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MINISTRY OF SUPPLY

R. & M. No. 2818 (10,180)

A.R.C. Technical Repor~

Swept

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

Wings " i n

M E M O R A N D A

a . . I . . . . . . . . . .

Supersonic ~gy

Flight

S. B. GATES, M . A .

Crown Copyrig/Jt Reserved .':ix

•., ! ( '

L O N D O N : H E R M A J E S T Y ' S S T A T I O N E R Y O F F I C E

I954

THREE SHILLINGS NET

Swept Wings in Supersonic Flight

S. B. GATES, M . A .

COMMUNICATED BY THE PRINCIPAL DIRECTOR O~ SCiENTifiC RESEARCH (AIR), MINISTRY OF SUPPLY

Reports avd Memoravd No. 28 8

December, 946 { O0, T19g4

~ ~ . .

Summary.--Opinion seems still unsettled on the aerod3mamic merit of swept wings in supersonic flight. To elucidate this, Ackeret's theory of two-dimensional wave reaction is here extended to include sweep. The formulae so derived. are used to compare the performance of a straight wing with one swept through 45 deg, making some allowance for frictional drag.

' As the wave form-drag varies as (thickness) = it is this part of the drag which causes most trouble. A straight wing of given thickness/chord ratio can be swept through an angle ~p either by yav4ing it or by shearing it. Jn both cases the critical M is increased from 1 to sec w and the favourable lift/incidence effects above the critical M are the same. But the form-drag of the yawed wing begins to be less than that of the straight wing soon after M = sec v~ and is reduced in the ratio cos 2 w : 1 at large M ; while the form-drag of the sheared wing always exceeds that of the straight wing. Thus to make the best use of sweep in a supersonic speed range beginning at M.=sec ~ the straight wing thickness which must be tolerated should be yawed through an angle W.

1. Introduction.---In A.R.C. 88061 M c K i n n o n W o o d goes a iong way to r e m o v e some of the confus ions which exist in assessing the a e r o d y n a m i c mer i t of s w e p t wings. Yet this pape r still seems to leave open t he a r g u m e n t be tween those who t h i n k of the wing sec t ion a long the di rect ion of flight and those who t h i n k o f it as pe rpend icu la r to {he leading edg< Nor does it ful ly r ebu t t he no t u n c o m m o n opin ion t h a t as sweep mere ly delays the crit ical Mach n u m b e r its usefulness at supersonic speeds is doubt fu l . As Ackere t ' s t h e o r y of wave reac t ion at supersonic speeds can be s imply e x t e n d e d to include sweep on the lines sugges ted by M c K i n n o n ~vVood, I have worked this ou t in the hope of throveing fu r the r l ight on the subject . The c a l c u l a t i o n g iven be low follows Taylor ' s out l ine of Ackere t ' s t h e o r y (R. & M. 1467 ~) w i t h some changes of no ta t ion .

2. Ackemfs Theory Extended to Include Swe@.~Consider an infini te u n s w e p t wing of un i t • chord m o v i n g at speed V (Mach n u m b e r M > 1) and incidence c~ referred {o t he chord joining t he sharp leading and t ra i l ing edges. If t he wing is now yawed th rough an angle W in ;the p lane of its edges w i t h o u t a l ter ing its mot ion , its speed and incidence in a p lane pe rpend icu la r to its edges are respect ive ly V cos ~o and c~ sec W (Fig. !). I t is the m o t i o n in this p lane :which p roduces t he p lane sound waves spr inging f rom the surface, and so we ca lcu la te the wave reac t ion by using the app ropr i a t e Mach angle # in this p lane and s u b s t i t u t i n g V cos ~0 for V and c~ sec ~0 for ~ in Taylor ' s equat ions . The g e o m e t r y is ske t ched in F ig . 2.

