A method for determining the free-stream characteristics of ship skeg-rudders

A M E T H O D F O R D E T E R M I N I N G T H E F R E E - S T R E A M C H A R A C T E R I S T I C S O F SHIP S K E G - R U D D E R S

by

A . F . Molland*

Summary

A m o d i f i e d l i f t i n g line theory is developed w h i c h supports the f o r m o f skeg-rudder experimental free-stream data obtained previously.

I t is demonstrated that satisfactory predictions o f the f o r m o f the spanwise loadings f o r d i f f e r e n t skeg and rudder angles can be made using l i f t i n g Hne theory w i t h the e f f e c t o f the skeg being incorporated as local inciden-ce reduction and the effects o f the midspan and t i p traihng vortiinciden-ces being incorporated as twist corrections to the local incidence. The correct magnitude o f the distributions is obtained b y applying suitable empirical correc-tions. Chordwise centre o f pressure is derived empirically.

The t h e o i y is used to provide a Hmited extension to the experimental data. Predictions using the m o d i f i e d theory show that, f o r f i x e d aspect ratio and taper r a t i o , changes i n the skeg depth can have a significant influence on the p r o d u c t i o n o f H f t whilst changes i n the skeg chord and rudder sweep have a marked e f f e c t on the stock p o s i t i o n , balance area and hence torque.

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

Extensive experimental results o f tests t o determ-ine the free-stream characteristics o f semi-balanced ship skeg-rudders are reported i n References [ 1 , 2, 3 and 4 ] .

A theory has been developed t o provide some theoretical evidence f o r the f o r m o f the experimental data, and t o allow an extension o f the experimental results.

I n the theoretical analysis, H f t i n g Hne theory, m o d i f -ied t o include the specific features o f the skeg rudder and t o account f o r the differences between theory and experiment, is used t o predict the spanwise load dis-t r i b u dis-t i o n s . Chordwise cendis-tre o f pressure is derived f r o m the apphcation, across the span, o f local centres o f pressure f r o m section and experimental data.

A n outline o f the theoretical analysis is given, together w i t h some comparisons w i t h experimental data. Examples are presented o f a parametric study w h i c h uses the theory t o investigate variations i n skeg particulars.

A f u l l account o f the development o f the t h e o r y , its comparison w i t h experiments and the f u l l para-m e t r i c study are given i n References [ 5 ] and [ 6 ] .

2. Skeg rudder characteristics

Ship rudders may be considered as l i f t i n g surfaces o f relatively l o w aspect r a t i o w i t h applications l y i n g m a i n l y i n the effective aspect r a t i o range o f 3 t o 4.

The principal features o f the skeg r u d d e r are shown i n Figure 1 and the n o t a t i o n o f the skeg and movable r u d d e r angles is given i n Figure 2.

*) University of Southampton, U.K.

Experimental data (References [ 1 , 2 , 3 ] ) obtained f o r skeg mdders w i t h square tips have indicated particular physical properties which need t o be i n -corporated i n the theoretical analysis. These a m o u n t t o :

(i) early separation a f t o f the skeg ( w h i c h is i n f l u e n -ced by gap f l o w )

(ii) a strong trailing t i p vortex ( w h i c h has a m a r k e d influence on the spanwise d i s t r i b u t i o n o f load

I

Figure 1. Principal features of skeg rudder.

near the t i p ) and

( i i i ) a trailing vortex at the break between the skeg and all-movable parts ( t e r m e d the mid-span vor-tex).

3. M o d i f i e d l i f t i n g line t h e o r y

M o d i f i c a t i o n s and additions t o basic l i f t i n g line t h e o r y are r e q u h e d due to the particular physical properties o f the skeg r u d d e r described i n Section 2. These entail i n c o i p o r a t i n g incidence correction due t o the presence o f the skeg (before and after separation), downwash c o n t r i b u t i o n s f r o m the t i p and mid-span traihng vorrices and any empirical corrections neces-sary t o take account o f the differences between theoretical h f t curve slopes and those derived f r o m ex-periments.

Assumed vortex model:

The ideahsed v o r t e x m o d e l assumed f o r the analysis is shown i n Figure 3 and comprises:

a) A trailing v o r t e x sheet due t o the basic l i f t i n g line. b) A superimposed t r a i l i n g t i p v o r t e x , w i t h a solid

core, whose strength is assumed t o be a f u n c t i o n o f a, T wdAR.

c) A superimposed trailing v o r t e x , w i t h a sohd core, at the break between the skeg and all-movable parts, whose strength is assumed to be a f u n c t i o n o f a and 6.

The b o u n d v o r t i c i t y associated w i t h b) and c) is as-sumed to be concentrated o n the h f t i n g line. T h e general f o r m s o f the expected distributions o f d o w n

-c h a n g e i n r due to t i p v o r t e x

c h a n g e in r due to mid v o r t e x

Figure 3. Basic vortex model.

3\

Skeg distribution due _{to l i f t i n g l i n e}

semi-span S_

wash due t o these three component parts are shown diagrammatically i n Figure 4.

4. O u t l i n e o f the basic l i f t i n g line theory 4.1. General

The m e t h o d used f o r deriving the basic spanwise load d i s t r i b u t i o n is generally along the lines o f that due to Glauert [ 7 ] . I n the present analysis the skeg rudder is considered as a special case o f a twisted a e r o f o i l . I n this case Glauert shows theoretically that the l i f t curve slope is independent o f the t w i s t . Ideahy, therefore, i t w o u l d be assumed that the e f f e c t o f the skeg w o u l d be t o change the angle o f incidence (measured f r o m the n o - l i f t angle) and to leave the slope o f the l i f t curve unaltered. F r o m the results given i n Figure 5, where experimental data f r o m References [ 1 ] and [ 2 ] have been re-plotted f o r constant values o f 5, these assumptions are seen t o be acceptable f o r small angles and where no separation occurs a f t o f the skeg. Figure 5 also indicates that i t is n o t unreasonable t o take a similar approach f o r the a range where early separation aft o f the skeg has occurred although, i n this case, some (constant) m o d i f i c a t i o n to the h f t curve slope has taken place and requires i n c o r p o r a t i o n i n the analysis. o o 6= 5 ° + H- 5 =10° a A 6= 1 5 ° O • 5 =20° V ^ 6 = 25° Refs.1,2 I 10 15 20 ANCLE OF ATTACK a Idegl

J

30

The rudder geometry used i n the analysis is given i n Figure 6. The e f f e c t o f the r e f l e c t i o n plane is repre-sented b y an image m d d e r and vortex system. T h e rudder is assumed t o have a t o t a l span, including its image, o f 2S and a taper ratio iCj,/Cj^) = T.

