Aerofoil theory for swallow tail wings of small aspect ratio

REPORT No. 41

2 4. Nov 1950

Kluyverweg i - .

THE COLLEGE OF AERONAUTICS

CRANFIELD

'Jtv^

-AEROFOIL THEORY FOR SWALLOW TAIL WINGS

OF SMALL ASPECT RATIO

by

A. ROBINSON, M.Sc, Ph.D., A.F.R.Ae.S. of the Department of Aerodynamics

TTifs Report must not be reproduced without the permission of the Principal of the College of Aeronautics.

2 4.

Nov 1950

REPORT NO. k^

T H E C 0 L L 2 G E O F A E R O N A U T I C S C R A N F I E L D

Aerofoil theory for swallow tail ivings of small

aspect ratio

-by-A. Robinson, M . S c , Ph.D., -by-A.F.R.Ae.S.

S U li I.i A R T

A method is developed for the calculation of the aero-dynamic forces acting on a 'swallow tail' wing of suiall aspect

ratio. Lift, induced drag, and aerodynamic centre position of

simple swallo\Y tail v/ings (Pig.l(b)) are computed as an applica-tion. For a given incidence, lift and induced drag are, Y/ithin the limits of the theory, proportional to aspect ratio and inde-pendent of speed. The chordwise lift distribution rises linearly from, zero at the apex, drops rapidly in the region of the root chord trailing edge, and then decreases gently to zero.

oOo

1, Introduction and discussion of results.

In a remarkable paper published some years ago (ref.1), R.T.Jones put forward a theory for low aspect ratio pointed wings at all speeds, both belo\7 and above the speed of sound. The theory was based on the idea, due to Munk (ref.2), that the in-duced velocity field set up by a slender moving body in a trans-onic plane, is essentially two-dimensional. In spite of the simplification involved in this idea, the results of ref.1 were borne out in a striking manner, both by relatively more exact theories and by experiment. The theory was extended by

H.S. Ribner (ref.3) to permit the calculation of the stability derivatives of low aspect ratio v/ings. It has also been used for the calculation of the effect of controls of different types by J. Deyoung (ref.4) and ii.D. Hedges (ref.5). G-.l], "Tard has applied an equivalent method to problems of wing body interfer-ence at supersonic speeds in a paper v/hich also includes a rigorous justification of the basic assumption mentioned above for the case under consideration (ref.6).

In the present paper, we are concerned r/ith the calcu-lation of the aerodynamic forces acting on small aspect ratio

' SY/alloY/ tail' Y/ings such as depicted in Figs. 1 (a) and l(b). It is assumed that the outline of the Yving planform varies mono-tonicaHy from a pointed nose to pointed tips (i.e. -r^ < 0 along ABC in Pig. •l(a) and '^ > 0 along A F E ) , This case is outside the scope of the methods yfhich are given in the papers mentioned above although for one particuls.r planform, a solution for a mathematically equivalent problem is described in ref.7.

Numerical results have been calculated for the case of a 'simple sv/alloY/ tail' wing (Fig.l(b)). The results depend on tYTO parameters, (i) the aspect ratio, (span)/area, or in fact the ratio of any tvro typical lateral and longitudinal dimensions, and (ii) the ratio c/c , v/here c is the root chord of the vidng,

' ' o' o ' and c is the longitudinal coordinate of the tips, meastu-ed from

the apex (Fig. 2). As in the case of the delta vdng (rcf.-l) it is found that the aerodynamic forces acting on the T/ing, for given air density, speed, area, incidence, and for given ratio c/c are actually proportional to the aspect ratio so that the results can be represented as functions of the ratio c/c only.

Figs. !+{a.) and 4(b) show the chordv,dse lift distribu-tion, i.e. the pressure difference integrated in spanv/ise dir-ection for a given chord position, for tvro values of the psirameter c/c . It will be seen that there is a discontinuity in the '

ohordv/ise .lift distribution (and indeed of the pressure) across the coordinate of the v/ing root trailing edge. Although such discontinuities may occur also under the conditions of ref.1, they are impossible- in a real fluid, but they reflect a rapid vaxiation of the pressure in the region under consideration.

