e v e l o c i t i e s c a l c u l a t e d w i t h use o f / 2 , 2 , 2 / ~ / 2 . 2 . 4 / and / 3. 1 / f o r m u l a e . Bxamnle: . Dp = 5 , 8 m , = 0 , 3 1 , KQ = 0 , 0 3 3 , m = 0,5 s~', t h e p r o p e l l e r a x i s i s 5 m above t h e b o t t o m 8 f t h e b a s i n : b o t t o m v e l o c i t y i n t h e p r o p e l l e r a x i s 3 0 m f r o m t h e p r o -p e l l e r d i s c has been measured, t h e r e c o r d e d v a l u e vms

1 , 1 , m/a,

- t h e c a l c u l a t e d v a l u e f o r t h i s p o i n t v/as V„ = 0 , 9 m/s.

4 , ADDITIONAL PARAMETERS AFFECTING THE SCREW RACE JET VELOCITIES 4 . 1 . I n f l u e n c e o f t h e r u d d e r A c c o r d i n g t o t h e r u d d e r t h e o r y t h e r u d d e r b l a d e a c t i o n , bes i d e bes o f c h a n g i n g t h e bes h i p d i r e c t i o n , i n c r e a bes e bes t h e p r o p e l -l e r t h m i s t and e x e r t s a r e c t i f i c a t i o n e f f e c t o n t o t h e screw r a c e j e t .

The f o l l o w i n g e f f e c t s o f t h e r u d d e r b l a d e upon t h e screw r a c e p r o p a g a t i o n can be f o u n d e x p e r i m e n t a l l y : " t h e r u d d e r b l a d e s i t u a t e d i n t h e screw r a c e , by i t s shape p r o p e r t i e s , r e c t i f i e s t h e r o t a t i n g j e t , t h e r u d d e r b l a d e d i v i d e s t h e j e t i n t o t h e b o t t o m and s u r -f a c e p a r t s , - t h e j e t a x i s v e l o c i t y changes a r e t h e same f o r s h i p s w i t h r u d d e r and s h i p s w i t h o u t i t , - f o r s h i p s w i t h the r u d d e r t h e j e t t o u c h e s t h e b o t t o m a t s h o r t e r ' d i s t a n c e . Trom t h e p r o p e l l e r d i s c / b o t t o m s t r e a m / , t i i e r u d d e r a n g l e change by a b o u t 45' f r o m t h e m i d s h i p s p o a i t i o n /<5=^/ s h i f t s t h e j e t d i r e c t i o n a c c o r d i n g t o t h e r u d d e r b l a d e d i r e c t i o n , ^ t h e l e a s t b o t t o m v e l o c i t y I j i t h e s h i p a x i s o c c u r s a t r u d -d e r a n g l e e q u a l t o 45 t o w a r d s t h e open w a t e r s i d e . The i n f l u e n c e o f t h e r u d d e r upon t h e v e l o c i t y d i s t r i b u t i o n and t h e screw r a c e j e t shape is c l e a r l y v i s i b l e on F i g s 5

and 9, Fig.5- shows t h e b o t t o m and s u r f a c e s t r e a m s whereas on F i g . 9 o n l y one-stream i s v i s i b l e .

4.2,, I n f J u a n o e o f s h i n d r a u g h t

The changes o f t h e s h i ? d r a u g h t , i f t h e p r o p e l l e r i a p a r t i a l -l y emerged, a f f e c t c o n s i d e r a b -l y t h e screw r a c e j e t v e -l o c i t i e s . These changes have t h e f o l l o w i n g e f f e c t s :

r e d u c t i o n o f t h e p r o p e l l e r h y d r a u l i c s e c t i o n , w h i c h i n -creases t h e l o a d and d e c r e a e e s t h e p r o p e l l e r e f f i c i e n c y , - a i r i s sucked by t h e p r o p e l l e r , t h e p r o p e l l e r h y d r a u l i c s e c t i o n i s f u r t h e r r e d u c e d and m o r e o v e r t h e medium d e n s i -t y d e c r e a s e s ,

- energy l o s s e s due t o t h e s u r f a c e wave g e n e r a t i o n ,

- i n c r e a s e o f t h e p r o p e l l e r b l a d e p r o f i l e d r a g when c u t t i n g t h e w a t e r s u r f a c e .