S k e g c :

Figure 6. Rudder geometry used in analysis.

The rudder is considered as being replaced by a l i f t i n g line o f length 2S. The co-ordinate y is replaced by the angle Ö defined asy = —Scosd.

A t any p o i n t on the rudder, c = C ^ ( l -kcosB) where k = {l - T ) and, at any p o i n t , the incidence o f the skeg rudder, considered as a h f t i n g hne w i h be:

a = a - 7 ( 1 ) where

a = incidence o f the movable r u d d e r part

y = decrease i n incidence over the skeg part, at some rudder angle a, to produce the same all-movable value, as shown diagrammatically i n Figure 7, i.e.

where

f ( e ) = 0 f r o m e = 0 to e = i// = 1 f r o m e = i// to e = w/2

and 7(6,0!) can be derived f r o m theoretical and ex-p e r i m e n t a l data w i t h n o seex-paration on the flaex-p and f r o m experhnental data when gap f l o w and separation are present.

Figure 7. Application of skeg incidence reduction y.

The circulation r at a p o i n t 8 on the rudder may be expressed as a F o u r i e r series:

r = 4S V X Afi sinne (2) and since the p l a n f o r m is symmetrical about the m i d

p o i n t (7T/2), only odd values o f n w i l l occur i n the series.

The induced velocity at a p o i n t 9 is given b y :

CO = F ( S « ^ 7 j s i n ? 7 e ) / s i n 9 ( 3 ) T h e section experiences a l i f t force corresponding to

twodimensional m o t i o n at the e f f e c t i v e angle o f i n cidence (a o j / V ) where co/V is the induced d o w n -wash angle. Local l i f t c o e f f i c i e n t

q = m(a - oj/V) ₍₄₎

where m is the two-dimensional l i f t curve slope, al-l o w i n g f o r thickness and viscosity effects.

Hence at any p o i n t L

r =-q - c

(5) I n c o r p o r a t i n g the value o f induced v e l o c i t y f r o m equation (3)

( 2 « ^ H s i n « 0 ) / s i n e ) ( 6 ) equating (2) and ( 6 ) and setting ii^mCj^ /8S leads t o

An sinne ()in + sine _{H • a • sint (7)} -A: cose

T h i s f u n d a m e n t a l equafion must be satisfied at all p o i n t s o n the rudder between 0 and 7r/2.

L o c a l l i f t

L =pVr = pVi4SVi:Ansinne) = 4pSV^iX An sinne) hence the local l i f t c o e f f i c i e n t

• — XAn smne c

C „ ( l ~k • COS0) 2 Aiisinne and local induced drag c o e f f i c i e n t

(8)

(9) T h e f u n d a m e n t a l equation ( 7 ) has t o be satisfied at all p o i n t s along the rudder. A large n u m b e r o f c o n t r o l p o i n t s is desirable since the analysis has t o include the f a c i h t y to vary a along the rudder ( w i t h a discon-finuity at the end o f the skeg), and t o superimpose vordces at the t i p and at the break between skeg and

all-movable parts. I n order t o o b t a i n a suitable spacing o f stations near m i d span 2 0 c o n t r o l points were chosen w h i c h allowed skeg depth/span ratios o f 0.36, 0.43, 0.50, 0.57 and 0.63 t o be investigated. This approach does n o t , o f course, represent a discontin-u i t y o f incidence at the skeg t o all-movable break b discontin-u t corresponds t o a continuous change o f incidence over a small region o f the order o f 6% to 7 / 2 % o f span.

4.2. Derivation of skeg incidence reduction, 7

I n i t i a l 7 values were derived f o r each rudder and skeg angle b y adjusting 7 i n the l i f t i n g line analysis u n t i l satisfactory agreement w i t h the spanwise dis-t r i b u dis-t i o n derived f r o m dis-the pressure measuremendis-ts o f Reference [ 3 ] was achieved.

The 7 values so obtained are shown i n Figure 8, and suitable equations representing these results ( f o r an average effective flap ratio o f 0.70) are

u p t o separation a f t o f the skeg:

7 = 0.236 ( 1 0 ) a f t e r separation:

7 = 0 . 3 Q : - 3 + 0.396 ( 1 1 ) I n order t o c o n f i r m these values, and t o p r o v i d e

m o r e general data f o r d i f f e r e n t f l a p sizes, the derivat-i o n o f 7 values f r o m alternatderivat-ive sources was derivat- invest-igated.

Considering, f i r s t l y , angles up t o the onset o f f l a p separation the theoretical results o f Reference [ 8 ] were investigated and suitably adapted; i n this

refer-• refer-• R e q ' d v a l u e s t o c o r r e l a t e w i t h p r e s s u r e m e a s u r e m e n t s O O C f / c = 0 . 5 0 R e f . l 1 C3 10 C f / c = p.ao R e f . 1 2 10 15 R U D D E R A N G L E 20 6 I d e g ) 25 3 0

ence Glauert appHes t h i n aerofoil theory t o a t w o -dimensional f l a p p e d f o i l .

I n the n o t a t i o n o f the present w o r k

dC, dC, '

L9|3 36 .

9 a

(12)

L3(3 ^ ' 9 5

Since, i n the unseparated range, lines o f constant 6 to a base o f a w i l l be parallel and 7 w i h be independent o f a, 7 can be obtained direct f r o m the result at = 0. Hence replacing a by 7 at C, = 0 9C, and 7 = - ( 7 - 6 ) = £ §

ac,

96 1 95 (13)

where — is c o m m o n l y t e n n e d the flap effectiveness

as

r a t i o .

