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LIb r RY

R O Y A L A I ~ , ~ ,~-~-, , : :-f L S F A B L I S - - ~ ~ i c - , * 1 ~

a

R . M . N o . 3 0 7 7

1 9 , 8 ~ 0 )

A R O Technical Report

M I N I S T R Y O F S U P P L Y

AERONAUTICAL RESEARCH COUNCIL

REPORTS AND MEMORANDA

Methods for Determining the Wave

Drag o f N on L ifting W ing Body

Combinations

y

L. M. SHEPPARD

Crown Copyright z9 58 .

L O N D O N

HER MAJESTY S STATIONERY OFFICE

1958

T E N SHIL L I NGS NE T


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i s a p p l i c a b l e t o t h e t r a n s o n i c r a n g e , c a n b e f o u n d . F o l l o w i n g t h e s u g g e s t i o n s o f L o r d s, t h e s o n i c

w a v e d r a g o f a w i n g - b o d y c o m b i n a t i o n m a y b e in t e r p r e t e d p h y s i c a l l y a s a m e a s u r e o f t h e s u d d e n

t r a n s o n i c d r a g r i s e of t h e c o n f i g u r a t i o n , w h e n s u c h a d r a g r is e c a n b e d e f in e d .

I t is w o r t h y o f n o t e t h a t t h e u s e f u ln e s s of e q u a t i o n (1) is g r e a t l y e n h a n c e d b y t h e f a c t t h a t

t h e l i n e a r i s e d t h e o r y p r e d i c t s z e ro l i f t - d e p e n d e n t w a v e d r a g a t M = 1. T h u s e q u a t i o n (1 ) i s

n o t r e s t r i c t e d t o n o n - l i f t in g c o n f i g u r a t i o n s s i n c e i t g i v e s t h e t o t a l s o n i c w a v e d r a g o f a li f t i n g

c o n f i g u r a t i o n ( fo r a d i s c u s s io n o f l i f t - d e p e n d e n t w a v e d r a g , see f o r e x a m p l e , L o m a x a n d H e a s l et la ) .

2 .12. D e s i gn fo r l ow w a v e d r a g . - - T h e d e s i g n o f a w i n g - b o d y c o m b i n a t i o n w i t h l o w s o n i c w a v e

d r a g r e q u ir e s t h e d e t e r m i n a t i o n o f o p t i m u m a r e a d i s t ri b u t i o n s w h o s e a s so c i a te d w a v e d r a g ,

g i v e n b y e q u a t i o n (1 ), i s a m i n i m u m . L o r d a n d E m i n t o n h a v e in v e s t i g a t e d t h i s p r o b l e m a n d

p r e s e n t e d o p t i m u m a r e a d i s tr i b u t i o n s fo r a n u m b e r o f s p ec if ie d c o n d i ti o n s . T h u s i n o r d e r t o

o b t a i n l o w s o n ic w a v e d r a g i t i s n e c e s s a r y t o a l t e r t h e b o d y , a n d p o s s i b l y t h e w i n g a l so , s o t h a t

t h e c r o ss - se c t io n a l a r e a d i s tr i b u t i o n b e co m e s a n o p t i m u m , o r n e a r o p t i m u m . A si m p l e e x a m p l e

o f a n o p t i m u m a r e a d i s t r i b u t i o n i s t h a t o f a s o -c a ll e d S e a r s - H a a e k b o d y , w h i c h i s p o i n t e d a t.

b o t h e n d s a n d h a s m i n i m u m w a v e d r a g fo r g i v e n l e n g t h a n d v o l u m e .

2.2. A r e a R u l e . - - 2 . 2 . 1 . E s t i m a t i o n o f w a v e d r a g . - -- T h e a r e a -r u l e m e t h o d o f e s t i m a t i n g w a v e

d r a g i s d u e t o J o n e s 5 a n d W h i t c o m b . H o w e v e r , t h e d e r i v a t i o n o f t h e a r e a r u l e re q u i re s t h e u s e

o f a n a s s u m p t i o n e v e n t h o u g h t h e w i n g - b o d y c o m b i n a t i o n i s r e p r e s e n t e d b y s o u r c e d i s t r ib u t i o n s

o n l y . T h e a s s u m p t i o n i s t h a t t h e e ff ec t o f t h e i n t e r fe r e n c e v e l o c i t y p o t e n t i a l o n th e w a v e d r a g i s

n e g l i g i b l e ,

i . e . ,

t h e p e r t u r b a t i o n v e l o c i t y p o t e n t i a l of a w i n g - b o d y c o m b i n a t i o n i s a s s u m e d t o b e

e q u a l to t h e p e r t u r b a t i o n v e l o c i t y p o t e n t i a l of t h e is o l a te d e x p o se d w i n g t o g e t h e r w i t h t h e

p e r t u r b a t i o n v e l o c i t y p o t e n t i a l o f t h e i s o l a te d b o d y .

I t i s c o n v e n i e n t t o de f in e a n e l e m e n t a l a r e a d i s t r i b u t i o n . C o n s i d e r t h e f a m i l y o f p a r a l l e l M a c h

p l a n e s in c l i n e d a t t h e M a c h a n g l e to t h e b o d y a x i s a n d s u c h t h a t t h e o r t h o g o n a l p l a n e c o n t a i n i n g

t h e b o d y a x i s i s i n c l i n e d a t t h e a n g l e 0 t o t h e ' p l a n e o f t h e w i n g ' . A g i v e n f a m i l y o f M a c h p l a n e s

c u t s t h e w i n g - b o d y c o m b i n a t i o n o b l i q u e l y a n d t h u s d e fi n es a n o b l iq u e s e c t i o n a l ar e a d i s t r i b u t i o n

s ( x , O , M ) ,

w h e r e M i s t h e M a c h n u m b e r u n d e r c o n s i d e r a t io n . T h e c r o s s - se c t i o n a l a r e a d i s t r i b u -

t i o n S ( x , O , M ) = s ( x, O, M ) s i n i f, w h e r e ff i s t h e M a c h a n g l e , i s d e f i n e d t o b e a n e l e m e n t a l a r e a

d i s t r i b u t i o n . I t s a s s o c i a t e d w a v e d r a g is g i v e n b y e q u a t i o n (1) a s

q - - 2 ~ S ' (x ~ ,O , M ) S ( & , O , M ) l o g ] x ~ - - [ d & d & ; . . . . ( 2 )

w h e r e

S ( x , O , M )

d e n o t e s

~ 2 S ( x , O , M ) / ~ x ~

a n d i t i s a s s u m e d t h a t

S ' ( x , 0 , M )

i s c o n t i n u o u s

e v e r y w h e r e . W h e n M = 1 i t i s s e e n t h a t a l l t h e e l e m e n t a l a r e a d i s t r ib u t i o n s a r e t h e s a m e , b e i n g

e q u a l t o t h e c r o s s - s e c t i o n a l a r e a d i s t r i b u t i o n .

H a y e s T a n d H e a s l e t, L o m a x a n d S p r e i te r 16 o b t a i n e d a r e s u l t w h i c h a p p l i e s t o w i n g s o n l y b u t

i t s a p p l i c a t i o n to w i n g - b o d y c o m b i n a t i o n s f o ll ow s i m m e d i a t e l y f ro m t h e a s s u m p t i o n t h a t t h e

e f fe c t of t h e i n t e r f e r e n c e v e l o c i t y p o t e n t i a l o n t h e w a v e d r a g is n e g l ig i b le * . T h e w a v e d r a g o f

t h e c o n f i g u r a t o n t h e r e f o r e c a n b e e x p r e s s e d i n t h e f o r m

D = 2 7 j o D (O , M ) dO , . . . . . . . . . . . . . . . .

(3)

i . e . ,

t h e w a v e d r a g o f a w i n g - b o d y c o m b i n a t i o n a t a M a c h n u m b e r M i s t h e m e a n o f t h e w a v e

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

* I n a p r i v a t e c o m m u n i c a t i o n L . E . F r a e n k e l h a s s h o w n t h a t t h i s a s s u m p t i o n i s v a l i d f o r s m a l l v a lu e s o f t h e r a t i o o f

b o d y r a d i u s t o w i n g c h o r d . T h i s c o n d i t i o n m a y b e d e r i v e d b y c o n s i d e r i n g a c o n f i g u r a t i o n w i t h a c i r c u l a r c y l i n d r i c a l

b o d y a n d t h e n o b s e r v i n g t h a t t h e i m p o r t a n t p a r a m e t e r o f a n i n t e rf e r en c e f l ow fi el d i s BR,/c w h e r e B = ~/ M 2 -- 1),

R , i s a t y p i c a l b o d y r a d i u s a n d c i s t h e w i n g - r o o t c h o r d . T h u s t h e i n t e r f e re n c e v e l o c i t y p o t e n t i a l c a n b e n e g l e c t e d

f o r s m a l l v a l u e s o f

BR,/c

b e c a u s e i t s e f f e c t v a n i s h e s w h e n R , = 0 a n d w h e n B - ----0 i .e ., M -- -- 1 ). W h e n t h e a b o v e

c o n d i t i o n i s n o t s a t i s fi e d t h e e r r o r i n t r o d u c e d b y t h e n e g l e c t o f t h e i n t e r fe r e rf c e v e l o c i t y p o t e n t i a l i s n o t k n o w n .

H o w e v e r , i t is p os s i bl e t h a t t h e e r r o r w i l l n o t b e l a r g e b e c a u s e t il e i n t e rf e r e n c e v e l o c i t y p o t e n t i a l c a n b e n e g l e c t e d

a l s o w h e n t h e r a t i o BR,/c i s l a r g e , a n y i n t e r f e r e n c e e f f e c ts v a n i s h i n g w h e n c = 0 o r B = o o.

3

71666) A*


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area rule originally given by Jones 5. Ward 8 has derived the same result by using operati onal

methods, while Graham, Be ane and Licher17 have also given this result. When M = 1, equa tions

(2) and (3) reduce to the sonic area-rule result, equation (1).

It is important to note that the derivation of the area rule (equation (3)), required all the

elemental area distributions to be such tha t S ( x , O , M ) is continuous everywhere. Nevertheless,

provided that almost all the elemental area distributions satisfy this requirement the wave drag

ma y still be given by equati on (3). This condition is much less restrictive tha n the original one*.

An elemental area distribution of a t hin-wing-slender-body combination can be written in the

approximate form S ( x , O , M ) = S ( x ) + S w ( x , O , M ) , where S ( x ) is the cross-sectional area distri-

bution of the body and Sw (x , O , M ) is the contr ibutio n from the exposed wing. The wing is take n

to lie in the plane z = 0 ( s ee Fig. 1) and its th ickness at the point. (x, y) is de noted by T ( x , y ) .

Consider any Mach plane cutting the axis at the point x = xl. Then this Mach plane cuts the

wing obliquely along an area which, whe n project ed on to t he plane x = xl, defines the area

Sw (x , O , M ) . This projection may be regarded as a projection on to a plane normal to the wing

followed by a project ion on to the plane x = xl. The former projection defines an area equal

to the wing thickness integrated along the line in which the Nach plane cuts the wing, i . e . , -

x -- xl = y ta n v where tan ~ = cot/~ cos 0. It follows that

S w ( x , O , M ) = f T ( x + Y l tan v, Yl) dyl .

T O

Heaslet, Lomax and Spreiter 16 obtaine d this result in an equiva lent form.

The above result applies to a wing lying in the plane z = 0. However, its extension to a more

general wing is not difficult and follows immedia tel y from the definition of an eleme ntal area

distribution. The general result, which includes the previous result as a special case, can be

written in the form

S w x , O , M ) = f T f f ) d r ,

T O

where an element of the wing plan-form is denoted by d x d r , a general point in the co-ordinate

system of Fig.. 1 is deno te d by R = (x, r), R = (x + Be0 r, r), where B = cot , ro is th e uni t

vector (cos

O,

sin 0) and 0 is defined by y = r cos 0, z = r sin 0.

