Slipstream Theory

8/17/2019 Slipstream Theory

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R . & H . N o . 2 ~ 8 8 '9 4 8 0 ,10,258 & 9262)'

A.R.C. Teclmi cal epor~

SEP ~95~

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

A E R O N A U T I C A L R E S E A R C H C O U N C I L

R E P O R T S A N D M E M O R A N D A

alculat ion of the Effect of Sl ipstreamon Lif t and Induced Drag

y

H . B. S~mR~, M .A . , and W . CHESTER, B.Sc.

rown opyright Reserved

L O N D O N : H I S M A J E S TY S STAT I O N E RY O F F IC E

x95o

F I V S H I L L I I ~ I G S N T


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a lcu la t ion o f th e E ffec t o f S l ips tream on L i f tI n d u c e d D r a g

C O M M U N I C AT E D B Y T H E P R I N C I PA L D I R E C TO R O F S C I E N T I F IC R E SE A R CHA I R ) ,M I N I S T RY O F S U P P LY

and

Reports and Memoranda No. 23 68

Octoaer 1945

PA RT I

W ing of nfini te Span

Summary. Thel i f t d is t r ibu t ion a long a wing of inf in i te span wi th a cent r a l je t of h igher ve loc i ty i s ca lcula ted bys t and a rd m e thod s o f ae ro fo i l t heo ry fo r s eve ra l va lues o f j e t ve loc i t y / st r eam ve loci t y ) and o f j e t d i ame te r /w ing cho rd ) .The l i f t increment and the induced drag are de termined and t i le appl ica t ion of the resul t s to prac t ica l cases i s d iscussed .

1. ln t roduc t ion . Thee f fe c t o f s l ip s t r e a m o n i n d u c e d d r a g i s o f i n t e r e s t i n c o n n e c t i o n w i t ht h e r e d u c t i o n o f d r a g o f a i r c r a f t c r u i s in g a t f a i r l y h i g h l i f t c o e ff ic i en t s a n d t h e p r e s e n t c a l c u l a t i o n sh a v e b e e n m a d e t o g e t s o m e i d e a o f t h e m a g n i t u d e o f t h i s ef fe c t. T h e g e n e r a l p r o b l e m o f t h ew i n g o f f i n i t e s p a n w i t h a j e t p a s s i n g o v e r i t i s c o m p l i c a t e d a n d t h e p r e s e n t c a l c u l a t i o n s d e a lo n l y w i t h t h e i n f in i t e - s p a n w i n g . T h e r e s u l ts h a v e a s f a r a s p o s s ib l e b e e n g i v e n in a f o rm w h i c he n a b l e s t h e m t o b e a p p l i e d i n p r a c t i c e t o w i n g s o f f i n i te s p a n t o e s t i m a t e t h e i n d u c e d - d r a gi n c r e m e n t d u e to s li p s t r e am . B u t i t i s d e s ir a b l e t o o b t a i n t h e c o r r e c t s o l u t io n fo r t h e f i n i te s p a nw i n g f o r c o m p a r i s o n ; D r. K . M i t c h e ll h a s m s d e c o n s i d e r a b le p r o g re s s w i t h t h i s p r o b l e m a n dh a s k i n d l y l e n t u s s o m e a u x i l i a r y t a b l e s f o r u s e i n t h e p r e s e n t c a l c u l a ti o n s .

2 . Mathemat ica l An a ly s i s . Th ew i n g is o f i n f i n i t e s p a n a n d h a s a c o n s t a n t c h o r d c a n d c o n s t a n ti n c i d e n c e ~ a l o n g t h e s p a n . T h e q u a r t e r - c h o r d l i n e o f t h e w i n g is a d i a m e t e r o f t h e c o n t r a c t e d

s l i p s tr e a m w h i c h is a c i rc u l a r j e t o f v e l o c i t y v a n d r a d i u s R . T h e s t r e a m v e l o c i t y o u t s i d e t h isj e t i s d e n o t e d b y V.

* Three repor ts :

R.A.E. Repor t No. Aero . 2083A, A.R.C. 9480, October, 1945 Par t I ) .R.A.E . Rep or t No. Aero . 2167, A.R.C. 10 ,256, Nov ember, 1946 Par t I I ) .R.A.E. Repor t No. Aero . 2083B, A.R.C. 9262, October, 1945 Par t I I I )

a r e co l lec t ed he re t oge the r unde r one cove r fo r conven ience . Each r epo r t i s s epa ra t e a s t he no t a t i ons and t r ea tme n t sare necessar i ly d i fferent .

89049) A


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L i f t i n g - l in e t h e o r y i s a s s u m e d t o b e v a l i d , s o t h a t t h e c i r c u l a ti o n r o u n d t h e w i n g i s g i v e n bTg : - -

2 k - - - - m c ( v ~ - - w ) , for I x l < R , 7

2 K : m c ( V c ¢ - - W ) , f o r [ x l > R , ~ • . . . . . . . . . (1)

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

k , K d e n o t e t h e c i r c u l a t io n s r o u n d t h e w i n g a t a n y s t a t i o n i n s i d e a n d o u t s i d et h e s l i p s t r e a m r e s p e c t i v e l y,

w, W d e n o t e t h e d o w n w a s h e s a t p o i n t s i n s id e a n d o u t s i d e t h e s l i p s t r e a mrespec t ive ly,

an d m i s t he l i f t - cu rve s lope o f t he ae ro fo i l s ec t ion .

W e s h a l l n o w c a l c u l a t e t h e d o w n w a s h a t a p o i n t P (x ) o n t h e w i n g d u e t o a s e m i - in f i n i te tr a i l i n gv o r t e x * o f s t r e n g t h p , w h i c h s t a r t s f r o m t h e p o i n t Q( x ) o n t h e w i n g a n d e x t e n d s d o w n s t r e a m .T h i s is a n i m a g e p r o b l e m w h i c h h a s b e e n s o l v e d b y K a r m a n a n d B u rg e r s ~ a n d b y K o n i n g 2.F o u r c a s es h a v e t o b e d i s t i n g u i s h e d a s s h o w n i n F i g . 1 ; t h e r e s u l t s a r e a s f o ll o w s : - -

C a s e I . P a n d Q b o t h i n s i d e t h e s li p s t r e a m .

T h e d o w n w a s h a t P is t h a t d u e t o a v o r t e x o f s t r e n g t h I a t Q( x ) p l u s t h a t d u e t o av o r t e x o f s t r e n g t h 12P a t t h e i m a g e p o i n t Q '( R 2 / x ) ,w h er e 2~2 = (v 2 -- V 2 ) / (v 2 + V ~) .H e n c et h e d o w n w a s h v e l o c i t y i s : - -

= P 1 + R ~ / ( 2 2 _ x )w 4-~ X - - x

Case I I . P i n s ide s l i p s t r eam , Q ou t s ide s l i p s t r eam.

T h e d o w n w a s h a t P (x) i s t h a t d u e t o a v o r t e x o f s t r e n g t h 2~ P a t Q ( x ') , w h e r e~ = 2 v V / ( v 2 +V 2 ) . H e n c e t h e d o w n w a s h v e l o c i t y i s :

w = ~ LX -- XJ

C a s e I I I . P o u t s id e s l i p s t r e a m , Q i n s i d e s l i p s t re a m .

T h e d o w n w a s h a t P (x ) i s d u e t o a v o r t e x o f s t r e n g t h Z iP a t Q ( x ') t o g e t h e r w i t h a v o r t e xa t t h e o r i g in . T h e l a t t e r c a n b e i g n o r e d s in c e w e a re a l w a y s c o n c e r n e d w i t h p a i r s o f v o r t i c e so f o p p o s i t e s ig n , a n d f o r a n y s u c h p a i r t h e i m a g e s a t t h e o r ig i n c a n c el . H e n c e t h e d o w n w a s hve lo c i ty i s : - -

W = ~ - - x

C a s e I V. P a n d Q b o t h o u t s i d e t h e s l i p s t r e a m .

The dow nw ash a t P (x) i s t h a t due to a vo rce_~ o f s t r e ng th F a t Q( x ) t o g e t h e r w i t h av o r t e x o f s t r e n g t h - - ,~ 2P a t t h e i m a g e p o i n t Q '( R ~ / x ~a n d a v o r t e x a t t h e o r ig i n . A sb e f or e , t h e v o r t e x a t t h e o r i g i n c a n b e ig n o r e d a n d t h e d o w n w a s h v e l o c i t y i s : - -

W = ~ x - x ( R 2 /x ) - x

T h e c o u n t e r - c l o c k w i s e d i r e c t i o n i s t a k e n a s p o s i t i v e .

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T h e s t r e n g t h o f t h e t ra i l in g v o r t e x s y s t e m a t a n y p o i n t Q(x ' ) i s equ a l to - k ' (x ' ) p e r u n i tl e n g t h i n s i d e t h e s l i p s t r e a m a n d t o - -K ' ( x ' ) p e r u n i t l e n g t h o u t s i d e t h e s l i p s t r e a m , w h e r e t h ed a s h d e n o t e s d i f f e r e n ti a t io n , a n d t h e t o t a l d o w n w a s h i s o b t a i n e d b y i n t e g r a t i o n a l o n g t h e w i n gs p a n . F o r a p o i n t in s i d e t h e s l i p s t r e a m w e o b t a i n

R ~ c o

= - L ÷ x ] I f 2I n t h e f i r s t i n t e g r a l w e p u t

x - = B y , x ' = R y ' ,

a n d i n t h e s e c o n d i n t e g r a l

x = R y , x ' = R / Y ' .

We a l s o p u t

k (y ) = 4~ R v o~ (y ) 7

K ( Y ) = 4 ~ R V ~ F ( Y ) ~ . . . . . . . . . . . . . . . . . 3)

W i t h t h e s e S u b s t i t u t i o n s 2 ) b e c o m e s

vo~ - , f ' ( Y ' ) ' - - -@ 1 - - y y ' J v - , 1 - - y Y ' . . . . .

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

4 z ~ W (x ) : - - f 2 R k ( x ) [x , 2 ~ 1 _ _ x ] d x' - - [ 5 1 2 +f T R ]K ' ( x ' ) I x, I G d xg s )- - x R / x - - x j

W e m a k e t h e s u b s t i t u t io n s 3) a n d i n th e f ir s t i n t e g r a l p u t

x = R / Y , x ' =- R y ' ,

a n d i n t h e s e c o n d i n t e g r a l

x = R / Y , x ' = R / Y ' .

