Wing body interference at supersonic speeds

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TECHNISCHE HOGESCHOOL VLIEGTUIGBOUVv'KUNDE

LüSHv ...L^ ^ 2 !u!i1950

Kluyverweg 1 - 2629 HS DÊÖAT_NO i J I A p r i l , J 9 4 7

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

C R A N P I F L D

Wine Body I n t e r f e r e n c e a t S u p e r s o n i c Speeds

S . K i r k b y , B . S c . , P h . D . , A . P . R . A e . S . , and A. Robinson, M.Sc., A . P . R . A e . S . • o O o 8UMARY -The i n c r e m e n t i n l i f t due t o T^dng-body I n t e r f e r e n c e a t s u p e r s o n i c speer^s i s c a l c u l a t e d a p p r o x i m a t e l y for an u n t a p e r e d v'ing, '••ithout sweepback.

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!•! Introduction

To estimate the increment in wing lift due to the presence of a body, it has been assumed that the body is represented by a cone travelling, vertex foremost, with its axis at incidence "U' . The velocity field due to the cone may then be considered to be

generated by an equivalent doublet distribution, along the axis pf the cone, for which the induced velocitj)-is determinable. In particular, the upwash velocity is evaluated along the mid-chord line of ohe wa.ng, to which Ackeret's theory is applied for the estimation of the increment in wing lift due to body interference.

. Some calculations are given to show that the use of Ackeret's theory is, foi- all practical purposes, permissible,

While it is believed that the calculations contained in this report represent an acceptable

approximation, further investigation of the subject is desirable.

1.2 Notation

P - air dönsi-üy

V - free stream velocity M - Mach number

M - Mach angle

€ - semi-vertical angle of cone

(3-/

M^ - 1

J - (cote )/Q

):] - wing span, tip t(> tip c - wing chord

Q - distance of wing leading ed^^e aft of vertex 1 of cone. c^ - distance of wing trailing edge aft of vertex

of cone. c' - distance of mid-chord line aft of vertex of

•s n cone,

A -(il?/2c'

-^ L - lift inoreij^ent due tp wing-]:?ody interference j^Cy - increment j.n lift coefficient, based on gross

wing area,;, due to wing-bod^^ interference di A L / ^ O V be

A c - i n c r e m e n t iin l i f t c o e f f i c i e n t , based on n e t win^3 are§L ( o r area, of wing ovüX^hang)

wioriin t u e iii§,ch cone of t h e body, due co win£,-boa^ i n t ' e r f e r e n c e

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1,2 riotatioxi ( o o n t d . )

X - chordwisö co-ox'uiiiate (measured from v e r t e x of cone agaijnst the d i r e c t i o n

of flow)

y - spanwisb co-ordinate (measured from centre line of cone and positive to starboard)

z - normal no-ordinate (measured from centre line of cone perpendicular to xy-plane and positive downwards)

Ov - wing incidence (in radians)

\lf " incidence of cone centre Ij.ne (in radians)

</-, - induced velocity potential duü to wing pC, - induced velocity potential due to body w - downwash vel^citv induced bv body

c

£^ p - pressure increment induced by body r, 9 - cylindrical co-ordinates

2. Analysis

I t 'has booii. enown by i s i e n (ref . i , e q . ' i } thao f o r a cone c r a v e l i i r i ^ a t supersonic speed the; maacod v e l o c i t y p o t e n t i a l may be exjortissed i n tho form

_ L " ^ cosh^(x/pr) * ^

-where \S =\/M — 1 and (x, r, 0) are cj/linarioal co-ox\iinatos referred to the vertex and axis of tho cone (fig,l),,

According to Tsien the appropriate function is given ty

f(-x -Q^r. cosh u) = K.("X - Q r . cosh u ) ,

where K. i s a c o n s t a n t . Hence, integi-atixig, as in r e f . l , eq. 4a, j ^

/• X y X \

/ = K A CO S 6 Ij - - I 1 - f

The Qj};:^nventipnal rectang-ular cartesi-an

o c - p r d i n a t e s for 'an a i r c r a f t ( f i g s . l , 2) ar^^ r e l a t e d to t h e above c y l i n d r i o a l c o - o r d i n a t e s bv the equatjons

y = r , sin Q, z = r . cos 0 ,

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-4-Hence the downwash velocity, w , due to the above doublcit distribution, at a point in thv:. x-y plane

(z = 0) is

* » « « i » \ ^ - L y

Sinc^:-' this downwash velocity, in the x-y plane, due to the cone is constant along a line x/y = constant, the

chordwise average of downwash velocity is approximately equal t® its value at mid-chord. It will now be assiAmed ttiat the lift distribuGion along a specified chord can be (estimated by Ackeret's formula, independently of conditions elsewhere along the span ('strip theory m e t h o d ' ) . In addition it will be assumed that the incidence is in fact

constant along any given chord, and is given b y its actual value at mid-chord where

X = - 3(0 t c ) = - 0' . 1 2

Thus the increment in wing lift ^ L resulting from the induced upwash, - w^, of the body v/ill, at supersonic sp.-ied, be given b y

i l ^ l = ieXc (- We),

r^-)

where c is the wing chord. The validity of tliLs approximation is discussed in the apx-'endix.

