The effect of a nozzle on steering characteristics

THE EFFECT OF A N O Z Z L E O N STEERING CHARACTERISTICS*)

L . A . v a n G u n s t e r e n * " ) a n d F . F . v a n G u n s t e r e n ' * ' ) .

A b s t r a c t .

A method is presented f o r p r e d i c t i n g how the s t e e r i n g c h a r a c t e r i s t i c s of a ship are affected by f i t t i n g a f i x e d nozzle. The presence of a nozzle u p s t r e a m appears to have a s i g n i f i c a n t e f f e c t on the r u d d e r f o r c e s . F u l l - s c a l e manoeuvring t r i a l s c a r r i e d out w i t h two t w i n - s c r e w tugs, one w i t h open p r o p e l l e r s and the other equipped w i t h nozzles, c o n f i r m the p r e d i c t e d t r e n d s . It is concluded that p r o p e l l e r , nozzle and rudder should be designed i n an integrated way to ensure that an o p t i m u m solution is obtained w i t h r e g a r d to both p r o p u l s i v e and s t e e r i n g q u a l i t i e s .

Introduction.

A f i x e d nozzle not only a f f e c t s the p r o p u l s i v e c h a r a c t e r i s t i c s of a ship but also i t s s t e e r i n g p r o p e r t i e s . A c c o r d i n g l y , when a nozzle is considered f o r i t s w e l l k n o w n p r o p u l s i v e f e a t u r e s -f or instance, an increase i n t h r u s t at low speeds-i t speeds-is speeds-i m p e r a t speeds-i v e to pay c a r e f u l attentspeeds-ion to the e f f e c t this w i l l have on the manoeuvring cha-r a c t e cha-r i s t i c s o f t h e ship. The pucha-rpose of this papecha-r i s to provide a means of p r e d i c t i n g how s t e e r i n g c h a r a c t e r i s t i c s w i l l be a f f e c t e d by f i t t i n g a nozzle, assuming the behaviour without a nozzle to be already known. P r o p e l l e r , nozzle and rudder can then be designed i n an integrated manner, taking account of both p r o p u l s i v e and manoeuvring r e -quirements.

Although the theory can also be used to p r e d i c t t r a n s v e r s e f o r c e s on s t e e r i n g nozzles w i t h or without s t a b i l i z e r s , we s h a l l confine ourselves here to f i x e d nozzles. F r o m a manoeuvring point of v i e w , the f i t t i n g of a nozzle i s comparable to i n c r e a s i n g the l a t e r a l area of skegs and the l i k e , t h i s having an adverse e f f e c t on the t u r n i n g d i a -m e t e r and a f a v o u r a b l e influence on response t i m e and s t a b i l i t y on a s t r a i g h t course.

T h i s can b e and g e n e r a l l y should b e c o m -pensated f o r by decreasing the l a t e r a l area of the a f t e r b o d y , p a r t i c u l a r l y d i r e c t l y i n f r o n t of the nozzle. A second e f f e c t of the nozzle is that i t influences the f l o w at nearby l i f t i n g s u r f a c e s , especially the f l o w and consequently the f o r c e s on the rudder downstream.

*) Paper presented at the Second International Tug Conference, London, October 25 - 28, 1971.

•*) Lips N . V . Propeller Works, Drunen, Holland. ***) Sea Transport Engineering N . v . , Amsterdam, Holland.

Since i t is not our intention to p r e d i c t the actual manoeuvring c h a r a c t e r i s t i c s , but only to d e t e r -mine the d i f f e r e n c e s i n s t e e r i n g c h a r a c t e r i s t i c s due to the presence of a nozzle, a r e l a t i v e l y s i m p l e m a t h e m a t i c a l model can be used, r e f e r e n c e [ 1 ] . The m o t i o n i n the h o r i z o n t a l plane is d e s c r i b e d by the l i n e a r i z e d equation of m o t i o n r e g a r d i n g moments w i t h respect to the centre of g r a v i t y (Nomoto's equation). The moments are s p l i t up i n t o :

1. Moments due to t r a n s v e r s e f o r c e s on the p r o p e l l e r -nozzle - r u d d e r c o n f i g u r a t i o n . 2. A l l other moments. A v a i l a b l e theory f o r the p r e d i c t i o n of l i f t f o r c e s on nozzles and r u d d e r s is l i m i t e d to an i s o l a t e d nozzle (without p r o p e l l e r , r e f e r e n c e [2] [ 3 ] , or w i t h p r o p e l l e r , r e f e r e n c e [4]) or r u d d e r . Since i t may be expected that coupling e f f e c t s between the nozzle and the rudder are s i g n i f i c a n t , a l i n e a r -ized theory has been developed which allows f o r these e f f e c t s . The theory is applied i n a computer p r o g r a m f o r p r e d i c t i n g l i f t f o r c e s on a n o z z l e -r u d d e -r c o n f i g u -r a t i o n at a -r b i t -r a -r y angles of attack. The r e s u l t s of calculations i n w h i c h the coupling t e r m s are d i s r e g a r d e d , are i n agreement w i t h known r e s u l t s of i s o l a t e d nozzles, r e f e r e n c e [ 3 ] , and r u d d e r s , r e f e r e n c e [ 5 ] . F u l l s c a l e m a n o e u v r -ing t r i a l s w i t h two t w i n - s c r e w tugs, one w i t h open p r o p e l l e r s and the other equipped w i t h nozzles, i l l u s t r a t e that the procedures given are indeed adequate f o r analysing how s t e e r i n g c h a r a c t e r -i s t -i c s are a f f e c t e d by f -i t t -i n g a nozzle.