T h e s e l e a d t o t h e f o r m u l a

W ( Y ) _ 2 1 v ( ~ Y d y ~ 1 F ( Y ) [ Y Y ' 2 ~ Y ] d Y ' . ( 6 )Voc V ~ - ~ T ( Y ) I : Y y ~ -~ Y - - Y I - -Y Y ' d

W e n o w a s s u m e t h a t t h e c i r cu l a ti o n f u n c t i o n sf ( y ) a n d F ( Y ) c a n b e r e p r e s e n t e d b y s e r i e se x p a n s i o n s o f t h e f o r m : - -

f ( y ) = a + b [2 log 2 - - 1 - - y ) log 1 - - y ) - - 1 + y ) log 1 + y ) ]

- - ~ . c . ( 1 - - y + ~ )o 2 n + 2 . . . . . . . . . . . . . . . . 7a)

F ( Y ) = A + B [ 2 1 o g 2 - - ( 1 - - Y ) l o g 1 - - Y ) - - 1 + Y ) l o g 1 + Y )]

- ~ . c ~ 2 - Y ~ , ~ ~ )o 2n + 2 . . . . . . . . . . . . . . 7b)

T h e f o r m o f th e s e e x p a n s i o n s i s s u g g e s t e d b y t h e s o l ut i o n o f t h e p r o b l e m o f t h e w i n g s p a n n i n g af r e e j e tL W i t h t h e s e e x p r e s s i o n s

B

f ( y ) = b l o g ~ + c W o +

( I - - Y )F ' ( Y ) = B l o g i + Y + C Y 2 + I

889o4o) A 2


5/28

S u b s t it u t io n i n ( 4 ) a n d

w _ f7 )~ -1+ 21V

V

(6 ) g ives

[ ] E ylog 1 y c v '2 + 1 1 =y1 + y * + '~ - - + d y '' - - y 1 - ~ y '

o 1 - - y Y '

WVc~

- - a l v ~ 1 [ b l o g 1- - Y ' + 1] Y d y 'v J - 1 1 4 - ~ + ~o c o y ' ~ 1 - Y y '

1 1 - - Y '- - f - 1 [ B lo g l + y , + ~ C , ~ y '2 ,,+ 1] I y y Y ' y0

To e v a l u a t e t h e i n t e g ra l s w e n o t e t h a t 3

f l l o g ( ~ - - y ' ) d y '

; ( 1o g y ' d y ' _- 1 + [y~ 1 - - y y 'Wi t h t h e s e r e s u l t s a n d t h e r e l a t i o n s

2 + ½ o w v i - 2 - -~ /

log , 1 + y3k ] ~ ] •

2 s Y d Y '1 - - - Y Y ' A

. . 8a)

. . (Sb)

y y ' _ y ~ ( f , 1 ) + y , y = y + y S ( y ' )Y ' - - y - - y 1 - - y y ' 1 - - y y '

w e o b t a i n1 - y ~ y y ' d y ' _ y Sf xo g ( ~ + ~ / 0 7 7 - - ~ [ - -~zs~ } 1 lo g sk ~ / J ( 1y '~]

f l - y d y ' _ ½ l o g s ( 1 + y h .

To d e t e r m i n e t h e o t h e r i n t e g r a l s w e w r i t e , a s i n R e f 3 ,= (~ y, , , , + l g y , = f , y, ~ . , + s a y ,

J ~ ( Y ) J -1 - y - r _ - y , K ' d Y ) J - 1 - 1 - y y '

A l s o

-~ ~ - y y (~ + 1 : } j ) ~ Y: y ~ K , x y ,

5 1 J 2 n + 2 J d Y l _ _ y 2 f ly t 2 n + X y l~ l )-1 y ' - - y 1 ,~-~ -~ • ay ' = y= j , , (y ) .

Wi t h t h e s e v a l u e s ( 8 a ) a n d ( 8 b ) b e c o m e

- ~ Y [ ~ l o g ~ ; 1 + Y ~ - ~ < ~ , x y ) ]v L ~ ~ ~ x '

W 21v

I ,7~2 2- - B 2

. . ~ a )

4


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W e m d s t e n s u r e t h a t t h e d o w n w a s h d o e s n o t b e c o m e i n f i n it e a t t h e s li p s t r e a m b o u n d a r y, w h i c his g iv en b y y = 1 = Y. P u t t i n g y = 1 - - ~ i n ( 9 a ) a n d Y = 1 - - ~ i n ( 9 b ) , w h e r e ~ i s a s m al lq u a n t i t y, a n d e q u a t i n g t h e c o e f fi c ie n t s o f l o g ~ ~ t o z e ro g i v e s

_ _ b ( 1 _ ~ ) _ ~ Y B - - 02 v 2 '

v . b B ( I + ) = 0V 2 2

_ 2 v V _ v ~ - - V ~W i t h ~'~ C + V ~ ' v~ + V 2 , t h e s e t w o e q u a t i o n s a r e c o n s i s t e n t a n d g i v e

B = - - b . W i t h t h i s v a l u e o f B ( 9a ) a n d (9 b ) b e c o m e

w(y) __ b '~ (.1 -- y2 (1 + y'~'] __v ~ 2 [ ~ - - - ~ ) l°g = \ ~ - - ~ / j ~ c,, [ J . ( y ) + K . ( y ) ]

+ V ~ c,,+ C , , ) K . y ). . . . . . . . . . . . . (lOa)7 o

2 [~2y~ _f_ 1 - Y ~ ) l o g2 ( i ÷ Y ) ] - y 2 ~ C~ K, ,0

w Y )V ~

+ y ~ , (c,, + c~) K,,(Y),

w h e r e u s e h a s b e e n m a d e o f t h e r e l at i o n s ,

1 - - ~ - - & V 1 + & - - ' ~ l vV J g °

A s y a n d Y t e n d t o u n i t y ( J~ + K , ) t e n d s t o z er o a n d K , t e n d s t o i n f i n i t y.w h e r e v i s s m a l l ,

- - K . y ) = l o g ( 7 / 2 ) + 2 2n+- -- --1 + 2 n ~ F- . . . . . + ~ + ½ + 1 ,a n d h e n c e

c , , +o o o \ 2 n + 1 /

I n o r d e r t h a t t h e d o w n w a s h s h a l l b e f in i t e a t t h e e d g es o f th e j e t t h e r e l a t io n

~ . c . + c~) = o . . . . . . . . . . . . . . . .

m u s t t h e r e f o r e b e s a ti s fi e d . W h e n t h i s h o l d s w e h a v e t h e r e s u l t

- - ~ , ( c .+ C . ) K . ( 1 ) = 2 [ Z( +_ C 2o o \ 2 n + 1 / . . . . . . . .

W e a r e n o w r e a d y t o s u b s t i t u t e i n th e b a s i c e q u a t i o n s (1) f r o m e q u a t i o n (3 ), a n d w e g e t

~ Y)v~ = 1 - e l y ) 1

w Y )

w h e r e ff - -8 ~ R , a n d w, W a r e g iv e n b y ( 10 ) a n df , F a r e g i v e n b y ( 7 ) .m

5

. . ( lOb)

Fo r y = 1 - - ~ ,

(11)

1 2 )

1 3 )


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3 . Boundary cond i t ions . - -Atl a rg e d i s t a n c e s f r o m t h e s l i p s t r e a m t h e c i r c u l a t i o n t e n d s t o ac o n s t a n t v a l u e a n d th e d o w n w a s h t e n d s t o z e ro . T h e d o w n w a s h a t in f i n it y i s o b t a i n e d b y p u t t iY equ a l to ze ro in (10b) an d i t w i l l be seen tha t a ze ro va lue i s ob ta ined .

A t t h e e d g e o f t h e s l i p s t r e a m t h e f i r s t c o n d i t i o n t o b e s a ti s f ie d i s t h e c o n t i n u i t y o f t h e l it h i s i s a c o n s e q u e n c e o f t h e r e q u i r e m e n t o f t h e c o n t i n u i t y o f p r e s s u r e . H e n c e

k v = K V f o r y - - 1 = Y ,r

v ~f 1 ) = V ~ F 1 ) ,

whence , f rom (7 ) ,

v a = V M . . . . . . . . . . . . . . . . . . . . . (14)We s e e t h a t t h e c i r c u l a ti o n a n d t h e d o w n w a s h a r e d i s c o n ti n u o u s a t t h e b o u n d a r y.

S o m e o f t h e c o e f fi c ie n t s h a v e b e e n r e l a t e d t o e n s u r e t h a t t h e d o w n w a s h i s f i n it e a t t h e e dof the s l ip s t r eam . The fu r the r cond i t ion th a t (13 ) i s s a t i s f i ed fo ry = 1 = Y wi l l a l so be imp ose

b = 1 - ~ a 1 5 )v 1 \ 2 n + 12 . . . . . . . . . . .

2x v + G 5 = 1 - e A . . . . . 1 6 )~ ; b V o \ 2 n - } - 1 / . . . . . .

t h i s g ives

m a k i n g u s e o f ( 1 2 ) .

F r o m (1 5) a n d ( 16 ) w e o b t a i n

b = 2 ( v ~ - - V ~ ) 2~ v 2 + V~ = ~-~22 . . . . . . . . . . . . . . . (17)

_ v w [ o + _ c , k l~ a v ~ = I ~ AV ~ v~ + V~ 1 + 2 ~ . \ 2 n + 1 / / . . . . . . . . . . . (18)

Subs t i tu t ing f rom (10) , (17 ) and (18) , equa t ions (13) become

+ , ~ , V ~ (c , + C ,) K , ( y )o

2 Z ~ { 2 1 o g 2 _ ( l _ y ) lo g (l _ y ) _ (1 + y ) t o g (1 + y ) }

_ ~ . c , , ( I - y + ' ) ]o 2 n 2+7~ j . . . . . . . . . . . . . . . . (19)

a n d

T h e q u a n t i t i e s a a n d A

+ y2 ~ c. + C.) K~ Y)

---- 1 - - t t [ A 2 ' t ' {21o g2 Y) log Y) log (1- ~ - - 1 - - 1 - - Y ) - - 1 + + Y ) }

_ ~ c,~ 1- Y,~+,)]2n -[- 2 . . . . . . . . . . . . .

a r e g iven by (18) ; a l so (1 l ) m us t be sa t i s fi ed .