P i g . 10 shows t h e r e s u l t s o f measurements f o r v a r i o u s d r a u g h t s o f t h e s h i p . The p r o p e l l e r emerges a t d r a u g h t e q u a l t o 7 cm /see t h e d r a w i n g l e g e n d / .

4« 3 . I n f l u e n c e o f t h e b a s i n d e p t h

The maximum v e l o c i t i e s a t t h e b a s i n b o t t o m decrease q u i c k l y w i t h t h e i n c r e a s i n g d e p t h . T h i s f a c t i s a r e s u l t o f t h e j e t t h e o r y . The j e t volume i n c r e a s e s w i t h t h e i n c r e a s i n g d i s t a n c e f r o m t h e source hy c a p t u r i n g t h e w a t e r f r o m t h e e n -v i r o n m e n t t h e energy b e i n g d i s s i p a t e d a t t h e same t i m e . P i g . 11 shows t h e changes o f maxim.um v e l o c i t i e s n e a r t h e b o t t o m f o r v a r i o u s b a s i n d e p t h s r e g 3 . s t e r e d d u r i n g one o f t h e l a b o r a t o r j r e x p e r i m e n t s , 4 . 4 . I n f l u e n c e o f a s t e r n r o t a t i o n o f t h e p r o p e l l e r The p r o p e l l e r o p e r a t i o n and j e t p r o p a g a t i o n c o n d i t i o n s a r e q u i t e d i f f e r e n t when t h e p r o p e l l e r r o t a t e s a s t e r n . Due t o d i s a d v a n t a g e o u s p i t c h d i s t r i b u t i o n and f l o w c o n d i -t i o n s -t h e p r o p e l l e r c h a r a c -t e r i s -t i c s f o r a s -t e r n m o -t i o n a r e i i o r s e t h a n f o r f o r w a r d m o t i o n . The p r o p e l l e r e f f i c i e n c y i s r e d u c e d by a b o u t 50%* I t has been f o u n d d u r i n g t h e l a b o r a t o -r y test,:;, t h a t t h e t -r o u g h d e p t h i s about t h -r e e t i m e s l e s s f o r t h o p r o p e l l e r r o t a t i n g a s t e r n t h a n f o r t h e f o r v / a r d r o -t a -t i o n / -t h e r u d d e r a n g l e b e i n g i m c h a n g e d / and -t h e g r e a -t e s -t t r o u g h d e p t h o c c u r s i n t h e s h i p l o n g i t u d i n a l a x i s , 4 . 5 . I n f l u e n c e o f t h e n o z z l e D u r i n g t h e t e s t s made w i t h a t u g e q u i p p e d w i t h p r o p e l l e r n o z z l e s i t has been f o u n d , t h a t t h e phenomenon o f t h e screw r a c e j e t p r o p a g a t i o n and t h e b o t t o m e r o s i o n i s s i m i l a r t o t h a t f o r t h e p r o p e l l e r s w i t h o u t n o z z l e s . There i s , however, some d i f f e r e n c e between c a l c u l a t i o n p r o c e d u r e s o f t h e gene-r a t e d v e l o c i t y f o gene-r t h e p gene-r o p e l l e gene-r w i t h t h o n o z z l e and t h e p r o p e l l e r w i t h o u t i t .

5. BuTTOM EROSION TESTS

The b o t t o m e r o s i o n t e s t s were made i n t h e l a b o r a t o r y as w e l l as i n t h e n a t u r a l c o n d i t i o n s .

