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i n t hi s p a p e r t h e f o u r m a i n r e q u i re m e n t s f o r a p r o p e l l e r a r e d e a l t w i th . T h e s e re q u i r e *ment s conce rn e ff i c i ency, cav i t a t ion , p rop e l l e r- e xc i t e d fo rces and s topp ing ab i l i t ie s . In ap r o p e l l e r d i a g r a m t h e c h a r a c t e r i s t i c e f f i c i e n c y c u r v e s f o r d i f f e r e n t c o n d i t i o n s a r e e x -p la ine d . A compar i son o f the op t imum e ff i c i enc ies fo r va r ious types o f propu lso r s i sg i v en , a n d th e a p p l ic a t io n s o n a 1 3 0 , 0 0 0 - d w t t a n k e r a r e c o n s i d e re d . C a v i t a t io n - i n -cep t ion cu rves bo th fo r a spec if ic p ro pe l l e r an d fo r sys t emat ic p ro pe l l e r s e r ie s a re d i s -c u s s e d . P r e d i c t e d t o r q u e a n d t h r us t f lu c tu a tio n s , b a s e d o n m o d e l * te s t d a t a , a n d t h eresu lt s o f measuremen t s on the fu l l - s i ze sh ip a re co mp are d . F ina l ly a quas i - s t ead y te s t ingt e c h n iq u e , d e v e l o p e d t o a n a l y z e d i f f e r e n t t y p e s o f s t o p p i n g m a n e u v e r s , i s d e s c r i b e d .

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s h i p d e s i g n .T h e m a i n r e q u i r e m e n t s f o r a s h i p p r o p e l l e r a r e :

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m a n e u v e r a b i l i t y .6 D e p e n d a b i l i t y - - m i n i m u m v u l n e r a b i l i t y .7 L o w i n i t i a l a n d m a i n t e n a n c e c o s t s .I n t h e f o l l o w i n g s e c t i o n s t h e a u t h o r h a s r e s t r i c t e d h i m -

s el f t o a d i s c u s s i o n o f t h e r e q u i r e m e n t s m e n t i o n e d u n d e r

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b l a d e e l e m e n t a t a c e r t a i n r a d i u s r i s g i v e n . T h e t h r u s t -i n g a c t i o n o f t h e s c r e w i n d u c e s v e l o c i t i e s i n t h e f l u i d .T h e m a g n i t u d e o f t h e s e in d u c e d v e l o c i ti e s d e p e n d s o n t h es c r e w l o a d i n g . I f t h e s p e e d o f a d v a n c eVa (o r sh ips p e e d Vs) i s d e c r e a s e d , w h i l e t h e r o t a t i v e s p e e d i s h e l dc o n s t a n t , t h e s c r e w l o a d i n g w i l l i n c r e a s e a n d t h e i n d u c e dv e l o c i t ie s w i l l i n c r e a s e a t a r a t e p r o p o r t i o n a l t o t h e i n -c r ea se i n l i f t f o r ced L a n d t h e e f f e c t i v e a n g l e o f a t t a c kO Z l .

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d i a m e t e r r a t i od h / D a r e f i x ed f o r t h i s se r ie s . T h e r e s u l t so f o p e n - w a t e r t e s t s f o r s u c h a s c r e w s e r ie s a r e g i v e n i n t h eK r - K Q - J d i a g r a m s , F i g . 3 .

T h e p r o p e l l e r e f fi c ie n c y ~ c a n b e e x p r e s s e d i n t e r m s o ft h e s e n o n d i m e n s i o n M c o e f f i c i e n t s a s f o l l o w s :

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B y i n t e r p o l a t i o n i n t h eK~.-I£~-J d i a g r a m o f a s c r ews e r ie s m o s t p r o b l e m s , w h i c h a r i s e w h e n d e s i g n i n g o ra n a l y z i n g s c r e w p r o p e l le r s , c a n b e s o l v e d .

T i l e m o s t w i d e l y e n c o u n t e r e d d e s i g n p r o b l e m i s t h a tw h e r e t h e s p e e d o f a d v a n c e o f t h e f lu i d in t o t h e s c r e wd i s k Va , t h e p o w e r t o b e a b s o r b e d b y t h e s c r e w P a n dt h e n u m b e r o f r e v o l u t io n s n a r e g i v e n . T h e d i a m e t e r Di s t o b e c h o s e n s o t h a t t h e g r e a t e s t e f f ic i e n cy c a n b e o b -t a i n e d . T h i s is d o n e as f o l lo w s :

B y c h o o s in g d i s c r e te v a l u e s o f t h e d i a m e t e r D , t h ec o r r e s p o n d i n g v a l u es o f t h e a d v a n c e r a t i o J a n d t h et o r q u e e o e t t lc i e n t K ¢ c a n b e c a l c u l a t e d . F r o m t h e K ~ -K Q - J d i a g r a m , F i g . 3 , t h e c o r r e s p o n d i n g p i t c h r a t i o sP / Da n d t h e e f f ic i e n cy ~ ; c a n b e r e a d o f f f o r e a c h d i a m e t e rc h o s e n . P l o t t i n g t h e v a l u e s o f ~ a s a f u n c t i o n o f t h ed i a m e t e r w i l l a l l o w t h e d i a m e t e r l e a d i n g t o t h e o p t i m u me t f i c i eney t o be chosen .

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- NPV ~ - 33 .08 (ICQ'~V~B~ Va 2V \ j; ]

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(b ) O p t i m u m r,~ , f o r J = c o n s t . T h i s c u r v e g ot h r o u g h t h e p o i n t s o f c o n t a c t b e t w e e n t h e c u r v e s o f 3 c o n s t a n d t h o s e f o r s ~ = c o n s t . T h e s e o p t i m u m sv a l u e s c o i n c i d e w i t h t h o s e o n t h e e n v e l o p e o f t h e e f f ic i e nc u r v e i n t h e K , r-KQ-J d i a g r a m , F i g . 3 .

(c) O p t i m u m ~ f o r t h e m o s t f a v o r a b l e d i a m e t e r DT h i s c u r v e c o n n e c b s t h e p o i n t s o f c o n t a c t b e t w e e n t hc u r v e s o f s s = c o n s t a n d t h e i r v e r t i c a l t a n g e n t s ( B p c o n s t ; P, N a n d V a a r e g i v e n ) .

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Fig . 6

TANKERSP

TWIN-SC.REW SINGLE SC REW CO AS TE RS TRAWLERS TUGSS H I P S i C A R G O S H I P S . . . . .

PROPELLERS

0 6 0 S E R I E S 4 - 7 0 I f

' PROPELLERS [ ;

~ ~ PROPELLERS '

FU LLY C A VITA TIN B P R O PE LLE R S 3 ~ 5 ~ " ~J " ~" I

V E RT IC A L A X I S P R O P E L L E R 5

o I II0 15 20 25 30 40 50 6 70 80 90 fGO I25 ~50 200

8p

C o m p a r i s o n of optimum efficiency values for differentt ype s of p ropu l so r s

c u r v e s o n w h i c h P, D a n dVa h a v e c o n s t a n t v a l u e s .T h e s e c u r v e s c a n b e c o n s t r u c t e d e a s i l y s t a r t i n g f r o m ac e r t a i n v a l u e o f B ~ a n d ~ ( fo r i n s t a n c e p o i n t s o n t h e ~ , -o p t i m u m c u r v e fo r o p t im u m d i a m e te r ) a n d r e d u c in gt h e r o t a t i v e s p e e d b y, s a y 1 0 , 2 0 , 3 0 , 4 0 a n d 5 0 p e r c e n t •T h e B , a n d ~ - v a l u e s w i l l t h e n a l so b e r e d u c e d b y 1 0 , 2 0,3 0 , 4 0 a n d 5 0 p e r c e n t .

T h e f o u r t y p i c a l c u r v e s c o i n c i d e a t o n e p o i n t i n t h e l ef tupper c o r n e r o f t h e d i a g ra m . T h i s p o i n t c o r r e s p o n d st o t h e m a x i m u m o f t h e e n v e l o p e o f t h e e f f ic i en c y c u r v e si n t h e o p e n - w a t e r c h a r a c t e r i s t i c s , F i g . 3 .

I f b o t h t h e s c r e w d i a m e t e r D a n d t h e r o t a t i v e s p e e da r e f r e e t o b e c h o s e n f o r a g i v e n p o w e r P a n d s p e e d o f

a d v a n c e Va , t h e o p t i m u m p r o p e l l e r w i ll c o r r e sp o n d t ot h i s o p t i m u m p o i n t . A s a r u l e , h o w e v e r , e i t h e r t h er e s u l t i n g d i a m e t e r i s t o o l a r g e o r t h e r e s u l t i n g n u m b e ro f r e v o l u t i o n s t o o s m a l l f o r p r a c t i c a l p u r p o s e s .