6

. . (20)


8/28

4 . R a n g e o f S o l u t i o n s . - - T h ea b o v e e q u a t i o n s h a v e b e e n s o l ve d a p p r o x i m a t e l y b y r e t a i n i n gon ly th e c oeffic ients Co, cl, c2, Co, C1, C2, C3, th e h igh er coe fficients being p ut eq ua l to zero. T hes ec o e ff ic i en t s w e r e d e t e r m i n e d b y s a t i s f y i n g e q u a t i o n ( 1 9) a t y - ~ 0 , 0 . 6 ' a n d 0 - 9 , a n d e q u a t i o n ( 20 )a t Y ~ - 0 , 0 . 6 , a n d 0 . 9 ; in a d d i t io n , t h e e q u a t i o n s a r e s a ti sf ie d f o r y = 1 a n d Y---- 1. T h ev a l u e s o f ~ c o v e r e d w e r e 1 . 0 , 2 . 5 a n d 5 . 0 , c o r r e s p o n d i n g a p p r o x i m a t e l y t o a s p e c t r a t io s o f t h ep a r t o f t h e w i n g i n t h e s l ip s t r e a m o f 0 . 5 , 1 . 2 5 a n d 2 . 5 . T h e v a l u e s o fv / V cons ide red were

1 .2 , 2 .0 a nd ~ (V - - 0 ).T h e a c c u r a c y t o w h i c h (1 9) a n d ( 20 ) w e r e s a ti s f ie d a l o n g t h e w i n g w a s d e t e r m i n e d b y c a l c u l a t i n gt h e v a l u e s o f t h e f u n c t i o n s a p p e a r i n g a t a la rg e n u m b e r o f p o in t s . I n a l l c a se s e x c e p t o n e t h ee r ro r s were eq u iva len t to change s o f wing inc idence eve ryw here l e s s th an 1½ pe r cen t . ; fo rl~ = 1. O, v /V ---- 2 . 0 , t h e e r r o r a m o u n t e d t o a n i n c i d e n c e v a r i a t i o n o f 5 p e r c e n t . o v e r a s m a l lp a r t o f t h e s p a n , b u t n o r e a s o n f o r t h e l a rg e r e r r o r i n t h i s c a se h a s b e e n d i s co v e r e d .

5 . R e s u l t s o f c a l c u l a t i o n s . - - T h ev a l u e s o f t h e c i r c u l a t i o n f u n c t i o n sf ( y ) a n d F ( Y ) a r e g i v e nin Tab le 1 . Th e spanw ise d i s t r ib u t ion s o f l i f t , c i r cu la t ion , and dow nw ash a re sh ow n in F igs . 2,3 , 4, i n w h i c h t h e v a l u e s o f th e l i f t a n d c i r c u l a t io n g i v e n b y s i m p l e s t r i p t h e o r y a r e a l so s h o w n .To s h o w t h e r e l a t i 6 n b e t w e e n t h e p r e s e n t c a l c u l a t i o n s a n d , th o se o f K o n i n g 2, t h e l i f t v a r i a t i o na l o n g t h e s p a n , p l o t t e d a s a p r o p o r t i o n o f t h e l i ft in c r e m e n t g i v e n b y s i m p l e s t r i p t h e o r y, i sshow n in F ig . 5 fo r ~ = 1. I t w i l l be seen th a t Ko n ing ' s ca l cu la t ions f i t i n we l l w i th thos e o ft h e p r e s e n t r e p o r t .

T h e t o t a l l if t i n c r e m e n t d u e t o t h e j e t, d e n o t e d b y AL , i s g iv e n b y

A L ~ p ( v k - - ½ e V2 m c~ ) d x + + p ( V K - - ~ eV ~ mo ~) d xR R

J - - i

T h e l i f t n c r e m e n t A L 0 w h i c h w o u l d b e o b t a i n e d o n a s i m p l e s t r i p t h e o r y b a s i s i s g i v e n b y :

: A L o ~ ½ p (v - - V ' ) 2 e R m s4. , ,

T h e r a t i o A L / A L ow a s d e t e r m i n e d f o r a l l t h e c a se s a n d t h e r e s u l t s a r e g i v e n i n Ta b l e 2 a n d F i g . 6 .I n d r a w i n g F i g . 6 , u s e h a s b e e n m a d e o f K o n i n g ' s r e s u l t t h a t , f o r s l i p s t r e a m v e l o c it i e s w h i c h

d i ff e r o n l y s l i g h t l y f r o m t h e s t r e a m v e l o c i t y, t h e l i f t i n c r e m e n t i s e q u a l t o t h a t g i v e n b y s i m p l es t r i p t h e o r y, s o t h a tA L / A L o - - >1 as v--+ V.T h e i n d u c e d d r a g A D~ i s d e f in e d b y t h e r e l a t i o n : - -

; I f : : f ; ]D~ = p w k dx + + p W K dxR= 4 z ~ R ' p v ' ~ ' - , - ~ - f ( y ) d y + - ~ - , V ~ y d Y

: 4z~R ' pv ' c~' 1(1 - - ~ f ) f d y + - ~ - , Y ' d Y ,

m a k i n g u se o f (1 3). T h e i n t e g r a l s w e r e e v a l u a t e d n u m e r i c a l l y a n d t h e v a l u e s o fA D J 4 z R 2 p v 2 ~a re g iven in Ta b le 3 . F ig . 7 shows the va lues o f( A D ~ ) T / ( A L ) ~ ,w h e r e T i s t h e j e t t h r u s t , d e f i n e das T ----- pv (v - - V) ~R 2. I t sho uld be n ot ed t ha t , in dra w ing th e cur ve in Fig . 7 , no dis t in c t io ni s m a d e b e t w e e n t h e r e s u l t s f o r d i f f e re n t v a l u e s o f/ ~. T h e i n d u c e d d r a g i s th e d i f f e r en c e b e t w e e nd r a g i n s i de t h e j e t a n d t h r u s t o u t s i d e i t ; t h e s e f o rc e s t e n d t o e q u a l i t y a s t h e j e t a n d s t r e a mv e l o c it i es t e n d t o e q u a l i t y. C o n s e q u e n t l y a v e r y h i g h s t a n d a r d o f n u m e r i c a l a c c u r a c y i s r e q u i r e dt o d e t e r m i n e t h e i n d u c e d d r a g c o r r e c tl y. A s u f fi c i en t l y h i g h s t a n d a r d h a s n o t b e e n a t t a i n e di n t h e p r e s e n t c a l c u l a t i o n s t o d o m o r e t h a n d r a w a m e a n c u r v h t h r o u g h t h e p o i n t s i n F ig . 7 .

6 . D i s c u s s i o n . - - T h ec a l c u l a t i o n o f t h e l i f t d i s t r i b u t i o n o f a w i n g o f f in i t e s p a n w i t h s l i p s t r e a mp r e s e n t i s d i ff i cu l t. W e a r e t h e r e f o r e f ac e d w i t h t h e p r o b l e m o f a p p l y i n g t h e c a l c u l a t i o n s f o rthe in f in i t e wing as f a r a s poss ib le to the p rac t i ca l case o f the f in i t e wing .

7


9/28

We n o t e a t t h e o u t s e t t h a t a c l o s 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 f o r t h e l i f ti n c r e m e n t d u e t o s l i p s t re a m i s n o t n o r m a l l y t o b e e x p e c te d . T h e p r i n c i p a l re a s o n s fo r t h i s a r e : - -

(1 ) The j e t i s no t sha rp ly de f ined , due to m ix in g a t t he edges , a l so i t con ta ins p e r iod ic com-p o n e n t s a n d r o t a t i o n f o r o r d i n a r y p r o p e l le r s ;

(2 ) T h e a s p e c t r a t i o o f t h e p a r t o f t h e w i n g in t h e s l i p s t r e a m i s t o o s m a l l f o r t h e a s s u m p t i o n s

o f l i f t i ng l i ne theo ry to be r ea l ly va l id .( 3 ) I n m a n y c a s e s t h e r e i s a n i n c l i n a t i o n b e t w e e n t h e p r o p e l l e r a x i s a n d t h e w i n d d i r e c t i o n ,

w h i c h i n t r o d u c e s a n i n c l i n a t i o n i n t o t h e j e t .

B u t t h e v a l u e o f t h e a b o v e i n v e s t i g a t i o n d o e s n o t w h o l l y d e p e n d o n t h e a c c u r a c y t o w h i c ht h e li f t i n c r e m e n t c a n b e p r e d i c t e d . F o r a g i v e n l i ft i n c r e m e n t , w h i c h c a n b e d e t e r m i n e d b ym e a s u r e m e n t o r f r o m g e n e r a l i se d d a t a ( R . M . 1 78 84 ), t h e d i s t r ib u t i o n o f t h i s i n c r e m e n t f o rc o n t r a - r o t a t i n g p r o p e l le r s w il l b e s im i l a r i n s h a p e t o t h e d i s t r i b u t i o n g i v e n b y c a l c u l a t i o n . I np a r t i c u l a r t h e i n d u c e d d r a g p a r a m e t e rA D i . T / AL) ~ w i l l p r o b a b l y b e n e a r l y c o r r e c t , a n d h a v ethe same va lue fo r w ings o f f in i t e span a s fo r w ings o f i n f in i t e span .

I f t h e c a l c u l a t e d v a l u e o f t h e i n d u c e d d r a g h a s a s i g n if i c a n t i n f lu e n c e o n t h e p e r f o r m a n c e o fa n a i r c r a f t , t h e n i t is w o r t h w h i l e t o c o n s i d e r w h e t h e r t h i s i n d u c e d d r a g c a n b e e l i m i n a t e d .F o r t h e c a s e o f t h e w i n g of i n f i n it e s p a n , w h i c h h a s b e e n c o n s i d e r e d i n d e t a i l a b o ve , t h e i n d u c e dd r a g c a n b e r e d u c e d t o z e r o b y a d j u s t i n g t h e i n c i d e n c e o f t h e p a r t o f t h e w i n g i n t h e s l i p s t r e a m ,s o t h a t t h e l o a d i n g i s c o n s t a n t a l o n g t h e w i n g ; t h e r e a r e t h e n n o t r a i l i n g v o r ti c e s a n d t h e i n d u c e ddrag i s i den t i ca l ly ze ro .