The l a b o r a t o r y , t e s t s were made v / i t h 1:20 s c a l e model o f B-20 t r a w l e r , ^ ' a r t i c u l a r a t t e n t i o n has been p a i d t o t h e r e l a t i o n between t h e b a s i n d e p t h and t h e e r o s i o n d e p t h and t o t h e e r o s i o n d e p t h r a t e . P i g , 1 2 shows t h e r e s u l t s o f t h e t e s t s made f o r v a r i o u s b a s i n d e p t h s w i t h use o f 1:20 s c a l e model o f B~20 t r a w l e r . Por t h e e r o s i o n d e p t h r a t e see P i g , 1 3 . I t s h o u l d be n o t e d , t h a t q u i c k r e d u c t i o n o f the e r o s i o n d e p t h w i t h i n c r e a s i n g b a s i n d e p t h i s i n a c c o r d a n c e w i t h t h e b o t t o m v e l o c i t y changes f o r i n c r e a s i n g b a s i n d e p t h s as shown on P i g . 1 1 .

D u r i n g t h e s e t e a t s a sigïiificant i n f l u e n c e o f t h e r u d d e r a n g l e upon t h e s i s e and p o s i t i o n o f t h e b o t t o m e r o s i o n v/as o b s e r v e d , Tho t e s t s have ehovwi t h a t :

t h e p o s i t i o n o f t h e t r o u g h depends on t h e r u d d e r a n g l e , the g r e a t e s t d e p t h s o c c u r alv.'ays a t t h e l e f t hand s i d e from'"the r u d d e r b l a d e , - t h e gf'eatest d e p t h o f t h o t r o u g h o c c u r s f o r m i d s h i p s po¬ f; i t i o n o f t h e r u d d e r , - t h e s m a l l e s t d e p t h o f t h e t r o u g h o c c u r s f o r the r u d d e r p o s i t i o n 4 5 ^ t o w a r d s t h e open w a t e r s i d e . ? i g , 14 shows t h e r e s u l t s o f l a b o r a t o r y t e s t s made i n t h e e r c d a b l e b o t t o m c h a n n e l f o r v a r i o u s r u d d e r a n g l e s .

Bottmv, e r o s i o n f i e l d i n v e s t i g a t i o n s were made f o r B-20 t r a w l e r and f o r B74 t h i n k e r , / r a t e d power 1500 HP, maximum r o -t a -t i o n a l pp^oil 325 r.p.m, D,,=^2,74 m, H=^2,08 m/. Tho f o l l o -w i n g f l p l d tepp:i -wore Tnada:'"

B-'IO t r a w l e r . D u r a t i o n o f t h e t e s t s - 10 h a t 170 r.p.m, d r a u g h t Ppual t o 4,9 KI, t h e p r o p e l l e r was submerged 2,7 m u n d p r the -.ater c u r f a c e . 'i'he average g r a n u l a t i o n s o f t h e b o t t o m were d^p^=^-0,05 mm a n d dg^, =0,2 mm. P i g . 15 shows t h e r e s u l t s of sOTMding b e f o r e and a f t e r t h e e x p e r i m e n t .

B'-74 t a n k e r . Draught a f t - 3,66 rn, p r o p e l l e r 2 m u n d e r t h e • ato?r p u r f a c e . Tho t e s t s were made w i t h the f o l l o w i n g r o -t a -t i c n a , ! sposds o f -t h e p r o p e l l e r :

- 60 r.P.m^ - i h 50 m i n , - 80 r.o.m. - 3h 35 m i n , - 150 r.p.ra, - U h 35 rain.

Thi. b.etton g r a n u l a t i n n s as i n t h e f o r m e r example. P i g . 16 chows t h P r e s u l t s o f t h e s o u n d i n g b e f o r e and a f t e r t h e e x -p e r i m e n t .