I t i s o f g r e a t i m p o r t a n c e t h a t t h e p r o p e l l e r d e s ig n e rm a k e s h i m s e l f t h o r o u g h l y f a m i l i a r w i t h o n e t y p e o f d e si g nc h a r t . I t i s a d v i s e d , th e r e f o r e , t h a t t h e p r o p e l l e r d e -s i g n e r r e s t r i c t s h i m s e l f t o t h e o p e n - w a t e r c h a r a c t e r i s t i c sKr-K~-J d i a g ra m s ) a n d t h e B ~-8 d i a g r a m s w h e n m a k i n g

use o f , sy s t ema t i c s c r ew-se r~es da t a . I f r J eces sa lT, t heo p e n - w a t e r c h a r a c t e r i st i c s c a n b e t r a n s f o r m e d i n s u c h am a n n e r t h a t t h e r e q u i r e d c a l c u l a t i o n s f o r a n y g i v e n d e -s ig n r e q u i r e m e n t s a r e r e d u c e d to a b a r e m i n i m u m . E x -a m p l e s a r e t h e B , - 8 d i a g r a m s f o r g i v e n v a l u e s o f t h es p e e d Va, r p m N a n d t h r u s t T a n d t h e t* - ~ d i a g r a m s f o rt h e c o n s t r u c t i o n o f t h e t o w - f o r c e d i a g r a m o f a t u g [ 1 ]. aT h e a d v a n t a g e o f t h e r e d u c e d a m o u n t o f c a l c u l at i o n sw h e n u s i n g a sp e c i a l d i a g r a m d o e s n o t o u t w e i g h t h e d i s -a d v a n t a g e s a r i s i n g f r o m t h e u s e o f a n u n f a m i l i a r d i a g ra m •

I n Ta b l e 2 a s u r v e y o f t h e s y s t e m a t i c s c r e w s e ri e st e s t e d b y t h e N S M B i s g i v en . R e s u l t s a re a v a il a b l e i nt h e f o u r f o n n s d i s c u s s e d p r e v i o u s l y [ 1 - 3 ] .

a Num bers in brackets designate References at end of paper.

0 , 8 0TA N K E R : D I S P L = 1 5 6 .0 0 0 m 3

_ d w = 1 3 0 ,0 0 0 t o n I I ~ ~ ) O . 7 , 0 0 m

R . . . ,o o I IB o, T s - - Vs .1 s .8 k ° . - - ~ ~ - _ .- 3 , ~ , ~ D , 6 7 4 m

S H P = 2 7 7 2 0 h p < I~ T~ ~ . - 6 1 4 2 m

W . . .. / - - - - ~ 5 . 7 8 m O . 7 /, 7,D = 8 . 0 0 m 1 ~ I . ( ~

I 0 = 7 . 6 0 m ~ ; I 0 ,

E o . e s o . 7. 2 o m ~ - - ~ ? ( - -

O . 6 0 L - - ~Z , 7 C R . O . P Tw i nTripte

sc r, s c r.- - N U M B E R O F B LA DE S

3-3 3 - 5 - 3

Fig. 7 Prop ulsive coefficients for a 13 0,000-to nt a n k e r

T a b l e 2 S u r v e y o fWa g e n i n g e n - S e r i e s

N U M B E RO F

BLADES

2

3

4

5

6

7

B L A D E A R E A - R AT I O

0.30 0.38

0.36 0,50 0.65 0,80

0.40 055 0.70

0.45 0,60 0.75

0.50 0.65 0.80

0.56 0.70 0.85

0.85 1.00

1,05

O p e n - w a t e r t e s t s h a v e b e e n p e r f o r m e d w i t h s y s t e ms e r i e s f o r t y p e s o f p r o p e l l e r s o t h e r t h e n t h e c o n v e n t i o n as c r e w . S o m e o f t h o s e r e s u l t s a r e s u m m a r i z e d i n F i g

I n t h i s f i g u r e c u r v e s a r e g i v e n s h o w i n g t h e h i g h eo b t a i n a b l e e f f i c i e n c y ~ p f o r d i f f e r e n t t y p e s o f p r o p e l l e ra s a f u n c t i o n o f B j~ . A t t h e t o p o f t h e f i g u r e t h e r a n g eo f B ~ - v a l u e s t y p i c a l f o r d i f f e r e n t s h ip t y p e s a r e i n d i c a t e dT h e l i g h t l y l o a d e d s c r e w s o f f a s t s h i p s a r e a t t h e l e f th a n d s i d e , w h i le t h e h e a v i l y l o a d e d p r o p e l l e r s o f t o w i n gv e s se l s a r e a t t h e r i g h t . S u c h a d i a g r a m c a n g i v e a q u i c ki n d i c a t i o n w h i c h t y p e o f p r o p e l l e r w i ll g i v e t h e b e s te f f i ci e n c y f o r a g i v e n t y p e o f s h i p.

I n p a r t i c u l a r , i t c a n b e s e e n f r o m t h e d i a g r a m t h a tp r o p e l le r s i n no z z le s a r e to b e r e c o m m e n d e d f o r h e a v yl o a d in g s , s u c h a s o c c u r i n t u g s , t r a w l e r s a n d l a rg e t a n k e r sF o r h e a v y t o w i n g a l o n g n o z z l e w i t h a c h o r d - d i a m e t e rr a t i o o f 0 .8 3 i s p r e f e r a b l e t o a s h o r t n o z z l e w i t h a c h o r dd i a me te r r a t i o o f 0 •50 [4 ].

F o r f a s t s hi p s, c o n t r a r o t a t i n g p r o p e l l e r s a p p e a r t o g i v ea h i g h e r e f f i c ie n c y t h a n c o n v e n t i o n a l sc r e w s , s u c h a st h e B 4 . 7 0 o r b r o a d - b l a d e d s c r ew s su c h a s t h e G a w n 3 -110.

I n f o r m a t i o n o n t h e o p t i m u m e f f ic i e nc y o f fu l l y e a v i t a t -i n g a n d v e r t i c a l - a x i s p r o p e l l e r s h a s b e e n i n c l u d e d i n t h edia gr am [5] , [6 ] .

F i g . 7 g i v e s t h e r e s u l t s o f c a l c u l a t i o n s o f t h e p r o p e l l e rf o r a t a n k e r w i t h a d e a d w e i g h t o f 1 3 0 , 0 0 0 t o n s , a s h i p

A P R I L 1 9 6 6 1 6 1


5/14

Ta b l e 3 R e q u i r e d P o w e r f o r a Ta n k e r

• Displacement volume = 156,000 m3; deadweight = 130,000 tonsRPM = 100; V, = 15.8 knots; 100 percent SHP = 27,720 hp.

D = 7.20 m 7.60 m 8.00 m

Conventional screw z = 4, percent .. .. 104 100 96z = 5, percent .. .. 103 100 98z = 6, percent ... . 103 100 100

z = 7, percent ... . 102 101 103Counterrotating propellersz = 4 5 ;D = 6.57 -- 5.78 m 95%

Propeller with nozzleD = 7.00 mResistance increase due to 89%ttogner stern and nozzle attachment.. .. 6 percent

Twin-screw shipz = 3 -- 3; D = 7.47mResistance increase due to bossings 100%.. .. 5 percent

Triple-screw shipz = 3 - - 5 - - 3 ;

D = 6.74 -- 6.42 mResistance increase due to bossings 94%.. .. 4 percent

speed of 15.8 knots and an installed power of 27,720 shp.The rotative speed of the propeller has been fixed at 100rpm. The propulsi ve coefficient, which is a measur e ofthe efficiency of the propeller including the hydr odyna micinteraction of the ship's hull and the propeller, is plottedvertically.

At the left-hand side of the diagram results are givenfor conventional ship propellers with 4, 5, 6 and 7 blades.The calculations have been carried out for three different

diamet ers: D = 7.20, 7.60 and 8.00 m. In order to givea complete survey of the potentialities of a conventionalscrew propeller it is necessary to carry out calculationsfor a range of number of blades, diameters and rotativespeeds. In this diagram the rotative speed has beenfixed at 100 rpm and it is obvious that for the highernumber of blades (6 and 7) a screw diameter of 8.00 mis too large; the hi ghest efficiency is found for a diameterof 7.60 m. For the lower num ber of blades the efficiencycontinues to increase with increasing diameter.

It might be the question if a diameter of 8.00 m givesalready the optimum for a 4-bladed propeller for thisship. For lower rota tive speeds, for instance 80 rpm f o r

turbi ne-dr iven ships, these tendencies are intensified.To the right, results are given for a set of contra rotati ng

propellers (CR) an d a propeller in a nozzle (DP = duct edpropeller). Results are also given for a twin-screwarrangement of two 3-bladed propellers and a triple-screw ship with two 3-bladed and one 5-bladed pro-peller. The increase in resistance (EH P) due to thenozzle attachment or due to the brackets for the twin-screw or triple-screw configurat ion are specified to be 6, 5and 4 percent, respectively. For each configuration theoptimum diameter, at 100 rpm, is indicated.

Fig. 8 Arrangement of propeller in nozzle for a tanker model

In Table 3 the required power for each case is given,using 27,720 shp as 100 percent.

The large saving in SHP which can be obtained usinga shrouded propeller sho uld be noticed.

As tanker sizes continue to increase, the advantageof shrou ded propellers will become greater. In Fig. 8,the installation of a propeller in a clear-plastic shroudon the afterbody of a tanker model is shown.