Glauert derived the theoretical expression 9^ 1 2 _ i

— = 1 cos

96 IT

1 - i

1

9

+ sin-' -L- ( 1 4 ) Due t o viscous effects and section thickness, ex-p e r i m e n t a l values w i h be smaller than theoretical values. M u c h data is b r o u g h t together ( i n c l u d i n g References [ 1 1 ] and [ 1 2 ] ) i n Reference [ 9 ] t o demonstrate this.

A mean line o f 7 = 0 . 2 3 5 , equation ( 1 0 ) , satisfac-t o r i l y represensatisfac-ts satisfac-the correlasatisfac-tion besatisfac-tween satisfac-the l i f satisfac-t i n g hne analysis and ^ e pressure measurements f o r an ef-fective f l a p r a t i o -J- o f 0.7. Based o n this, the Glauert theoretical result requires m o d i f i c a t i o n t o :

96 TT

0.48

9

1 - ^ -I- sin (15)

This represents a decrease i n the theoretical effective-ness r a t i o o f a p p r o x i m a t e l y 15%.

The relationship f o r 7 ( f o r angles up t o the onset o f separation) thus becomes:

7 = 6 1 - 1 0.48 V — ( 1 - 5 l U s m

c \ c ( 1 6 )

Figure 8. Values of skeg incidence reduction 7.

I t can be n o t e d t h a t this change i n 7 w i t h change i n f l a p size is supported b y the t r e n d o f the results f r o m References [ 1 1 ] and [ 1 2 ] shown i n Figure 8. T h i n a e r o f o i l t h e o r y suggests t h a t the f l a p effectiveness r a t i o should be independent o f aspect r a t i o . E x -p e r i m e n t a l and theoretical results re-produced i n Re-ference [ 9 ] c o n f i r m that o n l y very small variations

occur f o r aspect ratios d o w n t o about 2. Consequently, up to the onset o f separation, i t is assumed that 7 is independent o f aspect r a t i o .

Extensive pubhshed experimental data are avail-able f o r small flaps and f l a p angles. The need f o r data f o r large flap sizes ( 6 0 % - 75% chord) and large f l a p deflections (up t o 3 5 ° leading t o separation on the flap) severely l i m i t e d the published data suitable f o r the present investigation. Detailed experimental measurements are, however, available f r o m References [ 1 0 ] , [ 1 1 ] and [ 1 2 ] f o r flap sizes o f 30%, 50% and 80% c h o r d , and these were used to assess 7 values up to and a f t e r the onset o f separation o n the flap. The results are f o r flapped f o i l s w i t h sealed gaps i n t w o -dimensional flow. Such f o i l s do display separation a f t o f the hinge and discontinuities i n t h e i r l i f t curves, although its onset occurs at higher 6 angles than w i t h gaps open.

The 7 values derived f r o m References [ 1 1 ] and [ 1 2 ] show very shnilar trends t o those derived b y correlation o f the h f t i n g hne t h e o r y w i t h the pres-sure meapres-surements (Figure 8) although a f t e r separation they are up t o 1° l o w e r . Results i n Reference [ 1 3 ] w h i c h investigated the i n f l u e n c e o f gap size f o r a 30% chord flap, indicate t h a t w i t h gaps open and w i t h separation on the flap, 7 w o u l d be increased b y between 1° and 1Y2°. Thus the effects o f gaps w o u l d be to raise the results o f the gap closed data i n Figure 8 to approximately the same level as the pressure cor-related results as given b y equation ( 1 1 ) .

References [ 1 1 ] and [ 1 2 ] also gave an i n d i c a t i o n o f the influence o f flap size o n 7 a f t e r separation. This led t o a suitable c o r r e c t i o n f o r flap size being added t o equation ( 1 1 ) .

T h e relationship f o r 7 ( f o r angles a f t e r the onset o f separation) thus becomes

7 = (0.3a - 3 + 0.39S) + OAiCJc - 0 . 7 ) ( 2 8 - 5 ) (17) N o data could be f o u n d , f o r large flap sizes, t o determine the i n f l u e n c e o f aspect r a t i o o n 7 i n the separated c o n d i t i o n . I t is therefore assumed that 7 is independent o f aspect r a t i o a f t e r the onset o f separat-i o n as well as pre-separatseparat-ion f o r the l separat-i m separat-i t e d range o f aspect r a t i o considered i n the present investigation.

Thus i n the basic l i f t i n g line analysis the skeg i n -cidence r e d u c t i o n 7 is represented by equation ( 1 6 ) pre-separation and equation ( 1 7 ) a f t e r the onset o f separation, b o t h relationships a l l o w i n g the i n f l u e n c e o f flap c h o r d size ( a f t o f the skeg) t o be i n c o r p o r a t e d i n the analysis.

4.3. Angle of attack for onset of separation

The experimental results o f References [ 1 ] and [ 2 ]

indicated that gap flow promotes separation a f t o f the skeg and this process takes place over an angle o f attack range o f t w o t o five degrees. The precise angle at w h i c h the onset o f complete separation occurs is not clear, although the discontinuities i n the l e f t and drag curves give a broad i n d i c a t i o n .

Tl;e angles at w h i c h the d i s c o n t i n u i t y i n the l i f t curve occurred were derived f r o m References [ 1 ] and [ 2 ] and the mean values are shown i n Figure 9. Data f r o m References [ 1 1 ] and [ 1 2 ] are also included i n Figure 9 w h i c h indicate, as w o u l d be expected, some delay i n separation f o r the gaps sealed c o n d i t i o n ; these data also indicate t h a t the influence o f flap size is very smah, hence corrections were n o t considered necessary.

Figure 9 indicates that the onset o f separation can be satisfactorily represented b y : a = 4 + 0 . 4 ( / 3 - H 5 ) ( 1 8 ) • • F r o m Refs.1,2 O B 5 0 % Flop Ref. 11 ^ V 8 0 % F l a p Ref. 12 0 I 1 1 1 I I - 1 5 - 1 0 - 5 0 5 10 15 SKEG ANGLE p ( d e g )

Figure 9. a for onset of separation aft of skeg.