Figs. 3, 6 and 7 show the lift curve slope, the posi-tion of the aerodynamic centre, and the induced drag coefficient respectively, as functions of the parameter c/c . The numerical values used for their construction are given in Table I below.

(in that table, A denotes the aspect ratio, and h denotes the distance of the aerodynamic centre from the iving apex iii frac-tions of the chord c).

c/c^

1.0

1.5

2.0

TABLE

rv^

1.571

1.012

0.739

I

h

0.667

0.555

0.504

0.3173

0.3379

0.3836

2, Analysis

',Te talce the origin at the ape;: of the wing, vfith the X-axis parp.llel to the direction of f lev/, the y-axis pointing to starboard, and the z-axis pointing upwards, so that the coord-inate system is right-handed. Let ^ be the total velocity potential and 0 the induced velocity potential so that

= Vx + 0, v/-here V is the free stream velocity, ".7e denote the induced velocity components by u, v, \7, u =

_ M

M

ex ' ^ ~ dy '

w = T^ ,_{O z} so that the total longitudinal velocity equals V + u. By Bernoulli's equation, the pressure difference at a point of the aerofoil, /H^, is related to the longitudinal induced velocity

(1) A P = - 2pVu = - 2pV 1^

Y/here p is the air density. In accordance v;ith the introduc-tion, the partial differential equation for 0 is taken as

(2)

ii

il

_

= 0

dy dz

The boundary c o n d i t i o n a t t h e a e r o f o i l i s

(3)

w=

'4 = - V a

C O R R I G E N D U M

The form for the velocity potential assumed in the second equation of (k) on page U of the original report leads to an anti symmetrical pressure distribution with respect to the xz-plane. The correct pressure distribution should be symmetrical with respect to the xz-plane, and the corresponding velocity potential is

0 (x, V-,) = .R fvC/

0

Aill

tx'X.

y t (*)]'-'

dt + A(t)g(t,ri)dt 'O

for

c^ < X < c , where

!Z^(t,Ti) = f (K'-E

"' .E.. (ra(t)]^-[b(t)]^)K'^

ds

0 [a(t)j2-s2

J([.{^-s')is'-^^(t)Y^)

where K' and E ' are the complementary complete elliptic integrals of modulus k = b ( t ) / a ( t ) . With this modification, the analysis can b e continued as before and a d e

-tailed revised version will be published in the near future. The corresponding numerical results are given in Table I below which replaces Table I of the original report.

T A B L E C / C Q

1.0

1.5

2 . 0

T>

1 . 5 7

1. 19

1. 10

h

0 . 6 6 7

0. 600

0 . 5 9 9

0 . 3 1 8

0.14-20

O.I4.56

YiThere a is the local incidence, positive in the nose-up sense. Y/e shall consider only the case of a flat v/ing, a = const.

YIe xjTxtc y = _r a = + a(x) for the spanvrise coordi-nates of the leading edges, 0<x-^c, and y = + b = + b(x) for the spanv/ise coordinates of the trailing edges, c ^ x $ c , so that a(o) = b(c ) = 0 , a(c) = b(c). 'He also introduce the complex variable "o = y + iz, so that 0 may be regarded as a function of the tvro variables x and r\, 0 = 0{x,'r\). 0 may be vo'itten as the real part of a complex function of x and ri, v^hich

is an analytic function of "1.

In addition to satisfying Equation (2) and the boundary condition (3), 0 must be such that Zip, and hence -v^ , vanish at the trailing edges of the aerofoil (Joukov/ski-condition), while these quantities may become infinite at the leading edge. Finally, since the aerofoil does not peneti-ate any transverse plane x = const,<r0, it follows that 0 must be constant, and may be assumed to vanish for such x. All these conditions

suggest that 0 can be represented in the form

r

fVX

(R

(4) 0(x,r)) =/

A(t)

d t

for x < 0

for O-cx^c

^A

U oJlalt)]" -

r]-• , - _ A ( t )

L :

^°4-(t)]

d t + 2 2 - r\

Ut)l

[a(t)J" - T i '

for c < x<:c

o

where V<- denotes the real port of a complex number as usual, and A is a real function of its argument, which remains to be determined. In fact, differentiating (4) vdth respect to x and putting T) - y, v/e see that at the aerofoil

(5)

r

A(x)

u = <3x \

y&(x)]' - /

for 0<Cxdo

for c < X < c

o ^

This shows that the pressure difference becomes infinite at the leading edges and vanishes at the trailing edges, in accord-a,nce with the Joulco\Yski condition.