For the very big tanker the contrarotating propellerleads to a power reduction of 5 percent compared toconventional screws. The adva ntage of this type of

propulsion for large tankers lies primarily in the smallerallowable screw diameters and thus a possible reductionin the danger of vibration caused by the fluctuatingpropel ler-for ce field. As shown by Fig. 6, the contra~rotating propeller may become a serious competitor ofthe conven tiona l screw propeller for fast cargo ships [7 ].

C a v i t a t i o n o f th e S c r e w - B l a d e S e c ti o n s

The lifting-line theory for ship screws gives an idea ofthe induced velocities, pressures and forces along thedifferent radii of the screw blades. This inform ationforms the basis of considerations of cavitation andstrength.

Fig. 9 gives schematically the screw with a boundvortex or circulation F at the radius r and trailing heli-coidal vortices. The circul ation F of a screw-blade sec-tion is defined as the line integral of the flow field alonga closed curve aroun d this profile. This line integralis the integration of the product of a line element d s andthe component of velocity tangent to this line element.By choosing the closed curve along which the line integralis to be evaluated, in the manner indicated in Fig. 9, arelation can be derived easily between the circulation Fof a screw-blade section and the tangential-induced

1 6 2 M A R I N E T E C H N O L O G


6/14

Fig. 9

2trrc t .t z r

. . 7

i

Relation between circulation F and tangential inducedvelocity ct

1.00

0 .75 - -

0,50

CL

0.25i

Fig. 11

0.S 1.0 1.5 2.0 2.5& pq

CL -- t/l - - Ap/q diagram for circular-arc profiles

ADDITIONAL UNSTEADY

Fig. 10 Oosterve ld's hydrodyna mic model for screw blade asvortex sheet

v e l o c i t y c~ a t t h e s a m e r a d i u s r. T h e c u r v e s h o w n i nF ig . 9 cons i s t s o f a coax i a l cy l i nde r, o f r ad iu s r, cu t op ena l o n g a l i n e p a r a l l e l to t h e a x i s. F a r f o r w a r d o f t h ep r o p e l l e r c~ = 0 , s o t h a t t h a t p a r t o f t h e b o u n d a r y h a sz e r o c o n t r i b u t i o n t o t h e l i~ l e i n t e g r a l. T h e t w o l o n g i -t u d i n a l b o u n d a r i e s m a k e e q u a l b u t o p p o s i t e c o n t r i b u t i o n sa n d , h en c e , c a n c e l e a c h o t h e r. F a r a f t, t h e c o n t r i b u t im ~is 2retch. W h e n t h e a x i a l c y l i n d e r i s f l a t t e n e d o u t , a s a tt h e r i g h t - h a n d s i d e o f F i g . 9 , t h e c h o s e u b o u n d a r y s u r -r o u n d s t h e b o u n d v o r t i c e s o f t h e z - sc r e w b l a d e s a n d t h u s ,a c c o r d i n g t o S t o k e s ' l a w, w e f i n d

zF = 2 r r r c ,

Fo r a f i n it e num be r o f b l ades , s ay 4 o r 5, t he va lu e c ,/ 'o r t h e t a n g e n t i a l i n d u c e d v e l o c i t y f a r a f t t h e s c r e w w i l lo c c u r o n l y a t t h e h e li c o i d al v o r t e x s h e e t s . B e t w e e nt h e s e v o r t e x s h e e t s t h e t a n g e n t i a l i n d u c e d v e l o c i t i e s w i l lb e l es s. O n a c i rc l e w i t h r a d i u s r t h e t a n g e n t i a l i n d u c e dv e l o c i ti e s w i ll h a v e a n u n d u l a t i n g m a g n i t u d e w i t h a m a x i -n m m c , a t t h e f r e e v o r t i c e s a n d a m i n i m u m i n b e tw e e n .T h e r a ti o b e t w e en t h e m e a n a n d t h e m a x i m u m i n d u c e dt a n g e n t i a l v e l o c i t y is g i v e n b y t h e G o l d s t e i n r e d u c t i o nf a c t o r :

{~trnean ~ get

0 . 8

0.7

0 . 6 ~ / ~

0.5 /

0

0,3 /CLI 0,2

o . ,o / / ; "0 0.02 0.04 0.06 0.0~ 0.I0

t / t

" • .80

/ f

I0.12 0.14 O 6 O.'le

Fig. 12 Relat ion betwee n lift coefficient CL, ratio t / l andcavitation numb er ~r, respective ly, press ure coefficient Ap/q for

circular-arc profiles

I t f o l lo w s f o r p r a c t i c a l s c r e w s t h a t f o r th e c i r c u l a t i oa r o u n d t h e s c r e w b l a d e w e h a v e

Z r = 27rrCtme~n = 27rVKCt

T h e p r o d u c t o f t h e l if t c o ef f ic i en t a n d c h o r d I o f b l a d e s e c t i o n i s a b a s i c p a r a m e t e r i n t h e c a l c u l a t i o n ot h e c a v i t a t i o n p r o p e r t i e s of t h e b l a d e s e c t io n . A c c o r d i nt o t h e la w o f K u t t a - J o u k o w s k i t h e l i f ti n g f o r ce o f

s c r ew -b l ad e e l emen t , s ee F ig . 2 , isd L = p V F d r

w h e r e V is th e r e s u l t a n t v e l o c i t y o f t h e s c r e w - b l a d e s e ct i on . By de f i n i t i on

d L = C L } p V 2 l d r

a n d h e n c e

Cd - 2 r _ 47rrKc, _ 47 rD @x ) -}c~_V z V z V

A P R I L 1 9 6 6 1 6 3


7/14

F i g . 1 3

2 V2V V A p V m = , - 1 -1M~=v-~, /c- : T = v ' M T

Lo• T T ~ NACA - S E I ~ I E S I/ ~ . : o E S , G N C L - O

/ / " . \ " ? " lMcr /

. . . . o i , ~ / / - -

' i 0 . 2 1 / L ~j ~ ~ .....-04 -02 0 02 04 0 6 08 1Jo

- - C L

R e l a t i o n b e t w e e n c r i ti c a l M a c h n u m b e r a n d p r e ss u r ec o e f f i c i e n t Ap q

w h e r e z = n o n d i m e n s i o n a l r a d i u s = r / / ~ .W i t h t h e a i d o f F i g . 2 a g e o m e t r i c r e l a t i o n c a n b e

d e r i v e d b e t w e e n l c , / V a n d t h e h y d r o d y n a m i c p i t c hang le s 3 and 3~

~-C t- s i n 3 , t a n ( 3 , - 3 )

V

F o r a k n o w n v e l o c i t y d i s t r ib u t i o n t h e f o r c es o n th es c r e w - b l a d e e l e m e n t c a n n o w b e c a l c u l a t e d e a s i l y.

T h i s p a p e r w i l l b e r e s t r i c t e d t o t h e s e e l e m e n t a r y r e -m a r k s b a s e d o n t h e s t e a d y l i f t i n g - l i n e t h e o r y f o r s h i pp r o p e ll e rs . F u r t h e r d e t a i l s c a n b e f o u n d in t h e a v a i l a b lel i t e r a t u r e , s u c h a s [3 ]. F o r a c l e a r i n s i g h t i n t o t h e p r o -p e l le r a c ti o n , t h e c o n s t r u c t i o n o f m o r e c o m p l i c a t e d h y -d r o d y n a m i c m o d e l s m a y o ft e n b e n e c e s s ar y. T h e o r i e sh a v e b e e n d e v e l o p e d i n w h i c h t h e s c r e w b l a d e s a r e r e -g a r d e d a s v o r t e x s h e et s , b o t h f o r s t e a d y m i d u n s t e a d yp h e n o m e n a . F i g . 1 0 g i v e s O o s t e r v e l d ' s h y d r o d y n a m i cm o d e l f o r t h e s c r e w b l a d e a s a v o r t e x s h e e t. T h e a d d i -t i o n a l u n s t e a d y v o r t i c e s a r e c r e a t e d w h e n t h e s c r e wp a s s e s a r e g i o n o f l o w s p e e d o f a d v a n c e , s u c h a s o c c u r sw h e n t h e b l a d e p a s s e s t h e s t e r n p o s t .

T h e c r e a t i o n o f l i f t a t t h e d i f f e r e n t r a d i i o f t h e s c r e wb l a d e c a n b e a c c o m p l i s h e d b y c a m b e r a n d b y a n g l e o fa t t a c k . T h e i m p o r t a n c e o f t h e d i s t ri b u t i o n o f l if t b e -t w e e n t h e s e t w o f o r g o o d c a v i t a t i o n p r o p e r t i e s c a n b ed e t e r m i n e d f r o m :

( a ) C h a r a c t e r i s t i c s f o r t w o - d i m e n s i o n a l p r o f i le s g i v -i n g t h e r e l a t i o n b e t w e e n t h e l i f t c o e f f i c i e n t C L , t h e a n g l eo f a t t a c k a a n d t h e c a v i t a t i o n n u m b e r % o r t h e p r e s s u r e

coe ff i c i en tAp/q , see F igs . 11 , 12 and 13 .(b ) C a v i t a t i o n - i n c e p t i o n c u r v e s s u c h a s a, ¢ - K T d i a -g r am fo r a sh ip s c r ew, F ig . 14 .

(c) C a v i t a t i o n - i n c e p t i o n c u r v e s s u p e r i m p o s e d o n aB~-8 s c r ew-se r i e s d i ag r am , F ig . 1 5 .