Mathematical model of t u r n i n g .

hi o r d e r to p r e d i c t how t u r n i n g qualities are a f -fected by f i t t i n g a nozzle, we may describe the manoeuvre by the l i n e a r i z e d equation of moments about the centre of g r a v i t y i n the h o r i z o n t a l plane reference [ 1 ] :

I - r + Np • r = N g - 5 (1)

where:

I = mass moment of i n e r t i a , including h y d r o -dynamic e f f e c t s

r = r a t e of t u r n i n g

Ng • 5 = moment, due to the l a t e r a l f o r c e s on r u d d e r s and nozzles

NJ, • r = a l l other moments i n a steady t u r n . As explained i n the next section the r u d d e r -nozzle moment can be w r i t t e n :

Ng - 5 = C i a R - C 2 P (2)

where:

ctpj = angle of attack of the rudder

p = - OQ (p is positive i n steady turning) ajy = angle of attack of the nozzle

5 = rudder angle

If necessary, a c o r r e c t i o n f o r the s t r a i g h t e n i n g e f f e c t of the hull can be i n t r o d u c e d by taking a s m a l l e r value f o r C3 in equation (4) than f o l l o w s f r o m equation (5).

F o r the present purpose, X p c a n b e a p p r o x i m a t e d by:

X p = 0 . 75 L (6) where:

L = length of ship

Substitution of equations (3), (4) and (2) into equation (1) gives:

] r+r =[ „ . , „ „ . „ ] 5 Nr + ( C i + C 2 ) C 3 N r + ( C i + C2)C3

(7) T h i s is equivalent to Nomoto s equation:

T r + r = K 5 (8) where: I Nj, + ( C i + C 2 ) Cg (9) A c c o r d i n g to F i g u r e 1 , we have: 5 = aj^ + P (3)

The angle of attack at the rudder or r u d d e r -nozzle c o n f i g u r a t i o n i s p r o p o r t i o n a l to the r a t e of t u r n :

(4) The c o e f f i c i e n t C3 depends on the distance, X p , f r o m the r u d d e r - n o z z l e c o n f i g u r a t i o n to the pivotal point at which the centreline of the ship is perpendicular to the radius of t u r n , and on the axial velocity U at the r u d d e r n o z z l e c o n f i g u r a -t i o n , including -the e f f e c -t of -the p r o p e l l e r :

X p

^ 3 - I T (5)

K = C l (10)

Nr + ( C i + C 2 ) C3

When the rudder is moved f r o m z e r o to a c o n -stant angle SQ, the solution of equation (8) gives the t u r n i n g response as a f u n c t i o n of t i m e (see F i g u r e 2) :

- t / T

r = K 5 o ( l - e ) (11)

Figure 1. Definition of symbols.

The t u r n i n g m o t i o n is d e t e r m i n e d by two q u a n -t i -t i e s :

K , r e p r e s e n t i n g the t u r n i n g capacity

i . e . the r a t e of steady t u r n per degree of rudder angle, w h i c h is r e c i p r o c a l to the t u r n -ing d i a m e t e r .

T , r e p r e s e n t i n g the response t i m e

i . e. the t i m e taken to r e a c h 63% of the steady r a t e of t u r n .

A method w i l l now be given f o r c a l c u l a t i n g these two t u r n i n g q u a l i t i e s , ' t u r n i n g c a p a c i t y ' and 'response t i m e ' , f o r any r u d d e r n o z z l e c o n -f i g u r a t i o n considered, assuming that K or T i s known f o r an e x i s t i n g r u d d e r n o z z l e or n o n -shrouded c o n f i g u r a t i o n on a s i m i l a r h u l l .

T u r n i n g c a p a c i t y .

If the t u r n i n g capacity of a c e r t a i n rudder -nozzle c o n f i g u r a t i o n , K , is known the t u r n i n g capacity of a new r u d d e r - n o z z l e c o n f i g u r a t i o n . K . canbe calculated by applying equation ( 1 0 ) to both c o n f i g u r a t i o n s .

For the new c o n f i g u r a t i o n we have;

K

C l

(.12)

N,, + ( C i + C2) Cg

The t o t a l l e s s - r u d d e r - n o z z l e - m o m e n t coef¬ f i c i e n t , N j , , and Cg may be assumed to be i n -dependent of the rudder -nozzle configurations , at least f o r a f i s h i n g vessel h u l l design, r e f e r e n c e

[ 5 ] :

N , = N ,

C3 "C3

( 1 3 )

( 1 4 )

If necessary, c o r r e c t i o n s canbe introduced f o r d i f f e r e n c e s in hull design. The new t u r n i n g capa-city is obtained by substitution of equations ( 1 0 ) and ( 1 2 ) into equation ( 1 3 ) ;

-I K = K [ -_ -_

C l + K i Cg ( C l + Cg) - Cg ( C l + C2) i The c o e f f i c i e n t s C i and C2 can be calculated f o r both configurations w i t h the method given i n the next section.

R e s p o n s e t i m e .