2.2.2. D e s i g n f o r l o w w a v e d r a g . - - T h e problem of reducing the supersonic wave drag of a

wing-b ody combination with a given wing was exam ined first by J ones 5. At sonic speed, a bod y

may be modified so tha t the equival ent body, which has the same cross-sectional area distrib ution

as the wing-body combination, is an optim um with the same total volume,

i .e. ,

the modified body

is formed by shaping the opt imum equ ivalen t body in the region of the wing. At higher speeds

the wing may be influenced by, an d may itself influence, a larger part of the body. Thus t he

corresponding optimum equivalent body shaping extends outside the region of the wing and

may be shown to be over tha t part of the body withi n the Mach diamon d of the wing. Details

of this drag-reduction procedure are given in section 2.4.2. where the transfer rale is discussed.

Another problem of interest is that of reducing the wave drag of a general configuration and

this has been discussed by Jones5 in a qualitative form. All the ele mental area distributions

should be such that their associated wave drags are as low as possible. The design of combinations

* Since this report was completed, Lock has shown that the wave drag of a rectangular wing can be found using

the area rule (equations (2), (3)). Thus the smoothnesscondition that S (x, O, M ) be continuous everywhereappears

to be unnecessary for the wing of a configuration.


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satisfying this requir ement appears to be very difficult except in certain special cases ; for example,

at low supersonic speeds only a small number of elemental area distributions are required and

so an iterati on met hod can be used to obtain a modified configuration with low wave drag.

In the case of an isolated wing it is interesting to use the area rule to determine the optimum

thickness distribution for minimum wave drag, the plan-form and volume of the wing being

given. A particular case of this problem was first exami ned by J ones1 who showed that if the

plan-form is elliptic then the wing section must be parabolic-arc biconvex with thlckness/chord

ratio proportional to local chord. However, Graham, Beane and Licher17 have u sed the area

rule to obtain the same result and hav e shown that all the el ementa l area distributions represent

Sears-Haack bodies

( i .e. ,

bodies of min im um wave drag for given lengt h and volume.) The

result for the w ave-drag coefficient of the opt imu m wing of elliptic plan-form is

~ A ~ B ~ ~ A ~ B ~ ~

where CD is base d upon the wing plan- form area

A is the aspect ratio of the wing

*0 is the t hickness/ chord ratio at the wing root.

Also o =

32V)/ ~Ac3),

where-V is the vol ume of the wing and c is the wing-r oot chord.

2.3.

M o m e n t o f A r e a R u l e . - -2 . 3 . 1 . E s t i m a t io n o f w a v e d r a g . - - T h e

mome nt of area-rule met hod

is due to Baldwin and Dickey~, who expanded the wave drag in a series o.f powers of

B~ -- (M s - 1). The results presented later in this section are similar and were obta ined

indepen dently of Baldwin and Dickey~ whose results will be described first.

T h e first step towards a series solution in ascending powers of B ~ was made by Adams and

Sears ~ who develope d a not- so-s lend er wing theory by ob taining an expansion up to the first

power of B ~, th e te rm indep enden t of B ~being the slender-wing solution. These results emphasise

the fact that the series solution is actually an expansion in ascending powers of B 2 k ~ where k is

the slenderness para met er in troduce d by Ada ms a nd Sears ~. For example, in some wing problems

k may be defined as the aspect ratio.

Before proceeding with a description of the m ome nt of area rule developed by Baldwin and

Dickey11, it is necessary to obta in an al ternati ve form of equa tion (1). Using the subs titution

x = (1 -- cos ~), 0 ~< x ~< 1, S ( x ) is expressed as a Fourier sine series in th e new variable ~ ;

namely

S ( x ) = ~ A , ~ s i n n ~ , 0 ~< ~ ~< ~.

Substituting this series in equation (1) shows that the wave drag is given by

D ~ oo

~. nA,,~ .

q

Therefore, using similar substitutions equation (3) can be written as

q s o ,~=1

where

A , ( O . M )

is the F ourier coefficient appropriate to th e ele mental area distribu tion

S ( x , 0 , M ) .

Baldwin and Dickeyn expanded the Fourier coefficient

A, ,(O, M )

in a series of powers of

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B ~ = ( M ~ - 1) a n d i t is fo u n d t h a t , w i t h i n t h e u s u a l o r d e r o f a p p r o x i m a t i o n , t h e w a v e d r a g

o f a c o n f i g u ra t i o n w i t h t h e w i n g in t h e p l a n e z = 0 (see F i g . 1 ) d e p e n d s s o l e l y u p o n t h e m o m e n t s

o f a r e a d e f i n e d b y

= s x ) + t

T ( x ,

Y )

dy

d

T.~ 0

a n d

M , ( x ) = | T ( x , y ) y i d y , i even , i

>~

2 ,

d

T O

w h e r e S(x ) i s t h e c r o s s - s e c ti o n a l a r e a d i s t r i b u t i o n o f t h e b o d y a n d , i n t h e n o t a t i o n o f F i g . 1 ,

T ( x , y ) i s t h e w i n g t h i c k n e s s a t t h e p o i n t ( x, y ) . C l e a r ly , Mo(x) i s e q u a l t o t h e c r o s s - s e c t i o n a l

a r e a d i s t r i b u t i o n o f t h e w i n g - b o d y c o m b i n a t i o n a n d M~(x) i s e q u a l to t h e d i s t r i b u t i o n o f t h e

c r o s s - s e c t i o n a l p o l a r m o m e n t o f i n e r t i a o f t h e w i n g .

T h e f i n a l r e s u l t fo r t h e w a v e d r a g m a y b e w r i t t e n i n t h e f o r m D /q = ao + a=B~ - /O ( B ~) , w h e r e

a 0 d e p e n d s u p o n Mo(x) a n d a ~ d e p e n d s u p o n b o t h Mo(x) a n d M~(x) . T h e h i g h e r t e r m s i n t h i s

s e r i e s a r e m o r e c o m p l e x a n d s o , a s B a l d w i n a n d D i c k e y *t h a v e s t a t e d , t h e m e t h o d i s s i m p l e t o

a p p l y o n l y a t l o w s u p e r s o n i c sp e e ds . N e v e r t h e l e s s , w i t h i n t h i s s p e e d r a n g e i t d oe s o ff e r a s i m p l e

q u a l i t a t i v e m e t h o d f o r r e d u c i n g w a v e d r a g ; a us e f u l r e q u i r e m e n t is t h a t t h e m o m e n t o f a r e a

d i s t r i b u t i o n s b e a s s m o o t h a n d ' s l e n d e r ' a s p o s s ib l e . I t i s s u g g e s t e d t h a t , i n g e n e r a l , i t w i ll b e

s u f f ic i e n t t o c o n s i d e r o n l y Mo(x) a n d M~(x) .

B e g i n n i n g f r o m e q u a t o n s (2 ) a n d (3 ) i t h a s b e e n s h o w n t h a t , f o r a c o n f i g u r a t i o n o f u n i t l e n g t h ,

t h e w a v e d r a g is g iv e n b y

q - o o lo

+ - ~ { ~ , ~ (& )} ~ -~ { ~ , ~ (& )} lo g Ix ~ - - x~[

d d + O ( B ~ ) , . . . .

(4)

w h e r e % , , (x ) , J i , a(x ) a r e g e n e r a l i s e d m o m e n t s o f a r e a d e f i n e d b y

1 Ri+ 2(x, O) cos 40 dO

- i o

1 f f~R ~ '~(x , 0 ) s in 40 dO

0

w h e r e r = R ( x , 0) i s t h e e q u a t i o n o f t h e c r o s s - s e c t io n a t t h e s t a t i o n x . I t i s a s s u m e d t h a t t h e

s e ri e s i n e q u a t i o n (4) i s c o n v e r g e n t a n d t h a t a l l n e c e s s a r y c o n t i n u i t y a n d d i f f e r e n t i a b i l i t y c o n -

d i t i o n s a r e sa ti sf ie d . F u r t h e r m o r e , s i n c e t h e b o d y i s s l e nd e r i t is i m p o r t a n t t o n o t e t h a t t h e

e x p r e s s i o n s f o r t h e g e n e r a l i s e d m o m e n t s o f a r e a c a n b e s i m p l if i ed . W i t h i n t h e u s u a l o r d e r o f

a p p r o x i m a t i o n , t h e c o n t r i b u t i o n o f t h e b o d y c a n b e n e g l e c t ed f o r a l l % ,~ , ~ , ~ ( i 0 ). T h u s i n

e q u a t i o n (4 ) t h e b o d y c o n t r i b u t e s t o t h e w a v e d r a g o n l y i n s o f a r a s d o e s S(x ) , t h e c r o s s - s e c t io n a l

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

0 = O, ~/2 (see F i g. 1) th e g e n e r a l is e d m o m e n t s o f a r e a c a n b e w r i t t e n i n t h e s i m p l e r f o r m s

~'i ,a(x) = 0 , Z odd ,

Y,,a(x) ---- O , a ll ,~,

a n d s i n c e ~ , 0 (x ) i s t h e i t h m o m e n t o f a r ea , i t f o ll o w s t h a t ~ , 0 (x ) = M~(x)~


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T h u s e q u a t i o n ( 4 ) b e c o m e s

B 2

D = D { S } -J ---4 [ D { S + M ( ' } - - D { S } -

D{M('}] -}-

O (B 4 ) , . . . . . . (5)

Wahere

D { S }, D { M ( ' }, D { S + M ( ' }

d e n o te t h e w a v e d r a g o f ' b o d i e s ' w i t h th e c r os s- se c ti on a l

r e d ' d i s t r i b u t i o n s S ( x ) , M ( ' ( x ) , S ( x ) + M ( ' ( x ) r e s p e c t i v e l y a n d i t h a s b e e n a s s u m e d t h a t

M(H (x )

i s c o n t i n u o u s e v e r y w h e r e . A t f i rs t s i g h t t h i s c o n d i t i o n s e e m s t o l i m i t v e r y s e r i o u s l y t h e

a p p l i c a b i l i t y o f e q u a t i o n (5 ) b u t i t is n o t u n d u l y r e s t r i c t i v e s i nc e l fo r e x a m p l e , i t i s s a t i sf i e d b y

a n u n s w e p t , t a p e r e d w i n g w i t h s t r a i g h t e d g e s a n d b i c o n v e x s e c ti o n o f c o n s t a n t t h i c k n e s s ra t io .

M o r e o ve r, t h e c o n d i t i o n i s s a ti s fi e d b y t h e o p t i m u m s e c o n d - m o m e n t d i s t r i b u t i o n g i v e n i n S e c ti o n

2 .3 .2 , a l t h o u g h i t is n o t s a t is f i e d b y t h e J o n e s e l l i p t ic w i n g d i s c u s s e d i n S e c t i o n 2. 2. 2. W h e n

t h e w i n g is d e f in e d n u m e r i c a l l y , e q u a t i o n (5) h a s t h e s e r io u s d i s a d v a n t a g e t h a t t h e w a v e d r a g

c a n n o t b e d e t e r m i n e d v e r y a c c u r a t e l y s i n c e i t is n e c e s s a r y to c a r r y o u t t w o s u c c e s si v e n u m e r i c a l

d i f f e r e n t i a t i o n s in o r d e r t o c a l c u l a t e

M2 (x) .