I n F i g . 11 t h e m a x i m u m p r e s s u r e c o e f f i c i e n tAp/q =( V 2 . . . V 2 ) / V 2s g ive n a s a func t ion o f t he l i f t coe ff i c i en tC L f o r f o u r c i r c u l a r - a r e p r o f i le s w i t h t h i c k n e s s - c h o r d r a t i oo f 0.04 , 0 .07 , 0 .11 and 0 .15 .

T h r e e a r e a s c a n b e d i s ce r n e d :( a ) A n a r e a w i t h lo w C L - v a l u e s , w h e r e a s m a l l c h a n g e

kD

F i g . 1 4

\ F ' A C E O R / / B A C K O R

\ V I S I B L E / /\ ' ° /

IL

. . . . . .

- - I H R U S T C O E F F I C I E N T K T

C h a r a c t e r i s t i c c u r v e s in a a - K T d i a g r a m

i n C L ( a n d , h e n c e i u a n g l e o f a t t a c k a ) g i v e s a l a rgc h a n g e i n A p / q ( p r e s s u r e - s i d e c a v i t a t i o n ) .

( b) A n a r e a w h e r e t h e l o w e s tAp/q-values arer e a c h e da n d a c h a n g e i n C ~ g i v e s a l m o s t n o c h a n g e i nA p / q ( t h ea r e a o f s h o c k - f r e e e n t r a n c e } .

( c) A n a r e a o f h i g h C L - v a l u e s w h e r e a s m a l l c h a u gi n C L c o r r e s p o n d s t o a l a rg e c h a n g e i nA p / q ( s u c t i o n - s i d ec a v i t a t i o n ) .

I n F i g . 1 2 t h e d s J ;a o f F i g . 11 h a v e b e e n r e p l o t t e d i nm a n n e r , m o r e i n s t r u c t i v e t o th e s c r ew d e si g n er . T h if i gu re i nd i ca t e s , f o r a ce r t a in va lue o f z (o rAp/q) t h et h i c k n e s s - c h o r d r a t i ot / l w h e r e a m s x i m u m v a r i a t i o n i nt h e l i f t c o e ff i c ie n t C'L c a n b e s u s t a i n e d w i t h o u t c a v i t a t i oT h e p o i n t a t t h e e x t r e m e r i g h t - h a n d s id e of' t h eC~-t/ll o o p f o r a g i v e nAp/q-va luei s t l he po in t whe re a i s abou t

0 d e g a n d w h e r e s h o c k - f r e e e n t r a n c e c h a n g e s i n t o p r es u r e - s i d e c a v i t a t i o n . F o r c a l c u l a t io n o f p r o p e l l e r c a v i tt i o n a c c o r d i n g t o s t e a d y l i f t i n g - l i n e t h e o r y, a m a rg i n os a f e t y a g a i n s t c a v i t a t i o n i s g e n e r a l l y u s e d i n t h e c a l c u lt i o n s , f o r i n s t a n c e~xp/q = ~ - - 20 pe rce n t .

F i g. 1 2 sh o w s t h s, t t hi s r e d u c t i o n i n c a v i t a t i o n n m n b ea f f ec t s o n l y t h e d e t e r m i n a t i o n o f t h e o p t i m u m t h i c k n e sc h o r d r a t i o t / 1 i n o r d er t o o b t a i n a m a x i m u m C ~ v a r it i o n a t a g i v e nAp/q.

E x p e r i m e n t a l a i ~( t t h e o r e t i c a l d a t a , s u c h a s g i v e n F igs . 11 and 12 , ~ re s ca rce . An e x t r a sou rc e fo r if o r m a t i o n , h o w e v e r , i s g i v e n b y t h e c a l c u la t i o n s f o r thi n c e p t i o n o f s u p e r s ( ta l c p h e n o m e n a i n a e r o d y n a m i c s [8T h e r e i s a s im p l e r e? a t i on b e t w e e n t h e m a x i m u m p r e s s ucoe ff i c i en tA p / q o f :L p r o f i le a n d t h e c r i t i c a l M a t h n u mb e r, a s i s s h o w n i n F i g . 1 3. T h e c r it i c a l M a t h n m n b e r t h e r a t io b e t w e e n t h a t a d v a n c e v e l o c i t y V, w h e n a t s o mp o i n t o n t h e p r o f i l e t h e v e l o c i t y o f s o u n d c h a s b e er e a c h e d , a n d t h a t Ye l o c i t y o f s o u n d :

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F i g . 1 3 g i v e s ai t e x a m p l e o f t h e r e l a t i o n b e t w e e n t hc r i t i c a l M a t h n u m b e r M ~ , a n d t h e l i ft c o e ff i c ie n t C~ o f



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p r o fi le w i t h o u t c a m b e r b u t w i t h s ix t h i c k n e s s - c h o r dra tio s [8 ] .

F o r t h e i n v e s t i g a t i o n o f sc r e w m o d e l s in t h e c a v i t a t i o nt m m e l i t i s i m p o r t a n t t o s h o w t h e d i f f e r e n t r e s u l t s i n acha rac t e r i s t i c d i ag ram . F ig . 14 g ives , i n an in s t ruc t ivem a n n e r, t h e o n s e t o f t h e d i f f e r e n t t y p e s o f c a v i t a t i o n .S u c h a d i a g r a m c a n b e m a d e f o r e v e r y sc r ew m o d e l b ys y s t e m a t i c a l ly v a r y i n g t h e c a v i t a t io n n u m b e r ~ a t c e r-t a i n v a l u e s o f t h e t h r u s t e o e f f lc i e n t K ~ . F r o m s u c h ad i a g r a m i t c a n b e a s c e r t a i n e d i f t h e s c r e w m o d e l i s t o on e a r t h e i n c e p t i o n o f p r e s s u r e -s i d e c a v i t a t i o n a t t h ed e s ig n c on d i ti o n . B y r e d u c i n g th e c a m b e r s o m e w h a t

a n d c o m p e n s a t i n g f o r t h i s c a m b e r r e d u c t i o n b y a p i t c hi n c re a s e , t h e c u r v e s f o r t h e o n s e t o f p r e s s u re - s i d e a n ds u c t i o n - s id e c a v i t a t i o n c a n b e s h i f t e d t o t h e l e f t, m a k i n gpres su re - s ide cav i t a t i on l e s s l i ke ly.

Te s t s i n t h e c a v i t a t i o n t u n n e l w i t h s y s t e m a t i c a l l yv a r i e d s c r e w s e ri e s c a n g i v e d a t a a b o u t t h e o n s e t o f p r e s -su re - s ide o r suc t ion - s ide cav i t a t i on . Bur r i l l [ 9] hass y s t e m a t i c a l l y t e s t e d s o m e s c r e w s e ri e s a n d h a s i n d i c a t e dthe l i nes fo r cav i t a t i on incep t ion in a B~-6 d i ag ram , s eeF i g . 1 5. F r o m t h i s d i a g r a m i t i s e v i d e n t t h a t t h e c u r v e

f o r o p t i m u m d i a m e t e r f o r a g i v en p o w e r P, r o t a t i v e s p e eY a n d s p e e d Va , l i e s i n t he r eg ion o f suc t ion - s ide cav i t a -

t io n . B y a s l ig h t c h a n g e in c a m b e r o f t h e b l a d e s e c t i o nt h e a r e a m o r e f a v o r a b l e fr o m t h e v i e w p o i n t o f c a v i t a t i oc a n b e m o v e d t o w a r d t h e p o s i t io n o f t h e D o p c e u rv e .

Vi b r a t i o n G e n e r a t e d b y P r o p e l l e r

U s u a l l y t h e v a r i a t i o n s o f t h e f l ow f ie l d a t t h e s c r e wcan be sp l i t up in to tw o co mp onen t s , i . e. :

( a ) Th e r ad i a l va r i a t i on , e spec i a l ly o f t he ax i a l ve loe it ie s . T h i s v a r i a t i o n d o e s n o t l e a d to u n s t e a d y p h en o m e n a a t t h e s c r ew. A p r o p e l le r w o r k i n g i n s u c h v e l o c i t y f ie ld h a s a s t e a d y fl o w a n d f o r c e p a t t e r n . M o r eo v e r, t h e p r o p e l l e r c a n b e a d j u s t e d t o t h i s r a d i a l l y n o nu n i f o r m f l o w b y a n a p p r o p r i a t e d i s t r i b u t i o n o f p i t c h a nc a m b e r, a n d o p t i m u m e f fi c ie n c y a n d c a v i t a t i o n p r o p e rt i e s may be expec ted in such cases .

(b ) T h e c i r c u m f e r e n t i a l ( a t a g i v e n r a d i u s) v a r i a t i o no f b o t h a x i a l a n d t a n g e n t i a l v e lo c i ti e s . T h i s n o n u n if o r m i t y i s t h e o r i g i n o f t h e p e r i o d i c a l l y fl u c t u a t i n g f o r cp a t t e r n a n d t h e u n s t e a d y p r e s s u r e d i s t r i b u t i o n s a l o n g t hb l a d e c h o r d s , a n d d e t e r m i n e s t h e d y n a m i c p r e s s u r ef l u c t u a t i o n s in d u c e d b y t h e p r o p e l l e r o n t h e s t e r n .