Since the interference of the body will only exist in its 'after-cone', which is of semi-vertical angle _'•o-'

1^'-= sin" (lA) = cot" p

and has its vertex and, approximately, its axis coinciding with those of tho cone, it is necessary to consider two

cases when evalimting the increment in wing lift: namely (i) when the wing extends |?eyond the Mach cone of the )50dy and (ii) when the wing tips lie within this Mach cone. Then, for a rectangulai' wing' of span b and body cone of semi-vertical angle 6 , the lift increment is

..c'/8

A 4pV(3 j

. A L =; — ^ i (-wjdy (3a)

[0 NJ p.' t a n ^

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pb/2

é

L

4 p Vc

B

G' tcai Ê (•" w ) dy (3b) when thü wing tips lie witiiin tho Mach conu of thd body.

Indefinite integration of equations (3a) and (3b) along the lifting line at x - - c' yields, jifter

substituting from equation (1) ana wri'cing u = V^y/c' «r. 1, 4 P Vc

SJ

Ö

2pVEcc' (- w j d y

X

S-

1 C U nj U 2 1 - cosh"-^ f - W d^ u

r I

2

= »2pVKcc'y^:ï Il~— -^ u cosh""^ / ^ V

'i-s;

2 sin~^uV )

^ constant. (4)

N o w the value of K is given (ref.l, eq.5b) bv the boundary condition at the surface of the cone

where the velocity component along the negative z-di.rection is V sin "jL 30 that

' 4^I.^!?f:. i_„„_..._., (5)

K =

where ƒ = (cot€ )/^i .

Define ij'15/20' = "X .

. . . ( b )

. . . ( 7 )

After s u b s u i t u t i n g from equations (4) - (?)

i n equations (3a) and (3b) i t i s found, on

rea-x-raTigernent.-t h a rea-x-raTigernent.-t rea-x-raTigernent.-the incremenrea-x-raTigernent.-t êi C-^ i n l i f rea-x-raTigernent.-t c o e f f i c i e n rea-x-raTigernent.-t based on

g r o s s wing area (= be) may be w r i t t e n

Ac.

aL

ifVbc

" ^ i ^ ' '^ ( i ^ ^ sin-\l/.g) - i r ) ^. (öa)

for a wing extending beyond the Mach cone of the body,

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6

-or

sin If 4

when the '^'in,a; tips lie '"ithin the Mach cone of the ^O'-^y.

The gross '"ing areo employed in evaluatlno; /}« Gj includes the rin.q' centre section, pass! no: through the ôdy, "'hich Hoes not contribute to the lift and upon -hlch there is no vôdy intf^rf erence. It may therefore be üreff-raiî" in int-rpretino; the results to base the increment A C ' in lift coefficient on the net --'ing area rithln the Mach cone of the body, i.e., a-êa of -ing overhang "'ithin the Mach cone of the body, (= be - Sc'tani^ .c); and this method has been adopted in presenting the results (figs. 3 - 4 ) . The increment in lift coefficient A G ' is in gener-al

A G

_ _A_L ^ „4_°ii_..

^ ~ - I f V^(>, _ s c ' t a n ^ ^c 1 - l / X

-s

T h e r e f o r e

A G'L -_ £Hi.:kL . __J (l ^ L i i i ^ L i i I l _ a . . . ( ° a )

ft A - i/-t\^^ 'S /V" - i+-cosh-i^

for a '^ring extending beyond the Mach cone of the body, and

A c'

_L

sin\> . 4

2 sin-l(l/-S)->/(lA)^- 1 -Acosh-^(l/;\)-2 sin"'"" ;\\

-f /rj2™'"~ ^ , -1 ^

l.Aoh)

^ V J - 1 +• cosh ^ 3' • • ••• t-rhen the ving tips lie --'ithin the Mach cone of the body. The

previous result, equation (9a), may of course be deduced from, this last result by substituting p\^ = 1 T,7ithin the curly brackets in equation (9b).