10/28

P

c

c¢

q44R

x

V

W

w

K

k

A L

ALo

T

A D~

Y

Y

21

tz

f y)F Y) )

N O T AT I O N

d e n s i t y o f f l u i d

c h o r d o f a e r o f o i l

a e r o f o il i n c i d e n c e m e a s u r e d f r o m z e r o l if t d i r e c t io n

s lope o f l i f t cu rve o f ae ro fo i l s ec t ionr a d i u s o f j e t r e p r e s e n t i n g s l i p s t r e a m

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

s t r e a m v e l o c i t y o u t s i d e j e t

v e l o c i t y i n s i d e s l i p s t r e a m j e t i n c o n t r a c t e d c o n d i t i o n

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

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

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

c i r c u l a t i o n a t s e c t io n o f w i n g i n s i d e j e t

l i ft i n c r e m e n t d u e t o s l i p s t r e a m

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

j e t t h r u s t , e q u a l t o p v v - - V ) z R 2

i n d u c e d d r a gx/R for x < RR/x for x > R

2vV

v 2 + V

V ~ _ _ V

v + V

8~R

~ c

d e f i ne d b y e q u a t i o n 7 ).

N o . A u t h o r

1 K~rm{n and Burgers

2 Komng . .. .

3 Squire . . . .

4 Smelt and Dav ies ..

R E F E R E N C E S

Title etc.

Aerodynamic Theory (Dnrand). Vo l. I I Div. E., pp. 242-245. JuliusSpringer. Berlin. 1935.

Influence of the Propeller on other parts of the Airplane Structure. Aero-dynamic Theory (Durand). Vol. 4, Div. M. Ju li us Springer. Berlin.1935.

The Lift and Drag of a Rectangular Wing spanning a Free Jet of CircularSection. Phi l . 2Vfag. Vol. 27, 1939, p. 229.

Est imat ion of Increase in Lif t due to Slipstream. R. M. 1788. 1937.


11/28

T A B L E 1

Circulation Functions

V v = o

1.0 2 .5 5 .0

y, Y f y) F y) f y) F Y) f y) F Y)

00.20 .40 . 60 . 81 0

0 17080.16670.15360.1295

0.08860

0 13780 13490-12560 10680 07620

0.10380.10210.09630.084550.06230

( b ) V v = 0 . 5

g 1 0 2 . 5 5 . 0

y, Y f y) F Y) f y ) F Y) f y ) F Y)

00.2

0 .4

0 . 60 . 8

1 0

0 '4180

0 41520.4066

O- 3907O 3640

O 3074

1-0

1.0186

1. 0642

1.1159I' 1643

1.2294

0.2237

0.22190.2158

0.20440.18~1

0.1362

0.4000

0 40840.4298

0.45710-4895

0-5446

0.13750-1364

0.1326

0.12530.1113

0.0725

0.2000

0.2035

0.2127

0.22590.2461

0.2901

( c ) V v - -- - O 8 3 3

tt 1 0 2 5 5 0

y, Y f y) F Y) F Y) f y ) F Y)

00 . 2

0 .40-6

0-81.0

0.78720-78630-7833

0-77790-7689

0-7503

f y)

0 0.3394C0 68 0 33880233 0'33680419 0 3330

0587 0.32650805 0.3111

0 40000-40290-41020-4194

0 43020-4480

0.17940.17900.17780.1755

0.1710

0 1589

0.20000.20120.20420.2085

0.21500.2288

1


12/28

TABLE

Va l ue s o f A L A L o

y v

00 .50.833

1.0

0.12570.30620.5685

/2

2.5

o 2 6 2 s0.48800.7861

5.0

0.40810.62250.8863

t 4TABLE 3

Va lu es o f A D~ 4z~R ~ pv ~ c¢2

v / v

00.5O. 833

1 0

0 ' 1 0 8 70 13970 0797

2 5

0 18460 12540-0329

5 0

0 22310.11720 0151


13/28

CASE I P ~ (~ BOTH INSIDE 3 E T .

0

CA~E ~_ P I N S I D E . lE T

E A S E ~ I

Q OUTSIbE- JET.

P OUTSIDE J E T ) Q I N S I D E .TET.

CASE 1'9 P ~ Q B O T H OUTSIDE SET.

U N C O M P E N S A T E D , C O N P E N S A T E D .

F I G . 1. E f f e c t a t a P o i n t P o f a Vo r t e x a t Q .

X , I

9

T~

4-C

3 5

3 .0

1-5

t

S T R I P T H E O R Y IVALUE

fI

IIIiIIII

\

s.o 1 ~O 0 - 5 t . o I , e - o 3 C / R 2

W / % r = 0 - 5

' 3 _ _ , M . , , . = 5 . 0

X.L,= 2.5. 2 ~

.4.,0=1-0

|.1

1 00

t -4 I

STR F THEORY. I

VA L U E . I

0-5 t.o l.s 2.o X- /R ~.~• /tr = - 8 3 3 .

F I G 2 L i f t D i s t r i b u t i o n a l o n g S p a n

~ . 0


14/28

,-,-~, c V o{

~ T R t P TI EORYVALUE

I-0

0

I

I

II ~_1,0

~2 5- S . O

O-S J 0 I - 5

~ / t r = o s

2 .0 . ,V, /R 2-5 3 . 0

1.2

4 ~ K S T R IP T H E O R YVA L U E .y2 qrn c VoL

I . I.¢.L, = 5 .0

d t o = 2 5

1 0 ~

- U . , = I - 0

O-gi

0 .80 O.S

i

2-,5S . O

I ' 0 1 5 2 , 0 ~ / R 2 5X/ / 'Lr = ' O B~ 5 ,

F I G 3 C i r c u l a t io n D i s t r i b u t i o n a l o n g S p a n

} ' 0

0 . 5LU W

~ V ~ - -

- O - S

- I . 0

0 - 3

O-I

-0 1

-0 2

,,I.~ = 1-0 ~

O;ff hO I .E 2 .0

~ -,5.O- 2 - 5

l 0

~ / 2 J = 0 - 5 ;

~ = l . OJ

J

t ~ i i s 2~o z /R 2

I ' 0

5 3

V / , t r = O - 8 3 ~ .

F I G 4 D o w n w a s h D i s t r i b u t io n a l on g S p a n


15/28

),,,,,,t

0 ' 9

0 3

Z u 0 . 2

z

0 - 4

O 5 I , o 1 5 ~ . O X / R Z - ~

~ . t ~ - | - O

L i f t I n c r e m e n t g r a d i n g a l o n g t h e S p a nIG 5

3 0

I - 0

0 . 8

A L .

& L o

0 . 6

0 . 4 .

0 . 2f

0 2 0 . 4 0 6 8

FIG 6 Lif t Increment Due to S l ips t ream

I ° 0

0 . 8

0 . 7

0 - 6

O ' S

0 - 4

0 6

0 2

0 1

x A ~ = l . O0 / 4 ~ - - 2 . 5

÷ ~ = 5 o

0 2

FIG 7

0 . 4 - 0 - 6 V / 2 j 0 - 8 I .O

I n d u c e d D r a g D u e t o S l i p s tr e am


16/28

PA R T I I

Slipstream otat ion

Summary. Calculationsa r e m a d e b y l i ft i n g l i n e t h e o r y o f t h e d i s t r i b u t i o n o f c i r c u l a t i o n a l o n g a w i n g o f i n f in i t es p a n w h i c h i s l o c a t e d i n a r o t a t i n g s l i p s tr e a m . T h e w i n g t o r q u e a n d i n d u c e d t h r u s t a r e e v a l u a t e d . T h e n u m e r i c a lv a l u e s o f t h e w i n g t o r q u e a r e n o t e n t i r e l y r e l ia b l e b e c a u s e o f t h e a p p r o x i m a t i o n s i n t r o d u c e d i n t o t h e a n a l y s is . T h ei n d u c e d t h r u s t p o w e r is c a l c u l a t e d t o b e b e t w e e n 3 0 p e r c e n t. a n d 4 0 p e r c e n t . o f t h e r o t a t i o n a l p o w e r i n p u t t o t h ePrOpeller.

1 . I n t r o d u c t i o n . - - T h i si s t h e s e c o n d p a r t o f a n i n v e s t i g a t i o n o f t h e e f fe c t o f s l i p s t r e a m o n l i f ta n d i n d u c e d d r a g a n d d e a l s w i t h t h e e f fe c t o f s l i p s t r e a m r o t a t i o n . W e m a k e t h e f o l l o w i n g a s-s u m p t i on s : - -

(1) T h e w i n g i s o f i n f i n i te s p a n a n d h a s a c o n s t a n t c h o r d a n d i n c id e n c e .7 , .

(2) L i f t i n g - l in e t h e o r y m a y b e a p p l i e d a l t h o u g h t h e r a t i o o f p r o p e l le r d i a m e t e r t o w i n gchord i s no t l a ige in p rac t i ce .

(3) T h e a x i s 5 f t h e c y l i n d r i c a l s l i p s t r e a m i s p a r a l l e l t o t h e d i r e c t i o n o f t h e u n d i s t u r b e ds t r e a m a n d p a s se s t h r o u g h t h e q u a r t e r - c h o r d p o i n t o f t h e w i n g .

(4) T h e a x i a l v e l o c i t y i n t h e s l i p s t r e a m i s e q u a l t o t h e v e l o c i t y in t h e f re e s t r e a m .

(5 ) Ahead o f the wing the s l ip s t r eam ro ta t e s a s a r ig id cy l inde r.

(6) T h e e f fe c t of r o t a t i o n i s e q u i v a l e n t t o a c h a n g e i n t h e s t r e a m d i r e c t i o n a t t h e w i n g ,g i v i n g r i se t o a n e f f e ct i v e ly l i n e a r v a r i a t i o n o f w i n g i n c i d e n c e i n s id e t h e s l i p s tr e a m .