The response t i m e can be calculated f o r a new r u d d e r - n o z z l e c o n f i g u r a t i o n f r o m one of the two known t u r ning q u a l i t i e s , K or T of a c e r t a i n c o n -f i g u r a t i o n by substitution o-f equation (7) or ( 8 ) into equation ( 1 3 ) , w h i c h gives;

If T is known; f = T [ ] ( 1 6 ) I + T i C g ( C i + C2) - C g ( C i + C 2 ) l If K i s known; I K C l + K ICg ( C l + C2) - Cg ( C l + C2)! ( 1 7 )

For the qualitative p r e d i c t i o n of the response t i m e , the mass moment of i n e r t i a is assumed to be independent o f t h e rudder-nozzle c o n f i g u r a t i o n .

R u d d e r - n o z z l e f o r c e c o e f f i c i e n t s ( C l , C 2 ) .

The r u d d e r - n o z z l e moment can be w r i t t e n as; Ns - S = C L , ^ ^ ^ • y . p U 2 ( D l + b c ) • X f ( 1 8 ) i n which; ' L T O T ^R L _D (19) V , p U 2 ( l D + b c )

l i f t f o r c e on the rudder (positive i n t u r n i n g c i r c l e ) .

l i f t f o r c e on the nozzle (negative i n t u r n i n g c i r c l e ) . 1 = length of nozzle. D = diameter of nozzle, b = height of r u d d e r , c = chord length of r u d d e r . X f = distance f r o m the r u d d e r - n o z z l e f o r c e to the centre of g r a v i t y of the ship. U = velocity at rudder -nozzle c o n f i g u r a t i o n

including e f f e c t of p r o p e l l e r (see next section).

P = density.

The f o r m u l a f o r C^^.^^^ can be w r i t t e n i n the - ] ( 1 5 ) f o r m ; C L T O T " ^ ""1^^ """o " ^ i " R ^ where; d C LR be d C LD bc + l D H I D d a j 3 b c + l D ( 2 1 ) ( 2 2 ) d C Here- LR d a (b/c) and d C LD d a (1/D) are the l i f t D

gradients f o r the isolated r u d d e r and nozzle r e -s p e c t i v e l y , and can be obtained f r o m r e f e r e n c e -s [ 5 ] and [ 3 ] . The c o e f f i c i e n t s a i and b i r e p r e s e n t down-wash t e r m s due to the i n t e r a c t i o n between nozzle and r u d d e r . The i n f l u e n c e of the d o w n -s t r e a m l i f t i n g -surface on the u p -s t r e a m l i f t i n g surface i s s m a l l , but the r e v e r s e i s not the case. Consequently, b i may be neglected i n approximate c a l c u l a t i o n s . The c o e f f i c i e n t a i depends on the aspect r a t i o of the duct and the distance to the r u d d e r , but the influence of both these p a r a m e t e r s is s m a l l (see Table 1 ) . A c c o r d i n g to equations

Table 1

Down-wash coefficients a^ and bi-Fiudder aspect ratio

b/c =1. 52 b/D =1. 00

Nozzle aspect ratio

1/D _^1

.40 . 73 . 0945 2 a / D = .1183 . 50 . 80 . 0935 . 60 . 86 . 0905 2a/D = l . 00 . 50 . 85 . 020

( 2 ) , (18) and ( 2 0 ) the c o e f f i c i e n t s C i and C2 a r e : C i = X f • y , p u 2 ( l D + b c ) ( a o + b o b i ) - N R ( 2 3 ) C 2 = X f • V . p U 2 ( l D + bc)(bo-aoao) • N Q ( 2 4 ) where; N R = number of r u d d e r s N Q = number of nozzles r . K ^ . ( l . -''s

^

_^

Figure 2. Turning rate as a function of time.

P r e d i c t i o n of l i f t forces on rudder and n o z z l e .

In the t h e o r e t i c a l p r e d i c t i o n of the t r a n s v e r s e f o r c e s on p r o p e l l e r , nozzle and r u d d e r , the f o l l o w i n g assumptions are made (see also r e f -erence [ 7 ] ) :

- The f l u i d i s i n c o m p r e s s i b l e and n o n - v i s c i d . - Body f o r c e s are neglected.

- The f l o w i s steady.

- Thickness e f f e c t s may be neglected.

F o r our purpose we may neglect the p r o p e l l e r -nozzle i n t e r a c t i o n and use the s u p e r p o s i t i o n model. Then the l i f t of the duct i s o v e r - e s t i m a t e d , but t h i s i s compensated f o r by neglecting the

t r a n s v e r s e f o r c e on the p r o p e l l e r , thus y i e l d i n g an approximately c o r r e c t t o t a l l i f t (see r e f e r e n c e

[ 4 ] ) .

This reduces the p r o b l e m to the p r e d i c t i o n of the l i f t f o r ces on a nozzle without p r o p e l l e r at an angle of attack a-^ (= - (3) due to the yaw and d r i f t of the ship, and a r u d d e r at an angle of attack O R , w h i c h r e p r e s e n t s the rudder angle c o r r e c t e d f o r the e f f e c t of yaw and d r i f t . The m a i n s t r e a m v e l o c i t y U is taken to be; U = V S ( 1 - W ) - ( 1 + F ( C T ) ) ( 2 5 ) where; Vg = speed of ship, w = wake f r a c t i o n . F (C^Y) = c o r r e c t i o n on the m a i n s t r e a m a l l o w ing f o r the a x i a l v e l o c i t y due to the p r o -p e l l e r .