&. PRACTICAL REMARKS Oil THE APPLICATION OP THE DISCUSSED CALCULiiTION A N D BOTTOM REINPORCING METHODS

I t has been f o u n d , d u r i n g t h e p r a c t i c a l a p p l i c a t i o n o f t h e d i s c u s s e d above c a l c u l a t i o n methods, t h a t t h e w a t e r v e l o c i -t i e s n e a r -t h e bo-t-tora compu-ted w i -t h uae o f 2.71 and 3.1 f o r mulae a r e a good i n d e x f o r t h e d e t e r m i n a t i o n o f t h e n e c e s -s a r y b o t t o m r e i n f o r c e m e n t . The a r e a , w h e r e t h e j e t t o u c h e -s t h e b o t t o m s h o u l d be c o n s i d e r e d a s t h e most e n d a n g e r e d p l a c e T h i s a r e a o f t h e a p p r o x i m a t e w i d t h e q u a l t o t h e s h i p * s b r e a d t h e x t e n d s f r o m t h e p o i n t o f i n t e r s e c t i o n o f t h e b o t tom D i a n e w i t h t h e s t r a i g h t l i n e p a s s i n g t h r o u g h t h e p r o -p e l l e r c e n t e r and i n c l i n e d b y 25 i n r e s p e c t t o t h e h o r i z o n -t a l p l a n e -t o -t h e p o i n -t o f i n -t e r s e c -t i o n o f -t h e s i m i l a r l i n e i n c l i n e d by 15 / R o b a k i e w i c z 1966-2/. Numerous v o r t i c e s a p p e a r i n g i n t h i s a r e a b e s i d e s o f t h e h o r i z o n t a l v e l o c i t i e s , a r e s t r o n g enough t o s h i f t o r c a r r y away t h e r e i n f o r c e m e n t o f c o n s i d e r a b l e w e i g h t .

F a s c i n e - a t o n e m a t t r e s s e s w i t h a d d i t i o n a l l o a d o f b i g b l o c k s a r e g e n e r a l l y u s e d as the b o t t o m r e i n f o r c e m e n t . . I t has been o b s e r v e d ha.wever, t h a t f i n e m a t e r i a l i s o f t e n washed away from, under t h o m a t t r e s s e s o r f r o m t h e gaps between t h e matt r e s s e s and matt h e quay. F i l l i n g matt h e gaps o r c o v e r i n g matt h e r e -j.nforcement w i t h c o n c r e t e i s no good remedy becouse t h e con C r e t e c r a c k s . A l s o wa d o n ' t racomimend t h e f l a t c o n c r e t e b o t toi.i becouse o f i t s s m a l l roughness and, as a r e s u l t , s t r o n g e r o s i o n a t t h e r e i n f o r c e m e n t edge. P l a s t i c f o i l i s r e c e n t -l y used as t h e s e a -l u n d e r t h e m a t t r e s s whereas i t s t o p i s c o v e r e d w i t h b i t u m i c m a t e r i a l s h a v i n g a c e r t a i n degree o f f l e x i b i l i t y . The b e s t way of p r o t e c t i n g t h e b o t t o m a g a i n s t e r o s i o n i s i n c r e a s i n g t h e b a s i n d e p t h . F o r i n s t a n c e f o r a r i v e r pusher, f o r w h i c h t h e no:raal d e p t h i a e q u a l t o 2,5 m, t h e c o n d i t i o n s have i m p r o v e d c o n s i d e r a b l y a f t e r i n c r e a s i n g the b a s i n d e p t h t o 3,5 m. I n f e r r y b o a t t e r m i n a l s , where the n o r m a l d e p t h i s e q u a l t o 6 m, t h e d i f f i c u l t s i t , u a t i o n can ba s i g n i f i c a n t l y i m p r o v e d by i n c r e a s i n g t h e d e p t h t o 9 m. Por t h e a r e a s n e a r t h e f e r r y t e r m i n a l j e t t i e s , w h i c h are p a r t i c u l a r l y v u l n e r a b l e , we recommend t h e t r a p f l o o r s f o r a t t e n u a t i n g t h e screw r a c e j e t e n e r g y . R e c e n t l y we met t h e phenomenon o f t h e quay e r o s i o n caused by bow t h r u s t e r j e t s . T h i s p r o b l e m has n o t been r e c o g n i z e d as y e t .