T h e i m p o r t a n c e o f e q u a t i o n (5 ) l ie s i n t h e f a c t t h a t i t re d u c e s t h e c a l c u l a t i o n o f t h e w a v e d r a g

o f a ' n o t - s o - s l e n d e r ' c o n f i g u r a t i o n t o t h e c a l c u l a t i o n o f t h e w a v e d r a g o f t h r e e ' b o d i e s '. T h u s

n u m e r i c a l e v a l u a t i o n o f t h e w a v e d r a g i s c o n s i d e r a b l y s im p l i fi e d a n d c a n b e c a r r i e d o u t u s i n g

t h e m e t h o d s o f S e c t io n 4.2 . H o w e v e r , i t s h o u l d b e e m p h a s i s e d t h a t b o t h S ' ( x ) a n d M ( ( x ) m u s t

b e c o n t i n u o u s e v e r y w h e r e b e fo r e e q u a t i o n (5 ) c a n b e u s e d . T h e r e f o r e t h e F o u r i e r - se r i e s m e t h o d

d e s c r i b e d e a r l i e r i n t h i s S e c t i o n m a y b e u s e d , t h e s u b s t i t u t i o n s b e i n g

x = ( 1 - - c o s g ) , 0 ~ < x ~ < 1 ,

S ' ( x ) = ~ A, , s in n9

n = l

a n d

M21;r(x) = ~ B s i n n 9

E q u a t i o n (5) n o w b e co m e s

D

q 9 1: n = l V T ~ = 1

I t w i l l b e n o t i c e d t h a t o n l y t h e c o e ff ic i e nt s B ,, f o r w h i c h t h e c o r r e s p o n d i n g c o e f fi c ie n t A ,, i s n o n -

z e r o a p p e a r i n t h e w a v e d r a g . T h u s t h i s f o r m o f t h e w a v e d r a g i s u s e f u l w h e n m o s t o f t h e A , ,

v a n i s h . I t is e s p e c i a l l y u s e f u l w h e n t h e a r e a d i s t r i b u t i o n

S (x )

i s a n o p t i m u m d i s tr i b u ti o n w i t h

a f i n i te F o u r i e r se r ie s . F o r e x a m p l e , t h e S e a r s - H a a c k b o d y h a s A ,, = 0 , n # 2 .

2.3.2.

D e s i g n f o r l o w w a ve d r a g . - - I t

h a s a l r e a d y b e e n e m p h a s i s e d t h a t t h e m a i n u s e f u ln e s s of

t h e m o m e n t o f a r e a ru l e i s a s a q u a l i t a t i v e d e s i g n p r o c e d u r e f o r l o w s u p e r s o n i c s p ee d s . B a l d w i n

a n d D i c k e y 1~ h a v e e x a m i n e d t h e u s e o f t h e m e t h o d i n d e s i g n i n g c o n f ig u r a ti o n s w i t h m i n i m u m

w a v e d r a g u n d e r s p e ci fi e d c o n d i ti o n s . T h e p r o c e d u r e u s e d i s a s te p - b y - s t e p o p t i m i s a t i o n t e c h n i q u e

w h e r e b y s u c ce s s iv e t e r m s i n t h e s e r ie s s o l u t i o n a r e o p t i m i s e d o n e a f t e r t h e o t h e r . T h u s a u n i q u e

s e t o f m o m e n t d i s t r i b u t i o n s i s f o u n d .

T h e o p t i m u m s e t o f m o m e n t d i s t r ib u t i o n s f or m i n i m u m w a v e d r a g , t h e le n g t h s a n d t h e

' v o l u m e s ' b e i n g f ix e d, i s g i v e n b y B a l d w i n a n d D i c k e y 11. I n t h i s c o n t e x t t h e ' v o l u m e i s

t h e p t h m o m e n t ' v o l u m e ' * o f t h e c o n f i g u r a t io n d e f in e d b y

V ~ = f P M / x ) d x ,

* T h e p t h m o m e n t v o l u m e d o e s n o t h a v e a n y p h y s i c a l s i g n if i ca n c e u n l e s s 15 ~ 0 w h e n i t i s t h e a c t u a l v o l u m e o f

t h e c o n f i g u r a t i o n .

7


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w h e r e l p i s th e l e n g t h o f t h e p t h m o m e n t d i s t r i b u t i o n , i .e ., l0 ---- t, t h e l e n g t h o f t h e c o m b i n a t i o n

an d, fo r p ~> 2, Ip = l ~ , t h e l e n g t h o f t h e p r o j e c t i o n o f t h e w i n g o n t h e b o d y a x is . T h e r e s u l t is

t h a t t h e o p t i m u m m o m e n t d i s t r ib u t i o n s w h i c h v a n i s h a t x = 0 ,

I p ,

a r e

[

/

Z

z , / 2

] J

\

p e v e n , O ~ < x ~ < I p

a n d t h a t t h e w a v e d r a g o f t h e o p t i m u m c o n f i g u ra t i o n i s g i v e n b y

_ _ F M I , 1_ _ / 2 ) _

D _ 4 1 1 s [ M o ~

l l 2 ) - ] + 1 6 5 3 7 . 5

L

l~ ~ j B ~ I + O B ~ ) .

q

I t w i l l b e n o t i c e d t h a t n o t e r m i n B ~ a p p e a r s i n t h i s r e su l t.

S i n c e a l l t h e t e r m s i n t h e a b o v e e x p r e s s i o n f o r t h e w a v e d r a g a r e p o s i t iv e , i t h a s b e e n p o i n t e d

o u t b y B a l d w i n a n d D i c k e y 11 t h a t t h e w a v e d r a g o f a c o n fi g u r a ti o n , w h i c h i s o p t i m u m f o r t h e i r

c o n d i ti o n s , m u s t i n c r e a s e m o n o t o n i c a l l y w i t h i n c r e a s in g M a c h n u m b e r a n d b e v e r y la r g e w h e n

t h e s e c o n d t e r m i s v e r y m u c h g r e a t e r t h a n t h e f i rs t te r m , i .e .,

M o Z / 2 ) Z Z

B 0

]

T h u s t h e M a c h n u m b e r i n d i c a t e d b y t h i s r e l a t i o n c a n b e u s e d t o f i n d a n o b v i o u s u p p e r l i m i t t o

t h e r a n g e o f a p p l i c a b i l i t y o f t h e o p t i m u m t e c h n i q u e . T h i s e m p h a s i se s o n c e a g a i n th e r e s t r ic t i o n

o f t h e m e t h o d t o l o w su p e r s o n ic s p e e d s a n d t h e r e f o r e t h a t i t i s n e c e s s a ry t o c o n s id e r o n l y t h e

f ir s t f e w te r m s o f t h e s e ri es so l u ti o n . B a l d w i n a n d D i c k e y 11 h a v e p o i n t e d o u t t h a t t h e s t e p - b y - s t e p

o p t i m i s a t i o n t e c h n i q u e f a i l s a t h i g h e r s u p e r s o n i c sp e e d s b e c a u s e i t e l i m i n a t e s a l l e x c e p t p o s i t i v e

d e f in i t e t e r m s i n t h e e x p r e s si o n fo r t h e w a v e d r a g . I t a p p e a r s t h a t a m o r e re a l i st i c o p t i m i s a t i o n

p r o c e d u r e , b a s e d o n e q u a t i o n (5 ), w i ll b e o b t a i n e d i f t h e t e r m i n B ~ i s n e g a t i v e i n s t e a d o f z e r o

a s i n t h e a b o v e p r o c e d u re . T h e f a c t t h a t t h e t e r m i n B 2 c a n b e n e g a t i v e i s s h o w n r e a d i l y b y

c o n s i d e r i n g a n u n s w e p t w i n g , s u c h a s t h e J o n e s e l l i p t ic w i n g d i s c u s se d i n S e c t i o n 2 .2 .2 .

E x p e r i m e n t a l c o n f i r m a t i o n o f t h e u s e f u ln e s s of t h e m o m e n t o f a r e a ru l e fo r w i n g - b o d y c o m -

b i n a t io n s h a s b e e n g i v e n b y B a l d w i n a n d D i c k e y 11. F i r s t l y , t h e o p t i m u m a r e a d i s t r i b u t i o n w a s

f o u n d a n d t h e n a n o p t i m u m s e co n d m o m e n t d i s tr i b u ti o n w a s o b t a i n e d b y a d d i n g b o d ie s o f

r e v o l u t i o n n e a r t h e w i n g t i p s w h i le m o d i f y i n g t h e b o d y t o r e t a i n t h e o p t i m u m a r e a d i s tr i b u t io n .

T h e t e s t s i n v o l v e d a n u n m o d i f ie d c o n f i g u r at io n , o n e w i t h t h e o p t i m u m a r e a d i s t r i b u t io n a s w e l l

a s o ne w i t h b o t h t h e o p t i m u m a r e a d i s t r ib u t i o n s a n d , i t w i l l b e n o t e d , a n o p t i m u m , l e n g t h e n e d

s e c o n d m o m e n t d i s t ri b u t i o n . B y c o n s id e r in g t h e d i ff e re n c e b e t w e e n t h e w a v e d r a g o f t h e w i n g -

b o d y c o m b i n a t i o n a n d t h e w a v e d r a g o f t h e b o d y a lo n e th e a g r e e m e n t b e t w e e n t h e o r y a n d

e x p e r i m e n t w a s f o u n d to b e g o o d o v er t h e M a c h - n u m b e r r a n g e M = 1 . 0 t o M = 1 . 4 , f o r a n

e l l i p ti c w i n g o f a s p e c t r a t i o 2. A t M = 1 . 4 t h e m o m e n t o f a r e a - r u l e m o d e l h a d a s l i g h t l y l o w e r

w a v e d r a g t h a n t h e u n m o d i f i e d c o n fi g u r at i o n . T h u s t h e s o n ic a r e a r u le is e x t e n d e d t o l o w

s u p e r s o n i c s p e e d s b y u s i n g t h e m o m e n t o f a r e a ru l e . H o w e v e r , b e c a u s e 12 h a s b e e n i n c r e a s e d

w i t h o u t c h a n g i n g M ~(0 ), th e s e e x p e r i m e n t a l r e s u l ts d o n o t j u s t i f y a n y t h i n g m o r e t h a n t h e u s e

o f s m o o t h , s l e n d e r s e c o n d m o m e n t d i s t ri b u t io n s .

I t w i l l b e n o t e d t h a t t h e a b o v e t e c h n i q u e r e q u i r e s t h e a d d i t i o n o f b o d i e s t o t h e w i n g . T h i s is

i n c o n t r a s t t o t h e m e t h o d d e s c r ib e d i n S e c t i o n 2. 2. 2, w h e r e b y t h e m a i n b o d y o n l y w a s s h a p e d

f o r m i n i m u m w a v e d r a g . T h u s i t w o u l d b e d e s i ra b l e t o e x a m i n e c o m b i n a t i o n s w i t h s u b s i d i a r y

b o d i e s n e a r t h e w i n g t i p s b y u s i n g t h e m e t h o d s a p p l i c a b l e a t a n y s u p e r s o n i c s p e e d , n a m e l y ,

t h e a r e a r u le a n d t h e t r a n s f e r ru l e. T h e p r o b l e m o f s h a p i n g s u c h s u b s i d i a r y b o d i e s i s e x a m i n e d

i n S e c t i o n 5 . 2 . 2 .


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2 . 4 .

W a r d s, 9, ~0, w h o h a s b a s e d h i s c o n s id e r a t io n s u p o n th e d ra g o f so u rc e d i s t r i b u t io n s .

o b t a i n e d b y W a r d I is t h a t

D 1

q - - 2= ; f T ( R 1 )T ( R 2 )c o sh - * ( [ & - & ]

** -

. q

f f , > V c l , e x=

T r a n s f e r R u l e . - - 2 . 4 . 1 . E s t i m a t i o n o f t he w a v e d r a g . - - T h e t r a n s f e r r u l e m e t h o d i s d u e t o

T h e r e s u l t

w h e r e d ~ , d ~

R J ~

P I ~ [ 2

T

. . . . . .

6 )

d e n o t e e l e m e n t s o f a r e a o f t h e w i n g s u r f a c e

d e n o t e p o i n t s o n t h e w i n g s u r fa c e (i.e., us i ng F ig . 1 , R i s th e v ec tor (x, y , z ))

d e n o t e t h e v e c t o r d i s t a n c e o f t h e e l e m e n t s d ~ ,, d ~,~ f r o m t h e b o d y a x i s (i.e.,

us i ng F ig . 1 , r is the v ec to r (y , z ))

d e n o t e s

a T/ax , T

b e i n g t h e w i n g t h i c k n e s s ,

a n d t h e i n t e g r a t i o n s a r e t o b e t a k e n o v e r r e a l v a l u e s o f t h e i n t e g r a n d s . W a r d ~ h a s n o t p l a c e d

a n y r e s t r ic t i o n s o n t h e v a l i d i t y o f e q u a t i o n (6) b u t i t i s v a l i d o n l y i f t h e e f fe c t o f t h e i n t e r f e r e n c e

v e l o c i t y p o t e n t i a l o n t h e w a v e d r a g i s n e g l ig i b le .