To find the normal induced velocity, v/e have to differ-entiate (4) virith respect to z. And, since for any analytic function f (il),

where J denotes the (real) coefficient of the imaginary part of a complex number, as usual, vre obtain

(6)

_dz

A(tl

TÏÏÊ^'-ri' ' '

f o r x < 0 f o r 0 < x < ' c A ( t )

nx

/t(^'-V

A ( t ) / n ! z M i ! ^ a t

[a(t)]^-

••n f o r c < xKc o

Vfe now have t o deten'nine A l t ) from the c o n d i t i o n t h a t

lim 3 ^

rf^

ri-^j èT] U A ( t )

(7)

=v4(t)]

d t 2 '' - -n""

lim 3

6_ 0^1

A(t) ,^ a

yföt)]

2 2 - Tl f o r 0 ' C x < c >= Va f o r c -«r- x ^ r c o ^ f o r any p o i n t ( x , y , 0 ) on t h e a e r o f o i l . 17e n o t i c e t h a t l i m Tl-

Jö'o

7

= n , cfl^ /nl^L^ML'

a t a l l p o i n t s (x, y , O) on t h e a e r o f o i l , so t h a t by (6) t h e / n o r i a a l , . , = 0

normal induced v e l o c i t y i s constant along the chords of the

aero-f o i l aero-for any choice oaero-f A ( t ) . I t s magnitude i s determined

en-t i r e l y by en-the values of en-the i n en-t e g r a l s i n (6) ahead of, and i n

the region of, the leading edge.

I t i s easy to show t h a t the f i r s t equation i n (7) i s

s a t i s f i e d by

(8) A(t) = - Va a(t) a' (t) ,

0<.t<Cc

In fact

f- Va a(t) a' (tjl ' ^^ • • = - Va x/[a(t)]^-'

/ r ^, Ni 2 2

•-yii(t)j%

and so, for points (x,y,0) on the a e r o f o i l ,

t=x

t=0 = Va

iri ->y(a(x)] -ri'

r]^y

r Va a(t) a' (t")]

dt _{= Va}

_É

U o

7&(t)j-n2 L vfööjV

JL^-

_{= Va,}

as required.

I t v d l l be seen t h a t the solution implied by (8) agrees vdth

r e f . 1 . for the p a r t i c u l a r case considered there.

The l e f t hand side of the second equation i n (7) now

becomes, vfith the specified value of A(t) for 0 < , t < . c ,

lim 3

•n-^y ërr\

(j Va a(t) a' (t)J

U d t OX

s/^(¥^'

on

A(t) pLLMt^ll ^,

Ca(t)32 - r , 2

l i m . Va •n-^y ^

. v/fM"

-ri

+ lim. y

•n-j^y

a-n

fT^

A(t)

^r- - rb(tli'

d t = Va -f

_{—. : r r r + liin U ^}

Va y dr]

A(t) A ' - f c w j ; , ,

[a(tr-

Tl

where we confine ourselves to p o i n t s of the a e r o f o i l for v/'hich

y ^ a ( c ) . The condition for A ( t ) , c < t < ' c , no?/becomes

(9) lixn

Tl-»y

a-n

La(t)j'-il^

= - Va "

for points (x,y,O) of the a e r o f o i l such that x > c , y > a ( c ) .

r \ a

Yfe may modify the l e f t - h a n d s i d e of (.9) s l i g h t l y by r e p l a c i n g -^r by -T" , s i n c e TI = y •+• i z . I n t e g r a t i n g Y/ith r e s p e c t t o y on b o t h s i d e s of (9) we t h e n o b t a i n an a l t e r n a t i v e c o n d i t i o n f o r A ( t ) , v i z .