The c o r r e c t i o n on the m a i n s t r e a m extends the v a l i d i t y of the method to m o d e r a t e ' p r o p e l l e r loadings. In the case of a nozzle, the v e l o c i t y f a r downstream i s taken to be r e p r e s e n t a t i v e and according to the a x i a l m o m e n t u m t h e o r y d e s c r i b -ed i n r e f e r e n c e [8] t h i s y i e l d s ; F ( C T ) = - 1 ( 2 6 ) 2 ( - 1 + V 1 + T C T ) where: C T = t h r u s t c o e f f i c i e n t ; T T C T = 9 TT o y , p U ^ - D ^ 4

T'p -- total t h r u s t of p r o p e l l e r and nozzle. T = r a t i o of p r o p e l l e r t h r u s t to t o t a l t h r u s t .

As f a r as the r u d d e r is concerned t h i s i s a good a p p r o x i m a t i o n , but w i t h r e g a r d to the nozzle approximately half t h i s value w o u l d be c o r r e c t . An estimate of the e r r o r s i n t r o d u c e d by a l l o w i n g f o r p r o p e l l e r loacUng i n t h i s r a t h e r crude way may t h e r e f o r e be made by r e c a l c u l a t i n g w i t h F

half as l a r g e as i n equation ( 2 6 ) . I t i s only i f l i t t l e d i f f e r e n c e i s found that the r e s u l t s are t r u s t -w o r t h y . I n the case of an open p r o p e l l e r -we have

(see r e f e r e n c e [ 9 ] ) :

X / R

F ( C ^ ) =V2(-1+ V l + C-p)(l + -)Fj^(x/R) V 1 + ( x / R ) 2

where:

X = downstream distance to p r o p e l l e r plane F , = f a c t o r according to F i g u r e 3

I , , . .

O -,5 - IO - 15 - 2 0

Vn

Figure 3. Factors for calculation of axial velocity U.

The l i f t f o r c e s on nozzle and r u d d e r are c a l -culated w i t h the quarter point method i n which the l i f t i n g surface is r e p l a c e d by a single, c o n -centrated bound v o r t e x at the % c h o r d point, and the l i n e a r i z e d boundary condition i s s a t i s f i e d at the % c h o r d point. In o r d e r to estimate the a c -curacy of the method, we assume i n our computer p r o g r a m that the l i f t i n g surface r e p r e s e n t i n g the r u d d e r is s p l i t up into an a r b i t r a r y number of v e r t i c a l s t r i p s w i t h a concentrated bound v o r t e x at the '4 c h o r d point, and p i v o t a l points i n which the boundary condition is s a t i s f i e d at the c h o r d point of each s t r i p . F o r the sake of s i m p l i c i t y the d e r i v a t i o n is given f o r the case of only one concentrated bound v o r t e x on the r u d d e r . F r o m the l i n e a r i z e d theory of ducts i n oblique flow, r e f e r e n c e [ 3 ] , i s known that the l i f t c o e f f i c i e n t per degree of angle of attack on the duct depends only on the c h o r d diameter r a t i o . The camber and d i f f u s e r angle have no e f f e c t on the net l i f t f o r c e . The p r o b l e m can now be f o r m u l a t e d as f o l l o w s .

G i v e n :

D = diameter of c y l i nd er r e p r e s e n t i n g the duct; D = 2 R .

1 = c h o r d length of the duct, b = span of the rudder, c = chord of the r u d d e r .

a = distance f r o m t r a i l i n g edge of duct to leading edge of rudder (d = a + % 1 c ) .

a j ) = angle of attack of duct. OR = angle of attack of r u d d e r . Required: ^ L D l^^t c o e f f i c i e n t of duct; _ l i f t on duct C L D H p U ^ l R Cj^R = l i f t c o e f f i c i e n t of r u d d e r ; l i f t on rudder ^ L R = ^ y, p c b

In order to f i n d these, the strength of the bound v o r t e x d i s t r i b u t i o n s must be calculated. The r e q u i r e d l i f t c o e f f i c i e n t s then f o l l o w f r o m the K u t t a -Joukowsky l a w . The rectangular coordinates x , y, z, and the c y l i n d r i c a l coordinates x , r , e , are introduced as indicated i n F i g u r e 4. The bound c i r c u l a t i o n of the duct is expressed i n a F o u r i e r s e r i e s . The s e r i e s contains only cosine t e r m s having angles 6, 3e, 5 6 . . . , as can be d e r i v e d f r o m s y m m e t r y considerations:

r D ( 0 ) = 2 1 U ^ 2 ^ A i n C O S [ ( 2 m - l ) 9 ] (28)

S i m i l a r l y , the c i r c u l a t i o n of the rudder can be w r i t t e n as: r j ^ ( H ; ) = 2 c U ^ | ^ B j j ^ s i n [ ( 2 m - l ) H ^ ] (29) where: HJ=Glauert coordinate; b y = : - - c o S H ^ (30)