I t i s 0 t in t e r e s t t o n o t e t h a t i f t h e t h r e e t e r m s o f e q u a t i o n (6) a r e d e n o t e d b y D ~ , D w ~ a n d D B

r e s p e c ti v e l y , t h e n e q u a t i o n (6) c a n b e w r i t t e n i n t h e f o r m D = D w + D w B q - D B , w h e r e D w i s

t h e i s o l a t e d e x p o s e d w i n g w a v e d r a g , D wB i s t h e i n t e r f e r e n c e w a v e d r a g a n d D z i s t h e i s o l a t e d

b o d y w a v e d r a g . F o l l o w i n g W a r d 1 t h e i n t e r f e r e n c e w a v e d r a g D w~ c a n b e e x p r e s s e d i n a s i m p l e r

f o r m a n d a u s e f u l a l t e r n a t i v e t o e q u a t i o n (6 ) d e r i v e d . P u t

1 d V , . . . . . . . .

7 )

A ( x ) = ~ V { B = r , = _ ( x - -

&)2}, . . . .

w h e r e d V , = T ( R1 ) d ~ , i s a n e l e m e n t o f v o l u m e o f t h e w i n g a n d t h e i n t e g r a t i o n is o v e r t h a t p a r t

o f t h e w i n g f o r w h i c h I x - - x ~ l ~ < B [ r , ] . T h i s s h o w s t h a t A ( x ) v a n i s h e s o u t s i d e t h e M a c h

d i a m o n d w h i c h e nc lo s es t h e w i n g . F u r t h e r m o r e , W a r d s s h ow s t h a t e q u a t i o n (7) c a n b e u s e d f or

w i n g s c a r r y i n g s l e n d e r b o d i e s (e .g. , t i p t a n k s ) .

U s i n g e q u a t i o n (7) a n d a s s u m i n g t h a t ~ A / ~ x i s c o n t i n u o u s f o r a l l v a l u e s o f x , e q u a t i o n ( 6 ) c a n

b e w r i t t e n a s

1 I x 1 - x 2 1

q

1 f f A ( x l) A ( x2 ) l o g

( [& - - x 2 l )

d & d &

N

[ S t ( X l ) - { - A t t ( X l ) ] ~ . ~ t t ( X 2 ) @ A / t ( ~ 2 ) ] l o g ( [ . 5 ~ - - X 2 ] ) d x l d X 2 ( S )

~

w h e r e A ( x ) d e n o t e s ~2A/~x2. T h i s r e s u l t e n a b l e s t h e i n t e r f e r e n c e w a v e d r a g t o b e e x p r e s s e d i n

t h e s im p l e r f o rm

D w .

- - I t X - -

o )

q

T h u s D wB d e p e n d s u p o n M a c h n u m b e r s i n c e A (x) d o e s.

9


11/30

T h e t r a n s f e r r u le is t h e n a m e s u g g e s te d b y W a r f o r t h e w a v e d r a g r e s u l t g i v e n b y e q u a t i o n (8 ).

T h e r e f o r e

A ( x )

w i l l b e r e f e r r e d t o a s t h e t r a n s f e r r e d a r e a d i s t r i b u t i o n o f t h e w i n g . I t f o l lo w s

f r o m e q u a t i o n (7) t h a t t h e t r a n s f e r re d a r e a d i s t r i b u t io n c a n b e d e t e r m i n e d b y s p r e a d i n g e a c h

e l e m e n t o f v o l u m e o f t h e w i n g o v e r t h e x a x i s a c c o r d i n g t o t h e l a w dV~ / ~ / { B~ r ~ ~ - - ( x - - xd~}.

H e n c e a v o l u m e e l e m e n t a t t h e p o i n t (xx, r~) c o n t r i b u t e s t o t h e t r a n s f e r r e d a r e a d i s t r i b u t i o n o n l y

w i t h i n t h e r a n g e x ~ - - B I r l I < x < x ~ + B I r~ l- I n a d d i t i o n , t h e v o l u m e o f a b o d y w i t h t h e

c r o s s - se c t i o n a l a r e a d i s t r i b u t i o n A (x ) i s e q u a l t o t h e v o l u m e o f t h e w i n g .

T h e w o r k o f J o n e s 5 s u g g e st s a n a l t e r n a t i v e m e t h o d f o r c a l c u l a ti n g

A( x ) .

A t t e n t i o n i s c o n -

c e n t r a t e d u p o n a p o i n t o n th e x - a x i s r a t h e r t h a n a p o if it o n t h e w i n g . T h e n i t is i m p l i c i t i n th e

w o r k o f J o n e s 5 t h a t

A( x )

m u s t b e e q u a l t o th e m e a n w i n g e le m e n t a l a r e a d i s t r ib u t i o n . T h i s

r e s u l t c a n b e v e r i f ie d f o r a w i n g l y i n g t h e t i l e (x , y ) p l a n e b u p u t t i n g dV1 = T(R I) dx~dy~ a n d

c h a n g i n g t h e v a r i a b l e o f i n t e g r a t i o n x~ i n e q u a t i o n (7) f r o m x~ t o 0 = c o s -~ { ( x ~ - x) /By l} ,

0 ~< 0 ~< = . T h u s t h e d o u b l e i n t e g r a t i o n i n e q u a t i o n (7 ) i s r e p l a c e d b y a n i n t e g r a t i o n w i t h r e s p e c t

t o y , f o l l o w e d b y a n i n t e g r a t i o n w i t h r e s p e c t t o 0 a n d , u s i n g S e c t i o n 2 .2 .1 , i t c a n b e s h o w n t h a t

1 ST X , 0 M ) dO

x ) = , ,

w h e r e

Sw (x, O, M )

d e f i n e s a w i n g e l e m e n t a l a r e a d i s t r i b u t i o n .

N o w , r e t u r n i n g t o a c o n s id e r a t io n o f t h e i n t e r f e re n c e w a v e d r a g Dw~ , t h e w a v e d r ag , g i v e n b y

e q u a t i o n (8 ), is w r i t t e n i n t h e f o r m

D = D w + D { S + A } - D { A } , . . . . . . . . . .

1 0 )

w h e r e D{ S + A} , D ( A } d e n o t e t h e w a v e d r a g , g i v e n b y e q u a t i o n (1 ), a s so c i a t e d w i t h t h e a r e a

d i s t r i b u t i o n s S(x) + A(x) , A(x) , a n d D w i s t h e w a v e d r a g o f t h e e x p o s e d w i ng . E q u a t i o n (1 0)

s h o w s t h a t t h e i n t e r f e r e n c e w a v e d r a g c a n b e e x p r e s s e d a s Dw B = D { S + A } -- D { S } - ,D{A}.

F o r a s l e n d e r w i n g ,

D w = D { A }

t o t h e o r d e r of a p p r o x i m a t i o n c o n s i d e re d a n d s o D =

D{S -1- A},

v c h ic h i s t h e s t a n d a r d r e s u l t o f s l e n d e r - b o d y t h e o r y .

H o w e v e r , e q u a t i o n ( 1 0 ) i s n o t i n a f o r m s u i t a b l e f o r c o m p u t a t i o n s i n c e

D w

i s t h e w a v e d r a g

o f t h e e x p o s e d w i n g , n o t t h e w a v e d r a g o f s o m e s u i t a b l y c h o s e n g r os s w i n g . D e n o t e a s u i t a b l y

c h o s e n g r os s w i n g b y W 0 a n d t h e p o r t i o n b l a n k e t e d b y t h e b o d y b y A W = W 0 - - W . F u r t h e r -

m o r e , d e n o t e t h e b o d y s o u rc e r e p r e s e n t a t i o n b y Q ~, t h e w i n g so u r ce r e p r e s e n t a t i o n b y

Qw

a n d

s o o n . T h e n , t h e f u n d a m e n t a l a s s u m p t i o n t h a t t h e i n te r f e re n c e v e l o c i t y p o t e n t i a l c a n b e n e g l e c t e d

s h o w s t h a t t h e s o u r c e d i s t r i b u t i o n r e p r e s e n t i n g t h e w i n g - b o d y c o m b i n a t i o n i s Q =

Qw + Q~.

T h e r e f o r e Q = Qwo + QB -- Q~w, w h e r e Q~w i s t h e s o u r c e d i s t r i b u t i o n r e p r e s e n t i n g t h e p o r t i o n o f

t h e g r o ss w i n g b l a n k e t e d b y t h e b o d y . T h u s Q~w d o e s n o t n e c e s s a r i l y r e p r e s e n t a s le n d e r b o d y

a l t h o u g h A W is a l o w - a s p e c t - r a t i o w i n g . B u t , QAw c a n b e i n c o r p o r a t e d w i t h t h e s l e n d e r - b o d y

s o u r c e d i s t r i b u t i o n Q B, i .e . , f o r t h e p u r p o s e s o f w a v e d r a g e v a l u a t i o n Q ~w m a y b e r e p l a c e d b y t h e

e q u i v a l e n t s l e n d e r - b o d y s o u r c e d i s t r i b u t i o n

Q~A,

w h e r e

A A

i s t h e ( t ra n s f e r re d ) a r e a d i s t r i b u t i o n

o f t h e b l a n k e t e d w i n g A W . T h e w a v e d r a g o f t h e c o n f i g u r a ti o n c a n n o w b e s i m p l if ie d a n d

e c ~ u a ti o n ( 1 0 ) b e c o m e s

D = Dwo + D { S + A } - - D {A + A A } , . . . . . . . . (11)

i t b e i n g a s s u m e d t h a t

A A ( x )

i s c o n t i n u o u s e v e r y w h e r e . T h i s r e s u l t i s a d v a n t a g e o u s w h e n

Dwo i s k n o w n f r o m e x i s t in g r e s u l ts i n s u p e r s o n i c w i n g t h e o r y .

A n a l t e r n a t i v e m e t h o d o f c h a n g i n g e q u a t i o n ( 1 0) i n t o a fo r m s u i t a b l e fo r c o m p u t a t i o n i s t o

r e p l a c e

D w

b y

DWN,

t h e w a v e d r a g o f t h e n e t w i n g . T o c o n s t r u c t t h e n e t w i n g , t h e s e g m e n t s o f

t h e e x p o s e d w i n g a r e p l a c e d t o g e t h e r t o f o rm a c o n t i n u o u s w i n g . T h i s p r o c e d u r e i s n o t u n -

r e a s o n a b l e b e c a u s e Dw = DWN a t M -= 1 , oo. T h e f a c t t h a t Dw ~ DWN a t o t h e r M a c h n u m b e r s

s e r i o u sl y l i m i t s t h e a c c u r a c y o f a n y r e s u l ts o b t a i n e d b y s e t t i n g

Dw = Dw~v

i n e q u a t i o n ( 1 0 ) .

10


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I t h a s b e e n p o i n t e d o u t t h a t t h e m o m e n t - o f - a r e a r u le i s a p p l ic a b l e a t l o w su p e r s o n ic sp e e d s a n d ,

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

t h e r e s u l t s g i v e n i n t h e p r e v i o u s d i s c u s s i o n i t w i ll b e s e e n t h a t a s i m p l e r e x p r e s s i o n f o r t h e

t r a n s f e r r e d a r e a d i s t r i b u t i o n o f t h e w i n g s y s t e m is r e q u i r e d . T h i s is r e a d i l y f o u n d a t M = 1 s i n c e

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

A t l o w s u p e r s o n i c s p e e d s t h e t r a n s f e r r e d a r e a d i s t r i b u t i o n A (x ) c a n b e r e p l a c e d b y a d i s t r i b u t i o n

d e r i v e d a s f o l l o w s .