(10)

lim 9

•n-^y

Mt) h—^^^^,

dt = - Va

Vy^ -

[a(c

)]^ +

c o n s t . I n f a c t , a s o l u t i o n of (10) f o r any v a l u e of the c o n s t a n t on t h e r i g h t hand s i d e vrould l e a d t o a s o l u t i o n of ( 9 ) , b u t t h e c o n s i d -e r a t i o n of th-e limitin"-- cas-e x - ^ c , y - ^ a ( c ) - shows t h a t vr-e

^ o o

raust t a k e O a s the a p p r o p r i a t e v a l u e of the c o n s t a n t . '7c may noYf l e t r\ t e n d t o y ( i . e . l e t z t e n d t o O) on t h e l e f t hand s i d e of (lO) b e f o r e i n t e g r a t i o n , and t h e n l e t (:;:,y,0) t e n d t o the l e a d i n g edge so t h a t x and y a r c l i n k e d by the

r e l a t i o n y = a ( x ) . Then ^ fpc

A(t) A ^ J M ^ dt 4

C a ( t ) ] ^ - /

nx

A ( t )

[a(t)f -i:a(x)]2

so t h a t (10) becomes J c d t

W

.(11)

A(t) y 4 ( x ) ] ' - & ( t ) ] ' ^, ^ V a , > 4 ^

y [a(x)J^-(a(t)]2 ^^^

2 2 - a 0 Y/hcre a = a ( c ) , T h i s i s an i n t e g r a l e q u a t i o n of V o l t o r r a ' s t y p e . The f o l l o w i n g simple nuinerical method f o r i t s s o l u t i o n a v o i d s

any d i f f i c u l t y v/hich might be c a u s e d by the f a c t t h a t t h e i n t e g r a l on the l e f t hand s i d e of ( l l ) becomes i n f i n i t e a t the upper l i m i t of t h e i n t e g r a l .

F o r any x, x ' , x " , c j ^ x'<;; x " ; ^ x < c , v/e liavo, by a mean v a l u e theorem of the i n t e g r a l c a l c u l u s

nx

(12)

. y&w]'-ra(t)]' .

rs^

-ff|)viw]=-§(uT

nx

i' ( t ) d t a ' ( t ) d t

yÉGi"^^^i(Ip

la(x)j'^-|b(ë)]'

U X

r . -1 a(x")

/sxn • 7 s - oj-ii —T—\ L a(.x) a ( x )

-1 aC

fO

Yirhere t. is some intermediate value between x' and x",

YIe

shall

accept the approximation that

B,

is midway between x' and x" ,

i:

= i ( x ' + x " ) .

To solve (11), we divide the interval < c , c > into m

equal sub-intervals, <ic = x , x . > , < x , , Xp>•,...,< x..^__^,

X = c > ,

"lie

then satisfy (1I) for x = x., x = x^,...,

r^ =

x ,

evaluating the integral on the left hand side approximately by

applying (I2) to all the sub-intervals in < c , x ^ . In this

Y/ay vre obtain the follovdng triangular system of m linear

equa-tions for the unknovms A, =

A ( L ) , t,, =^{x,

+ x , _ . ) , k = 1,2, ...,m.

(13) 2 1 I T V T I ^ Ö ^ ^ K ^ r^""'

k=1 ^ ^=^k^ / ^ L

- s m

-1 ^ ^ \ - /

K-Ya^^ixJ\ - a^ , n = 1,2, . . . ,

Por a si...,do swallovy-tail v/ing ( F i g . l ( b ) ) , a ' ( x ) = const.

= — = tan x> where y i s the semi apex angle of the vdng.

Hence, p u t t i n g B(t) = ~—'^•^ , B, - A-VVa tan YJ we obtain the

m

system of equations

Va tan y

(^4) ^&afe„)J'-D={sJ]' [.

s m —r " A - s m •1 ^ ^ ' . -1 »(=T£-1 a-(x J n

airrjpc

=/&M'-a n = 1, 2 , . . .,m

For the calculations on Y/hich Figs. 4-6 are based,

<:;^c , c ^ vfas divided into five sub-intervals, ra = 5. To obtain

an idea of the accuracy of the solution, similar calculations vrere

made for m = 3, and m = 4, for a value of — = 2 . The results

are shovm in Fig.3, from which it appears that there is good

agreement betv/een the quantities obtained for m = 4 and m = 5,

Y/hile the calculations based on m = 3 are inadequate.