The experimental data o f References [ 1 ] and [ 2 ] indicate that there is a smaU delay i n the e f f e c t o f separation on the drag curve. I t was, therefore, assum-ed that the r a p i d increase i n drag values occurs at:

«sep = 5 + 0 . 4 ( / 3 + 1 5 ) ( 1 9 ) I t should be n o t e d t h a t these f o r m u l a e represent

a p r e d i c t i o n o f suitable locations o f the discontinuities i n the H f t and drag curves rather t h a n a precise i n d i c a t i o n o f either the start o f gap flow or the onset o f c o m -plete separation a f t o f the skeg, although t h e y can be taken as a reahstic estunate o f the latter.

5. I n c o i p o r a t i o n o f the superimposed vortices i n the analysis

5.7. General

L i f t i n g line t h e o r y requires that at any section, f r o m equation ( 4 )

q = i?2(a - co/V)

= m(a - a.)

where a. is the downwash angle due t o the v o r t e x sheet.

I t is i n i t i a l l y assumed t h a t the i n f l u e n c e o f the ad-d i t i o n a l trailing vortices is t o superimpose a ad-downwash or up wash Uj. o n that due t o the v o r t e x sheet, e.g. when the local m o d i f i c a t i o n Uj. leads to an increase i n

q = « 2 ( a - ( a . - f f i j , ) ) ( 2 0 ) = m ( a - o : . j ) ( 2 1 ) where a.j = a. - and represents the net downwash

i n c l u d i n g the e f f e c t o f t h e superimposed vortices. Since the c o r r e c t i o n a^, cannot be retained directly as a downwash c o r r e c t i o n i n the h f t i n g line equations, equation ( 2 0 ) is r e - w r i t t e n i n t h e f o r m :

q = m ( 5 + a ^ ) - « . ) ( 2 2 ) and f o r the s o l u t i o n o f the f u n d a m e n t a l equation ( 7 )

the c o i T e c t i o n is treated, i n e f f e c t , as equivalent local t w i s t Uj, applied t o the l o c a l angle o f attack.

3.2. Assumptions for the downwash induced by the superimposed vortices

I n order t o provide general guidance on the f o r m o f the downwash m o d i f i c a t i o n s , and hence the equiv-alent t w i s t t o be applied t o the l i f t i n g line equations, the superimposed v o r t e x systems are assumed t o b e ideahsed as single traihng vortices w i t h solid cores.

A t the r u d d e r t i p a vertical v o r t e x sheet is generated w h i c h increases i n length w i t h increase i n incidence. T h e development o f such a sheet has been described, f o r example, b y K ü c h e m a n n , Reference [ 1 4 ] . The t i p v o r t e x sheet m i g h t b e expected, and is hence assumed, t o r o l l u p i n t o a conical v o r t e x , w h i c h i n t u r n is assum-ed t o be replacassum-ed b y a concentratassum-ed V o r t e x ' ( w h i c h increases i n strength downstream f r o m its apex). Such a replacement has, f o r example, been made b y Cheng, Reference [ 1 5 ] , and others, i n the development o f delta w i n g t h e o r y . Cheng p o i n t s o u t t h a t this s o l u t i o n also exhibits certain general characteristics o f edge separation observed.at l o w speed f o r low AR aerofoils.

The m i d v o r t e x is i n i t i a t e d as a v e r t i c a l v o r t e x sheet springing f r o m the abrupt d i s c o n t i n u i t y between the skeg and all-movable parts near t h e leading edge; this

may be considered as a 'partspan v o r t e x ' i n the t e r m -i n o l o g y o f Reference [ 1 4 ] and t h o u g h t o f as a con-t i n u a con-t i o n o f con-the bound vorrices w h i c h do n o con-t con-travel o u t b o a r d . T h e m i d v o r t e x is also assumed t o roU u p and be replaced b y a concentrated conical v o r t e x .

I n the analysis, each v o r t e x is t h e n considered as being represented by a single traihng v o r t e x o f constant ' m e a n ' strength i n the n e i g h b o u r h o o d o f t h e r u d -der. Thus the downwash induced b y each trailing v o r t e x is initiaUy assumed t o have the general f o r m o f ' the n o r m a l velocity d i s t r i b u t i o n f o r a v o r t e x w i t h a sohd core. T h e strength o f the t i p v o r t e x is assumed t o be a f u n c t i o n o f incidence, t i p c h o r d (hence taper r a t i o ) and aspect r a t i o , whilst t h a t o f the m i d - v o r t e x is assumed t o be a f u n c t i o n o f incidence.

T h e spanwise pressure d i s t r i b u t i o n s were used t o locate the p o s i t i o n o f the vortices and their approx-imate ' m e a n ' core sizes, and n o a t t e m p t was made t o theoretically predict these properties. T h e actual downwash distributions were obtained b y suitably adjusting the superimposed t w i s t i n t h e basic analysis u n t i l the f o r m o f the d i s t r i b u t i o n o f load agreed w i t h the e x p e r i m e n t a l results.

5.3. Tip trailing vortex

I t is assumed t h a t the influence o f the t i p v o r t e x is responsible f o r the non-linear c o m p o n e n t o f l i f t nor-m a l l y e x h i b i t e d b y l o w aspect r a t i o H f t i n g surfaces, as was concluded by K ü c h e m a n n , References [ 1 4 ] and [ 1 6 ] .

Based o n the results i n References [ 1 7 ] and [ 1 8 ] and others, i t is reasoned i n Reference [ 1 6 ] that the non-linear increment o f l i f t decreases w i t h increasing aspect r a t i o and increases w i t h increasing taper r a t i o , and is a p p r o x i m a t e l y a quadratic f u n c t i o n o f inciden-ce.

T h e non-linear component has been derived semi-empirically as bemg a f u n c t i o n o f a'^ b y several invest-igators, such as References [ 1 9 ] and [ 2 0 ] . These are generally based o n the hypothesis o f Betz, [ 2 1 ] , i n w h i c h the non-linear ( a ^ ) c o m p o n e n t arises f r o m cross flow.

Reference [ 1 9 ] assumes the non-linear c o m p o n e n t to be p r o p o r t i o n a l to l/AR, w h i c h tends t o zero as

AR tends t o i n f i n i t y as is expected e x p e r i m e n t a l l y .

T h e procedure adopted i n Reference [ 1 8 ] also amounts t o an inverse f u n c t i o n ofAR.