C o n s i d e r a n a r r o w s t r e a m w i s e s t r i p of t h e w i n g a t a d i s t a n c e R f r o m t h e a x i s a n d w i t h t h e

c r o s s - se c t i o n a l a r e a d i s t r i b u t i o n ~ ( x ) . I t h a s b e e n s h o w n t h a t A ( x) is e q u a l t o t h e m e a n w i n g

e l e m e n t a l a r e a d i s t r i b u t i o n a n d s o, d e n o t i n g t h e c o n t r i b u t i o n o f t h e s t r e a m w i s e s t r i p t o A (x ) b y

bA (x) , i t f o l lows t h a t

d A (x ) = ~ ~ ( x - - B R cos 0) dO

o r

2 ~ a ~

a A ( x ) = ~ J o , { ~ ( x - - B R cos O) + ~ ( x + B R cos 0)} dO .

T h u s , w h e n B R = 0 , dA (x) ----- V(x) an d A (x) i s equ a l t o t he c ros s - sec t io na l a r e a d i s t r ib u t io n o f

t h e w i n g . - W h e n

B R

i s s m a l l t h e i n t e g r a n d i s a l m o s t c o n s t a n t a n d m a y b e r e p la c e d b y

{~(x - - B R cos ~/4) + N ( x + B R c o s ~ /4 ) }, s in c e 0 = ~ / 4 i s a m e a n v a l u e o f 0 i n t h e r a n g e o f

i n t e g r a t i o n , i . e . ,

B R

a n d , i f 7x ( x ) i s c o n t in u o u s e v e r y w h e r e ,

aA x) _ +

( t h i s l a s t r e s u l t m a y b e d e r i v e d b y e x p a n d i n g A (x ) i n a s c e n d i n g p o w e r s o f B~R~).

T h e r e f o r e , ~ it l o w s u p e r s o n i c s p e e d s w h e r e

B R

i s s m a l l , t h e s i m p l i f i e d p r o c e d u r e f o r c a l c u l a t i n g

A (x ) i s t o t r a n s f e r a n e l e m e n t o f w i n g c r o s s -s e c t io n a l a r e a d ~, s i t u a t e d a t t h e p o i n t ( x l , r l ) b y

p l a c i n g e l e m e n t s

d~ 112

a t t h e p o i n t s I x - - x l[ = B i t 1 ] X /2 . T h i s p r o c e d u r e c a n b e a p p l i e d e a s i l y

a n d q u i c k l y .

2.4.2. D e s i g n f o r l ow w a v e d r a g . - - T h e p r o b l e m o f d e t e r m i n i n g t h e o p t i m u m f u s e l a g e w h e n t h e

w i n g i s g i v e n h a s b e e n e x a m i n e d b y W a r d 9,10. F r o m e q u a t i o n (1 0), i t fo l lo w s t h a t t h e w a v e d r a g ,

a t a g i v e n M a c h n u m b e r , w i ll b e a m i n i m u m w h e n t h e w a v e d r a g, D { S - / A } , o f t h e b o d y w i t h

t h e c r o s s - s e c t i o n a l a r e a d i s t r i b u t i o n S ( x ) - k A ( x ) is a m i n i m u m . T h u s t h e p r o b l e m h a s b e e n

r e d u c e d t o a n e q u i v a l e n t s l e n d e r - b o d y p r o b l e m w h i c h c a n b e t r e a t e d u s i n g t h e m e t h o d s g i v e n b y

L o r d a n d E m i n t o n . F o r a g i v en w i n g a n d M a c h n u m b e r , t h e o p t i m u m b o d y c a n b e f o u n d b u t

i t w i l l b e n o t i c e d t h a t t h e n e w e x p o s e d w i n g u s u a l l y d i ff e rs s l ig h t l y f r om t h e g i v e n w i n g .

F o l l o w i n g W a r d 1 t h i s d if fe r en c e is t a k e n i n t o a c c o u n t b y c o m b i n i n g i t w i t h t h e b o d y . H o w e v e r ,

s i n c e t h e e f fe c t o n t h e b o d y s h a p e i s s m a l l , t h i s a d d i t i o n a l r e f i n e m e n t c a n b e n e g l e c t e d i f i t i s

d e s i r e d t o s i m p l i f y t h e a n a l y s i s .

T h e a p p l i c a t i o n o f t h i s p ro c e d u r e a t l o w s u p e r so n i c s p e e d s c a n b e c o m p a r e d w i t h t h e m o m e n t -

o f - a r e a r u l e m e t h o d , w h i c h is d e s c r i b e d i n S e c t i o n 2. 3. 2. T h e a r e a - r u l e a n d t r a n s f e r - r u l e m e t h o d s

c o n s i d e r c o n f i g u r a t i o n s w i t h f i x e d w i n g s a n d d e r i v e a b o d y s h a p e w h i c h , i t w i l l b e n o t e d , v a r i e s

w i t h M a t h n u m b e r . O n t h e o t h e r h a n d , t h e m o m e n t - o f - a r e a ru l e c o n si d e rs v a r i a t i o n s i n w i n g

g e o m e t r y , s u c h a s t h e a d d i t i o n o f b o d i e s n e a r t h e w i n g t ip s , a n d d e r iv e s o n e b o d y s h a p e f o r t h e

e n t i r e l o w s u p e r s o n i c M a c h - n u m b e r r a n g e .

A m o r e g e n e r a l m i n i m u m d r a g p r o b l e m a r is e s if , f o r e x a m p l e , o n l y t h e w i n g p l a n a r e a , s p a n

a n d v o l u m e a r e fi x e d a s w e l l a s t h e b o d y l e n g t h a n d v o l u m e . S u c h p r o b l e m s h a v e n o t b e e n

e x a m i n e d i n th e l i te r a t u re a l t h o u g h W a r d 9 h a s m a d e s om e r e l e v a n t c o m m e n t s , w h e n t h e M a c h

11


13/30

d i a m o n d e n c lo s in g t h e w i n g li es w i t h i n t h e b o d y l e n g t h . H e sh o w s t h a t t h e w a v e d r a g O f a

w i n g - b o d y c o m b i n a t i o n s u b j e c t t o t h e s e r e s t ri c t io n s c a n n o t b e re d u c e d t o t h a t o f t h e S e a r s - H a a c k

b o d y w i t h th e s a m e t o t a l v o l u m e a n d l e n g th . T h i s r e s u l t d o es n o t n e c e s s a r il y i m p l y t h a t t h e

w a v e d r a g c a n n o t b e r e d u c e d t o a l ow v a lu e .

I t i s su g g e s t e d b y W a r d 9 t h a t t h e f u se l a ge s h o u l d a l w a y s b e d e s i g n e d s u f f ic i e n tl y l o n g f o r t h e

w i n g s y s t e m t o b e i n c l u d e d w i t h i n i t s M a c h d i a m o n d , i . e . ,

S x)

i s c h o s e n s o t h a t i t i s n o n - z e r o

w h e r e v e r A (x ) is n o n - z e r o .

A n i m p o r t a n t c la s s o f o p t i m i s a t i o n p r o b l e m s h a s b e e n e x a m i n e d b y L o m a x a n d H e a s l e t 13, w h o

h a v e i n v e s t i g a t e d t h e o p t i m u m f u se l ag e d e s ig n fo r a r a n g e o f M a c h n u m b e r s , t h e w i n g b e i n g

f i x e d ( i t h a s b e e n s h o w n e a r l i e r i n t h i s S e c t i o n t h a t D{S + A} i s t h e n t h e o n l y t e r m d e p e n d i n g

o n t h e f u s e la g e ) . L o m a x a n d H e a s l e P 3 c o n s i d e r in e f fe c t t h e m i n i m i s a t i o n o f t h e i n t e g r a l

{ s + A } d i n ,

w h e r e M ~ ,


14/30

3 . C o n fi g u ra t io n s R e p r e s e n t e d b y D i s c o n t i n u o u s S o u r c e D i s t r i b u t i o n s . - - 3 . 1 . I s o la t e d l / V i n g s . ~

W a r d ~ h a s s h o w n t h a t t h e w a v e d r a g a s s o c i a t e d w i t h a d i s c o n t i n u o u s s u r f a c e s o u r c e d i s t r i b u t i o n i s

- - P c o s h - I d Q R I ) a s 2 d Q R ) . . . . 1 2 )

4 z ~ ~ I r , - r . I

W h e r e s0 i s t h e f r e e - s t r e a m d e n s i t y

p oQ (R ) i s t h e s o u r c e d e n s i t y a t t h e p o i n t R = ( x, y , z )

r i s t h e p o s i t i o n v e c t o r i n a p l a n e x = c o n s t a n t , i . e ., r = ( y , z)

d S d x d e n o t e s a g e n e r a l v o l u m e e l e m e n t ,

t h e c o - o r d i n a t e s

( x , y , z )

a r e c h o s e n a s i n F ig . 1 a n d t h e i n t e g r a t i o n i s o v e r r e a l v a l u e s o f t h e

i n t e g r a n d . F u r t h e r m o r e , s i n c e Q (R ) i s a d i s c o n t i n u o u s f u nc ti op , o f x t h e a b o v e i n t e g r a l is a

d o u b l e S t i e l t j e s i n t e g r a l w h i c h m u s t b e w r i t t e n i n th i s f or m . I t i s p o s s ib l e to r e p la c e d Q (R ) b y

Q' (R)

d x

o n l y w h e n Q (R ) is a c o n t i n u o u s f u n c t i o n o f x .

I t f ol lo w s d i r e c tl y f ro m e q u a t i o n ( 12 ) t h a t t h e w a v e d r a g o f a w i n g is g iv e n b y

0 U 2 / I x l - x 2 1

D - -

4 3 1 J ~ c s h - ~ \ B T r ( - - r o l ] d T ' ( R 1 ),

(13)

w h e r e d x d r d e n o t e s a n e l e m e n t o f a r e a o f t h e w i ng , T (R ) d e n o t e s t h e w i n g t h i c k n e s s a t t h e

p o i n t R , a n d t h e i n t e g r a t i o n i s o v e r t h o s e p a r t s o f t h e w i n g f o r w h i c h t h e i n t e g r a n d is r e a l. T h i s

r e s u l t h a s b e e n g i v e n b y W a r d W h e n t h e w i n g ' h a s r o u n d e d s u b s o n i c l e a d i n g e d g e s i t is m o d i f ie d

b y t h e a d d i t i o n o f a l e a d i n g - e d g e ~d r a g f o rc e . S u c h f o rc e s a r e e x a m i n e d i n S e c t io n 5 .

I t i s Of i n t e r e s t t o n o t e t h a t W a r d 1 h a s s u g g e s t e d t h a t e q u a t i o n s ( 1 2 ) a n d ( 1 3 ) s h o u l d b e

m o d i f i e d w h e n t h e t o t a l s o u r c e s t r e n g t h i s n o t z e ro , i .e . , t h e b o d y i s n o t c lo s ed . S u c h a m o d i f i c a -

t i o n is n e e d e d t o a ll o w f o r t h e b a s e d r a g t e r m . E q u a t i o n ( 1 3 ) g i v e s t h e e x t e r n a l w a v e d r a g o f

a w i n g , e x c e p t fo r t h e b a s e d r a g , w h e t h e r t h e t o t a l s o u rc e s t r e n g t h i s z e r o o r n o t . T h i s r e s u l t

c a n b e p r o v e d m o s t e a s i ly i f t h e s o u r ce s a r e i n c l u d e d in t h e e q u a t i o n o f c o n t i n u i t y a n d t h e

a e r o d y n a m i c f o r c e s a re c a l c u l a t e d b y t h e m e t h o d o f W a r d 21.

A p a r t i c u l a r f o r m o f t h e r e s u l t , e q u a t i o n (1 3), h a s b e e n g i v e n b y L i g h t h i l P 2 f o r a p l a n e w i n g

w i t h p o l y g o n a l s t r e a m w i s e s e c t io n s . T h e w a v e d r a g o f a w i n g l y i n g i n th e p l a n e z = 0 ( see Fig. 1)

i s f o u n d t o b e

D _

w h e r e t h e d i s c o n t i n u i t i e s i n T ' o c c u r a l o n g t h e l i n e s x = c , (y ) a n d a r e o f m a g n i t u d e 2 / U a , , ( y )

( n = 1 , 2 , 3 , . . . ) .