There are other \/a;^s of applying the above mean value

theorem. For example, instead of (12), we might have used the

folloYdng formvila

-(15)

A^

X,

>y £a(x)] -[a(t)J J ^ , a(t)a'(t)

t(r.)VS 'f)V

i(^)r-ü

(f)}'-= :,^êb-V&«J'-&fcO^

_{a(t)a' (ë)}

a(t)a' (t)dt

= '^;-:y&^^3'-r^fe)]' yB-)f-[e^u')y -ypü)j^-ra(x")j^

where g is again some intermediate value be tv/een x' and x" , but of course not necessarily the same as before. (15) can readily be used to find the exact value of A ( C + O ) , i.e. the limiting value of A(t) as t tends to c from above. Putting

0

•ri- _{= c and x" = x in (15) and substituting the result on the}

o l e f t hajid s i d e of (11), vre o b t a i n

; ^ y r a ( x ) J ^-|h(0] Vp(x)]^-La(CoO' = W[^U)f- a^

a ( 5 ) a ' (li) But a ( c ) = a and so o ' o 0 A(i;) = Va

„a(e)a'(ë)

Nov/ l e t X t e n d t o c from above. Then E t e n d s t o c a l s o ,

o ^ o ' and so in the limit

(16) A ( C + 0 ) = Va a' (c )

The corresponding value for B ( C ^ + O) is (17) B ( C + 0) = 1

o

Still considering the simple swallov/ tail vdng let r\ = .

O 0 I t i s of i n t e r e s t t o d e t e r m i n e t h e l i m i t i n g form of t h e integï>al e q u a t i o n ( I I ) a s A t e n d s t o 0 . We i n t r o d u c e t h e non-dimen-s i o n a l v a r i a b l e 3 by x = ( l + ^ non-dimen-s ) c . Then a ( x ) = a ( c ) — = a (1 + X 3 ) , b ( x ) = a (I +7\)s o so t h a t (11) becomes

(18)

7\

u

A' o

*(o-) / ( j l M p i I ^ S ao- = Va tan rJ^T^sf - 1

V ( l - H A s ) 2 - ( l + A c r ) 2

Y/here v/e have p u t t = (l + Acr)c , A (cr) = A ( t ) = Al(^+}sa•)c j , F o r s m a l l A , e q u a t i o n ( I 8 ) t e n d s t o t h e form

(19)

A—^

W s - cr do- = 2Va t a n y s/s ,

and we may verify directly that this equation is satisfied by (20) A*(o-) = ^^ ^-^ T •

On t h e o t h e r hand, f o r A — • O Ö , e q u a t i o n (18) becomes I s (21) A(a-)dcr = Va t a n Y s . . J o T/hich i s s o l v e d by (22) A(cr) = Va t a n y

3. Calculation of aerodynamic forces

Having computed the function A ( X ) , we are in a position to determine the aerodynamic forces which act on the \i±ng. By

(1) and (5), the pressure difference at a point (x, y, 0) of the aerofoil is given by - 2pV

J M

7&W]'

-

J (23) A P =

- 2pvA(x) / i L - ^ - ^ k i !

2 2 [ a ( x ) ] ' ^ - y :'or 0 < x < c f o r c < x < c , The l i f t p e r u n i t chord i s g i v e n by

rA(x)

(24) ^ ( x ) = ^ p dy J - a ( x )

for 0<|,x<;;^c . Thus, in that case

-f(x) = - 2pV

na(x)

A(x)dx

-a(x)y|a(x)7 ~

y^

.:-- - 2pYA(x) - -| a(x)

h ' '

^ J _a(x)

= - 27ï;pYA(x)

Or, taking into account (o),

(25) -f(x) = 27ip\r^a a ( x ) a' (x) ,

_{0 < x c c ^ .}

On t h e o t h e r hand, f o r c <_x<^c, p-'b(x) •t(x) = A p dy + U - a ( x ) u b ( x n a ( x ) A P <iy = 2 h ( x ) a p a ( x ) ^ p dy = - 4pVA(x) b ( x ) (^ r\a(x)

^y'-r^(xli'

2 2 b ( x ) ^ § ( x ) J - y L e t k' = t y a(2

j - , k = y - k'^

k , and i n t r o d u c e t h e v a r i a b l e (3-y = a ( x ) dn (i3.,k) /ïAiere . . .