F r e e r e c t i l i n e a r v o r t i c e s are shed downstream f r o m the duct having a strength per u n i t a r c length according to the law of c o n t i t u i t y of v o r t e x

strength: 1 clrj3(e) >'D(6) = - ^ de (31) w h i c h gives w i t h equation (28): (6) = - 21U 2 ( 2 m - l ) A m S i n [ ( 2 r a - l ) e ] ^ (32) S i m i l a r l y , the f r e e r e c t i l i n e a r v o r t i c e s shed

downstream f r o m the rudder have a strength per unit of span of:

d r ^ (33) r R ( y ) R d y ^ R ' ^ ^ = - ^ J^ ( 2 m - l ) B ^ c o s [ ( 2 m - l ) ^ ] (34) To apply the boundary condition at the duct, we have to consider the r a d i a l v e l o c i t y v^ ( 1 / 2 , R , 6) at the % c h o r d point of the duct induced by the e n t i r e v o r t e x s y s t e m . W i t h r e g a r d to the rudder the upwash Vg (d + c / 2 , y , 0) must be considered. Because of s y m m e t r y , only p i v o t a l points i n the f i r s t quadrant of the nozzle and i n the upper half of the rudder need be taken into account. A f t e r some m a n i p u l a t i o n , application of the B i o t and Savart law gives the f o l l o w i n g r e s u l t s . W h e r e necessary, coordinates r e f e r r i n g to v o r t e x elements are distinguished f r o m p i v o t a l point coordinates by the s u b s c r i p t v . The symbol <^ i n -dicates that the i n t e g r a l has to be taken i n the sense of the Cauchy p r i n c i p a l value.

Duct s e l f - i n d u c e d : 1+- 2R [ ( ^ ) ^ + 2 - 2 c o s ( e - e v ) ] ' ' ' ^ (36) s i n g u l a r i t y at: 9 = e v Rudder s e l f - i n d u c e d : ^ (d+ c / 2 , y . 0) U R c 2 b " s i n v ^ ^ ^ ^ B ^ s i n [ ( 2 m - l ) H > ] ^ ^ 0 [(c/2)2+y2+(^cosiv)2+ybcoS4j]3/2 (37) Vz ( d + c / 2 . y , 0 ) U yR TT 2 ^ ( 2 m - l ) Bjj^cos[(2m-l)^v](y+-coSMj) 2TT ^ 0

1+-[y2+ (—cos^j)2+y b cos^jj] 2 c/2 [ ( c / 2 ) 2 + y2 + (-c0S4;)2+ybC0Si|;f''= 2 d^i^ (38) s i n g u l a r i t y at: y = - -costp 2 Induced by duct on r u d d e r : V z ( d + c / 2 , y , 0 ) U D I R (d+c/2) T T/ 2 ƒ - T T/ 2 v , , ( l / 2 , R U D - 7 W ' ^ T (3 5) c o s ( e - e v ) A j ^ c o s [ ( 2 m - l ) e v ] [ ( ^ ) ^ + 2 - 2 c o s ( e - e v ) ] 3 / 2 V r ( l / 2 , R , 0 ) ^ ^ 2T s i n ( e - e y ) ^ ( 2 m - l ) A ^ sin [ ( 2 m - 1 ) 6^] [ l - c o s ( 9 - e v ) ] 2 A ^ c o s [ ( 2 m - l ) e l cose m = l ™ [(d+ c / 2 ) 2 + y 2 + R 2 _ 2 y R s i n e ] 3/2 de (39) Vz ( d + c / 2 , y , 0 ) U ^ D = T T/ 2 I ( 2 m - l ) A ^ s i n [ ( 2 m - l ) e ] ( y - R sine) J_ ƒ m = l ^ TT - T T/ 2 [ 1 + -(y2 + R 2 - 2 y R sinf (d+ c/2) ,de [(d+ c/2)2 + y 2 + R 2 - 2 y R sine]'''' (40) Induced by r u d d e r on duct:

V r ( l / 2 , R , e ) b ( d - 1/2)c TT U 4T T 2 ^ B j j ^ s i n [ ( 2 m - l ) e ] sinnjcosf m = l [ ( d - 1/2)2+R2+ cosiij)2+R b cosn;sine]3/2 dif (41) , (1/2 , R , 9) c b TT U 1 ( 2 m - l ) cos [ ( 2 m - l ) y ] cos^i^cosf m = l o b 2 [R + (-costal) + R b c o S 4 ; s m e | 2 ( d - 1 / 2 ) [ ( d - 1/2)2+(^coSHj)2+RbcoS4J sine]'''^ (42) We truncate the s e r i e s at N t e r m s : m = l , 2 N

N p i v o t a l points have to be selected i n the f i r s t quadrant of the nozzle and N p o v i t a l points on the upper half of the r u d d e r , f o r instance at:

m 2 N m m = 1,2 N f o r the duct (43) TT m - 1 v m = i ( i + - i r ) b y m = - 2 ° o ^ ^ m m= 1. 2 . . . . N f o r the rudder (44) The boundary concUtions g i v e :