2 . G e n er a l A n a l y s i s . - - T h ea b o v e a s s u m p t i o n s m a k e t h e p r o b l e m , a s f a r a s t h e c i r c u l a t i o nd i s t r i b u t i o n a l of i g t h e w i n g s p a n i s c o n c e r n e d , e q u i v a l e n t t o t h e p r o b l e m o f a w i n g o f in f i n i t es p a n w i t h a l i n ea r v a r i a t i o n of i n c id e n c e w i t h i n t h e s l i p s t r ea m b o u n d a r y a n d c o n s t a n t i n c id e n c eo u t s i d e t h e b o u n d a r y. S i n ce a c o n s t a n t i n c r ea s e o f i n c i d e n c e a l o n g t h e w h o l e s p a n c a n b e m a d ew i t h o u t a f f e c t i n g t h e s l i p s t r e a m r o t a t i o n e f f ec t, it m a y b e c o n s i d e r e d t h a t t h e w i n g h a s z e r o i n-c idence ou t s ide the s l ip s t r eam . He nce the w ing inc idence c¢ i s g iven by

c o x ) = ~ y / V , f o r - R < x < R , 7

= o , f o r I > R , . . . . . . . . 1 )

w h e r e x , i s m e a s u r e d a l o n g t h e s p a n f r o m t h e c e n t r e o f t h e j e t , R i s t h e j e t r a d i u s a n d ~o i s t h ea n g u l a r v e l o c i t y o f r o t a t i o n i n t h e s l i p s tr e a m .

To d e t e r m i n e t h e c o r r e s p o n d i n g c i r c u l a t i o n d i s t r i b u t i o n w e s t a r t w i t h P r a n d t l s r e s u l t 1 f o ra s i n u s o i d a l v a r i a t i o n o f i n c i d e n c e , w h i c h g i v e s f o r t h e i n c i d e n c e d i s t r i b u t i o n

c¢ = c¢0 s i n / ~ x ,

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

s i n

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w h e r e V is t h e s t r e a m v e l o c i t y, c is t h e a e r o f o il c h o r d , m i s t h e l i f t c u r v e s l o p e i n t w o d i m e n s i o n a lf lo w, a n d :~0 i s a c o n s t a n t . T h e d o w n w a s h w is r e l a t e d t o t h e c i r c u l a t i o n a n d i n c i d e n c e b y t h ee q u a t i o n

w _ = 2 K (2 )- - . . . . . . . . . . . .

o ~ - - V m c V

a n d h e n c e

w _ ~ K _ ~ So s i n ~ x .V 4V ~ + 8/mc)

W e g e n e ra l is e t h e a b o v e r e s u l t b y u s e o f F o u r i e r 's d o u b l e i n t e g r a l t h e o r e m .d i s t r i b u t i o n o: x),w h i c h i s a n o d d f u n c t i o n o f x , w e h a v e t h e f o r m u l a

F o r a n y in c i d e n c e

o~(x) ---- 2 sin # x d# c~(t) si n # t dt1 7 ~ 0 0

a n d , m a k i n g u s e o f t h e a b o v e r e s u l ts f o r a s i m p l e s in u s o i d a l v a r i a t io n , w e c a n w r i t e d o w n t h ev a l u e o f t h e c i r c u l a t i o n c o r r e s p o n d i n g to t h e a r b i t r a r y i n c i d e n c e d i s t r i b u t i o n ~ (x ) t o b e

K x) 2 f ~° 4V f~~ o /z + (8/mc ) s in ~x d/z o c~(t) s in/ ~td t .

W i t h t h e s p e c i f ie d l i n e a r in c i d e n c e d i s t r i b u t i o n i n s i d e t h e r o t a t i n g s l i p s t r e a m , g i v e n b y (1 ), w e g e t

~( t ) s in ~t dt = co ts i n tzt dto o

- - ~ 2 [ s i n ~ R - - ~ R c o s ~ R ] .

H e n c e t h e c i r c u l a t i o n d i s t r i b u t i o n c o r r e s p o n d i n g to (1 ) i s

8o~ ( * s i n ~ x [ si n # R - - ~ R c o s z R ]K x ) a t= J0

P u t t i n g x / R = y , t ,R = 2 , and - -8R = a , t h i s be c om es~ C

2 K ( y ) 2 a f ~ (s i n Z - - 2 c o s 2 )mc o)R -- ~ o 2 2 2 + a) s i n ( Z y ) d 2 , (3)

c o r r e s p o n d i n g t o t h e i n c i d e n c e d i s t r i b u t i o n

c ~ ( y ) - - V y f o r - - 1 < y < 1

c~ (y) = 0 for [ y I >(4)

T h e i n t e g r a l i n ( 3) c a n b e e x p r e s s e d i n t e r m s o f t h e s i n e a n d c o s i n e i n t e g r a l s , w h i c h h a v e b e e nt a b u l a te d b y J a h n k e a n d E m d e .

C a l c u l a t i o n s o f t h e d i s t r i b u t i o n s o f c i r c u l a t i o n a l o n g t h e s p a n h a v e b e e n m a d e f o r a = 0 5 ,1 . 0 , 1 . 5 , a n d 2 . 0 , c o r r e s p o n d i n g to t h e f o l lo w i n g r a t io s o f j e t d i a m e t e r t o w i n g c h o r d

2 R / c = O . 7 8 5 , 1 . 5 7 , 2 . 3 5 , 3 . 1 4 ,

t a k i n g t h e s e c t i o n l i ft - c u r v e s l o p e m t o b e e q u a l to 2 ~ . T h e r e s u l t s o f t h e c a l c u l a t i o n s a r e s h o w ni n F i g . 1 , w h i c h g i v e2K/mc ~oRa s a f u n c t i o n o fy / R f o r t h e v a r i o u s v a l u e s o f a o r2R/c .

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3 . To r q u e a n d T h r u s t o n t he Wi ~ z g . - - T h ea n t i - s y m m e t r i c l o a d i n g o n t h e w i n g i n d u c e d b y t h es l i p s t r e a m r o t a t i o n p r o d u c e s a t o r q u e w h i c h o p p o s e s t h e p r o p e l l e r t o r q u e . T h e m a g n i t u d e o ft h e t o t a l w i n g t o r q u e i s

Q = p v K ) x d x ,- -¢o

s ince t he l i f t pe r un i t span i s equa l t o pV K . I t c a n b e s h o w n t h a t t h e w i n g t o r q u e i s e q u a l t ot h e t o r q u e c a l c u l a t e d b y s t r i p t h e o r y, i g n o r i n g a l l d o w n w a s h e f fe c ts . T h e l a t t e r t o r q u e i s g i v e nb y

o = ~ pV ~ me x dx = ½ p mc R ~oV. R V

F u r t h e r t h e p r o p e l l e r to r q u e Q 1 i s e q u a l t o t h e r a t e o f i n c r e a s e o f a n g u l a r m o m e n t u m i n th es l i p s t r e a m s o t h a t

Q1 = ~ p R o V .

He nce , s ince Q = Oo,

Q _ Q o _ 2 m c _ 16- - _ _ _ _ .

Q1 O l ~ R 3 ~ aT h i s l e a d s t o t h e s u r p r i s i n g r e s u l t t h a t t h e t o r q u e o n t h e w i n g is g r e a t e r t h a n t h e p r o p e l l e r to r q u efo r 2 R / c < 8 /3 , i. e. fo r w i n g c h o r d s g r e a t e r t h a n 0 . 3 7 5 o f t h e s l i p s t r e a m d i a m e t e r. T h e t h e o r yi s h o w e v e r o n l y r e l ia b l e f o r v a lu e s o f t h e w i n g c h o r d w h i c h a r e l es s t h a n t h i s a m o u n t .

I n a d d i t i o n t o t h e t o r q u e o n t h e w i n g t h e r e i s a n i n d u c e d t h r u s t , b e c a u s e t h e s t r e a m i n s i d e t h es l i p s t r e a m i s i n c l i n e d t o t h e d i r e c t i o n o f m o t i o n o f t h e w i n g . I n s i d e t h e j e t t h e l if t v e c t o r o nt h e w i n g i s i n c l i n e d b a c k w a r d s f r o m t h e l o c a l s t r e a m d i r e c t i o n b u t f o r w a r d s r e l a t i v e t o t h eg e n e r a l s t r e a m d i r e c t i o n ; o u t s i d e t h e j e t t h e l i f t v e c t o r i s i n c l i n e d f o r w a r d s b e c a u s e o f t h e u p w a s ht h e r e . T h e i n d u c e d t h r u s t T is g i v e n b y t h e f o r m u l a

T = p ~ - - d x ,co

s i n c e t h e l i f t v e c t o r i s i n c l i n e d b a c k w a r d s a t t h e a n g l ew / V r e l a t i v e t o t h e l o c al s t r e a m d i r e c ti o n ,t h e l o c a l s t r e a m d i r e c t io n b e i n g i n c l i n e d f o r w a r d s a t t h e a n g l e 7 r e l a t i v e t o t h e g e n e r a l s t r e a md i r e c ti o n . S u b s t i t u t i n g f r o m ( 2) a n d m a k i n g u se of t h e s y m m e t r y of t h e j e t w e o b t a i n

o k m c m R / d y .

I t i s c o n v e n i e n t t o r e la t e t h e i n d u c e d t h r u s t t o t h e p o w e r e x p e n d e d i n g e n e r a t i n g t h e r o t a t i o no f t h e s l i p s t r e a m . T h e p r e s e n c e o f t h e w i n g h a s th e e f f e c t o f r e c o v e r i n g a p a r t o f t h i s p o w e ri n p u t . W e s h a ll t h e r e f o r e e x p r e s s t h e i n d u c e d t h r u s t p o w e r a s a p r o p o r t i o n o f t h e r o t a t i o n a lp o w e r i n p u t . * T h i s p o w e r i n p u t P 1 i s g i v e n b y

P 1 = 1 ~ Q 1 = ~ p R o ~ V .

H e n c e T V _ _ 32 f® (__2K___ ~ dxP 1 ~ a o \ m c ~o R /

where , as before , a = 8 R / m c . T h e v a l u e s o f t h i s q u a n t i t y f o r t h e v a r i o u s c a s e s a r e g i v e n i nTa b l e 1. I t w i l l b e s ee n t h a t t h e e f f i ci e n c y o f e n e rg y r e c o v e r y r e a c h e s a m a x i m u m v a l u e o f37 pe r cen t .

* The rotational p ower input P1 is normally abou t 3 per ce nt. of the tota l pow er input of a propeller.

17890~9) B*


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~

4 . D i s c u s s i o n . I fthe chord i s le ss than one- th i rd o f the j e t d iamete r and the angu la r ve loc i tyin the je t i s smal l there is good reason for supposing th at the m etho d given abov e is fa i r ly re l iable.Unfor tuna te ly the p rac t i ca l r ange cor responds more to wing chords abou t equa l to the j e tdiam eter and in th is region l i f ting line theo ry ceases to be an acceptable appro xim at ion. Thejust i f icat ion for working out examples in th is region is that they may give some rough guidanceunt i l l i f ting surface theory can be appl ied i f th is i s ever done.