r

Y/here dn(!3,k) i s t h e f a m i l i a r J a c o b i a n e l l i p t i c f u n c t i o n . Then 2 2 2 2, ana y 2 - j j ( x ) ] 2 = a ( x ) k^ cr/-(p,k) , [ ^ ( x ) ] ' - y'^ = a ( x ) k^ s n ' ( p , k ) i dy 1^ - a ( x ) k En(p,k) cn(f3.,k) Hence o(x)' ,y - & ( x j . ( x , 2

b(x)^ b^^y -y

cn^(;D,k) d[3 = - a ( x ) [E(k) - k ' ^ K ( k ) ] (x)

so that the expression for -^(x) becomes

KIV1

-êj

(26) -f(x) =- 4pVA(x) a ( x ) E

(fi)-i'i^^(f^}

y

F o r t h e case of a .ciir-ple sv/allov/ t a i l '.7ing vre have A ( X ) = Va t a n y B ( X ) , and so

(27) i ( x ) . - = 4pV^a a ( x ) t a n y B (x) E U/1

# ) -(^HMS/

F o r .•:': = £,_ , wc t a k e B t o be g i v e n 'X:>y B(i|, ) = 3, , a s de te rmin e d from (iii-)o Thv.i

(28) ^f (g, ) :. 4r7^a a ( x ) t a n y E, E

( • ^ ) - ( t f ^ ( ^

2 ^

The t o i ; ^ l l i f t , L, i s g i v e n by

(29) L -. ! -xOijdx = L^ -v I. . sav, vfhere

Jo ' " '

/TO Ï ' o r,c X

J

f(nr)d..

L r <J<=r ^ ( x ) 6 a /'o;r^ I n t e g r a t i n g (,2^);. .-.••o o b t a i n ii.'aTiediately ,2 r 2 2 (30) L^ =r T^fV a i a ( c ^ ) j = r.pV" a a

i n agreement w i t h r e f , 1. On the o t h e r hand, L^ must be o b t a i n e d by ?iu.;erical i n t e g r a t i o n , and i n vievr of the p r o c e e d i n g a n a l y s i s t h e use of the follov/i.ig simple formiila seems a p p r o p r i a t e

(31) L =

m 4pV^a t a n y ( c - c ) m

The p i t c h i n g moment round t h e apex, H, i s g i v e n by

(32)

ne

M =

-f(x)

_{dx = K „ + ï/i} f r

ne,

n. ^ ( x )

rP

dx ï,I = r X -^(x)dx

(33)

Thus, f o r t h e case of a simple swallow t a i l vdng,

M^ = 2 2 275: 2 2

X. 27i:pV a t a n y x d x = •=— p V a t a n y

U o

v/hile a n u m e r i c a l formula f o r i.i i s r (34) c - c m r m '^1 -a k •- k k=1 4pV a taji y ( c - c ) m m _K= t,, a(g, ) B, ^k k ' k

The d i s t a n c e of the aerodynamic c e n t r e from t h e apex of t h e a e r o f o i l i s t h e n given by d = 1V1/L.

F i n a l l y vre e s t a b l i s h a formtila f o r t h e induced drag of the a e r o f o i l , D . . D. i s t h e d i f f e r e n c e betv/een t h e s u r f a c e

' 1 1

p r e s s u r e drag D = La and the fonward s u c t i o n f o r c e D

* p s e x e r t e d on t h e l e a d i n g edges of t h e vdng.

(35) D . = D - D = L a - D

1 p s s

To calculate D Vire surround the leading edges of the vdng by small cylindrical surfaces S as given by the equation

(36) £ = r (ë, 9) = 'C ± + (+ a(^) + e cos 0) J_ + e sin O) k /v/here ...