V r ( 1 / 2 , R , 9in) on the duct: U ^ D ' ^ D ' ^ R ' ^ R ^ Vz(d+ c / 2 , y j ^ , 0 ) on the r u d d e r : U " D cos e m (45) ^ D ' ^ D - ' ^ R ' ^ ' R " - p (46) Equations (45) and (46) f o r m a set of 2 N l i n e a r equations f r o m w h i c h the 2 N unlcnown c o e f f i c i e n t s Aj^.^ and B^^ ( m = l , 2 . . . . N) can be obtained. B e -cause of the s i n g u l a r i t y at 9=eyin equation (36), a s m a l l i n t e r v a l around the s i n g u l a r i t y should be excluded f r o m the i n t e g r a t i o n :

e-A9 e+Ae 2TT

f . . . . d 9 v + ƒ . . . . d 9 y + ƒ . . . . d 9 y 0 9-Ae e+Ae

It can be shown that the middle t e r m does not contribute to the induced v e l o c i t y , p r o v i d e d A9 is taken s m a l l enough ( f o r instance Ae = 0.05 radians). Equation (38) should be evaluated i n a s i m i l a r way. Once the c o e f f i c i e n t s A m and B ^ i have been f o u n d , the l i f t on nozzle and r u d d e r f o l l o w f r o m the Kutta-Joukowsky l a w :

l i f t on duct:

2TT

L n = 2 p l u 2 R [ 2 A..^ cos [ ( 2 m - 1 ) 9 ] • cos9 d9

U • 1 I I I 0 (47) l i f t on r u d d e r : TT L R = p c U 2 b ƒ 2 B i n S i n [ ( 2 m - l ) y ] • sin^j dn^ _ m = 1 0 ₍₄₈₎

Only the f i r s t t e r m of the F o u r i e r s e r i e s c o n -t r i b u -t e s -to -the l i f -t , so -tha-t -the l i f -t c o e f f i c i e n -t s become; L D 'LR^ f u ^ l R = 4T TA I L R I n ^ b c rB-, (49) (50) Table 2

Comparison of calculations for isolated nozzles and rudders with known results.

1 / D = 0. 8 _{C L D / « D O}

reference [ 3 ] 0. 1 2 1 5 present method

(3 Fourier terms) 0 . 1 2 1 6

b/c = 1. number of lifting lines

reference [5] 0. 0 2 5

-present method 0. 0 2 5 1 4 1 (3 Fourier terms) 0. 0 2 5 5 1 2

0. 0 2 5 6 9 5

Nu m e n c a l res m i l s

A computer p r o g r a m based on the theory o f t h e preceeding section has been developed to p r e d i c t the l i f t f o r c e s on nozzle and r u d d e r . The input consists of the p a r a m e t e r s " R , , 1 / R , c / R , b / R , a / R , t h e number of F o u r i e r t e r m s and the n u m

-Figure 5. Influence of nozzle on rudder lift characteristics.

ber of concentrated bound v o r t i c e s on the r u d d e r . The output consists of the F o u r i e r c o e f f i c i e n t s

and B j ^ and the l i f t c o e f f i c i e n t s C L R and . Results of calculations f o r isolated nozzles and rudders are compared w i t h k n o w n r e s u l t s , r e f e -rences [ 3 ] , [ 5 ] , i n Table 2 which also shows the e f f e c t of the number of l i f t i n g l i n e s on the r u d -der. It can be concluded that the agreement is good and that the number of l i f t i n g l i n e s on the rudder has l i t t l e i n f l u e n c e . Two l i f t i n g lines have t h e r e f o r e been talcen throughout a l l the c a l -culations.

Results of calculations p e r t a i n i n g to the c o n -f i g u r a t i o n s o-f the -f u l l -scale tests d e s c r i b e d i n the next section are given i n F i g u r e s 5 and 6 . The i n t e r f e r e n c e e f f e c t s appear to be s i g n i f i c a n t . The response t i m e is c h a r a c t e r i z e d by the l i f t gradients at z e r o yaw given i n F i g u r e 7. It can be seen that the presence of the nozzle reduces the l i f t f o r c e of the rudder to some extent, but that the t o t a l l a t e r a l f o r c e is i n c r e a s e d , thus y i e l d i n g an i m p r o v e d response t i m e . A n i m p r e s -sion of the l a t e r a l f o r c e s on nozzle and r u d d e r

9, i n which the l i f t gradient of the i s o l a t e d rudder

has also been i n d i c a t e d . In o r d e r to v e r i f y our method f o r the p r e d i c t i o n of t u r n i n g qualities f o r v a r i o u s r u d d e r - n o z z l e c o n f i g u r a t i o n s , f u l l s c a l e manoeuvring e x p e r i -ments w e r e conducted w i t h two t w i n - s c r e w tugs, one w i t h open p r o p e l l e r s and the other equipped w i t h nozzles (see Figures 10 and 11).

Figure 9. Lift forces on rudder and nozzle for (3 = 1 0 ° .

Table 3 Particulars of the tugs.

Tug 1 Tug 2 Length o.a. [m] 37. 70 33.65 Length b. p. [m] 35. 00 30. 25 Breadth mid. [m] 9. 15 8. 90 Draught, mean[m] 4. 00 3.39 Power [HP] 2 X 1600 2 X 1400 Propeller diameter [m] 2.600 2. 135 Number of blades 4 4 Pitch ratio . 82 . 975

Blade area ratio . 55 . 58

Propeller rotational speed [r. p. m. ] 237. 5 260 during tests 150 170 Rudder area (1 rudder) [m^] 4.32 2. 98

Rudder aspect ratio 1. 565 1. 52

Number of rudders 2 2

Nozzle diameter [m] - 2. 28

Nozzle length [m] - 1. 37

T u r n i n g c i r c l e t e s t s

A f t e r putting the rudder i n a c e r t a i n p o s i t i o n , the f o l l o w i n g quantities were measured on b o a r d d u r i n g steady t u r n i n g at given t i m e i n t e r v a l s :

- c o u r s e angle i . e . d i r e c t i o n of the ship w i t h respect to an axis of r e f e r e n c e .