The Mach angle ~, is g iven by

s i n = V c o s - - M c o s w . . . . . . . . . . . . . . (1)

(60497) A

where

At the surface the component veloci ty of the air normal to the surface is equal to the veloci ty of the surface normal to itself. The surface condit ion is therefore

V c o s ~ . / 3 = u c o s (~ - / 3 )

if u is the wave veloci ty and /3 the incl inat ion of the t angen t of the surface to the und is tu rbed flow. /3 being small compared wi th # we have from this

u = T7/3 c o s ~p s e c # . . . . . . . . . . . . . . . . . (2)

But the wave pressure p is given by

jb ~ p g~t

----- ,oaV/3 cos W sec # from (2)

P V2 cos~ W = ~/'( ~I2 cos2 V' -- 1) /3 from (1) . . . . . . . . . . . (3)

If #1, #, are respect ively the slopes of the upper and lower surfaces referred to the chord, # is (-- c~ sec W + #,) along the upper surface and (c~ sec w -¢- f12) along the ]ower Surface.

Hence the upper surface pressure p , is given by

p V ~ cos 2 '~v P* = V ( M 2 cos 2 2# -- 1) ( - ~ sec w q- ill) , • . . . . . . . . . (4)

and for the lower surface

p V 2 COS = ~o

P2 = V ( M ~ c o s 2 ~ _ 1) (~ s e c v' + #2) . . . . . . . . . . . (5)

We can now integra te along the un i t chord to get the section coefficients as follows

1 , c ~ - - ~p v 2 fo (p~ - #1) d ~

4 = v ~ ( M ~ _ seo2 ,p) ( ~ - ~ o ) , • . . . . . . . . . . . (G)

f' ~0 = ~ c o s ~ (#~ - # 2 ) dx . . . . . . . . . . . . . . (7) 0

c ~ - ½ p V 2 o A ( / 3 1 - ~ s e c ~o) + P2(/3~ + ~ s e c ~) &

= ~ / ( M 2 - - s e c ~ ~o) 4c~2 + 2 c o s ~ ~ . o ( # # + / 3 # ) &

4cz ~

+ V ( M ~ _ s e c 2 ~) fo (#2 - #~) dx . . . . . . . . . . . . (8)

CM about leading edge

1 fl ~ p V o

' { ; } = V ( M 2 - - s e c 2 ~o) - 2 ~ + 2 o (/3~ - #~)~ d x

= - ½ c ~ + x / ( M 2 - - s e c 2 v,) o . - - (9)

2

Thus the aerodynamic centre is at half the chordl and

c o s ~ {2/f 1 f l } C.0 = ~v/(M~ _ sec 2 ~) ~ o ( ~ - - ~2)x d x - - o (t~ - - t~2) dx . . . . . (10)

}

In the above Co and CMO are referred to the yawed chord and the yawed span. We are more interested in their components C~ cos 9, CM o cos ~o referred to the direction of flight, and denot ing these by the suffix F we have

C~ F ---- ~ / (M ~ _ sec ~ ~o) 4c~2 3- 2 cos ~ ~ 0 (/31~ 3- fl99) d x

4~z cos ~ ;1 3- ~ / (M 2 - - s e c 9 ~o) 0 (132 -- ill) d x . . . . . . . . . . (8')

l ; ; } c ° s ~ 2 ( ~ - - ~ ) x d x c ~ o ~ = V ( M 9 - s ee 2 9) o - 0 ( ~ - ~9) d x . . . . (~0 ' )

3. B i c o n v e x S e c t i o n . - - E q u a t i o n s (6) to (10') give the wave reaction produced by an infinite yawed wing in the general case. To discuss the effects of the sweep ~o in more detail consider a biconvex section in which the edge values of/~1, /39 are/3., 2, /~.,, so tha t

/~1 /39 __ 1 -- 2x. ~.,1 tL, 2

If the thickness/ch0rd ratio is t and the camber of the centre-line is ~ we have

I t follows tha t (1 (~1 - ~9) dx = 0 d 0

0

= ~(~2 + 4~)