I n F i g . 1 6 a rt e x a m p l e i s g i v e n o f t h e w a k e d i s t r i b u t i obeh ind a s ing le - sc rew sh ip mode l . I n t h i s f i gu re on lyt h e a x i al v e l o c i t y c o m p o n e n t is g i v e n i n t h e f o r m o f t hl o c a l w a k e f r a c t i o nw = ( Vo - - V ~ ) / V , ,where V, i s t hesh ip speed .

A l t h o u g h o f t e n a t t e m p t e d [ 1 0] , t h e e x p e r im e n t M

d e t e r m i n a t i o n o f t h e t a n g e n t i M - v e l o e i t y c o m p o n e n t susua l ly me e t s m an y d if f icu l ti e s . Th e 5 -ho led sphe r i cap i t o t t u b e o f Va n d e r H e g g e - Z i j n e n s ti ll g i ve s n o c o n s i s t e nr e s u lt s f o r v e l o c it i e s b e l o w 1 m / s e e . T h e s e t a n g e n t i a lv e l o c i t y c o m p o n e n t s a r e m u c h s m a l l e r t h a n t h e a x i ac o m p o n e n t s b u t f o r a n a c c u r a t e t h e o r e t i c a l a n a l y s i s ot h e u n s t e a d y p h e n o m e n a a t t h e s c r e w p r o p e l l e r t h e c i rc u m f e r e n t i a l i n e q u a l i t y c a u s e d b y t h e t a n g e n t i a l v e lo c it i e s m a y n o t b e n e g l e c t e d i n t h e f u t u r e .

A s i m p l e q u a s i - s t e a d y a n a l y s is w i t h t h e a i d o f aK ~ - K o -

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F I G U R E S I N D I C AT E T F t EB L A D E A N G L E S I N D E G R E E S

T h r u s t e c c e n t r i c i t y c a l c u la t e d b y St u n t z P i e n H i n t e r -t h a n a n d F i c k e n

J d i a g r a m a s p r o p o s e d b y S c h u s t e r [11 ] m a y g i v e a q u a li -t a t i v e p i c t u r e o f t h e f o r c e s g e n e r a t e d b y a s c r e w in a c i r -c u m f e r e n t i a l l y n o n u n i f o r m f l o w f i e l d .

F o r e v e r y 5 o r 1 0 d eg o f t h e c i r c u m f e r e n c e a n i n s ta n -t a n e o u s e x a m i n a t i o n o f t h e b l a d e i s m a d e . T h e a x ia lw a k e v e l o c i t i e s a r e r e g a r d e d a s c o n s t a n t a t e a c h b l a d e

p o s i ti o n . W i t h t h e a i d o f t h e o p e n - w a t e r c h a r a c t e r i s t i c sK~-KQ-J d i a g r a m ) o f t h e p a r t i c u l a r s c r e w, th e t im eh i s t o r y o f t h e t h r u s t a n d t o r q u e c a n b e f o u n d . T h e p a t ho f t h e c e n t e r o f th e t h r u s t w i ll b e s y m m e t r i c a l w i t h r e -g a r d t o t h e l o n g i t u d i n a l c e n t e r p l a n e o f t h e s h i p w h e n t h et a n g e n t i a l w a k e v e l o c i t ie s a r e n e g l e c t e d . T h i s p a t h w i llb e s w e p t z - t i m e s e v e r y r e v o l u t i o n f o r a z - b l a d e d p r o p e ll e r.U s u a l l y th e r e g i o n o f m a x i m u m w a k e v e l o c i t y a b o v e t h ep r o p e l l e r a x i s w i l l b e b r o a d e r ( t h i c k e r ) t h a n t h a t b e l o wt h e p r o p e l l e r s h a f t . T h e c lo s e d p a t h o n w h i c h t h e c e n t e ro f t h r u s t i s m o v i n g w i l l l i e m o s t l y a b o v e t h e p r o p e l l e raxis.

W h e n t h e t a n g e n t i a l w a k e v e l o c it i e s a r e in c l u d e d , t h e

r o t a t i v e s p e e d o f t h e s c r e w b l a d e s w i l l b e s m a l l e r w h e ne n t e r i n g t h e p e a k o f t h e w a k e a n d l a r g e r w h e n l e a v i n gt h i s p e a k . T h i s w i ll c a u s e a s h i f t o f t h e p a t h o f th e c e n t e ro f t h r u s t t o s t a r b o a r d f o r a sc r e w t h a t r o t a t e s c l o c k w i s ea n d t o p o r t f o r a s c r e w t h a t r o t a t e s c o u n t e r c l o c k w i s e .T h e s h a p e o f t h e s e c t i o n s in t h e s h i p s a f t e r b o d y h a s ap r o n o u n c e d i n fl u e n c e o n th e p o s i t io n a n d f o r m o f th i spa th , F ig . 17 .

B e c a u s e o f t h i s e c c e n t r i c p o s i t i o n o f t h e t h r u s t , h o r i -z o n t a l a n d v e r t i c a l b e n d i n g m o m e n t s a r e c r e a t e d i n t h ep r o p e l l e r s h a f t .

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L o o k i n g m o r e c l o s e l y a t t h e v a r i a t i o n o f t o r q u e , i

i s obv ious tha t a d y n a m i c - f o r c e p a t t e r n is c r e a t e d i n t h ep r o p e l l e r s h a f t i n w a y o f t h e p r o p e l l e r b e c a u s e o f t hc i rc u m f e r e n i ;i a l i n e q u a l i t y o f t h e w a k e a n d , h e n c e , t ht o r q u e - g e n e r a t i n g F orc e. T h e s e h o r i z o n t a l a n d v e r t i ct r a n s v e r s e f o r c e s a n d b e n d i n g m o m e n t s h a v e t o ba b s o r b e d m a i n l yby t h e s t e r n t u b e a n d t h e s t e rn p o s t .

T h e f o r c e s a n d m o m e n t s a c t in g o n t h e p r o p e ll e r w o r ki n g i n t h e f l o w f i e l d b e h i n d t h e s h i p c a n b e d i v i d e d i n ts ix c o m p o n e n t s :

Axial. T h r u s t a n d t o r q u e .Transverse. A t r a n s v e r s e f o r c e , b e c a u s e o f t h e

c i r c u m f e r e n t i a l i n e q u a l i t y o f t h e t o r q u e f o r c e ( u n b a l a n co f t o r q u e ) ; a v e r ti c a l b e n d i n g m o m e n t d u e t o t h e t h r u

e c c e n t r i c i t y.Vertical. A v e r t i c a l f o r c e a n d a h o r i z o n t a l b e n d i n gm o m e n t f o r t h e s a m e r e a s o n s a s s t a t e d i n th e f o r e g o i nsee F ig . 18 .

T h e e x p e r i m e n t M d e t e r m i n ' ~ t io n o f t h e t h r u s t a nt o r q u e f l u c t u a t i o n s o f a s c r e w m o d e l b e h i n d a m o d e l o fs ing le - sc rew sh ip was ca r r i ed ou t succes s fu l ly fo r t he f i rt i m e b y K r o h n a n d We r e l d s m a [1 2 ]. T h e y c a r ri e d o ut h e i r m e a s u r e m e n t s o f th e h y d r o d y n a m i c f o rc e s c r e a teb y t h e p r o p e l l e r u s i n g a m e a s u r i n g s h a f t o f v e r y g r es t i ffness .

M a n y s y s t e m a t i c a n d i n d i v i d u a l e x p e r i m e n t s h a vb e e n d o n e u si n g t h e a p p a r a t u s o f K r o h n a n d W e r e l d s m[ 1 3, 1 4] . T h e s y s t e m a t i c e x p e r im e n t s g iv e i n f o r m a t ia b o u t t h e i n f lu e n c e o f n u m b e r o f b l a d e s, t h e s h a p e o f t ha f t e r b o d y a n d t h e p o s i t i o n o f t h e p r o p e l l e r s h a f t .

T h e f r e q u e n c i e s o f t h e p e r i o d i c f o r c e f l u c t u a t i o n s d ut o t h e p r o p e l l e r r u n n i n g i n t h e f l o w f ie l d b e h i n d t h e s h iw i ll b e e q u a l t o t h e n u m b e r o f r e v o l u t i o n s o f t h e p r op e l le r t i m e s t h e n u m b e r o f b la d e s ( t h e b l a d e f r e q u e n c yo r a m u l t i p l e t h e r e o f .

R e g a r d i n g t h e i n f l u e n c e o f t h e n m n b e r o f b l a d e s, t hc h a r a c t e r i s t i c d i f f e re n c e b e t w e e n p r o p e l l e r s w i t h e v ea n d o d d n u m b e r s o f' b l a d es m u s t b e m e n t i o n e d .