R E ™ ENGE S

1 . Biikov/alci J. 1970 Meohanika piynow P\71. Warszawa 2. Robakiev/icz 1966-1 T e o r i a i b a d a n i a h y d r a u l i c z n e

Gtoso?/ane p r a y a n a l i z i e z.jawirak s t r u m i e n i a z a a r u b o w e g O e Rozprawy H y d r o t e c h n i c z n e . Z e a z y t 19. 1966 r .

3. R o b a k i e w i c z \V. 19662 Wp3:yv' d z i a l i a n i a s t r u r a i e n i a z a -érubowego na dno p r z y s t a n o w i a k u p r o b na uv/i§zi na p r z y k l a d z i e badan modelowych i terenovvych z t r a u l e r e m B~20. Rozprawy h y d r o t e c h n i c z n e . Z e s z y t 19, 1966 r .

4. Proenak J,V/. 1970 Meohanika piynów PV/l. V/arszawa 5. S'-uliez V/, 1984 V/yznaczanie p o l a prpdkcéci

generowa-mgo p r s e z p o d n i k i statkc-w. P r a c a wevmgtrzna IBV/ PAK, Gdansk

6. Tuszkov/ska T., Koronc/vicz T,. 1971 O k r e s l e n i e c h a r a k -t e r y n -t y k g e o -t n s -t r y c z n y c h sru.by -t y l n e j ukladov; -tandem w s c ' o l b i e i n y c h , P r a c a v;eymetrzna IMP PAIT, Gdansk

Fi9.2 Propeller whirl (k-0)

mé ^om

whirl ,

Dêsignations.

15 20 25 30 40 45 50 55 50 Distance from the quay [cm]

Note : water velocity in cmj^

B^,^rO\sknbu^m

of water velocity horizontal components in the screw race je!

- ' • • • ' {.20m from the propeller)

1%"^ lÜstr&uém öf m t e r velocity horizontal components in the screw race jet

- - ' / \-~ sr.

{.50m trom the propeller)

60 ^ Distance from the quay [cm]

^ i i - : , ; i ^ ^ f ^ p ^ i ^ | l f t ^ velocity "horizontal components in the screw race |et

t.90m from the propeller)

3.64 Z lm]

Fig.7 M e o ^ t d surface stream water velocities and the water velocity rate curves acci^-ding

~-':f^ ' to the authors empincol formulae

^mj

114 154 . Z14 2.54 _ / 114 .154: !

Fig.S M ^ s u r e d bottom stream water velocities and ihe water velocity rate curves accorciirg

to the author's empirico! formulae

Distance from the quay Icml

3 ^

vetoctty hoftzoife^

m

the écrew röee

2.0 ,

Distance from the propeller

Fig. 10 Distribution of water velocity neor the bottom in the ship

centre line depending on the ship draught

maximum recorded velocities near the bottom ^

— — — — N* 3

o"55-26 24 22 20 18 16 14 12 • 10 • 8 6 ' 4 • 2 O' • •. \

\

\

„^.„«Itf ' 'JÖQQ f 'l1(K) rpm — — ^ 1200 rpm — — —»® 1300 f p m

\

V

32 Im]

5 2 3 4 S - e 7 B, 9 ^

r.

2- 3 4 5 5 7 8 "9 y^l :3 4 5 6 --^ .9

dimension of the channel

Bg.t3

Bolf©HI

f r ^ t e changes near the channel window, 800 r.p.m, scale of experiment,

Durot/on Qf the txpenment ~l hour PropeUer ahead Rudder angle 315° 33fl ^.5'^'340-' 314 328 3.0 f m 1 Propeller ahead Rucfcler angle 225" 347

am

m-020 018 QO&j 360. « . » Distance \Tvm thé propeller 334 327 334 ^31$ 342 331 3 ) 2 ^ ' 3 4 2 ^ ^ f 3 t O 32.7 311 34 310 32:9^5^317 333 J:^6 317 327 322 ZS 20 355-402—3?4« 402

sao

:Sl2

m

1343 3Sl4 323 311 soa , , ^ ^ 2 ^

317s^èïV3i4

317 m -F B ^ 3 4 3

Distance from th© propeller

R9.I4 Shape and dkmensions of the trough for various rudder positions and for astern

motion