F r o m an investigation o f the cross flow drag coeff i c i e n t values i n Recoefference [ 1 9 ] the nonlinear c o m -p o n e n t is f o u n d t o be -p r o -p o r t i o n a l t o T; t h i s is also impHed i n Reference [ 1 8 ] . However the i n f l u e n c e o f taper was r e q u i r e d t o be increased t o T^-^ i n order t o reflect the changes derived f r o m the experiments (References [ 1 ] and [ 2 ] ) .

Based o n the above reasoning the equation used i n the analysis f o r the downwash induced b y the t i p v o r t e x i s :

aj,^(d) = 0.645H^(6)a^T^^/AR ( 2 3 ) where H^ie) is the general f o r m o f the v a r i a t i o n o f

downwash across the span at a particular incidence, aspect r a t i o and taper r a t i o , and the constant is i n t r o -duced t o correlate the magnitude o f the load w i t h experiment at the d a t u m angle.

The f i n a l d i s t r i b u t i o n o f H^{B), f o h o w i n g adjust-m e n t by t r i a l i n order t o b r i n g the f o r adjust-m o f the dis-t r i b u dis-t i o n o f load i n dis-t o line w i dis-t h dis-the experimendis-tal results, is shown i n Figure 10. The downwash dis-t r i b u dis-t i o n is seen dis-to be asymmedis-trical w h i c h may be p a r t l y due t o the downstream g r o w t h o f the v o r t e x i n the neighbourhood o f the rudder, w i t h the o u t b o a r d side o f the v o r t e x being always approximately aligned w i t h the t i p . 0.5 • 0 . 5 2.0 1.0 H,(e) 0.5 -0.5 0.2 0.3 0.4 0.5 0.6 0.7 0.9 0.9 1.0 >'/s ( = - C O S e )

Figure 10. General form of functions i / i ( 0 ) and

/^jC^)-5.4. Mid span trailing vortex

Since the v o r t e x is generated b y f l o w t h r o u g h the h o r i z o n t a l gap, as well as the basic pressure d i f f e r -ential due t o the d i s c o n t i n u i t y between the skeg and aU-movable parts, the g r o w t h o f the vortex strength w i l l be dependent on a as well as the difference i n incidence ( 5 ) between the skeg and all-movable parts.

Based o n an inspection o f the experimental span-wise d i s t r i b u t i o n s the equation used i n the analysis f o r the d o w n w a s h induced by the m i d span v o r t e x is:

0.5 s 0.5 ₍₂₄₎

where H.^(_e) is the general f o r m o f the v a r i a t i o n o f downwash across the span at a particular indicence, and t h e constant is i n t r o d t l c e d t o correlate the mag-n i t u d e o f the load w i t h experimemag-nt at the d a t u m angle.

T h e f i n a l d i s t r i b u t i o n o f H^iO), f o l l o w i n g adjust-m e n t by t r i a l i n order t o bring the f o r adjust-m o f the dis--t r i b u dis--t i o n o f load i n dis--t o line w i dis--t h dis--the experimendis--tal results, is shown i n Figure 10. The required downwash d i s t r i b u t i o n is seen t o be a p p r o x i m a t e l y symmetrical b u t i t s p o s i t i o n set towards the r o o t ; this m a y be due to the spanwise pressure gradient causing the v o r t e x t o be swept i n t o a region o f l o w e r pressure as i t moves a f t across the chord.

6. Corrections to theoretical l i f t and drag values 6.1. Lift curve slope

As w o u l d be expected f o r relatively l o w aspect ratios, the l i f t i n g line analysis results i n h f t curve slopes i n excess o f experimentally derived values. The basic analysis, i n e f f e c t , derives w h a t m a y be as-sumed t o be a 'mean' induced downwash i n the neigh-b o u r h o o d o f the rudder, and neglects any change i n downwash along the chord. A l s o the f l o w is i n all places assumed parallel t o the Z - a x i s , whereas cross f l o w wUl occur near areas o f r a p i d pressure change, such as near the t i p . These influences become m o r e i m p o r t a n t w i t h increase i n chord relative to span, (i.e. decrease i n AR). I n the present analysis i t is assumed t h a t the f l o w remains parallel t o the Z-axis and t h a t the required decrease i n predicted b y the h f t i n g Ime analysis is caused b y an increase i n the 'mean' downwash.

The correction t o the h f t i n g line result is o f the f o r m :

q^ = q^-a(AR) (25)

where C^^ is the theoretical h f t c o e f f i c i e n t f r o m l i f t i n g line t h e o r y

C^^ is the corrected l i f t c o e f f i c i e n t .

a is the r a t i o o f the experimental t o the theoret-ical and is a f u n c t i o n o f aspect r a t i o .

I t is assumed t h a t the f o r m o f the spanwise dis-t r i b u dis-t i o n o f ü f dis-t remains dis-t h e same b u dis-t dis-thadis-t idis-ts magn i t u d e across the spamagn is reduced by the f a c t o r a. A l -t h o u g h -this approach is approxima-te, comparisons w i t h t h e experimental results o f Reference [ 2 2 ] f o r the ah-movable case of AR = 3.4, f o r example, indicate t h a t h f t i n g hne predictions m o d i f i e d b y this m e t h o d lead t o spanwise distributions w h i c h are very satis-f a c t o r y satis-f o r the purposes o satis-f t h e present investigation.

All-movable l i f t curve slopes were derived f r o m the h f t i n g line analysis f o r taper ratios o f 0.60, 0.80 and 1 and aspect ratios o f 2, 3 and 4. A two-dimensional section slope (rn) o f 5.5 was assumed i n the analysis, this being derived f r o m Reference [ 2 3 ] as apphcable f o r a section thickness ratio o f 20% w i t h L . E . rough-ness. T h e derived h f t curve slope values f o r taper

r a t i o 0.6 are adequately represented b y the equation:

dC,

= 2 7 r / 5 7 . 3 ( 1 . 1 4 + 2 M i ? ) (26)

da. J a = o

This satisfies the c o n d i t i o n that dq/da 5.5 as

AR ^ «>.

E x p e r i m e n t a l results indicated that the i n f l u e n c e o f taper ratio (T) derived f r o m the H f t i n g Hne analysis required some c o r r e c t i o n , and the basic all-movable slope ( 2 6 ) was therefore m o d i f i e d by a suitable em-p m c a l correction o f the f o r m 1.052 7*-^.