W a r d 8 h a s o b s e r v e d t h a t e q u a t i o n ( 13 ) i s e x a c t l y e q u i v a l e n t t o t h e w a v e - d r a g r e s u l t o b t a i n e d

. b y i n t e g r a t i n g t h e p r e s su r e d i s t r i b u t i o n o v e r t h e w i n g p l a n - fo r m . F u r t h e r m o r e , i n v i e w o f t h e

u s e f u l n e s s of t h e s o n ic , a r e a r u l e , t h e r e s u l t a s M - - > 1 i n e q u a t i o n ( 1 3 ) i s o f i n t e r e s t . I n m a n y

c a s e s t h e p r e s e n c e o f d i s c o n t i n u i t i e s i n T ' i m p l ie s t h a t t h e l i n e a r i se d t h e o r y y i e l d s i n f in i t e w a v e

d r a g a t M = 1 b u t t h e r e a r e c a s e s f o r w h i c h it y i e l d s f i n i te w a v e d r a g a t M = 1. T h i s p o i n t

h a s b e e n d is c u ss e d b y H e a s l e t , L o m a x a n d S p r e i t e r 16 a n d b y L o r d , R o s s a n d E m i n t o n I t

w o u l d a p p e a r t h a t t h e w a v e d r a g is f i n i te a t M = 1 w h e n t h e r a t e o f c h a n g e of t h e c r o s s - s e c t io n a l

a r e a d i s t r i b u t i o n is c o n t i n u o u s e v e r y w h e r e .

13

71666) B


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3.2.

I s o l a t e d B o d i e s . - - T h e

w a v e d r a g of a s le n d e r b o d y w h i c h c a n b e r e p r e s e n t e d b y a d is c o n-

t i n u o u s s u r f a c e so u r c e d i s tr i b u t i o n h a s b e e n g i v e n b y F r a e n k e l a n d P o r t n o y ~ . T h e s u r fa c e

s l o p e ~ i s d i s c o n t i n u o u s o n a f i n i t e n u m b e r o f c o n t o u r s C ~ ( i = 1 , 2 , . . . , ~ ) l y i n g i n p l a n e s x = &

a n d s e p a r a t e d b y d i s ta n c e s o f m a g n i t u d e 0 ( 1 ), t h e b o d y l e n g t h b e i n g 0 (1 ) . T h e n , w i t h i n t h e

u s u a l o r d e r o f a p p r o x i m a t i o n , t h e w a v e d r a g o f th i s b o d y i s g i v e n b y

D 1 1

1 A S , ) ~ l o g 2

w h e r e ~ ,

A , b , , ) ,

A ~

A S I

-]- ii= l

~ C i

[(~,,)~=~,_ A w - - ( A4 ~ ,) ~ n ,= ~ + ] d r , . . . . . (14)

i s ~ t h a t p o r t io n o f t h e s l e n d e r b o d y p e r t u r b a t i o n v e l o c i t y p o t e n t i a l w h i c h c a n

b e d e t e r m i n e d f r o m t h e t w o - d i m e n s i o n a l i n c o m p r e s s i b l e c r o s s - f l o w

i s t h e i n c r e a s e i n ~ a t t h e p o i n t o f d i s c o n t i n u i t y x = &

d e n o t e s t h e d i s c o n t i n u i t y in t h e s u r fa c e s l o p e a t t h e p o i n t x = x~

d e n o t e s t h e d i s c o n t i n u i t y i n S ( x ) a t t h e p o i n t x = &

d e n o t e s t h e f i n i te p a r t o f a d o u b l e S t i e l t j e s i n t e g r a l t a k e n o v e r a l l v a l u e s o f x l

a n d x2 f r o m j u s t u p s t r e a m o f t h e b o d y n o s e to j u s t d o w n s t r e a m o f t h e b o d y

b a s e

a n d t i le i n t e g r a t i o n i n t h e t h i r d t e r m is , o v e r t h e c o n t o u r C~.

F r a e n k e l a n d P o r t n o y 2~ ~ a v e s h o w n t h a t t h e w a v e d r ag , g i v e n b y e q u a t i o n (1 4), r e m a i n s

u n c h a n g e d i f t h e s t r e a m d i r e c ti o n i s r e v er se d . F u r t h e r m o r e , t h e v a r i a t i o n o f t h e w a v e d r a g

w i t h M u c h n u m b e r d e p e n d s s o le l y o n t h e d i s c o n t i n u i t ie s i n S ( x ) a n d d o e s n o t d e p e n d u p o n t h e

c r o s s - se c t i o n a l s h a p e . T h e c a s e w h e n t h e b o d y p r o f il e i s d i s c o n t i n u o u s h a s b e e n e x a m i n e d b y

F r a e n k e l a n d P o r t n o y ~-~ b u t n o a p p l i c a t i o n s o f t h e r e s u l t t o e n g i n e i n t a k e s h a v e b e e n g i v e n .

F i n a l l y , i t s h o u l d b e m e n t i o n e d t h a t t h e r e s u l t o b t a i n e d b y W a r d ~ d i ff e rs f r o m e q u a t i o n ( 14 )

a n d i s n o t c o r r e c t fo r a s l e n d e r b o d y o f a r b i t r a r y c r o s s -s e c t io n a l sh a p e .

T h e e f f e c t o f c r o s s - s e c t i o n a l s h a p e o n t h e w a v e d r a g o f a s l e n d e r b o d y c a n b e i n v e s t i g a t e d u s i n g

e q u a t i o n (1 4). T h e p r e s e n c e o f t h e t h i r d t e r m i n e q u a t i o n ( 1 4 ) s h o w s t h a t t h e c r o ss - s e c ti o n a l

s h a p e a t a n d w i t h i n t h e n e i g h b o u r h o o d o f a d i s c o n t i n u i t y in su r f ac e s l o pe i s i m p o r t a n t a n d

t h a t t h e r e f o r e t h e w a v e d r a g o f s l e n d e r b o d i e s d o e s n o t d e p e n d s o l e l y o n t h e c r o s s -s e c t i o n a l

a r e a d i s t r i b u t i o n . I t f o ll o w s t h a t t h e w a v e d r a g is i n d e p e n d e n t o f t h e c r o s s- s e c t io n a l s h a p e

a w a y f r o m t h e d i sc o n t i n u it i es . W a r d 1 h a s p o i n t e d o u t t h a t t h e w a v e d r a g i s a m a x i m u m f o r

t h e b o d y o f r e v o l u t i o n . B o d i e s of e l l ip t i c c r o s s- s e c ti o n h a v e b e e n e x a m i n e d b y F r a e n k e P 5 w h o

h a s s h o w n t h a t t h e w a v e d r a g o f t h e e q u i v a l e n t b o d y o f r e v o l u t i o n is g r e a t e r b y a n a m o u n t

g ~ i : l

w h e r e e i s t h e e c c e n t r i c i t y o f t h e e l l i p t i c c r o s s- s e c ti o n . I t w a s p o i n t e d o u t b y F r a e n k e P 5 t h a t i f

t h e c r o s s - s e c t i o n i s ' n e a r l y ' c i rc u l a r , i n t h e s e n s e t h a t e --- 0 ( # /~ ) , w h e r e t i s t h e t h i c k n e s s r a t i o

o f t h e b o d y , t h e n t h e w a v e d r a g , w h i c h is 0 ( # l o g t ), w i l l n o t c h a n g e t o t h i s o r d e r. H e n c e i t m a y

b e a s s u m e d t h a t m o s t s l e n d e r b o d i es o f p r a c t i c a l i m p o r t a n c e h a v e a w a v e d r a g i n d e p e n d e n t o f

c r o s s - s e c t i o n a l s h a p e .

14


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T h e f o r m o f e q u a t i o n ( 1 4) f o r a b o d y o f r e v o l u t i o n w a s g i v e n f i rs t b y L i g h t h i l P 6 a n d , f o r a b o d y

w i t h r a d i u s R ~ a t t h e p o i n t x~, i s f o u n d t o b e

- I - R ? ( d i ) l o g 2 . . . ( 1 5 )

~=1 B R ~ ' . . . . . .

w h e r e t h e i n t e g r a t i o n s a r e R i e m a n n i n t e g r a t i d n s i g n o r i n g t h e p o i n t s a t w h i c h S ( x ) i s u n d e f i n e d

a n d t h e f i r st t h r e e t e r m s a r e t h e e x p a n d e d f o r m o f t h e f i n i te p a r t o f t h e d o u b l e S t i e l t j e s i n t e g r a l

g i v e n in e q u a t i o n (1 4). T h e la s t t e r m i n e q u a t i o n ( 1 5) g i v e s t h e r e s u l t t h a t t h e w a v e d r a g o f a

s l e nd e r b o d y w i t h d i s c o n t in u i t i e s i n S ' ( x ) i s i n f i n i t e a t M = 1 a c c o r d i n g t o t h i s t h e o r y . A l s o , a s

a l r e a d y n o t e d , i t fo l lo w s fr o m e q u a t i o n ( 1 4 ) t h a t e q u a t i o n ( 1 5 ) g i v e s t h e w a v e d r a g o f a s l e n d e r

b o d y w h i c h h a s c i r c u l a r se c t io n s a t a n d w i t h i n t h e n e i g h b o u r h o o d o f t h e d i s c o n t i n u i t y c o n t o u r s C ,:.

3 .3 .

W i n g i B o d y C o m b i n a t i o n s . - - A n

e x t e n s i o n o f t h e a r e a r u l e a n d t h e t r a n s f e r r u l e t o c o n -

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

s e c t io n a l S h ap e o f t h e b o d y i s a r b i t r a i y .

C o n s i d e r t h e f l o w f ie l d o n a c i r c u l a r c y l i n d r i c a l c o n t r o l su r f a c e a t a l a rg e d i s t a n c e f r o m t h e

c o m b i n a t i o n . T h e n t h e w a v e d r a g is t o b e f o u n d b y c a l c u l a t in g t h e r a te o f f lu x of m o m e n t u m

a c r o ss th i s s u r f ac e . H e a s l e t , L o m a x a n d S p r e i t e r 16, f o r e x a m p l e , h a v e s h o w n t h a t t h e p e r t u r b a t i o n

v e l o c i t y p o t e n t i a l d u e t o t h e w i n g a l on e , a t a g i v e n a n g u l a r p o s i t i o n 0 o n t h e c o n t r o l s u rf a c e , is

t h e s a m e a s t h a t a s s o c ia t e d w i t h t h e a p p r o p r i a t e w i n g . e l e m e n t a l a re a d i s t ri b u t i o n . L e t t h e

e f fe c t o f a n y i n t e r f e r e n c e v e l o c i t y p o t e n t i a l o n t h e w a v e d r a g b e n e g l e ct e d ~ . T h e f l o w f i el d o n

t h e c o n t r o l s u rf a c e is t h e n t a k e n a s t h e s u m o f t h e p e r t u r b a t i o n v e l o c i t y p o t e n t i a l s d u e t o t h e

b o d y a l o n e a n d t o a n a p p r o p r i a t e w i n g e l e m e n t a l a r e a d i s t ri b u t i o n s o t h a t , a t a g i v e n a n g u l a r

p o s i t io n o n t h e c o n t r o l su r fa c e , t h e s o u rc e d i s t r i b u t i o n s r e p r e s e n t i n g t h e b o d y a n d t h e a p p r o p r i a t e

e l e m e n t a l a re a d i s tr i b u t i o n a r e s u p e r im p o s e d . T h e w a v e d r a g m a y b e w r i t te n a s

f l D { s s w x , o , M }

= - ~ . dO , . . . . . . . . . .