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,o I i / / ~ z . ,- 5" 7 ~ t / ~ f , ~ \ . (- ' ~ , ' / I / f ' t f ~ I / / ' Z = 6

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Ta b l e 4 F o r m u l a s A p p r o x i m a t i n g T r a n s ve r se F or ce s a n d B e n d i n g M o m e n t s ,Exci ted by Prope l le r( 1 5 - K n o t Ta n k e r )

F O R M U L A E A P P R O X IM AT IN G T H E T R A N S V E R S E F O R C E S A N D B E N D I N G M O M E N T S E X C I T E D B YAPROPELLER 15KNOTS TANKER)

1 ) H O R IZ O N TA L T R A N S VE R S E F O R C E: / F y O ' 7 D /- = 0 . 1 2 , 0 . 0 7 3 S I N ( 4 ~ 3 + 8 0 ° ) Z , 4 )

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= 0 . 1 2 + 0 . 1 5 0 S I N ( 51 3 ÷ 1 01 ° ) Z I 5 )

2 ) V E RT IC A L T R A N S V E R S E F O R C E : r ] / F x ' O 7 D / = 0 0 6 . 0 . 0 7 6 S J N ( 41 3 .1 2 6 ° ) Z = 4 )[ T z g e m j

= 0 0 6 , 0 1 3 0 S IN ( 5 p , 2 0 1 o ) ( Z = ' 5 )

3 ) H O R I Z O N TA L B E N DIN G M O M E N T: [ T x ] = 0 . 0 0 7 , 0 0 0 8 S IN ( 4 1 3+ 1 4 7 ° ) ( Z = 4 )L F g e m 0 . 7 D J

= 0 . 0 0 7 - 0 0 1 9 S IN ( 5 13 + 15 9 ° ) ( Z = 5 )

4 ) V E R T I C A L B E N D IN G M O M E N T : [ - T y ] o o . o 3 2 - o . o 1 2 S ~ N (4 ~ ,.1 3 0 °~ C Z - 4 >IF gem 0.7D]

= 0 0 3 2 + 0 0 5 3 S IN (5 1 3 + 1 5 5 ° ) ( Z = "5 )

F o r a s c r e w p r o p e l l e r w i t h a n e v e n n u m b e r o f b l a d e s ,t he f l u c tua t ing fo rces o f two oppos i t e b l ades w i l l g ive r i s et o a l a rg e r t o t a l t h r u s t a n d t o r q u e a m p l i t u d e b e c a u s e t w ob l a d e s p a s s s i m u l t a n e o u s l y t h e s t e r n a n d i t s a s s o c i a t e d

p e a k s i n w a k e v e l o c it ie s . T h e t ra n s v e r s e fo r c e a n db e n d i ng m o m e n t o f o n e b l a d e w ill b e c o m p e n s a t e d m o r eo r l e ss by thos e o f t he op pos i t e one .

F o r p r o p e l le r s w i t h a n o d d n u m b e r o f b la d e s , t h e b l a d e sw i ll pa s s, a l t e r n a t i n g l y, t h e u p p e r a n d l o w e r w a k e p e a k .T h e t o t a l t h r u s t a n d t o r q u e f l u c t u a t i o n s w i l l t h u s b es m a l l e r t h a n f o r a n e v e n - b l a d e d p r o p e ll e r. F o r a n o d dn u m b e r o f b l a d e s t h e t r a n s v e r s e f o r c e s a n d b e n d i n gm o m e n t s , t h e f a v o r a b l e m u t u a l c o m p e n s a t i o n e x p e ri -e n c e d b y t h e e v e n - b l a d e d p ro p e l l e r w i ll n o t o c c ur. F i g . 1 9

g i v e s a n i l l u s t r a t i o n o f r e s u l t s o f m e a s u r e m e n t s o n 4 7 a n d 6 - b l a d e d s c r e w m o d e l s i n t h e w a k e o f a sh i p m o d e l .

A s t a t i s t ic a l i n v e s t i g a t i o n o f t h e e x p e r i m e n t a l d a t ao n t h r u s t a n d t o r q u e f l u c t u a t i o n s o f s o m e 4 0 d i f fe r e n

s h i p m o d e l s , t e s t e d a t t h e N S M B , l e a d s t o t h e f o l l o w i nconc lu s ions :

1 N o s y s t e m a t i c r e l a t i o n c a n b e f o u n d t o e x is t b et w e e n t h e a m p l i t u d e s o f t h e f o r c e f l u c t u a t i o n s a n d t hp r i n c i p a l s h i p - s h a p e p a r a m e t e r s s u c h a s b l o c k c o e f f ic i enp r i s m a t i c c o e f f ic i en t a n d s c r e w d i a m e t e r - s h i p l e n g t h r a t i o

2 F o r p r i s m a t i c c o ef f ic i en t s o f t h e a f t e r b o d y b et w e e n 0 . 73 a n d 0 . 7 9 i t w a s a s c e r t a i n e d f o r 4 - b l a d e d p r op e ll e rs t h a t w i t h a p r o b a b i l i t y o f a b o u t 8 0 p e r c e n t t hf o l l o w i ng r e s u l ts w i ll b e o b t a i n e d : T h e a m p l i t u d e o f th

A P R I L 1 9 6 6 1 6 7


11/14

~ z l z * ~ z td :z *~z *~z + ; z C z =rz

1- ]Coeff ic ien t symbol i me asured

H y d r od y n am i c 1 7 6 1 0 _ ~ k g m s e c2 1mom ent o f ine r t i a i i

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F i g . 2 0 S c h e m e o f c o u p l e d d i f f e r e n t i a l e q u a t i o n s o f s c r e ws h a f t -t h r u s t b l o c k s y s t e m

f i rs t h a r m o n i c 4 o f th e t o r q u e f l u c t u a t i o n w i ll b e 6 ~ p e r -c e n t o f t h e m e a n t o t a l t o r q u e , t h e a m p l i t u d e o f t h e f ir s th a r m o n i c o f t h e t h r u s t f l u c t u a t i o n s w i ll b e 1 0 p e r c e n to f t h e m e a n t o t a l t h r u s t , a n d t h e a m p l i t u d e s o f t h e h i g h e rh a r m o n i c s w i l l b e s u b s t a n t i M l y l o w e r.

A n d l ik e w i s e f o r a 5 - b l a d e d p r o p e l l e r: T h e a m p l i t u d e so f t h e f i r st a n d s e c o n d h a r m o n i c s o f t h e t o r q u e f l u c t u a -t i o n s w i ll b e , re s p e c t i v e l y, 1 ~ a n d 1 p e r c e n t o f t h e t o t a lt o r q u e , a n d t h e a m p l i t u d e s o f t h e f i r s t a n d s e c o n d h a r -

mon ics o f t he t h ru s t f l uc tua t ions w i ll be , r e spec t ive ly, 2a n d lJ /~ p e r c e n t o f t h e t o t a l t h r u s t .D e v i a t i o n s f r o m t h e s e i n d i c a t i o n s l a rg e r t h a n 2 p e r -

c e n t a b s o l u t e d o n o t o c c u r.3 F in e -en ded vesse l s, wh ich inc ludes mo s t f a s t sh ips,

c a n h a v e s u b s t a n t i a l l y g r e a t e r f o r c e f l u c t u a t i o n s .I n T a b l e 4 a r e v i e w o f t h e f o r m u l a s w h i c h a p p r o x i -

m a t e t h e t r a n s v e rs e f o rc e s a n d m o m e n t s g e n e r a t e d b y a4 - b l a d e d a n d a 5 - b l a d e d p r o p e l l e r b e h i n d a 1 5 - k n o tt a n k e r is g i v e n . F o r t h e lo a d i n g o f t h e s h a f t in th e v e r -t i c a l d i re c t i o n b e si d e s t h e h y d r o d y n a m i c f o r c e s t h e w e i g h to f t h e p r o p e l l e r h ag t o b e t a k e n i n t o a c c o u n t .

T h e m e a n v a l u e o f th e p r o p e ll e r -g e n e r a t e d tr a n s v e r s ef o r c es c a n b e n e g l e c t e d c o m p a r e d t o t h e p r o p e l l e r w e i g h tf r o m a v i e w p o i n t o f s t a t i c s h a f t l o a d in g . T h e s t a t i cb e n d i n g m o m e n t , li ft s th e p r o p e l l e r u p a n d r e d u c e s t h ed e f l e c ti o n o f t h e s t e r n t u b e . R e c k o n i n g h a s to b e h e l d,h o w e v e r, w i t h a l a rg e b e n d i n g m o m e n t i n t h e s h a f t i nw a y o f t h e s c r e w p l a n e .

C o m p a r i n g t h e d y n a m i c b e h a v i o r o f a 4 - b l a d e d a n d5 - b l a d e d p r o p e l l e r , i t i s n o t e d t h a t :

( a ) T h e f l u c t u a t i o n s i n t h e t r a n s v e r s e f o r c e o f t h e 5 -

4 The first harmonic has the blade frequency.

Ta b l e 5 S t o p p in g M a n e u v e r s f o r a l O 0 , O 0 0 - d w t Ta n k e r f oH e a d r e a c h o f 4 k m ( 2 . 5 m i l e s )

Ini tia l sp ee d kn 10,1 11.7 13.3 14.6

S p e e d a t w h i c h t u g s a s s i s te f f e c t i v e l y i n b r a k i n g a n dk e e pt h e s h i p o n c o u r s e

R e a ch a t 2 0 R , R M . a h e a db e f or e t u g a s s i s : a n c e b e c om e se f f e c t i v e

kn 7 8 9 10

km 3.1 2 .9 2 .6 2 ,3

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H e a d r e a c h

km 0.9 1.1 1.4 1.7

km 4 .0 4 .0 4 .0 4 0

b l a d e d p r o p e l l e r ( a l t h o u g h u n i m p o r t a n t ) a r e t w i c e a sl a rge a s t hose fo r the 4 -b l ade d p rope l l e r.