- distance f r o m the ship to a buoy l a i d before the experiment; this distance was measured w i t h a coincidental distance m e t e r and by r a d a r . - heading r e l a t i v e to the buoy.

I k J

Figure 10. Stern arrangement of Tug 1.

i

Figure 12. Determination of response time. By p l o t t i n g the ship's p o s i t i o n as a f u n c t i o n of t i m e on polar paper, the r a t e of t u r n per degree of rudder angle, i . e . the p a r a m e t e r K , and the d r i f t a n g l e t u r n i n g r a t e r a t i o , C3, can be o b -tained. The response t i m e T was g r a p h i c a l l y d e t e r m i n e d by p l o t t i n g the course angle as a f u n c t i o n of t i m e .

Integration of equation (11) gives the yaw angle

- t / T M J = K 5 o ( t - T ) + K 5 Q T e A f t e r some t i m e , say: t > 3 T we have: K 5 Q ( t - T ) » K 5 Q T ! : and: H . = K 5 o ( t - T ) (51) - t / T (52) The g r a p h i c a l d e t e r m i n a t i o n of T is indicated i n F i g u r e 12.

It can be concluded that K , T and - p r o v i d e d the measurements are s u f f i c i e n t l y accurate - also C3 canbe obtained by s i m p l e t u r n i n g c i r c l e tests. The r e s u l t s of the t u r n i n g experiments are given i n Table 4.

Table 4 E x p e r i m e n t a l r e s u l t s . T u g Run Date Rudder

angle [degrees] Speed [knots] Rate of t u r n [ d e g r e e s / s e c ] K 1 1 1 4 / 7 / 7 1 5 SB 8. 60 0. 78 0. 155 1 2 - 10 SB 8.35 1. 55 2 3 2 4 / 9 / 7 1 7 PS 8. 8 0. 938 2 4 - 4. 5 SB 9. 0 0. 650 0. 138 2 5 - 7. 5 SB 8. 7 0. 915 0. 138 2 6 - 12. 5 PS 8.4 1. 840 Table 5

Comparison of measured and calculat-ed t u r n i n g q u a l i t i e s .

Tug 1 Tug 2

measured calculated measured

K ' 1.17 0. 68 0. 84

T' 2. 08 1.32 1. 62

D i s c u s s i o n .

The t u r n i n g qualities of tug 2, equipped w i t h nozzles, calculated according to the present procedure f r o m the test r e s u l t s f o r tug 1 w i t h open p r o p e l l e r s , are compared w i t h the f u l l - s c a l e test r e s u l t s f o r tug 2.

The influence of the ship's size is allowed f o r by using the non-dimensional c o e f f i c i e n t s :

V s T' K' = L C l T ( Y ) C' =-L V s C2 C^ c ^ L V 3 2 Vs C 3 ( T )

The r e s u l t s are given i n Table 5.

Although the e f f e c t of a nozzle on the t u r n i n g quality indices K and T i s o v e r - e s t i m a t e d by the calculations, i t can be concluded that the f u l l -scale manoeuvring t r i a l s c o n f i r m the p r e d i c t e d trends i . e . a considerable decrease i n t u r n i n g capacity K and response t i m e T .

A p p a r e n t l y , nozzles have the same e f f e c t on t u r n i n g capacity as p r o p e l l e r s and skegs: 'They tend to r e s i s t any t u r n i n g of the s t e r n , having a decided objection to being moved sideways , r e f e r e n c e [ 1 0 ] .

Conclusions.

1. A method is given f o r p r e d i c t i n g how the s t e e r i n g c h a r a c t e r i s t i c s of a ship are a f f e c t e d by f i t t i n g a f i x e d nozzle.

2. F u l l - s c a l e manoeuvring t r i a l s c o n f i r m the p r e d i c t e d t r e n d s .

3 . The f i t t i n g of a f i x e d nozzle r e s u l t s i n : - reduced t u r n i n g capacity

- better response t i m e

- increased s t a b i l i t y on a s t r a i g h t course 4. P r o p e l l e r , nozzle andrudder shouldbe

designed i n an integratdesigned way to ensure that an o p t i m u m solution is obtained r e g a r d i n g both p r o -pulsive and s t e e r i n g q u a l i t i e s .

Acknowledgement.

The authors w i s h to express t h e i r appreciation to the owner of the two tugs: T e r m i n a t e s , C . A . , M a r a c a i b o , Venezuela, to the s h i p b u i l d e r s : D . W . K r e m e r Sohn, E l m s h o r n , Germany, and especially to: Jonker & Stans, H e n d r i k Ido Ambacht, Holland f o r t h e i r cooperation i n the manoeuvring t r i a l s .

Nomenclature.