(16)

w h e r e

D { S + S w ( x , O ,

M ) } is i n t e r p r e t e d b y e x a m i n i n g t h e f lo w fi e ld n e a r t h e a x i s, i .e .,

D { S + S w ( ) , O ,

M ) } d e n o t e s t h e w a v e d r a g o f a c o m b i n e d s l e n d e r b o d y . T h i s c o m b i n e d

b o d y i s a s t r e a m s u r f a c e o f t h e d i s c o n t i l m o u s s u r f a c e s o u r c e d i s t r i b u t i o n Q~, r e p r e s e n t i n g t h e

i s o l a te d b o d y , a n d t h e c o n t i n u o u s s o u r c e d i st r i b u t i o n d e f in e d b y S w ' ( x , O , . M ) . S i n c e a l m o s t a l l

t h e

S w ' ( X , O , M )

a r e c o n t i n u o u s e v e r y w h e r e t h o s e d i s c o n r i n u o u s s o u r c e d i s t r i b u t i o n s a s s o c i a t e d

w i t h t h e d i s c o n t i n u o u s S w ' ( x , O , M ) a r e o m i t t e d f r o m t h e i n t e g r a l i n e q u a t i o n (1 6).

U s i n g e q u a t i o n ( 12 ) t h e w a v e d ra g of t h e c o m b i n e d b o d y c a n b e e x p r e s se d a s

D{ S +

S. , x , O, M ) } =

D { S } +

D } Sw x , O , M ) }

poU ( B IC I ' ) aS lag~(R.) Sw (~~,

o ,

M) d,x~.,

1 f c o s h - 1 ( 1 1

w h e r e D { S } , t h e i s o la t e d b o d y w a v e d r a g , is g i v e n b y e q u a t i o n ( 14 ) a n d D { S w ( x , O , M ) } , t h e W a v e

d r a g a s s o c i a t e d w i t h

S w ( x , O , M ) ,

i s g i v e n b y e q u a t i o n (1 ), w i t h i n t h e u s u a l o r d e r o f a p p r o x i m a t i o n .

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

b o d y a n d / o r t h e w i n g c ro s s - se c t io n a l a r e a d i s t ri b u t i o n ) i s d i s c o n t i n u o u s w i t h i n t h e r e g i o n o f t h e w i n g b e c a u s e

e q u a t i o n 1 4 ), d e r iv e d b y F r a e n k e l a n d P o r t n o y 24, c a n n o t b e o b t a i n e d b y n e g l e c t i n g t h e i n t e r fe r e n c e v e l o c i t y p o t e n t i a l .

T h e a s s u m ? t i o n i s v a l i d f o r a n y o t h e r t y p e o f s le n d e r w i n g - b o d y c o m b i n a t i o n .

15

7 1 6 6 6 ) B 2


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F o l l o w i n g W a r d ' s a n a l y s i s1 f o r s l e n d e r b o d i e s o f r e v o l u t i o n t h e i n v e r s e c o s h i n t h i s r e s u l t i s

e x p a n d e d i n t e r m s o f l o g a r i th m s . T h e u n k n o w n b o d y s o u rc e d e n s i t y Q B(R ) i s e l i m i n a t e d

b y n o t i n g t h a t t h e t o t a l s o u r c e s t r e n g t h i n a p l a n e x = c o n s t a n t is e q u a l t o U S ' ( x ) , w h e r e S ( x )

~s th e c ro s s - s e c t io n a l a r e a d i s t r i b u t io n o f t h e b o d y 1. S in c e

S~v'(x , O, M )

v a n i s h e s a t i t s e n d

p o i n t s , i t c a n b e s h o w n q u i t e s i m p l y t h a t t h e w a v e d r a g o f t h e ' c o m b i n e d ' b o d y b e c o m e s, w i t h i n

t h e u s u a l o r d e r o f a p p r o x i m a t i o n ,

D { S -~ - S w x , O , M } = D { S } 4 - D { S w x , O , M }

p o ~ _ f

2~ log I x , - - x 2 ] dS ' ( x l ) Sw (x 2 , O, M ) dx~ . . . . . (17)

E q u a t i o n s ( 1 6 ) a n d ( 1 7) p r o v i d e a g e n e r a l i s a t i o n o f t h e a r e a r u l e, e q u a t i o n s (2 ) a n d ( 3) , t o c o m -

b i n a t i o n s i n c o r p o r a t i n g b o d i e s w i t h d i s c o n t i n u i t i e s i n s u r fa c e s l op e .

T h e t r a n s f e r r e d w i n g a r e a d i s t r i b u t i o n A (x) is e q u a l t o t h e m e a n w i n g e l e m e n t a l a r e a d i s t r i b u -

t ion , i . e . ,

1 (==

A x ) = ~ J o Sw x , o , M ) d o .

T h u s , u s i n g e q u a t i o n (1 7), e q u a t i o n ( 1 6 ) b e c o m e s

= D { S } D w p o V f f l o g a S ( x , ) ( 1 8 )

2 ~ I ~

w h e r e

D { S }

is g i v e n b y e q u a t i o n ( 14 ) a n d

D w

b y e q u a t i o n ( 13 ), f o r e x a m p l e . T h i s r e s u l t i s t h e

e x t e n d e d f o r m o f t h e t r a n s f e r r u l e , e q u a t i o n s ( 8 ) a n d ( 1 0 ) .

E q u a t i o n ( 1 8 ) i m p l i e s t h a t t h e i n t e r f e r e n c e w a v e d r a g D w~ i s g i v en b y

w h i c h is t h e g e n e r a l i s a t i o n o f e q u a t i o n (9 ). I t s h o u l d b e n o t e d t h a t t h e i n t e r fe r e n c e w a v e d r a g

D w~ .is n d e p e n d e n t o f t h e c r o s s - se c t i o n a l s h a p e o f th e b o d y . T h e r e s u l t f o r t h e i n t e r f e r e n c e w a v e

d ra g c a n b e d e r iv e d d i r e c t ly f ro m e q u a t io n (1 2 ) w i th Q (R ) ---- Q B (R ) + Q w (R ) ; Q w(R) = U T ' ( R ) .

I t i s n e c e s s a r y t o u s e W a r d ' s i n t e g r a l e q u a t i o n ~ f o r t h e t r a n s f e r r e d a r e a d i s t r i b u t i o n A ( x ) .

I n S e c t io n 2.4 .1 i t w a s s h o w n t h a t t h e w i n g - b o d y c o m b i n a t i o n c a n b e r e p la c e d b y a n e q u i v a l e n t

c o m b i n a t i o n m a d e u p o f a s u i t a b l y c h o s e n g r os s w i ng W 0 a n d a m o d i f ie d b o d y w h i c h i s d e n o t e d

b y

B - - A A ,

w h e r e A A d e n o t e s th e ( t r a n s f e r r e d ) a r e a d i s t r i b u t i o n o f t h e p o r t i o n o f t h e g r o ss

w i n g b l a n k e t e d b y t h e b o d y . I t is a s s u m e d t h a t

S ' ( x )

i s c o n t i n u o u s w h e r e

A A ( x )

i s d e f i n e d a n d

t h a t A A ' ( x ) is c o n t i n u o u s e v e r y w h e r e a s o t h e r w i s e t h e s o u rc e d i s t r i b u t i o n r e p r e s e n t i n g t h e

m o d i f i e d b o d y i s n o t e q u a l ( in t h e w a v e - d r a g s en se ) t o t h a t r e p r e s e n t i n g t h e b o d y le ss b l a n k e t e d

wing, i .e . , Q(B-~A)-/-~QB-~w. T h e g e n e r a l i s e d f o r m o f e q u a t i o n ( 1 1 ) f o ll o w s f r o m e q u a t i o n ( 18 )

a n d c a n b e s h o w n t o b e

D : D wo + D { S + A } - - D { A + A A } , . . . . . . . . (19)

w h e r e D { S - / A } d e n o t e s t h e w a v e d r a g o f a ' c o m b i n e d ' b o d y , i . e . ,

D { S + A } = D { S } + D { A } P o g ~ ~ f

u J~ log Ix, -- x~[ dS'(x~) Sw (x~, O, M ) dx~ .

4 . N u m e r i c a l E v a l u a t io n o f W a v e D r a g . - - I n t h e p r e v i o u s s e c t io n s a t t e n t i o n h a s b e e n c o n c e n -

t r a t e d u p o n g i v i n g t h e t h e o r e t i c a l m e t h o d s r a t h e r t h e n t h e p r a c t i c a l a p p l i c a t i o n o f t h e s e m e t h o d s .

T h e p r e s e n t s e c t io n is i n t e n d e d t o g i v e a b r i e f d e s cr i p t io n o f t h e n u m e r i c a l m e t h o d s t h a t h a v e

b e e n d e v e l o p e d f o r e v a l u a t i n g w a v e d r ag . I t w i l l b e s e en t h a t t h e s e m e t h o d s h a v e b e e n d e v e l o p e d

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

d i ff ic u lt b e c a u s e t h e r e s u lt s o b t a i n e d f o r t h e w a v e d r a g o f w i n g - b o d y c o m b i n a t i o n s r e q u i r e o n l y

t h e e v a l u a t i o n o f w i n g a n d ' b o d y ' w a v e d r a g s.

16


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4 . 1 . I so la te d W i n g s . - - T h e r e

h a s b e e n a l a rg e n u m b e r o f i n v e s t i g a t i o n s i n t h e f i e l d o f s u p e r s o n i c

w i n g t h e o r y a n d n o a t t e m p t w i ll b e m a d e t o s u m m a r i s e t h e m h e re . H o w e v e r , i t i s s u ff ic i en t t o

n o t i ce t h a t t h e r e s u l t s fo r w a v e d r a g h a v e b e e n f o u n d b y i n t e g r a t i n g p r e s s u r e d i s t ri b u t i o n s .

A l so , t h e w i n g p la n - f o rm s a r e s t ra i g h t - e d g e d a n d t h e W i n g s e c ti o n s s i m p l e s h a p e s. A s u m m a r y

o f t h e r e s u l t s o b t a i n e d h a s b e e n g i v e n b y B i s h o p a n d C a n e ~7. S o m e r e s u l t s f o r s o n i c s p e e d h a v e

b e e n o b t a i n e d b y L o r d , R o s s a n d E m i n t o n . ~3 u s i n g t h e s o n i c a r e a r u l e .

F r o m t h e r e s u l t s g i v e n i n S e c t i o n 3.1 i t i s s u g g e s t e d t h a t e q u a t i o n ( 1 3 ) b e u s e d t o c a l c u l a t e

t h e w a v e d r a g p r o v i d e d t h a t t h e p r e s s u re d i s t r i b u t i o n is n o t r e q u ir e d . O t h e r w i s e it w o u l d

p r o b a b l y b e b e t t e r t o d e t e r m i n e t h e p r e s su r e d i s t r i b u t i o n fi rs t a n d t h e n t h e w a v e d r a g. I f t h e

p r e s s u r e d i s t r i b u t i o n i s r e q u i r e d i t c a n b e f o u n d u s i n g t h e m e t h o d o f T h o m s o n 28.

N e v e r t h e l e s s , i t i s a p p a r e n t t h a t t h e a r e a r u l e ( e q u a t i o n s (2) a n d ( 3 ) ) o f f e r s . a n a l t e r n a t i v e

m e t h o d f o r c a l c u l a t i n g t h e w a v e d r a g o f a n i s o l a t e d w i n g . H o w e v e r , a d i f f ic u l t y a r is e s i f a n y

s t r a i g h t l in e s , a lo n g w h i c h t h e w i n g p l a n a r s o u r c e d i s t r i b u t i o n i s d i s c o n t i n u o u s , a r e i n c l i n e d s o t h a t

t h e c o m p o n e n t o f f r e e - s t re a m v e l o c i t y n o r m a l t o t h e l i n e is s u p e r s o n i c ( f or e x a m p l e , a l o n g s h a r p

e d g e s t h e s o u r c e d i s t r i b u t i o n i s d i s c o n t i n u o u s ) . B u t , s i n ce a l m o s t a l l t h e w i n g e l e m e n t a l a r e a

d i s t r ib u t i o n s h a v e s o u rc e r e p r e s e n t a t io n s c o n t i n u o u s e v e r y w h e r e , t h e w a v e d r a g m a y b e c a l c u l a t e d

u s i n g t h e s e e l e m e n t a l a r e a d i s t r i b u t i o n s o n l y a n d i t m a y n o t b e n e c e s s a r y t o u s e t h e r e s u l t s o f

S e c t i o n 3 .2 fo r b o d i e s w i t h d i s c o n t i n u o u s s o u r c e r e p r e s e n t a t i o n s . L o r d ~9 h a s p o i n t e d o u t t h a t

a n u m e r i c a l a p p l i c a t i o n o f t h i s p r o c e d u r e t o w i n g s w i t h s u p e r s o n i c e d g e s i s n o t s a t i s f a c t o r y .