(b ) T h e fl u c t u a t i o n s i n b e n d i n g m o m e n t a re m u c hh i g h e r f o r a 5 - b l a d e d t h a n f o r a 4 - b l M e d p r o p e l le r.

R e a l i z i n g t h a t t h e s h i p d e s i g n e r g e n e r a l l y h a s a t h a n de f f e c t iv e a n d r e l a t i v e l y c h e a p m e a n s o f a v o i d i n g a x i as h a f t v i b r a t i o n s ( t o r q u e a n d t h r u s t ) a n d t h ai ; h e h a s t o

r e d u c e t h e e x c i t a t i o n o f h o r i z o n t a l h u l l v i b r a t i o n s t o am i n i m u m , t h e 5 - b l a d e d p r o p e ll e r i s t o b e r e g a r d e d a s a nu n f a v o r a b l e p r o p e l l e r c o m p a r e d t o a 4 - b l a d e d o n e .

F r o m r e s u l ts o f re c e n t s y s t e m a t i c t e s t s w i t h , a m o n go t h e r s , t h e Wa g e n i n g e n B - S e r i e s i t c o u ld b e d e d u c e d t h a ta 6 - b l a d e d p r o p e l l e r h a s a b o u t 3 p e r c e n t l e s s e f f ic i e n ct h a n t h e c o m p a r a b l e 4 - b l a d e d p r o p e l l e r [ 4] . T h e s m a l l es c r e w d i a m e t e r , t h e l a rg e r s c r e w c l e a r a n c e s a n d t h e v e r yf a v o r a b l e p a t t e r n o f t h e f l u c t u a t i n g f o r c e s ( se e F ig . 1 9a re , however, d i s t i n ,% advan tages , j u s t i fy ing t l he app l i cat ion o f 6 -b l ad ed p rope l l e r s fo r s ing le - sc rew sh ips .

T h e p r o p e l l e r s h i f t a n d t h e s h i p ' s s t e r n a r e n o t i n-f in i t e ly s t if f . Thu~ , be cause o f t he d esc r ibed fo rcep a t t e r n s , e l a s ti c d e f o r m a t i o n s w i ll o c c u r.

T h e t o r s i o n a n d t h e a x i a l d i s p l a c e m e n t o f t h e s c r e wo w i n g t o t h e e l a s t i c s h a f t g i v e r i s e t o h y d r o d y n a m i ccoup l ing be tw een tl '~e ax i a l dyn am ic sc rew fo rces ( t h rus ta n d t h e d y n a m i c t o r s i o n f o r c e s ( t o r q u e ) .

Th e de f l ec t ions ,o f t he p rope l l e r sha f t due to t heb e n d i n g m o m e n t s c r e a te g y r o s c op i c p h e n o m e n a a t t h ep rope l l e r.

A c e r t a i n v o lu m e o f w a t e r f o ll ow s th e u n s t e a d y m o v em e n t s o f t h e s c r e w b l a d es , m a n i f e s t i n g i ts e l f a s a n a d d e dl n a S S .

T h e u n s t e a d y c h a r a c t e r o f t h e s c r e w l o a d i n g w i l l ' i n -d u c e i n t h e s c r e w r a c e h e l ic o i d a l t r a i li n g v o r t e x p a t t e r n sv a r y i n g p e r i o d i c a l ly [ n s t r e n g t h , s e e F i g . 1 0. T h e e n e rg y

c a r r ie d a w a y b y t h i s v o r t e x s y s te m c a u se s h y d r o d y n a m id a m p i n g . A l l t h e h y d r o d y n a m i c q u a n t i t i e s o f t h e s c r e wa s a s o u r c e o f' v i b r a t i o n a r e s u m m e d u p n o w. I f it w e r ep o s s ib l e t o c a l c u l a te o r d e t e r m i n e e x p e r i m e n t M l y t h e s eh y d r o d y n a m i c q u a n t i t i e s , t h e n i t w o u l d b e p o s s i b l e t op r e d i c t t h e e x p e c t e d s t re s s e s in t h e s t e r n c o n s t r u c t i o n a n di n t h e p r o p e l l e r s h a f t r e s u l t i n g f r o m t h e u n s t e a d y f o r c eo f t he sh ip p rope l l e r.

F ig . 20 g ives t he :~cheme o f coup led d i f f e r en t i a l equat i o n s o f t h e s c r e w - s h a f t - t h r u s t b l o c k s y s t e m .



12/14

FI ,~ * ¢~PIEEE

F i g . 2 1

~ ~i~i~ii~i~i~i~i~ii~i~ii~i~i~i~ii~i~ii~i~i~i~i i~i~i~ii~ii~i~i~ ~i~i~i~ ~ii~ ~i~ii~ ~i~i~ ~;~i~i~iII ~ i~ ~ iii~i~ iiii i ii'iiiiii i ~i~i, ii iiii iiiiiiiiiiiili~iiiiii ili ii iiiiiiiiiiii iiiiiiii~i~ i i~iii i~i~ ii~ii~ iiiiiii~ i~ii ~ ~ i ~iii ii ~iii ii~iiiii ii~i~i ~

P r o p e l l e r e x c i te r f o r d e t e r m i n a t i o n o f co e ff i ci e nt s f o r h y d r o d y n a m i c m a s s a n d d a m p i n g a n d h y d r o d y n a m i cc o u p l i n g b e t w e e n t h r u s t a n d t o r q u e v i b r a t i o n s

, 5

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P r o p H [ e r p o s i t i o n e

M E A S U R E D - F U L L S I Z E S H I P

. . .. P R E D I C T IO N D E R I V E D F R O M M O O E L T E S T - R E S U L T S

F ig . 2 2 C o r r e l a t i o n o f m e a s u r e m e n t s o n f u ll si z e a n d p r e d i c t i o no f t o r q u e a n d t h r u s t v a r i a t io n s , b a s e d o n m o d e l - t e s t r e s ul t s

The hydro dynamic mass and damping and the hydro-dynamic coupling between thrust and torque vibrationshave to be determined for the prediction of the stressesin a given shaft configuration. Wereldsma has developeda propeller exciter, Fig. 21, to evaluate the coefficientsappearing in the left-hand side of the equations. Withthis exciter a given axial or torsional vibrati on can be im-posed on a model screw at a certain load KT or K~ and ata certain advance ratio J. In the column at the right in

Fig. 20 the values for a model of a specific single-screwcargo ship, as measured by Wereldsma, are given [15].

The results of Wereldsma's prediction of the torque

and thru st fluctuations based on model-test data and theresults of measureme nts on the full-size ship are comparedin Fig. 22. The good correlation between prediction andmeasurement indicates that a new area of ship-modeltesting has been opened as a service for the shipbuildingand ship operating industry.

Stopping of ShipsWith the aid of th e quasi-stead y velocities and forces

diagram acting on a screw-blade section an insight can begained into the force pattern around the screw duringstopping, Fig. 23.

When the rotative speed of the screw is reduced theangle of attack, and, hence, th e thrust , will decrease fromth at a t full-ahead power (phase 1). At abou t 70 per-cent of the normal ahead number of revolutions the t hrustwill become zero and the screw turns freely (phase 2).

As the RPM is further reduced, a negative angle ofatta ck will result in negative lift and thrust. A furt herreduction in RPM leads to such large negative angle ofattack that flow separation on the screw blades willoccur with an accompanied loss in lift. This separationstarts at ab out 30 percent of the R PM ahead (phase 3).

The decreasing lift causes a decrease in braking force

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d v d v d s d vK : m . a = m ~ - = m ~ - ~ - = m ~ d ~ V

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A = D i s p l a c e m e n t t o n

g = G r a v i t a t i o n a l a c c e l e r a t i o n m s e c - 2

v i = I n i t i a l s p e e d m . s e c - 1

v t = T e r m i n a l s p e e d m s e e - 1

K = B r a k i n g f o r c e t o n

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

u n t i l s u c h t i m e a s t h e s e p a r a t i o n o r p r o f i le d r a g b e c o m el a rg e e n o u g h t o p r e d o m i n a t e a n d t h e b r a k i n g f o r c e a g ai ninc rea se s (phase 4 ) .

T h e c o n t i n u e d i n c r e a s e o f p r o f il e d r a g w i t h t h e a s t e r no p e r a t i o n o f t h e p r o p e l l e r w i ll f u r t h e r i n c r e a s e t h e b r a k -i n g f o r e s ( p h a s e 5 ). A t a h i g h n u m b e r o f a s t e r n r e v o l u -t i o n s t h e p r o b a b i l i t y o f c a v i t a t i o n a n d o f d r a w i n g a i r i n t ot h e p r o p e l l e r i n c r e a se s . T h e o c c u r r e n c e o f o n e of t h e s em a y c a u s e a d e c r e a s e i n b r a k i n g f o r c e .