There is in each of the above cases an increment in --'ing drag associated -ith the increment in i-'ing lift due to body

interference '-hich is of amount

t^ % = / ^ C L t a n o ( . A C L . ^

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-7-3. 'Limiting Value of A C L for "Wings of Small Overhang

¥hen the wing overhang, b/2 - c'tanS , becomes small, tending to zercj), it is necessary to calculate the limiting value of A Cj . In this extreme case, the downwash velocity, w^, clue to the body is constant along the wing

overha.ng and eqi:ial to ibs value at the surface of the cone where

X = - c' , y = c' tci,n € , z = 0 and t h u s , from .equations (1) and ( 5 ) ,

2 KM w c = V sin \jy I "

y f Js^^ ^ - 1 - «osh" ''f )

. ' ' • ' ' "' •• ~—;•_ — ~ — " ' ' — ' - ' - /

c

Under bhese conditions b/2 - c'tan€-^^0 so t h a t , from equation (3b),

A 4 p Vc / b •'\ UL—? - A _ . . (- w^)/ - - c ' t a n e j

from which it f ollows that

-1

**Ö ï / f -^ -t- - * Ü'**

This formula gives the maximum value of A Or . \3/sin"'|^ for any given ƒ (= 1/X, under limiting conditions) and is plotted as the upper boundary in figure 3.

•i_p -^ Further Ac.. — ^ 4 sinU/vi asymptotically as J -——> OO ^ xiiis ^-^ is the absolute maximum incremexit in lifx coefficient A.C' for any given wing-body combination

with a 'slender body at incidence 'V/ and Mach niomber M

with a 'slender body a t incidence 'XL' and

4. Results

The intei'ference experienced by a wj..ng attached to a slender body is due to the upwash ge-iierated by the body in its 'after-cone' causing the wing to operate at an

increased effective incidence. The resulting increment in lift is propprtipnal tc^ sin 'U- whilst 'being a furxcticn pf the Mach number, the cone angle, the position and span

of the wing as shpwn in equati£)ns (8a) and (8b).

In general, for a given wing posi-cion, c' , and span, 2b, the increment in lift poeff icient,/l!iC'x, , deer ear-; ;-•>,.•=; with the an^le .of the cone, 2Ê ,' (figs. 3 - 4 ) although

this -apparently does not app],y to •'^'elatively thick bodies

( !ƒ < 3) with wings^almost spanning, the Mach cone of the body (i.e., for which /V =• ! ) • Neglecting this region, ^ <. ƒ < 5 ^'"^ ' ^''"' < A < 1> in which the

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8

-aj.;proximations a r e d o u b t f u l , i t i s see-n ( f i g . 3) t h a c t h e r e i s l e a . s t i n t e r f e r e n c e , w i t h A c ' . Q / s i n U- - 0 . 3 , f o r a s l e n d e r body w i t n a wiu^ a t l e a s t sparuiiiii^ one Mach cone of t h e boay. On t h e o t h e r m n d , maximum i n t e r f e r e n c e , w i t h .6 C^ . ^ / s i n l i - = 4 , i s t o be

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

I f t h e s e r e s u l t s a r e to be used f o r b p d i e s of slriape o t h e r t h a n c o n i c a l , h u t a p p r o x i m a t e l y s.o, cot € 3l:i,Quld be t a k e n t o be t h e r a t i o of t h e mean r a d i u s of a c h a r a c t e r i s t i c s e c t i o n of t h e body t o i t s d i s t a n c e a f t of t h e nose of t h e body. The mean r a d i u s may, f o r e.xample, 'be t a k e n t o be V A / T T " , whe,ie A i s t h e a r e a of t h e c r o s s - s e c t i o n c o n s i d e r e d . REFERENCES -No. _Author _{T i t l e , e t c .} 1 . _{H-S. T s i e n} S u p e r s o n i c Flow o v e r an I n c l i n e d Body of R e v o l u t i o n . J o u r n a l of A e r o n a u t i c a l S c i e n c e s , Vol. 5, 19'38..

A..tii. x'uckect Su_tJ^^r3om-G v«^ve vTa.^ of xxu.n •Airfoiils.

J o u r n a l of i i e r o n a u t i c a l S c i e n c e s , V o l . 13 J iS'^iO*

3. A. Rolpinson The Wave Dj-ag of Diamond-Siiaped Aerofoils at Zero Incidence. R.A.E. Tech. Note No. Aero.1784, 1946.

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-10-Now by the linearised Bernoulli equation the pi-essure increment at the point (x, y) is

A T

e

_Dï

V

" ^

1 =

_gj^(py -V q)

The resulting lift increment over a rectangular area

bounded by x = - o , x _{2 ' y = y i ' ^'^^ y ^ yg ' ^ i s "^^^^••^} A L

= 2

0 - G _'y A p . dx, dy sJ - 0 sj V 1 ^ 1 2CV . f * « - C 2 dx \J - c . ₁ ry^ 1 (py-V q)dy 2 p V c

r

_{p(yi t ^2)}

₎

-L _^ .^ q ^. s i n c e c - o = o = wing chord, 2 1 A l t e r n a t i v e l y t h e l i f t i n c r e m e n t e v a l u a t e d by