T h e a r e a -r u l e m e t h o d i s p a r t i c u l a r l y u s e f ul f or w i n g s w i t h c u r v e d p l a n - f o r m s a n d h a s b e e n

u se d b y L o r d a n d B e n n e t t . D e t a i l s m a y b e f o u n d in R e fs . 3 0 a n d 3 1.

4.2.

I so l a te d B o d i e s . -- - R e c e n t l y

a n u m b e r o f a u t h o r s h a v e i n v e s t i g a t e d n u m e r i c a l m e t h o d s f o r

e v a l u a t i n g t h e d o u b l e i n t e g r a l o f e q u a t i o n ( 1 ) ,

i .e . ,

m e t h o d s h a v e b e e n d e v e l o p e d f o r c a l c u l a t in g

t h e w a v e d r a g o f s l e n d e r b o d i e s w i t h o u t d i s c o n t i n u i t i e s i n s u r f a c e s lo p e . T h e m o r e g e n e r a l c as e

w h e n t h e s o u r c e r e p r e s e n t a t i o n i s n o t c o n t i n u o u s w i l l b e e x a m i n e d l a t e r i n t h e p r e s e n t s e c t io n .

T h e r e a r e t w o m a i n m e t h o d s f o r e v a l u a t i n g

D { S } ,

e q u a t i o n ( 1 ) , n a m e l y , t h e F o u r i e r - s e r i e s

m e t h o d a n d t h e o p t i m u m m e t h o d o f E m i n t o n 31. C o n s i d e ri n g a b o d y o f u n i t l e n g t h , t h e

F o u r i e r - s e r ie s m e t h o d r e li e s u p o n e x p r e s s i n g

S ( x )

a s a s i n e s e r i e s i n t h e n e w v a r i a b l e

= c o s -1 (1 - - 2 x) . T h e d i s a d v a n t a g e o f t h i s m e t h o d l ie s i n t h e d e t e r m i n a t i o n o f

S ( x ) ,

w h i c h

r e q u i r e s a n u m e r ic ~ il d i f f e r e n t i a t i o n o f t h e c r o s s -s e c t io n a l a r e a d i s t r i b u t i o n S ( x ). F u r t h e r m o r e ,

i t i s n o t c l e a r h o w m a n y t e r m s i n t h e F o u r i e r s e r i e s a r e r e q u i r e d t o g i v e r e a s o n a b l e a c c u r a c y i n

t h e a n s w e r f o r t h e w a v e d r a g .

T h e d i f f ic u l ti e s o f t h e F o u r i e r - s e r ie s m e t h o d c a n b e o v e r c o m e m o s t s a t i s f a c t o r i l y b y u s i n g t h e

m e t h o d o f E m i n t o n 4, 3~ f o r e v a l u a t i n g t h e d o u b l e i n t e g r a l o f e q u a t i o n (1 ). T h i s m e t h o d d e p e n d s

u p o n s o m e re s u l t s i n t h e t h e o r y o f o p t i m u m a r e a d i s t ri b u t i o n s a n d d e t a il s c a n b e fo u n d i n t h e

p a p e r s c i t e d 4,31. T h e g i v e n a r e a d i s t r i b u t i o n i s r e p l a c e d b y t h e o p t i m u m a r e a d i s t r i b u t i o n w i t h

m i n i m u m w a v e d r a g f o r a la r g e n u m b e r o f f ix e d a re a s a t e q u a l l y sp a c e d p o i n t s a l o n g t h e b o d y .

T h u s t h e e v a l u a t i o n o f t h e w a v e d r ag , g i v e n b y t h e d o u b l e i n t e g r a l o f e q u a t i o n (1 ), re q u i re s

o n l y a k n o w l e d g e o f t h e a r e a s a t a la r g e n u m b e r o f p o i n ts , u s u a l l y n i n e t e e n , a l o n g t h e b o d y .

T h e d e r i v a t i v e

S ( X )

i s n o t r e q u i r e d f o r t h i s m e t h o d .

S l e n d e r b o d ie s w i t h d i s c o n t i n u o u s s o u r c e r e p r e s e n t a t i o n s c a n n o t b e t r e a t e d i n g e n e r al , b u t

c a n b e t r e a t e d p r o v i d e d t h a t t h e c r o s s- se c ti o n is c i r c u la r a t a n d w i t h i n t h e n e i g h b o u r h o o d o f a

d i s c o n t i n u i t y i n s u r fa c e s l o p e ,

i .e . ,

t h e w a v e d r a g is g i v e n b y e q u a t i o n ( 15 ) a n d d e p e n d s u p o n t h e

c r o s s - s e c t i o n a l a r e a d i s t r i b u t i o n o n l y .

A n u m e r i c a l p r o c e d u r e w a s d e v e l o p e d f i rs t b y D i c k s o n a n d J o n e s a2, w h o u t i l i s e d a t r a n s f o r m a -

t i o n o f e q u a t i o n ( 1 5 ) w h e r e b y F o u r i e r s e r ie s c o u l d b e u s e d d i r e c t l y t o e v a l u a t e t h e i n t e g r a l s

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

a r e a d i s t r i b u t i o n s i f s u c h o p t i m a c o u l d b e . fo u n d f o r t h i s m o r e g e n e r a l c a se . E m i n t o n i s

i n v e s t i g a t i n g a n e x t e n s io n o f t h e o p t i m u m m e t h o d t o t h e p r e s e n t p r o b le m , t h u s d e t e r m i n i n g t h e

o p t i m u m a r e a d i s t r ib u t i o n w i t h m i n i m u m w a v e d r a g ( e q u a t io n (1 5)) f o r p r e s c ri b e d v a l u e s o f

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A S , at the points x -. x~where the cross-sectional area is S(x~) (i -- 1, 2 , . . . n). It appears

that a simple and illuminating extension may be possible and that the optimum method will

be exte nded to the evaluati on of equatio n (15). However, since the meth od of Dickson and

Jones 3~ seems to be the only publi shed m et hod and since thei r r eport 3~ ma y not be general ly

available, their method will be summarised here.

Consider a body of unit length and introduce the function

P(x)

defined by

/ . .

P ' ( , ) = S 6 ) - ~ ~ S , + x ~ & ' , x ; ~< x ~ . % 1 .

= I = 1

The function P(x) is chosen by Dickson and Jone s 3~ because P (x ) is continuous everywhere.

Now int rodu ce the Fourier-series vari able = cos -~ (1 -- 2x) and put

co

P (x) = ~ A, ,

sin m~,

so that equati on (15) becomes

D q2 _ ~ . \ ~ 1 z l & , ) '_

1 2

B R ,

-1- ~ mA . 2 / A S ~ [ _ ( 2 m ) 2 1 ]

i l

~ , t = l

-1- AS~' A,. cos m~

i = 1 1 1 1

+ - 1 - l o g ( 1 - + . , x ; }

Y g , = 1 j = l

+ I x , - '

where x~ = 1(1 -- cos %) and =R, 2 = S(x,). This result is in a somewha t different form from

that given by Dickson and Jonesa'.

4 . 3 . Wing-Body Combina t ions . - - In Sections 2 and 3 it has been shown that the calculation of

the wave drag of a wing-body combination can be reduced either to .a wing problem together

with two ' bo dy ' problems by using the transfer rule or to a numb er of ' bo dy ' problems by

using the area rule. Hence the numeri cal methods for wings and bodies given in Sections 4.1

and 4.2 enable the wave drag of a wing-body combination to be found. Procedures for calculating

the transferred area distribution of the wing are described in the Appendix.

5. Some Miscellaneous Rem arks .--5.1. Configurations with Roun d-No se Win g Se ct io ns.- -W ard 10

has pointed out that the results of linearised theory are incorrect if the Wing of the Wing-body

combin ation has roun ded edges where T'(R) is infinite. Nevertheless, the linearised-tlheory

result for wings with round ed subsonic leading edges can be obtaine d by allowing for the effect

of the leading-edge singularit y in the way suggested by Jones3L The result obta ined is tha t t o

the existing result (equation (6), for example), there must be add ed a leading-edge drag F per

unit length given by

F ~r cosaA

~ - - [1 - ( M c o s A T ] 1 '

where q is the kinetic pressure, A is the local sweepback angle of the leading edge, and r is the

local nose radi us..

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

i n t h e c u r v a t u r e o f t h e l e a d i n g e d g e. M o r e o v e r , t h e r e s u l t sh o w s t h a t t h e l i n e a r i s e d - t h e o r y w a v e

d ra g o f a w in g w i th a ro u n d e d l e a d in g e d g e i s i n f in i t e i f t h e l e a d in g e d g e is so n ic i .e ., M c o s A - - 1 ).

T h i s i n f i n i t e r e s u l t f o r a w i n g w i t h a s o n i c l e a d i n g e d g e is o b v i o u s l y in c o r r e c t a n d e m p h a s i s e s t h e

f a c t t h a t t h e l i n e a r is e d t h e o r y is n o t s a t i s f a c t o r y f o r w i n g s w i t h r o u n d e d e d g es . T h u s , i n t h e

a b s e n c e o f r i o n - l in e a r t h e o r y , e x p e r i m e n t a l r e s u l ts a r e r e q u i r e d f o r d e t e r m i n i n g t h e w a v e d r a g o f

s u c h w in g s. H o w e v e r , s in c e t h e t r a n s f e r ru l e s ep a r a t e s t h e w i n g w a v e d r a g fr o m t h e t o t a l w a v e

d r a g o f a w i n g - b o d y c o m b i n a t i o n , i t c a n b e u s e d t o e s t i m a t e t h e i n t e r f e r e n c e W a v e d r a g .

5.2. Sys t em s o f Wi ngs and Bod i e s . - - The d i sc u s si o n s o f a r h a s b e e n r e s t r i c te d t o w i n g - b o d y

c o m b i n a t i o n s , a l t h o u g h i t c o u l d a l so h a v e b e e n g i v e n f o r w i n g - s y s t e m - b o d y - s y s t e m c o m b i n a t i o n s .

T h e w i n g s y s t e m i n c l u d e s w i n g - l ik e e le m e n t s o f t h e c o n f i g u r a t i o n a s w e l l a s a n y s u b s i d i a r y b o d i e s

s i t u a t e d o n t h e s e e l e m e n t s w h i l e th e b o d y s y s t e m i s e s s e n t ia l l y a x i a l a n d m a y i n c l u d e fi ns a n d

t a i l p l a n e s . S i n c e t h e a r e a rule m o m e n t o f a re a ru le a n d t r a n s f e r r u l e a r e a l l e q u a l l y a p p l i c a b l e

t o s y s t e m s o f w in g s a n d b o d i e s a s t o w i n g - b o d y c o m b i n a t i o n s , n o e s s e n t i a l l y n e w p r o b l e m i s

i n t r o d u c e d .

5.2.1. Arrangements Of bodies . - -Arrangem ents o f b o d i e s h a v e b e e n e x a m i n e d b y F r i e d m a n

a n d C o h e n3~ a n d R e n n e m a n n 35. I n b o t h c a s e s t h e a n a l y s i s d o es n o t u s e a n y o f t h e m e t h o d s

g i v e n i n . S e c ti o n 2 b u t s u c h a n a n a l y s i s c o u l d b e c a r r i e d o u t u s i n g e i t h e r t h e a r e a r u l e o r t h e

t r a n s f e r r u le . R e n n e m a n n 35 h a s e x a m i n e d t h e s