A fu r the r weakness in the theory i s the assumpt ion tha t fo r the purpose o f ca lcu la t ing thel i ft d is t r ibut ion the rota t ion in the s lips t ream is equivalen t to a twis t in the wing. This assump-t ion wil l fa i l i f the dis tor t ion of the s l ips t ream bo un dar y becomes appreciable and i f the w ingchord is too large a propo rt ion of the je t d iam eter. He re again we can har dly expect the theo ryto be ful ly re l iable for wing chords greater than one- thi rd of the je t d iameter.

A compar i son wi th ad hoc exper imenta l da ta has no t been made . A re li able compar i sonwould requ i re a p rogramme of t e s t s spec ia l ly a r ranged to de te rmine the re levan t quan t i t i e s .

No. Author

1 Prandtl and Betz ..

R E F E R E N C ETitle etc.

Vier Abhandlangen zur Hydrod ynami k und Aerodynamik p.29.Wilhelm Institu te G6ttingen. 1927.

N O TAT I O NR radius of s l ips t reamy dis tance a long wing span m easured f rom t i le centre of the s lips t ream.V s t ream ve loc ityw downw ash ve loc ity~o ang ula r veloc ity in sl ipstr eamc wing chord

wing incidence to local s t ream direct ion

m Sect ion l i ft curve slope tak en to be equal to 2~ in the calcula t ionsK circula t ion a long the wing.x y / Ra 8 R / m cQ torque on wing

Q0 torqu e on wing calcula ted by s t r ip theo ryQ1 propeller torq ueT th rus t on wing

P1 rota t ion al pow er inp ut to propel ler

Kaiser

TA B L E 1l nduce d Thru s t Pow er on the W ing a s a F rac t ion o f the

Ro ta t iona l Power Inpu t by the P rope l l e r

~g

2R/cT V / P 1

0-5O. 785O. 280

1.01.570-350

1.52.350.369

2.03.14O. 367

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1 3 / ~ ~ s ~

FIG 1

~,0y R

Circulation Distribution along the Span

~.0

9

89049) C


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PA R T I I I

M inimum Induced ra g

Summary. Ap r o o f is g i v e n t h a t m i n i m u m i n d u c e d d r a g i s o b t a i n e d f o r c o n s t a n t d o w n w a s h a n g l e a l o n g t h e w i n gand th e ca lcu la t ion of the sp an load ing of a wing in th i s condi t ion i s cons idered . Ex am ple s o f wings wi th a cen t ra lj e t a n d w i t h a p a i r o f j e t s a r e gi v e n . I t i s c o n c l u d e d t h a t i n p r a c t ic a l c a se s t h e s p a n l o a d i n g in t h e m i n i m u m d r a gc o n d i t i o n i s n e a r l y e l l i p t i c a n d t h a t t h e m i n i m u m i n d u c e d d r a g i s n e a r l y t h e s a m e a s f o r a w i n g w i t h e l l i p t i c l o a d i n gi n a u n i f o r m s t r e a m .

1. Introduct ion. Calculat ionso f t h e i n c r e m e n t i n l i f t a n d o f t h e i n d u c e d d r a g f o r a s i n g l es l i p s t r e a m o n a n i n f in i t e w i n g w e r e g i v e n i n P a r t I . I t w a s th e r e p o i n t e d o u t t h a t t h e i n d u c e dd r a g f o r a n i n f i n i te w i n g c o u ld b e e l i m i n a t e d b y r e d u c i n g t h e i n c i d e n c e o f t h e p a r t o f t h e w i n gi n t h e s l i p s tr e a m , s o t h a t t h e l i f t s h o u l d b e u n i f o r m l y d i s t r i b u t e d a l o n g t h e w i n g s p a n . T h i s

s u g g e s ts (1) t h a t a s im i l a r t e c h n i q u e m i g h t b e u s e d t o r e d u c e t h e i n d u c e d d r a g d u e t o s l i p s t re a mf o r a w i n g o f f i n it e s p a n a n d (2) t h a t t h e s p a n l o a d i n g g i v i n g m i n i m u m i n d u c e d d r a g w i t h s l ip -s t r e a m s h o u l d b e i n v e s t i g a t e d . T h e l a t t e r i n v e s t i g a t i o n i s d e s c r ib e d i n t h i s p a r t o f t h e r e p o r t ;i n s e c ti o n 9, t h e c o n d i t i o n f o r m i n i m u m i n d u c e d d r a g i s d e t e r m i n e d a n d i n s e c t io n s 3 a n d 4 so m es p a n l o a d i n g s s a t i s f y i n g th i s c o n d i t i o n a r e c a l c u l a t e d .

T h e c o n c l u s i o n of t h e p r e s e n t i n v e s t i g a t i o n is t h a t t h e m i n i m u m i n d u c e d d r a g i n p r a c t i c a lc a se s is o b t a i n e d b y a d j u s t i n g t h e i n c i d e n c e to g i v e e ll i p ti c l o a d i n g a l o n g t h e s p a n a n d t h a t t h ei n d u c e d d r a g i s t h e n e q u a l to t h e m i n i m u m i n d u c e d d r a g w i t h s l ip s t re a m a b s e n t.

2 . Condit ion fo r Min im um Induced D rag . I th a s b e e n s h o w n b y K ~ i rm ~ n a n d Ts i e n 1 t h a t ,f or a w i n g in a s t r e a m o f n o n - u n i f o r m v e l o c i t y,m ini m um induced drag is obtained i f the downw ashangle is constant along the span. A s imple a l t e rn a t iv e p roof o f t h i s r e su l t, fo l lowing the p roc edureu s e d b y B e t z ~ f o r t h e p r o b l e m o f th e p r o p e l le r w i t h m i n i m u m e n e rg y l o ss , w il l n o w b e g i v e n :

S i n c e t h e i n d u c e d d r a g i s r e l a t e d t o t h e e n e rg y i n t h e w a k e f a r d o w n s t r e a m , t h e l i f t in ge l e m e n t s c a n b e m o v e d a l o n g t h e w i n d d i r e c ti o n w i t h o u t c h a n g e i n t o t a l d r a g p r o v id e dt h a t t h e i r i n c i d e n c e s a r e a d j u s t e d t o m a i n t a i n t h e l i f t o f t h e e l e m e n t s u n c h a n g e d ; t h i s i sM u n k s s t a g g e r t h e o r e m . I f a s m a l l e l e m e n t g i v i n g a l i ft b L i s a d d e d s o m e w a y d o w n s f l- e a mt h e i n d u c e d v e l o c i t y a t t h e w i n g i s n e g l ig i b le , a n d t h e i n c r e as e i n i n d u c e d d r a g i s e q u a l t oE 6 L, w h e r e e i s t h e d o w n w a s h a n g l e a t t h e s t a t i o n w h e r e t h e e l e m e n t i s p l a c e d ; a s u s u a lt h e d o w n w a s h f a r d o w n s t r e a m i s e q u a l t o tw i c e t h e d o w n w a s h a t t h e w i n g . I f t w o e l e m e n t sw i t h l i ft s ~ L 1 , ~ L 2 , a r e a d d e d t o a l i ft i n g s y s t e m t h e t o t a l l i f t w i ll b e u n c h a n g e d , p r o v i d e dt h a t

~L1 + ~L,. = 0 .

F o r a w i n g in t h e m i n i m u m d r a g c o n d i t i o n a s m a l l c h a n g e in t h e l o a d i n g d i s t r ib u t i o n a tc o n s t a n t t o t a l l i f t p r o d u c e s n o c h a n g e i n t h e i n d u c e d d r a g ; t h e c o n d i t i o n f o r t h i s i s

,15L1 + *~6L2 ---- 0 ,

w h e r e e l a n d , , a r e t h e d o w n w a s h a n g l e s d u e t o t h e w i n g f ie l d f a r d o w n s t r e a m f r o m t h el o c a t i o n s o f t h e t w o l i f t in g e l e m e n t s . T h e s e t w o e q u a t i o n s a r e o n l y c o m p a t i b l e f o r a r b i t r a r yl i f ti n g e l e m e n t s i f t h e d o w n w a s h a n g l e i s c o n s t a n t a l o n g th e w i n g sp a n , b o t h f a r d o w n s t r e a mand a t t he win g i t se l f . Th i s is t he r equ i r ed r e su l t .

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T h e f lo w r o u n d a w i n g in t h e m i n i m u m i n d u c e d d r a g c o n d i t i o n is t h e s a m e a s i f t h e w a k ewere ma de i n to a r ig id shee t ; t h i s i s , o f course , a l so t rue fo r a wing in a s t r eam of un i fo rm ve lo c i ty.H e n c e , b o t h w i t h a n d w i t h o u t s l i p st r ea m , t h e l if t d i s tr i b u t i o n fo r m i n i m u m i n d u c e d d r a g o n am o n o p l a n e w i n g i s o b t a i n e d b y s o lv i n g t h e t w o - d i m e n s i o n a l p o t e n t i a l p r o b l e m o f a n i n f i n it er i g id p l a t e i n a s t r e a m w h i c h i s n o r m a l to t h e p l a t e . A t t h e c y l i n d r i c a l b o u n d a r i e s of t h e s li p -s t r eams , the fo l lowing con d i t ion s mus t be sa t i s f i ed :3

v¢ ---- V ¢ , (co nt i nu i ty of pressure)

' 1 06 _ 1 0 ~ (c o n se rv a t io n o f m ass) ~ . . . . . . . . (1)v O n V ~ n

w h e r e v a n d V a r e t h e a x i a l v el o c i ti e s i n s i d e a n d o u t s i d e t h e s l i p s t r e a m r e s p e c t i v e l y, ¢ a n d ¢a r e t h e v e l o c i t y p o t e n t i a l s i n s i d e a n d o u t s i d e t h e s l i p s t r e a m r e s p e c t i v e l y, a n d n i s m e a s u r e dn o r m a l t o t h e b o u n d a r y o f t h e s l i p s tr e a m .