Y/here s is a small positive quantity, and the limits of varia-tion of g and Ö are O ^ ^ ^ c , 0:$r9$27r. By the momentum theorem the force exerted by the air on the portions of the vdng Y/hich are inside S, F(e) say, is given by

O

(37)

F(e) = - p dS - p

S O

2 (£ dS)

where p i s the p r e s s u r e , _£ i s the v e l o c i t y v e c t o r , £ = (V+u)i + V_J2 + v/ k, and dS i s the d i r e c t e d surface element

p o i n t i n g outiTards from the surface. Thus dS = (^aAUp) ^^ ^ > so t h a t (37) niay be replaced by

(38) F(e) = - p(£0Ai;^) do d:, - p

s

4 ^-e^-^l)

d o 61L

Y/here -f or - are to be taken in the expression for +;- , as given by (36), on starboard and. port respectively. '.7c have

rgAz^r = ("^ sin 0_si + e cos 0 k ) A ( i + a' (^)j.) = + a' (^i)e cos 6i + e sin 6 Thus, we obtain for the longitudinal component D(e) of P(e), in

Y/hich v/e arc chiefly interested,

(39) D ( £ ) =

r

p a' (il) cos 6 e dG d^-p

S U U ( + ua' (i.^) cos C +v cos 0

+ w sin e) e dO di^ To continue, wc require formulae viiich express the infinitesimal behaviour of the velocity components in the neigh-bourhood of the leading edges. Confining oiorselvcs to

star-board wc see that equation (4) yields, for fixed x =. t, and small e,

, ^ ( / ï _ ^

Afe)

_{^fy èML}

yia(^)J^- y^ »/|(^)J^-|a(ii) + e cxp (iO)]^ V^2a(c^)e cxp (ie)

or

©

+ oi i (40) _{u = -} A(£) s/2a(t^) and s i m i l a r l y (41)

N/2a(i)

1 s i n - + o / ^ \ for 0<::c<c^

yffeFifeö'^i =i-l^°(i)

for c ^ < ë < c Again, the unbounded ccmponcnts of v and v/

depend only on the value of the integrands neai- the upper l i m i t

X = E, of the i n t e g r a l s in (4). Thus, vre may replace these

i n t e g r a l s by

^

n^

_{A(g) . a ( t ) a ' . ( t l}

dt ar

ad(R.

r^

u

A(iiy5m!lii5I a.Ctja'.Ct)

^^'=^ ^' ^ y [ a ( t ) ] 2 _ r

dt

for 0<il^-<lc and c <£<Cc, r e s p e c t i v e l y . I n t e g r a t i n g v/ith

respect to t , and then d i f f e r e n t i a t i n g with respect to IT, v/e

obtain the follov/ing expressions for v and v/

(42) V = - - i ^ Ü - < ^ - - J

••©•-

. . A ( £ l .

_{N y ^ T E T ^}

^ " ^ '' a" (L;)

for 0-<^ë<c , since ri = a(ë) + ee , as before. For the

1 . 6

?3 s m

-"©

(43)

w =

same case,

isi_3

J l

^(«'='' ^5) ./^:feöv

(^)°'..r'

" + 0 - =

A(c^)

1

a ' ( t ) s / 2 a ( i . )

^

cos r + 0/ —

(i)

Similarly, for c <^ ^ < c

(44) V = A k i

a' (li)y2o.(E)

w = -

Afe)

y&teFËfer^-i-/D

a ' ( ë ) y 2 a ( i )

Yfó may summarise (40) - (4!4.) i n the follov/ing formula^

1 . . 0 . V 1 \ _

GOÜ.

1

(45) u = - Q(E)Jz s i n I + 0 / A

_{V = •—^•^- — s m - - + o}

. a ' C . ) ^ '^

©

G(IO. I_

0

Y/here

(46) G(ë) =

r

_{A k L}

(dv^

/ 1

+ 0/

-1^

>/2l(i)

for 0 < ë < c ,

y2a(ë)

By B e r n o u l l i ' s equation

p = P + i p [y^-(V+u)^ - v^ - x^2j

Y/here P i s the pressure a t i n f i n i t y , and so

(47)

p = -ip|>(,j]^p|.^^l.o(l)

S u b s t i t u t i n g the values of u, v, w, and p from (45)

and (47) i n (39), and talcing i n t o account the contributions from

both port and starboard, v/c obtain

D(s) = 2p

§(E)]

Ü

-d?

kT.

i / s i n " - +-^ ^ —g \ 0.' (E) COS

- 7 ;

f

(50'