Sy m b o l s

A F o u r i e r c o e f f i c i e n t s f o r the nozzle c i r c u l a t i o n ,

distance f r o m t r a i l i n g edge of nozzle to leading edge of r u d d e r ; a = d - y j - v , c l i f t gradient f o r the i s o l a t e d r u d d e r ; L _R a i B ^0 = • ~ p U 2 (b c + 1 D ) c o r r e c t i o n t e r m due to down-wash of nozzle F o u r i e r c o e f f i c i e n t s f o r the rudder c i r c u l a t i o n span of rudder

l i f t gradient f o r the i s o l a t e d nozzle;

c C C R D 1 , C 2 , C 3 UQ -% P U 2 (b c + 1 D )

c o r r e c t i o n term; due to down-wash of rudder

chord length of rudder c o e f f i c i e n t s of t u r n i n g l i f t c o e f f i c i e n t of nozzle; ^ L R " L D -y . p U ^ l R l i f t c o e f f i c i e n t of r u d d e r ; C L R ^R D F ( C T ) F l ( x / R ) K m N N D N R N r Ns R t U v Vs w x t h r u s t c o e f f i c i e n t ; T h r u s t O TT O H p U ^ - D ^ p r o p e l l e r diameter nozzle diameter

axial distance between quarter points of nozzle and rudder

f a c t o r f o r mean induced v e l o c i t y f a c t o r r e l a t i n g mean induced v e l o c i t y i n p r o p e l l e r s l i p s t r e a m to v e l o c t i y induced at the centreline by actuator disc

h o r i z o n t a l mass moment of i n e r t i a of the ship, including hydrodynamic e f f e c t s t u r n i n g capacity; length of ship length of nozzle l i f t f o r c e on nozzle l i f t f o r c e on rudder F o u r i e r t e r m index number of F o u r i e r t e r m s number of nozzles number of r u d d e r s

total less - rudder - nozzle - moment c o e f f i c i e n t rudder c o e f f i c i e n t p r o p e l l e r radius nozzle r a d i u s t u r n i n g r a t e r a d i a l coordinate response t i m e ; V . T' T ( f ) t h r u s t t i m e a x i a l v e l o c i t y at the r u d d e r - n o z z l e c o n f i g u r a t i o n induced velocity speed of ship wake f r a c t i o n downstream distance to p r o p e l l e r plane

distance f r o m the r u d d e r - nozzle f o r c e to the centre of g r a v i t y of the ship

X p distance between p i v o t a l point and r u d d e r - n o z z l e c o n f i g u r a t i o n

x , y , z C a r t e s i a n coordinates x , r , e c y l i n d r i c a l coordinates

a-Q angle of attack of the nozzle

apj angle of attack of the r u d d e r

(3 - «D (P i s p o s i t i v e i n steady t u r n i n g )

r c i r c u l a t i o n

y v o r t e x s t r e n t h per unit length

5 r u d d e r angle e angular coordinate P density T r a t i o of p r o p e l l e r t h r u s t to t o t a l t h r u s t y G l a u e r t coordinate; b y = - 2 c°SH^ yaw angle S u b s c r i p t s . D nozzle (duct) R r u d d e r V v o r t e x element

f j ) due to bound v o r t i c e s of duct

TR due t o bound v o r t i c e s of r u d d e r

y-Q due to f r e e v o r t i c e s of duct

y-p due to f r e e v o r t i c e s of r u d d e r

B a r r e d symbols i n d i c a t e new c o n f i g u r a t i o n . Symbols w i t h apostrophe indicate n o n d i m e n s i o n -a l c o e f f i c i e n t s .

A dot on a s y m b o l i n d i c a t e s t i m e d e r i v a t i v e .

R e f e r e n c e s .

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steering of ships', Technological University, Delft, 1967.

2. Weissinger, J . , ' Z u r Aerodynamik des Ringflilgels, l . D i e Druckverteilung dUnner,fast drehsymme-trischer Flügel in Unterschallströmung', Deutsche Versuchsanstalt fiir Luftfahrt, E . V . , Bericht Nr. 2, Mülheim, September 1955. 3. Morgan, W . B . and Caster, E . B. ,'Predictionof the

aerodynamic characteristics of annular airfoils', David Taylor Model Basin, Report No. 1830, January 1965.

4. Greenberg, M . D . , O r d w a y , D . E . and L o , C . F . , ' A three -dimensional theory for the ducted propeller at angle of attack', T H E R M Inc. T A R - T R 6509, December 1965.

5. Wicker, L . F . and Fehlner, L . F . , ' F r e e stream characteristics of a family of low aspect ratio control surfaces', David Taylor Model Basin, Report 933, May 1958.

6. Bardarson, R . R . , Wagner Smitt, L . and Chislett, M. S. , 'The effect of rudder configuration on turning ability of trawler forms. Model and f u l l -scale tests with special reference to a conversion to purse-seiners , Transactions of the Royal Institution of Naval Architects, V o l . I l l , 1969, p. 283-310.

T . G o r d o n , S . J . and Tarpgaard, P . T . ,'utilization of propeller shrouds as steering devices', Marine Technology, V o l . 5, No. 3, July 1968.

8. Manen, J . D . van, and Oosterveld, M . W . C . , 'Ana-lysis of ducted propeller design', Transactions of the Society of Naval Architects and Marine Engineers, V o l . 74, 1966.

9. Gunsteren, L . A . van, 'Eine Analyse des Einflusses der Dickenverteilung von FlUgelschnitten auf Kavitationseigenschaften', Schiffstechnik, Band 18, Heft 90, 1971.

10. Watts, P . , ' T h e steering qualities of the Y a s h i m a ' , Transactions of the Institution of Naval A r c h i -tects, V o l . 40, 1898.