T h e t y p i c a l S - c h a r a c te r i st i c , d e s c r i b in g t h e t h r u s tb e t w e en 1 0 0 p e r c e nt R P M a h e a d a n d 1 0 0 p e r c e n t R P Ma s t e r n o p e r a t i o n , w a s d e s c r i b e d f o r t h e f i r s t t i m e b yT h a u [ 1 6 ]. F o r t h e p r o p e l l e r a l o n e , t h i s c u r v e c a n b ed e t e r m i n e d f o r q u a s i - s t e a d y o p e r a t i o n , w h e n th e o p e n -w a t e r s c r e w c h a r a c t e r i s t i c s a r e a v a i l a b l e f o r t h e a h e a d a n da s t e r n r u n n i n g c o n d i t i o n [ 1 7 ] . T h e s e q u a s i - s t e a d y c o n -s i d e r a t i o n s a r e a t t h e s a m e t i m e t h e b a s i s f o r a m e t h o df o r c a l c u l a t i n g t h e h e a d r e a e h . I n F i g . 2 4 a n in t e g r a li s d e r i v e d c a l c u l a t i n g t h e h e a d r e a c h f r o m t h e b a s i c l a w o fd y n a m i c s , t h a t

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H e a d r e a c h : fo r a 1 0 0 0 0 0 - d w t t a n k e r ; i n i t i a l s p e e d1 4 k n o t s

T h e h y d r o d y n a m i c a d d e d m a s s h a s b e e n t a k e n i n t o acoun t by t i l e u se o f a f ae to r 1 .05 .

T h e v a l u e s o f t e e r a t i oV/K c a n b e d e t e r m i n e d f r o m am o d e l t e s t f o r e a c h s p e e d V a t d i f f e r e n t r o t a t i v e s p e e dF o r a g iv e n s h ip d i s p l a c e m e n t t h e b r a k i n g f o r c e K c a ll c a l c u la t e d f o r a n y c o m b i n a t i o n o f s p e e d V a n d t h e r o tl i v e s p e ed , a n d t h e i n t e g r a l : fo r t h e h e a d r e a c h c a n d e t e r m i n e d .

T h e r a t i o b e t w e e n d i s p l a c e m e n t a n d p o w e r A / S H P, a s wr i t t en i n t he i n t eg ra l o f F ig . 24 , t he r,~ t i oA/K, isv e r y i m p o r t a n t f o r t h e l e n g t h of t h e h e a d r e a e h . A l a rd i s p l a c e m e n t p r o p e l l e d b y a r e l a t i v e l y s m a l l p o w e r wg i v e a l o ng h e a d r e a c h ( l ar g e v a lu e o f th e r a t i o A / S Ht a n k e r s ) . A l o w v a l u e f o r t h e r a t i o A / S H P, a s f o r is t a n c e f o r d e s t r o y e r s a n d t u g b o a t s , w i l l g i v e a v e r y s h oh e a d r e a c h .

A n a l y s i s fo r a m a x i m u m a l lo w a b l e h e a d r e a e h o f 4 k( 2 . 5 m i l e s ) w e r e m a d e f o r a 1 0 0 , 0 0 0 - d w t t a n k e r a t d i f f

e n t i n i t i a l s p e e d s , w i t h t h e a s s u m p t i o n t h a i , t u g s w o ua s s i s t i n t h e s t o p p i n g m a n e u v e r. Ta b l e 5 i s a r e v i eo f t h i s a n a ly s i s . T h e o n l y p o s s i b l e m a n e u w ~ r i s t h e os t a r t i n g f r o m a n i > i ti a l s p e e d o f 1 0 .1 k n o t s . W h e n b r ai n g f r o m 1 0 .1 t o 7 k n o t s w i t h a r o t a t i v e s p e e d o f 2 0 r pt h e d i s t a n c e c o v e r e d is 3 .1 k m ( 2 m i l e s) . A t. 7 k n o t s tt u g s t a k e h o l d a n d e x e r t a n e x t r a b r a k i n g f o r c e o f 4 0 t oT h e r o t a t i v e s p ee d b e c o m e s 50 r p m a s t e r n a n d t h e t a n ks t o p s a f t e r a n o t h e r 0. 9 k m ( 0 .6 m i l e ). O p e r a t i o n s q u i r i n g t u g s t o n t s k e f a s t a t s p e e d s g r e a t e r t h a n 7 k nn m s t b e c o n s id e r e d a v e r y r i s k y u n d e r t a k i n g . F o rm a x i m u m a l l o w a b l e h e a d r e a c h o f zi k m t h e i n iti~ d s p eo f t h e 1 0 0 , 0 0 0 - to > t a n k e r m a y, h e n c e , n o t e x c e e d 1

k n o t s .T h e s e c o n c lu s io n s a r e b a s e d o n m o d e l te s ts . I t mb e p o ss i b le , d u e t o a c o n s e r v a t i v e i n t e r p r e t a t i o n o f ts c a l e e f f e c t , t h a t t h e s e r e s u l t s a r e s o m e w h a t p e s s i m i s tO w i n g t o t h e l a c k o f s u f i % i e n t d a t a f r o m f u l l - s c a l e t e sa c o r r e c t i o n o f t h e d a t a i n Ta b l e 5 fo r s c a le e f f e c ts is nposs ib l e .

I n F i g . 2 5 t h e c ~ d c u la t io n o f s t o p p i n g o f a 1 0 0 , 0 0 0 -t a n k e r i s g i v e n f o r a n in i t i a l s p e e d o f 1 4 k n o t s . Td i f f e r e n t p h a s e s i n t o w h i c h t h e w h o l e m a n e u v e r c a n d i v i d e d a r e i n d i c a t e d . A n e s s e n t ia l p a r t o f th .e m a n e u v

1 7 0 M A R I N E T I ' C H N O L O G


14/14

is the fact that the screw is to be stopped when the speedreaches 6 knot s and the tugs make fast. Otherwise thevessel will loose steerageway.

Finally, it may be noted that data and testing methodsas treated in this review are impo rtant resources neededin the choice of the type of propeller and the determina-tion of its dimensions for a given application.

R e f e r e n c e s

1 L. Troost, Open Wat er Test Series with Moder nPropeller Forms, Trans. NE CI,1950-51.

2 W. P. A. van Lammeren, L. Troost, and J. G.Koning, Resistance, Propulsion and Steering of Ships,TheTechnical Publishing Company, H. Stare, Haarlem, 1948.

3 J. D. van Manen, Fundamentals of Ship Resist-ance and Propulsion, Part B, Propulsion, InternationalShipbuilding Progress,1957.

4 J. D. van Manen, A Review of Research Activi-ties at the Netherlands Ship Model Basin, InternationalShipbuilding Progress,1963.

5 A.J . Taehmindji and W. B. Morgan, The Designand Estim ated Performance of a Series of SupercavRating

Propellers, Second Symposium on Naval Hydrod y-namics, Wash ington , 1958 (1960).6 J. D. van Manen, Ergebnisse systematischer

Versuche mit Propellern mit armb~hernd senkrechtstehender Achse, Jahrbuch STG,1963; Schip en We'll,1964.

7 J .B . Hadler, W. B. Morgan, and K. A. Meyers,Advanced Propeller Propulsion for High-Powered

Single-Screw Ships, Trans . SN AM E,vol. 72, 1964, pp.231-293.

8 I. It. Abbott, A. E. yon Doenhoff, and L. S.Stivers, Summary of Airfoil Dat a, NACA Report 824,1945.

9 L. C. Burrill and A. Emerson, ':Propeller Cavita-

tion: Further Tests on 16 in. Propeller Models in theKing's College Cavitati on Tun nel, Trans. NECI, 1962-63 ; International Shipbuilding Progress,1963.

10 J. D. van Manen, Dut ch die Schraube erregteSehiffssehwingungen, Schiffstechnilc,1965; Schip enWe~f,1965.

11 S. Schuster, Propell er in Non-Uni form Wake-Collection of Existing Work, Tent h ITTC, London,England, 1963, Report of Propulsion Committee, Ap-pendix 7.

12 J. Krohn and P~. Wereldsma, Comparative ModelTests on Dy namic Propeller Forces, International Ship-building Progress,1960.

13 (a) J. Krohn, Ueber den Einflusz der Propeller-belastung bei verschiedener Hintersehiffsform auf dieSehub- und Drehmomentsehwankungen am Modell,Sch~i~ und Hafen,1958.

(b) J. Krohn, Ueber den Einflusz des Propeller-durchmessers auf die Schub- und Drehmomentschwank-ungen am Modell, Sch~stechnih,1959.

14 J. D. van Manen and R. Wereldsma, Propell erExcited Vibratory Forces in the Shaft of a Single ScrewTanker, International Shipbuilding Progress,1960.

15 (a) R. Wereldsma, Dyn ami c Behaviour of ShipPropellers, Doctor 's Thesis, Technological Universi ty,Delft, 1965; Publication No. 255 of the NSMB.

(b) R. Wereldsma, Expe rime nts on VibratingPropeller Models, International Shipbuilding Progress,1965.

16 W. E. Thau, Propellers and Propelling Ma-chinery. Maneuvering Characteristics During Stoppingand Reversing, Trans . SNAME,1937.

17 It. F. Nordstr6m, Screw Propeller Character-istics, Meddelanden Statens SkeppsprovningsanstaltNo. 9, 1948.

A P R I L 1 9 6 6 1 7 1

the choice of the propeller

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