+ s m

_{— [ - a ' (4; s m —}

e / , /,.N . 0 „ J . 0

_COS

_{0 + —— s m — co£}

" \ . '^ a'(ii) ^

cos -r sin 0 ) d0 •+• o(l)

a' (H)

'- J

= 2p

JG(^)12 a' C-) d£; f 4 sin^ I COS0 ^

^ -^

V ^ Ta' 5 ) 1 '

O J -TH L ' ^ ü

2 sin^ I j dG + o ( l )

I n t e g r a t i n g

(48) D(e) = 27^p

Ü

a'(^0 \X^ J

1 ) dZ, + 0 ( 1 )

The suction force D as defined above equals ƒ - lim D ( e ) j .

V^ s - ^ o J

Also, since we are dealing v/ith the l i m i t i n g case of mngs of small

aspect r a t i o , v/e may assui"iie t h a t | a ' (?-S)j i s small compared vath

1, (For a simile sv/allow t a i l vdng a' (c^) = tan y i s proportional

to the aspect r a t i o , for given c / c . )

Hence

n°

(49)

D = 2r.p

&(a)1^

a ' ( 5 )

dë

For a simple sv/allow t a i l A/ing, the formula becomes

\o

(50) D^ = -^T^pV^ a^ of- + r.pv''^ a^ tan y j

| B ( L ) J

^ ^(g,)l ^ - ( ^ (^^)J ^ a^^

a (5)

For the special case of a delta vdng, the value of the induced. drag obtained vdth the aid of this formula agrees v/ith the result given in ref.1. For the general case, the second in-tegral in (50) G8J1 be evaluated numerically, as before.

LIST OP PJ]FSRENCi]S 1 . 2. 3.

4.

6.

7.

Author R.T, Jones M.id, ïiiunk H.S. R i b n e r J , Deyoung M.D. Hodges G.N. Ward M,A. H e a s l e t , H, Loma:c, and J . R . S p r e i t e r T i t l e , e t c . P r o p e r t i e s of l o v / - a s p c c t - r a t i o p o i n t e d wings a t speeds belov/ and above t h e speed of sound.

• N.A.C.A. R e p o r t No.835, 1946. The aerodjaicd-dc f o r c e s on a i r s h i p h u l l s , N.A.C.A.Roport No.184, 1924. The s t a b i l i t y d e r i v a t i v e s of lov/-a s p e c t - r lov/-a t i o t r i lov/-a n g i i l lov/-a x vdngs lov/-a t subsonic a.nd s u p e r s o n i c s p e e d s .

N.A.C.A. T e c h n i c a l Note No.1423, 1947. Span'.ïdse l o a d i n g f o r vdngs and

c o n t r o l s u r f a c e s of low a s p e c t r a t i o . N.A.C.A.Technical Note No.2011, 1950. Nose c o n t r o l s on D e l t a v/ings. College of A e r o n a u t i c s t h e s i s , 1950. (Unpublished). S u p e r s o n i c f lev/ p a s t s l e n d e r p o i n t e d b o d i e s , Qioartcrly J o u r n a l of Mech-a n i c s Mech-and A p p l i e d MMech-athemMech-atics, v o l . I I , 1949, pp. 7 5 - 9 7 . L i n e a r i s e d c o m p r e s s i b l e flov/ t h e o r y f o r s o n i c f l i g h t s p e e d s ,

F I G

I

F i e - 2 P»MEMSlOI»4S OP STA.niBOARe3>WIN© C - - r - 1 i . . . 1 • 1 ® t ® m « 3 m ^ 4-X m s 5 1

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t-5

V^c

2 ' 0

R G P O R T b J o 4-1 2 - 0 n o 4. cMCRDwjse w r T O I S T R I B U T I O M

1

A = A S P E C T R / V r i O F\ö' 5 V A R \ A T » O K * o r w » r T C Ü R V C S U O P E o - o C o ;

AC?

e>50 0 2 C O'tO

p i & c P O s y r i o r M off ACRooy»>aAMicS C;E^JTRK A T T OP» •W<tNI& A P E X , 4%. ( p R A C T i o t M S o r o)

1 0 r s a - ö c ^ P t O - r >/ARtAT>0»4 o r IMOÜCfeO * O R A o - c c e r n c i g M T