Equivalent experimental all-movable slopes were derived f r o m published data and the results f o r the 'gaps open' case i n Reference [ 1 ] . T h e gaps open case showed some decrease i n slope compared w i t h the gaps sealed cases, the loss being due p r i m a r i l y to the h o r i z o n t a l gap and the gap around the p i n t l e . Since such effects wUl n o t d i r e c t l y m o d i f y the t w o d i m e n -sional section slope i t is assumed t h a t this loss is i n e f f e c t accounted f o r b y a f u r t h e r relative increase i n downwash.

T h e gaps open results are f o u n d t o be adequately represented b y the e q u a t i o n :

dC,

da 1 = 0

1.757r/57.3(l - l - 3 . 9 M i ? ) (27) This retains the c o n d i t i o n that dq/da ^ 5.5 as AR CO _

Hence the overall c o r r e c t i o n t o the basic l i f t curve slope derived f r o m l i f t i n g line t h e o r y is assumed t o be represented b y the rario o f equations ( 2 7 ) and ( 2 6 ) as:

a= 1 . 0 5 2 r ° - i X 0.815{l.\4AR + 2)KAR+3.9)

(28)

6.2. Correction to downwasli and induced drag

Assuming that the decrease i n theoretical is caused by an increase i n 'mean' downwash, and the general f o i m o f equations ( 4 ) and ( 2 1 ) are preseived, then equation (21) can be w r i t t e n as:

C^^ = ?7i(a-a.^) ( 2 9 )

and

"il = « - ^ i c / ' " <;30)

where a . j is the f i n a l net d o w n w a s h i n c l u d i n g the ef-fects o f the superimposed vortices and the empirical correction t o Hft.

ttj.j is used t o c o m p u t e the i n d u c e d drag, and the local induced drag c o e f f i c i e n t can n o w be w r i t t e n as:

(31) T h e local drag c o e f f i c i e n t m a y be derived as:

^ f l

'-'Dp ^ ^Di (32)

the flapped (skeg) and all-movable parts may be derived f r o m suitable empirical data.

Spanwise numerical i n t e g r a t i o n o f the local l i f t and drag values is used t o derive t o t a l values.

L o c a l n o r m a l f o r c e c o e f f i c i e n t is derived as:

= C^^cosa -I- Cj^ sina (33)

where values f o r the p r o f i l e drag c o e f f i c i e n t f o r

T h e local n o r m a l force coefficients are used i n the subsequent derivation o f the chordwise and spanwise centres o f pressure.

7. P r o f i l e drag (C^^ ) and centre o f pressure

T h e present investigation required p r o f i l e drag data f o r t h i c k sections w i t h a large flap b o t h at l o w angles of attack and over the flap staUed range o f angles. A search o f related pubhshed papers did n o t reveal suitable p r o f i l e drag data w h i c h also i n c l u d e d the flap staUed c o n d i t i o n and the required i n f o r m a t i o n was, therefore, extracted f r o m the experimental results o f References [ 1 ] and [ 2 ] .

Spanwise centre o f pressure, CPs, is obtained f r o m a numerical i n t e g r a t i o n o f the l o a d d i s t r i b u t i o n .

L o c a l chordwise centre o f pressure, CPc, f o r the skeg (or flapped) p a r t f o r d i f f e r e n t flap sizes was derived f r o m the pressure measurements o f References [ 3 ] , [ 1 1 ] and [ 1 2 ] , and f o r the all-movable p a r t f r o m References [ 3 ] and [ 1 9 ] . Spanwise i n t e g r a t i o n o f the local products o f n o r m a l f o r c e and CPc is used to d e t e i T n i n e the t o t a l chordwise centre o f pressure

CPc.

8. Analysis c o m p u t e r program

T h e theoretical analysis, embracing the basic h f t i n g line t h e o r y together w i t h the necessary adjustments and empirical corrections, was incorporated i n a c o m -p u t e r -program. F o r given i n -p u t values o f as-pect r a t i o , taper ratio, skeg depth/span, leading edge sweep, mean flap size i n way o f skeg, skeg angle /3 and r u d d e r angle

a the program is capable o f o u t p u t t i n g the spanwise

d i s t r i b u t i o n o f l i f t and n o r m a l force f o r the r u d d e r plus skeg, together w i t h its i n t e g r a t i o n f o r t o t a l forces and spanwise CPs. The t o t a l force o n the movable rudder alone is also calculated together w i t h the chordwise CPc f o r the movable rudder alone.

A more detailed account o f the program is given i n Reference [ 6 ] .

9. Discussion o f the theoretical results

T h e results o f the spanwise load p r e d i c t i o n s , t o -gether w i t h the e x p e r i m e n t a l results derived f r o m the pressure measurements, are given i n Figure 1 1 . I t is seen t h a t the c o r r e l a t i o n between t h e o r y and ex-periment f o r the overah f o r m o f the d i s t r i b u t i o n s is

good. Agreement i n magnitude was achieved b y em-pirical con-ection.

The results i n Figure 11 show the skeg incidence reduction approach t o predict satisfactorily the meas-ured changes i n load d i s t r i b u t i o n due to change i n skeg

angle. The changes i n load near the t i p , w i t h change i n incidence, show that i t is reasonable t o assume the t i p vortex e f f e c t t o be a f u n c t i o n o f .