Aciveret' s t h e o r y y.ields an i d e n t i c a l r e s u l t v/ith t h e j j r e s e n t v e l o c i t y d i s t r i b u t i e , . . ! s i n c e pUutiin^, a s i n e q u a t i o n ( 2 ) ,

d {[^h)

dy

2 p V c

{^\ c

and i n t e g r a , t . i n g over t h e l i f t i n g l i n e equivci-ient to t]:ie apov^ rec-oangle g i v e s i n t h e p r e s e n t case

A

2|OVc

2g

VP

r

^2

(py-h ^)^i

J y

₁ ir ^1

(y - y )Üi^Z2) _

- ft ' H " "'^"'

another way of deriving this results is as follarwB".

Since the flo.'w of a compressible fluid past a l^iody of revolution produces small pertur'bations for which the induced velpcity potential / satisfies the linearised eqLiatipn of motion

{Vlk.

-ji'f.JÜif =0, (10)

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-11-it follows that if c/= f(x, z) is a solution of equation (10) then so is /'"=y.f(x, z ) .

We conclude that if ?/ is the induced velocity potential corresponding to a cer-cain incidence distribution

which is independent of y, g(x) say, then /' is the

induced velocity potential corresponding to an incidence distribution w' = y.g(x). It follows that in order to find the lift distribution corresponding to w'=y.g(x), we can in fact calculate the precise lift distribution corresponding to w = g(x), by Ackeret's theory, and multiply the result by y,

2). Inverse square variation of spaav/ise vele city distribution

We may express the spanwise velocity distribution relevant to the present problem and given equation (1) in the approximate form

Wc =

-P

"yS

-Q , (see fig.6)

Using this expression, it can be shown that

*^i =

-P

x+^l

sfP

^y2 - ( X + C-L)

- 5 (x-^c.)

2 ^

Ap = -pv

^ 2 i?/3 y

[jpV-(x.c,)2JV2 p j

which, on integrating over the same rectangular ai'ea as previously (see case 1 ) , yields an increment in lift ^l^

_ 2pVc

^

(y2- yi)

p ^

c(y2 - y i )

s i n

_•'{

fi'^i

fSyg

, / i

|3ï2;

_.^yJ

'•]

U Q

_J

which is such that for small values of c/Ay-, and c/Ay

A

L — > 2-oVc

_{(72 - y i )}

_{4.. Q}

y i y2

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-12-This limiting value is idenxical with that obtained by the aj^proxima-oe theory ^iven in the body of this repor-o which, with -one present velocity

distribution J ijreaicos a. lift, increment

A

L =

2 e

Vc

?^

py,

Jy

V

-^

--V^ Q

dy 2 /O Vc

£

G

(y -2 y ) / _{1 \} L y y \ 1 2 \ Q

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F I G U R e A.

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• 1

4 A<p

3 Z i » O ' t

v

c

t / / / ^

r

^ A R I A T I O N OF I N D U C E D W I N G L I F T C O E F F I C i e N T W I T H 0 ^ 4 £ A N G L E F O R D I F F E R E N T W I N G P O S I T I O N S . / ( ^ ( ^ \ ^ ~ -• ^ ^ « 1 ^ . ^ X N \ \ \ " ^ ^ ^ - ^ . ^ " ^ ^ - - . ^ ^

' V

. , \ . . ^ ^ " ^ ^ - v ^ ^ ^ " ^ ^ s 1 ^ ^ " ^ ^ ^ - - ^ ^ ^ ^ ^ ^ ^ -—

V

' ^ ^ . . ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^

^^iiii:;:;^

— — \ " - ^ - ^ ^ > . _ _ ^ - ^ — ^ - _ • - .

1

1 — ^ ^ " ^ . ^ . ^ ^ ^ ^ " ^ ^ " ^ ^ ^ ^ ^ - -' 2 4 6 8 lO 12 14 R G U R E . , ^

_5.

^ ^C^e)/^

0-zo

O'ZS O'hO 0 ' 4 0 O-75 , 16

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CURVES OF C O N S T A N T AC^^//OÜt^-v^

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1 ^o=^-C--o)/^

^ , ^ 0 ^,= ^^C-x--Co)/^ 3c^=-<:r ^ o ' ^ ' ^ S

FIGURE: .5.

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zeof SPANWt5EL VARIATION OF UPWASH VELOCITY _

tNDUCED A T WING M I D - C H O R D B Y T H E B O D Y