C o n s i d e r n o w a m o n o p l a n e w i t h a s i n g le c e n t r a s l i p s t r e a m o r w i t h a p a i r o f s l i p s t r e a m s a ss h o w n in F ig . 1 . Ta k e a x e s a s s h o w n i n t h e f ig u r e. L e t e b e th e ( c o n s t a n t ) d o w n w a s h a n g l ea l o n g t h e s p a n a t i n f i n i t y d o w n s t r e a m , s o t h a t o u r p o t e n t i a l p r o b l e m is t h e d e t e r m i n a t i o n o ft h e f l o w o f a s t r e a m w h i c h h a s a v e l o c i t y e V a t i n f i n i t y a n d w h i c h i s n o r m a l t o t h e s p a n o f t h ew i n g . T h e c o n d i t i o n s t o b e sa t is f i ed o n th e b o u n d a r y o f t h e s l i p s t r e a m a r e g i v e n i n e q u a t i o n s (1 ).

N o w t h e g r a d i e n t o f c i r c u l a t i o n a lo n g t h e w i n g s p a n i s r e l a t e d t o t h e v e l o c i t y p o t e n t i a l a sf o l l o w s :

d k = 2 ~¢ (inside),K _ 2 ~¢ (ou t s ide ) , and~-~ ~ ,d x ~ x

w h e r e K , k a r e th e c i r c u l a t i o n s o u t s i d e a n d i n si d e t h e s l i p s t r e a m r e s p e c ti v e l y. W e n o w i n t r o d u c es e c o nd h a r m o n i c f u n c t i o n s f , F w h i c h a r e r e l a t e d t o t h e v e l o c i t y p o t e n t i a l b y t h e f o r m u l a e

e V F = ¢ , (ou t s ide ) ,

s o t h a t w e h a v e , a l o n g t h e w i n g s p a n ,

d K 2 ~ V a F (outs ide)~ - - ~

H e n c e

K = 2 ~ V F ,

Va n d , V f = ~ ¢ , ( in si de ),

•V a f

and = 2~ - - ( in s ide ) :

y 2k = 2 ~ - - f ,

7

i f i t is a r r a n g e d t h a t F v a n i s h e s a t t h e w i n g t i p s , w h e r e t h e c i r c u l a t i o n v a n i s h e s , a n d w e n o t et h a t k a n d f m u s t v a n i s h t o g e th e r.

O n t h e j e t b o u n d a r y e q u a t i o n s ( 1 ) a r e r e p l a c e d b y

~ f _ ~ Ff = F a n d v ~ ~ n V 2 ~ n

A t i n f i n i t y w e s h a l l h a v e

F = y .

T h e w i n g l i f t i s g i v e n b y

L = ~ p ( vk , V K ) d x .s p n

S u b s t i t u t i n g fo r k a n d K i n t e r m s o f f a n dF , t h i s b e c o m e s

L = 2 p e V 2 f ( f , F) , ix . ~ o os p n

21

. . . . . . . . 2 )


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

D , = ½ ~ L , . . . . . . . . . . . . . . . . . . ~ )

s i nc e t h e d o w n w a s h a n g l e a t th e w i n g is o n e - h a l f t h e d o w n w a s h a n g l e a t in f i n i t y. E l i m i n a t i n gb e t w e e n 2 ) a n d 3) w e o b t a i n

L 2

D , = . . . . . . 4)4p V* Jspan ( f F ) d x . . . . . . . .

T h e p r o b l e m i s t h u s r e d u c e d t o t h e d e t e r m i n a t i o n o f t h e f u n c t i o n s f a n d F a n d t h e e v a l u a t i o no f t h e i n t e g r a l i n 4 ). T h e d i s t r i b u t i o n o f l i ft a lo n g t h e s p a n i s t h e s a m e a s t h e v a r i a t i o n o f fa n d F a l o n g t h e s p a n . I f n o s l i p s t r e a m is p r e s e n t m i n i m u m i n d u c e d d r a g i s o b t a i n e d w i t he l li p t ic l o a d i n g a l o n g t h e s p a n ; i t w il l b e f o u n d c o n v e n i e n t t o c o m p a r e t h e l o a d i n g a n d i n d u c e dd r a g w i t h s l i p s t r e a m p r e s e n t w i t h t h e s t a n d a r d c a s e o f e l l ip t i c l o a d i n g .

3 . S ing le Cen t r a l S l ip s t r ea m . - -Forc o n v e n i e n c e w e s h a l l t a k e t h e s e m i - s p a n t o b e u n i t y, w h i c hc a n b e d o n e w i t h o u t l o s s o f g e n e r a l i t y. I f w e w r i t e

f = a0 + a, cos2nO ,

1 ~ p ) ( r ~ ( R ~ 1

where r, 0 a re me asu re d a s show n in F ig . 1 ,

R i s t he j e t r ad ius ,

a0, a , a r e cons t an t s ,

a n d p s t a n d s f o rV~/v~,

t h e n f , F a r e h a r m o n i c f u n c t i o n s w h i c h s a t i s f y t i le b o u n d a r y c o n d i t i o n s a t th e j e t b o u n d a r y.F o r

c o

F = ao + ~ . a , cos 2no = f , [

o n r = R .0 fOF _ 2r ip ~ , a ,,cos 2n0 = p ~ .

Or

I f f a n d F a r e th e r e a l p a r t s o f c o m p l e x p o t e n t i a l sh , H , t h e nco

h = a o + ~ , a . ( z / R ) ~ , . . . . . . . . . . . . . . S )

i ~ 6 )

I t i s a l so p o s si b le to e x p r e s s t h e e x t e r n a l p o t e n t i a l H i n a n o t h e r f o r m ; w e m u s t h a v e F t e n d i n gt o y a t i n f i n i ty, f u r t h e r H m u s t b e r e a l o n t h e y a x i s a n d o n t h e x a x i s b e t w e e n + 1, 0 ) a n d

- - 1, 0 ), a n d p u r e l y i m a g i n a r y o n t h e x a x i s o u ts i d e t h i s i n t e r v a l . C o n s i d e r a t i o n o f t h e s e c o n -d i t i o n s a n d o f t h e s o l u t i o n w i t h s l i p s t r e a m a b s e n t s h o w s t h a t H m u s t b e o f t h e f o r m

H = 1 - - z ~ ) ~ /~ 1 - -B ~ z ~ - - B ~ z ~ + . . ) ,


24/28

where B~, B~, . . . a re constants .

H = ( 1 - - ½ ~ - - } z ~ - - ~ z ° + . . ) ( 1 - - B < ~ - - B ~ - ~ q - . . )

B1 B~ B~= l + ~ - + ~ f f + 1 ~ - - . •

B1 z_2 B2 B a

F o r I z I 1 w e c a n e x p a n d t h i s e x p r e s s io n i n t h e f o r m

B 3

{- • . • • • . . • • • • ° • •

E q u a t i n g c o e f fi c ie n ts i n t h e e x p a n s i o n s 6) a n d 7) w e o b t a i n

B2 B a- g + 1 - 6 - - '

+ + . . . ; a~ R e = - B l + - ~ - + 8

2 * *

,

B 1a 0 = 1 -+- - ff +

B~a l ( 1 ~ 2 ~ ) R _ _ 2 7 - - - 1 + 8 -

B1

a a 1 -+2 ~b )R-6 = -

7)

N e x t al, a~, a~a r e e l i m i n a t e d f r o m th e s e e q u a t i o n s a n dBI, B2, B8e v a l u a t e d b y s u c ce s si v e a p p r o x i -m a t i o n a s s e ri e s e x p a n s i o n s in R . R e t a i n i n g t e r m s u p t o R ~ o n l y w e g e t

B I = ½ qR 4 + ~ q R 8 l - q ) + i ½ ~ q R~2 1- q) 3 - q) ,

B ~ = ~ q R S + ¢ ~ q R ~ 1 - q ) ,

B 8 = - ~ q R I ~

1 - - p _ v ~ - - V ~where q

1 + p v ~ + V S

W ith these v a lues o f B~ , B , , B3 we ca n ca lc u la t e the va lues o f ao , a~ , a s, a , t o th e same o rde r o fa c c u r a c y a n d o b t a i n

ao = 1 + -~ q R ~ + -~:~q R 8 3 - 2 q ) + ~ q R 12 5 - 5q + q~) ,

a ~ = 1 + t > ) 1 - ~ q R ~ - ~ q R S 2 - q ) ,

a~ = 4 1 + t ) ) 1 - - ~ q R ~ - ~ q R ~ 9 - 4 q) ,

aa = S (1 + p)x

23


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i , : ° ,

A s a n e x a m p l e w e s h a l l c a l c u la t e t h e s p a n l o a d i n g g i v i n g m i n i m u m i n d u c e d d r a g f o r a w i n gf o r w h i c h t h e s l i p s t r e a m ' d i a m e t e r i s o n e - h a l f t h e s p a n a n d t h e v e l o c i t y i n t h e j e t i s t w i c e t h es t r e a m v e l o c it y. W e t h e r e f o r e p u t

a n d o b t a i n

VS 1 ' • 1 - - p _ 0 . 6 ,R = 0 . 5 , v / V = 2 . 0 , P - - v 2 - - ~ ' q - - 1 + p

a o = 1 - 0 0 9 5 ,

B 1 = 0 0 1 8 8 6 ,

a 1 = - - 0 . 1 9 9 1 .

B s = 0 . 0 0 0 2 9 ,

a ~ = - - 0 0 1 2 5 ,

B a = 0 . 0 0 0 0 1 .

a 3 = - - 0 . 0 0 1 6

T h e c a l c u l a t e d s p a n - l o a d i n g d i s t r i b u t i o n i s g i v e n in F i g . 2 a n d c o m p a r e d w i t h t h e e l l ip t i cd i s t r i b u t i o n . I t w i l l b e s ee n t h a t t h e d i f f e re n c e s a r e n o t l a rge , in s p i te o f t h e e x t r e m e v a l u e s o fr a d i u s a n d v e l o c i t y r a t i o w h i c h h a v e b e e n u se d . T h e i n d u c e d d r a g i s 2 . 7 p e r ce n t . g r e a t e r t h a nf o r t h e c a s e o f e l l i p t i c l o a d i n g w i t h s l i p s t r e a m a b s e n t .

I t m a y b e c o n c lu d e d f r o m t h e a b o v e t h a t m i n i m u m i n d u c e d d r a g f o r a si n gl e s l ip s t r e a m i so b t a i n e d i n a l l p r a c t i c a l c a se s b y m a k i n g t h e s p a n l o a d i n g e l li p ti c , a n d t h a t t h e i n d u c e d d r a gi s t h e n n e a r l y t h e s a m e a s f o r e l l ip t i c l o a d i n g w i t h s l i p s t r e a m a b s e n t .