Figures 12(a), (b) and (c) show the results o f the integrations o f the spanwise distributions f o r load and

C P s 30 40 20

-

J A 10 30 20 < ta 50 10 40 g 3 0 X "___iEs— — T P £ 20 O.t _

~ •

u

cr~ ——•—11

0.3 1 1 1 1 0.; 0.5 0.6 S K E B D E P T H / S P A N S A R = 3 ; T = 0 . 8 ; fli = 0 . 1 5 3 ; (3 = 0 ° c,;c = o.6o [,/t= 0.70 (Q1 (X = 1 0 °

Figure 13. Variation in skeg depth

T i 1 ' i r

ANOLE OF A T T A C K CC ( d e g l

(c) Skeg Angle p = + 4.75°; Taper Raiio=0.60 Figure 12 continued. OM 0.5 0.6 S K E G D E P T H / S P A N S i/ 5 A R = 3 ; T = 0 . 8 ; n i = 0 , 1 5 3 ; B = 0 ° c , / c - - o . 6 o C f / C = 0.70 ( b ) C C = 2 0 °

1 movable chord (sweep constant).

centre o f pressure, superimposed on the direct force measurements. As w o u l d be expected, f o l l o w i n g the good agreement w i t h the spanwise d i s t r i b u t i o n s , the l i f t predictions are also satisfactory. The p r o f i l e drag was estimated f r o m the experimental results and the satisfactory predictions o f t o t a l drag indicate, there-f o r e , that the induced drag predicted b y the analysis is o f the correct order o f magnitude. The CP predict-ions are generally reasonable and show the correct trends w i t h changes i n incidence, although CPc is f o r -w a r d o f the results f r o m the force measurements u p to about a = 15° and CPs is d e f i c i e n t f o r m o s t skeg angles. The deficiencies i n CPs are generally t h e same as those f o r the integrated pressure measurements (Reference [ 3 ] ) , and better agreement w i t h the CPs values f o r the force measurements [ 1 ] and [ 2 ] c o u l d , f o r example, have been achieved by using slightly larger y values and an increased h f t curve slope cor-r e c t i o n f a c t o cor-r .

10. Parametric study

A parametric study using the c o m p u t e r p r o g r a m was carried o u t t o provide a broad o u t l i n e o f the expected changes i n performance w i t h changes i n skeg and r u d -der particulars, and to highlight the problems faced when choosing suitable skeg p r o p o r t i o n s .

Figures 13(a) and ( b ) show the influence o f skeg d e p t h and movable c h o r d f o r a taper r a t i o o f 0.80. I t is seen that f o r constant C^, increase i n skeg d e p t h has a detrimental e f f e c t o n h f t c o e f f i c i e n t , p a r t i c u l a r l y at the larger angle o f attack; f o r example, increasing the skeg d e p t h r a t i o f r o m 0.45 t o 0.55 leads t o a 3% loss i n h f t at 10° and an 8% loss i n h f t at 2 0 ° ( 2 0 ° represents the case a f t e r separation a f t o f the skeg). Skeg depth also has an i n f l u e n c e o n CPc ( f o r constant cp, an increase i n skeg d e p t h f r o m 0.45 t o 0.55 leading to about 3% a f t m o v e m e n t o f CPc f o r b o t h a = 10° and 2 0 ° .

Figure 13(a) also shows t h a t change i n the movable chord size ( ( ^ ) has a significant e f f e c t on C^ and CPc at a = 1 0 ° . F o r b o t h angles o f attack. Figures 13(a) and ( b ) , the movable c h o r d size has a marked i n f l u e n c e o n the stock p o s i t i o n , skeg area and hence balance; f o r example, i n Figure 13(b), a change i n C y c f r o m 0.6 to 0.7 leads t o a f o r w a r d movement o f stock o f about 10% compared w i t h a f o r w a r d m o v e m e n t i n CPc o f o n l y about 2%.

I n Figure 14 one value o f skeg d e p t h is considered and the influences o f CJc and sweep are investigated

Figure 14. Variation in movable chord ratio and sweep (skeg depth constant).

f o r a = 2 0 ° . I t is n o t e d that b o t h C^/c and sweep have a marked influence o n the stock p o s i t i o n and balance area, F o r example, i f i t is assumed that CPc is to coin-cide w i t h the stock p o s i t i o n (at o: = 2 0 ° ) then f o r a sweep o f 0.153 rads the required C^/c is 0.655 and the required balance area is approximately 20.5% whereas f o r .0.070 rads sweep the required C^/c rises t o 0.695 and the balance r a t i o is again about 20.5%. S i m i l a r l y , f r o m Figure 1 3 ( b ) , i f C^ is increased by decreasing skeg d e p t h , and a large disparity between CPc and the stock p o s i t i o n is t o be avoided, the balance r a t i o has t o be held at a p p r o x i m a t e l y 20.5% and C^/c has to be i n -creased f r o m about 0.6 at S^/S = 0.63 t o about 0.7 at

SJS=036.

I t is thus clearly seen that i f an increase i n p e r f o r m -ance is t o be achieved b y decreasing skeg d e p t h t h e n this has t o be accompanied b y a decrease i n skeg chord ( o r an increase i n L . E . sweep i f structural requirements l i m i t the decrease i n skeg c h o r d ) .

Figure 15 shows that the influence o f aspect r a t i o o n at a particular skeg d e p t h is generally similar

50

0 . 4 0 . 5 0.6 SKEG D E P T H / S P A N S ] / s

T = 0 . 8 ; « 1 = 0 . 1 5 3 ; Cf/c = 0 . 6 5 ; 0 = 0° 0C = 10°

to that expected f o r ah-movable c o n t r o l surfaces. T h e influence o f skeg depth on is seen t o be indepen-dent o f aspect r a t i o . I t is also seen that as aspect r a t i o is decreased f o r constant values o f taper r a t i o , skeg depth r a t i o , f l a p c h o r d r a t i o and sweep, the skeg area ratio remains f i x e d , CPc moves f o r w a r d b y a small amount and the stock p o s i t i o n moves a f t b y a sig-n i f i c a sig-n t amousig-nt. Thus, f o r cosig-nstasig-nt sweep, asig-n isig-ncrease i n f l a p chord ratio (hence decrease i n skeg area r a t i o ) is required w i t h decrease i n aspect r a t i o i f the t o r q u e lever is t o remain a p p r o x i m a t e l y constant.

1 1 . Conclusions

(i) The earher experimental investigation demon-strated that a skeg rudder exhibits a number o f par-ticular characteristics w h i c h a f f e c t the p r e d i c t i o n or assessment o f its performance. The experimental results, supported b y the present theoretical analysis, showed t h a t changes i n the skeg angle, f o r a particular rudder angle, lead t o changes i n local l i f t i n way o f the skeg and at the same t i m e have a significant e f f e c t on the all-movable p o r t i o n o f the rudder. Reahstic predictions o f load are, therefore, n o t h k e l y t o be achieved b y assuming the skeg rudder t o be made u p o f separate f l a p p e d and all-movable parts.

(ii) The theoretical study demonstrated that satis-f a c t o r y predictions o satis-f the satis-f o r m o satis-f the spanwise load-ings f o r d i f f e r e n t skeg and rudder angles can be made using h f t i n g Une t h e o r y , w i t h the e f f e c t o f the skeg being incorporated as local incidence r e d u c t i o n and the effects o f the mid-span and t i p traUing vortices being incorporated as t w i s t corrections t o the local inciden-ce. The correct magnitude o f the distributions was satisfactorily reproduced b y applying, spanwise, a single empirical c o r r e c t i o n based o n the r a t i o o f the experimental and theoretical l i f t curve slopes.

( i i i ) The theoretical and empirical extensions t o the experimental w o r k demonstrated t h a t , f o r f i x e d aspect ratio and taper r a t i o , the p r o d u c t i o n o f sideforce is significantly i n f l u e n c e d b y the size o f the skeg depth and movable c h o r d (hence skeg chord) at l o w e r angles o f attack and b y the skeg d e p t h at larger angles o f at-tack. A decrease i n skeg d e p t h leads t o an increase i n h f t p r o d u c t i o n and i t foUows that the best h y d r o -dynamic performance wUl be achieved b y m i n i m i s i n g this dunension t o the l i m i t s o f structural requirements.

V a r i a t i o n i n skeg depth has a m a r k e d e f f e c t on balance area and CPc. F o r a particular skeg depth, the movable chord and sweep have a relatively small influence on CPc whereas t h e y have a large i n f l u e n c e on the stock p o s i t i o n and balance area, hence having a m a r k e d e f f e c t on t h e magnitude o f torque over the incidence range. A c a r e f u l choice o f the skeg and movable rudder p r o p o r t i o n s is, therefore, necessary.

N o t a t i o n

A t o t a l rudder area (movable rudder plus skeg) AR effective aspect ratio

a h f t curve slope correction f a c t o r

An coefficients i n Fourier series f o r spanwise l o a d d i s t r i b u t i o n

balance area ratio c chord

c mean chord ((C^, + C^ ) / 2 ) Cj. tip chord

C^ r o o t chord C^ flap chord

CPc local ( o r section) centre o f pressure chordwise, %c measured f r o m L . E .

CPc t o t a l centre o f pressure chordwise, %c, meas-ured f r o m L . E .

CPs centre o f pressure spanwise, %S, measured f r o m r o o t

C^ drag c o e f f i c i e n t {DlVipAV'^) Cjj . induced drag c o e f f i c i e n t

^Dp p r o f i l e drag c o e f f i c i e n t C^ h f t c o e f f i c i e n t (L jVip A F ^ )

CJy n o r m a l force c o e f f i c i e n t , n o r m a l t o rudder {NIVipAV'^)

D drag force i n f l o w d i r e c t i o n

H^(d) general f o r m o f downwash v a r i a t i o n i n d u c e d b y tip v o r t e x

H^id) general f o r m o f downwash v a r i a t i o n induced b y mid-span vortex

k defined as ( 1 - T ) m l i f t i n g line analysis L U f t f o r c e n o r m a l t o flow d i r e c t i o n m two-dimensional l i f t curve slope

N n o r m a l f o r c e , n o r m a l t o centreline o f movable rudder

5 r u d d e r span 5^ skeg depth 5 j skeg area ratio T taper r a t i o ( C ^ / C ^ ) V i n f l o w v e l o c i t y

X distance t o centreline o f stock f r o m L . E . o f rudder (%c)

y spanwise coordinate i n h f t i n g hne analysis a rudder angle relative t o flow (Figure 2) a. downwash angle

fty, equivalent local twist

a defined as ( a — 7 ) i n l i f t i n g hne analysis

«sep approx. angle w h i c h onset o f separation occurs a f t o f skeg

|3 skeg angle relative t o flow, or ship d r i f t angle at rudder (Figure 2)

7 skeg incidence r e d u c t i o n i n l i f t i n g line analysis r c i r c u l a t i o n

6 rudder angle relative t o skeg, or ship (Figure 2) 9 spanwise coordinate i n l i f t i n g line analysis

ft c o e f f i c i e n t i n l i f t i n g line analysis P mass density

\p value o f 8 at end o f skeg n J sweep o f leading edge CO induced n o r m a l v e l o c i t y

Skeg area ratio (S, ) ^ ^keg area (assumed to £ o f stock) t o t a l area (movable + skeg) Balance area ratio (B^ ) =

movable area f o r w a r d o f Ê o f stock t o t a l movable area

T o t a l movable area = t o t a l area (movable + skeg) -skeg area (assumed to £ o f stock)

References

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2. Molland, A.F., 'Further free-stream characteristics of semi-balanced ship skeg-rudders'. University of Southampton, Ship Science Report No. 2/78,1978.

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5. Molland, A.F., 'The free-stream characteristics of ship skeg-rudders', Ph.D.Thesis, Faculty of Engineering and Applied Science, University of Southampton, 1982.

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8. Glauert, H., 'Theoretical relationships for an aerofoil with hinged flap', A.R.C., R & M No. 1095,1927.

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10. Ames Jr., M.B. and Sears, RJ., 'Pressure - distribution investigation of an NA.C.A. 0009 airfoil with a

30-percent-chord plain flap and three tabs', T.N. No. 759 N.A.C.A. 1940.

11. Street, W.G. and Ames Jr., M.B., 'Pressure - distribution investigation of an N.A.C.A. 0009 airfoil whh a 50-percent-chord plain flap and three tabs', T.N. No. 734 N.A.CA. 1939.

12. Ames Jr., M.B. and Sears, R.L, 'Pressure - distribution investigation of an N.A.C.A. 0009 airfoil with an 80-per-c'ent-chord plain flap and three tabs'.T.N. No. 761 N.A.C.A. 1940.

13. Sears, R.L, 'Wind tunnel investigation of control surface characteristics. I - Effect of gap on the aerodynamic characteristics of an NA.C.A. 0009 airfoil with a 30-per-cent-chord plain flap'. Report L-377, N.A.C.A., June 1941. 14. Küchemann, D., 'Types of flow on swept wings, with

special reference to free boundaries and vortex sheets', J.R. Aero Soc, Vol. 57, November 1953.

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