4. P a i r o f S y m m e t r i c a l ly P l a c e d J e t s . - - W ea g a i n t a k e t h e s e m i - s p a n t o b e u n i t y a n d a d o p tn o t a t i o n a s s h o w n in F ig . 1. T h e n t h e f u n c t i o n s f a n d F a r e g i v e n b y

f = a o + ~ , a c o s nO ,

( + pF = a o + ~ , a . [ - - - ( R y + ( I ~ - . p ) ( R ) 1 n O,a n d t h e a s s o c i a t e d c o m p l e x p o t e n t i a l s a r e g i v e n b y

oo

h = ao + X S Y1 \ R /

H = ao + a . (& ' ) +1 \ R / ' ,

w h e r e Z l = z - b , p = V S / v i

a n d b i s t h e d i s t a n c e o f t h e j e t c e n t r e s f r o m t h e m i d - p o i n t o f t h e w i n g . A s b ef o r e w e c a n e x p r e s sH as

z S _ b s ( z s - b s ) s . . ,

w h e r e A 1 a n d A s a r e r e a l c o n s t a n t s . T h i s f o r m s a ti s fi e s t h e s y m m e t r y c o n d i t i o n s a n d t h ec o n d i t i o n s a t i n f i n i t y.

T h e p r o c e d u r e f o r t h e d e t e r m i n a t i o n o f t h e c o n s t a n t s f o ll ow s t h e s a m e l i n es as b e f o le , b u ti t i s m o r e c o m p l i c a t e d a n d w i l l n o t b e s e t o u t h e r e . T h e r e s u l t s o b t a i n e d f o r tw o e x a m p l e s w e r e : -

Case Ib = 0 . 3 , R = 0 . 2 , V / v = 0 . 5 ;

A 1 = 0 . 0 0 5 3 6 , A s = 0 . 0 0 0 0 4 2 .

a o = 0 97 6, a~ = - - 0 . 104, as 0 .0 37

C a s e I Ib = 0 . 5 , R = 0 . 2 , V / v . - 0 . 5 .

A , ----- 0- 01 73 8, A ~ = 0. 00 08 5 ,

a o = 0 . 8 8 8 . a l = - - 0 . : 1 8 4 , a 2 = - - 0 . 0 4 9

2 4


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T h e c o r r e s p o n d i n g c a l c u l a t e d s p a n l o a d i n g s a r e g i v e n i n F i g . 3 ~ n d 4 . I t w i l l b e se e n t h a t t h el o a d i n g s d o n o t d i f fe r g r e a t l y f r o m e l l i pt ic , a l t h o u g h t h e r a t i o o f j e t v e l o c i t y t o s t r e a m v e l o c i t y,w h i c h h a s b e en t a k e n e q u a l t o 2 . 0 i n th e e x a m p l e s , i s h i gh . F o r t h e f i rs t e x a m p l e w i t h t h es l i p s t r e a m s n e a r e r t o w a r d s t h e c e n t r e o f t h e w i n g ( F i g . 3 ) t h e i n d u c e d d r a g i s 1 . 5 p e r c e n t . le s st h a n f o r a p la i n w i n g w i t h e l li p ti c lo a d i n g , a n d f o r t h e s e c o n d e x a m p l e ( F ig . 4) t h e i n d u c e d d r a gi s 2 . 5 p e r c e n t . l e ss t h a n f o r t h i s s t a n d a r d c a s e.

I t m a y b e c o n c l ud e d a ls o fo r t h e c a s e of a p a i r o f j e ts t h a t m i n i m u m i n d u c e d d r a g i s o b t a i n e di n a ll p r a c t i c a l c a s e s b y m a k i n g t h e s p a n l o a d i n g e l l i p ti c , a n d t h a t t h e i n d u c e d d r a g i s t h e np r a c t i c a l l y t h e s a m e a s f o r t h e s t a n d a r d c a s e o f a w i n g w i t h e l l ip t ic l o a d i n g i n a u n i f o r m s t r e a m .

P

X

Y

Zr, 0

bZl

v

V

q

q~

k

K

L

D~

N O T AT I O Nde ns i ty o f f lu id ~

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

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

z i yp o l a r c o o r d i n a t e s m e a s u r e d f r o m c e n t r e o f j e td i s t a n c e o f c e n t r e o f j e t s f r o m c e n t r e o f w i n g

z - - b

r a d i u s o f j e t

ve loc i ty ins ide j e t

v e l o c i t y o f m a i n s t r e a mV v

1 pl + pv e l o c i t y p o t e n t i a l i n s i d e j e t

v e l o c i t y p o t e n t i a l o u t s i d e j e t

c i r c u l a t i o n a t s e c t i o n o f w i n g i n s id e j e t

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

d o w n w a s h a n g le f a r d o w n s t r e a m

wing l i f t

i n d u c e d d r a g

No. Author1 K ~ r m ~ n a n d Ts i e n

2 B e t z . . . .

3 K ~ I n ~ n a n d B u rg e r s

R E F E R E N C E STitle etc.

L i f t i n g L i n e T h e o r y f o r a Wi n g i n N o n - u n i f o r m F l o w.Quarter ly of App l iedMathemat icsVol. 3. Ap ri l , 1945. p. 1.

S c h r a u b e n p r o p e l l e r m i t g e r in g s t e m E n e rg i e v e r l u s t . Vi e r A b h a n d l u n g e nz u r H y d r o d y n a m i k u n d A e r o d y n a m i k . K a i s e r Wi l h e lm I n s t i t u t eGot t ingen . 1927 .

A e r o d y n a m i c T h e o r y e d. D u r a n d ) . Vo l.II, Div. E . , pp . 242-245 . Ju l iusSpr inger, Ber l in . 1935.

25


27/28

7

2

F I G 1 N o t a t i o n f o r S l i p s t r e a m C a l c u l a ti o n s

I . O

O ~

O.~ . . . . .

d

a:

Z<

0 . 2

O

R = O ~,'ur/~/= 2 O

FIG 2

WiTH ,~LI PSTR E A14.;

NO SLI PSTRF_. M

x

o x o 0 6 x 0 8 i o

L i f t D i s t l i b u t i o n f o r a S i n g le C e n t r a l J e t

0 . ~

0 . 6

2~

~( 04O_J

( ~ 0 - 2

R:O,~ '

g-: o. ~

• uc/v. 2. O

~LIPSTRF_AN.

0 2 0 4 0 6 ~ o B

FIG 3 Lif t Dis t r ibut ion f or a Pair of Je ts

Case I

I o

C1.6

Z

<

O~ 0.4-

0 . 2

0 8

O

~.: O' 2 WITH S L l l

\ , . ,\

STREAM.

~= 0. 5 .. .. .. NO ~L'~PSTREA'~I.

V~ P. ,O . . . .

F I G 4 L i f t D i s t r i b u t i o n fo r a P a i r o f J e t s

Case H

2 6

(8904 9) W~. 13/806 K 5 6~50 ]~w . PRINTED IN GREAT BRITAIN


28/28

(9480, 10,2~6 92e 2).R.~J. Tecl~dcal Relmr~

Pub] iCat ions o f th e

A e r o n utica R e s e ar c h C o m m i t e eO F T H E A E R O N A U T I C A L R E S E A R C HE C H N I C A L R E P O R T S

C O M M I T T E E - -

193 4-35 Vol. I .Vol . IL

I93 5-3 6 Vol. I .Vol. II .

i936 Vol . I .

go l . I I .I93 7 Vol. I .

go l . I I .

1938 Vol . L

Vol. II .

A N N U A L R E P O R T S O FC O M M I T T E E - -

x 9 3 3 - 3 4~934-35

Aerodynamics . 4os . (4os . 8d . )Seaplanes, Structures, Engines~ Materials , etc.

4os . 4os . 8d . ) iAe rodyn am ics. 3os. (3os. 7d.) IStructures, Flutter~ Engines, Seaplanes, etc.

3os. (3os. 7d.)Aerodynamics Genera l , Performance, Airscrews,

Flutter and Spinning. 4os. (4os. 9d.)

Stability and Control, Structures~ Seaplanes~En gines , etc. SOS. (5os. l od.)Aerodynamics Genera l , Performance~ Airscrews,

Flut ter and Spinning.4os (4os. 9d.)Stabil i ty and Control , Structures~ Seaplanes,

Engines, etc. 6os. (6IS.)Aerodynamics Genera l , Performance, Airscrews,

SOS. SIS.)Stabil i ty and C on tro l~ Flutter~ Structure s,

Seaplanes, W in d Tun nels, M aterials . 3os.(dOS. 9d.)

T H E A E R O N A U T I C A L R E S E A R C H

Is. 6d. (IS. 8d.)Is . 6d. (Is . 8d.)

Ap ri l I , 1935 to De cem ber 3 I , I936 . 4s . (4s. 4d . )I9 37 ~s, (zs. ~d.)x938 xs. 6d. (Is . 8d,)

I N D E X E S T O T H E T E C H N I C A L R E P O R T S O F T H E A D V I SO R YC O M M I T T E E O N A E R O N A U T I C S - -

Dece mber x , I936 m June 30 , 1939 .J u ly I , 1 9 3 9 - J u n e 3 0, 1 94 5.July I , 1945 m jun e 3 o , 1946.Ju ly I , 1946 - - D ecember 3 I , I946 .J a n u a r y I , I 9 4 7 - - J u n e 3 o , 1 9 47 .

R . M . N o . I 8 5 o .R . M . N o . I 9 5 o .R . M . No . 2050 .R . M . No . 215o .R . M . No . 2250 .

~ s . 3 d . i s . 5 d . )i s . i s . 2 d . )I s . i s . i d . )IS. 3 d. (Is. 4d.)i s . 3d , i s . 4d . )

Pric es in brackets include postage ]

Obtainable f rom [• • •

H is M aJesty s Sta t ion ery OfficeL o n d o n W. C . 2 : Yo r k H o u s e , K i n g sw a y

[Post Orde rs--P.O . Box No. 569 London 8.E.L]Ed inbu rgh 2 : I3A Cast le St ree t M anchester 2 : 39 Kin g StreetB i rmingham 3 : 2 Edm und St ree t Ca rd i ff : I S t. And rew ' s Crescen tBristol ~ : To w er La ne Belfast • 8o Chich ester Street

or through any booksel ler.

S.O. G~de No. 23-2368

Slipstream Theory

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