2) J 0

The results for a biconvex section are therefore

4~ C~ = V ( M ~ - - s e c ~ ~)

1 { , ( t9 4~9) } C . F = ~ / ( M 2 _ s e c 9~0) 4 ~ + - - + cos 9v

L D ~9 + {(t2 3- 4 v~) c os9 ~o t

C O S ~ ~v C M O t 7 _ _ 8

3 ~ / ( M 2 _ s e c 9 9 )

3

(11)

4. Symmetrical Biconvex Wing with Frictional Drag.=--The analysis can be made rather more realistic by applying it to the case of finite swept wings, and making some allowance for the skin friction of the wings and auxiliary surfaces, while still neglecting the aspect-ratio effects and any'other 'wave drag or interference drag which may be present. This amounts to increasing Cv~ by KC I where Cj is the skin-friction coefficient (assumed to be independent of sweep) and K is the ratio of total wetted area to wing area.

It will be useful to compare the performance of the swept with the unswept wing over a range of Mach number, wing loading, and altitude. The wing loading w is introduced by the relation

2w/r CL = 9,poM ~

where P0 is sea-level pressure (2110 lb/ff~), r is relative pressure P/Po, and ), is the ratio of the specific heats 0fair, taken constant at 1-4. The relative pressure r is shown as a function of altitude in Fig. 3. The equations for a symmetrical biconvex section are on these assumptions : - -

4¢z 2w/r CL = ~ / ( M ~ sec . ~f) -~ ),iboM 2 . . . . . . . . . . . . . . (12)

4 Cvr. = V ( M ~ __ sec . ~f) (z3 + ,~t ~ cos" ~) + KCI (13)

L

K . . . . . . . . . (14) D - - + . t2 cos v + _ s e c v ) C

The most useful basis of drag comparison appears t o be what may be called the specific drag D/Sr, i.e., the drag per unit area per relative pressure. This is o})tained from (12) and (13), by eliminating e, in the form

D 1 / w ' \ ~ , v / ( M 2 - - s e c 2 ~ , ) M ~ Sr -- 2),20 [ , r ) M 2 + ~ t~ cos~ v • 7Po • V ( M ~ _ sec ~ v)

1 2 • + ~:,poM KC~ . . . . . . . . . . . . . . . (15)

5. Performance Com parisons.--The characteristics given by (12) to (15) are surveyed in Figs. 4 to 9 where the following numerical values and ranges have been taken

M ----- Critical to 4.

KCj = 0.01. This can be only a very rough typical value, as K depends on the proportion of wing to body and C I depends on the Reynolds number, which varies between 107 and 108 in the range considered. In this range C I is of the order 0. 0020 to 0. 0025, and so the value of KC I chosen represents a design in which wetted area is four or five times wing area.

w/r w may vary from 20 to over 100, and r is less than 0.1 above 50,000 ft. A range of w/r from 100 to 1000 is covered.

and 0.1. t 0

~0 0 and 45 deg.

Lift and Incidence (Eqns. 12).--CL, dC~/dc~ and c~, which are independent of thickness, are plotted against M in Figs. 4 to 6. These diagrams demonstrate the low values of CL and e which suffice for supersonic flight in the stratosphere at wing loadings less than 100. Fig. 6 shows in

4

particular the maximum in incidence which is characteristic of supersonic flight at any given w/r. This is found from equation (12) as

a~x O. 0048 w = - cos ~ degrees (16) r ° ° • ° • • ° • ° • • •

and occurs at M = ~/(2) sec .¢.

An indication of the Co, e-r6gime is given by the following table which shows the values required to produce lg and 5g when W/S = 50 :

ground lg Sg

52,000 lg ft. 5g

C~

M = 1.5 2.5

0.008 0.003 0.040 0.015

0.08 0-029 0.40 0.145

%= (deg)

9 = 0 45 deg

0.24 0" 17 1 "20 0"85

2.4 1.7 12.0 8-5

The advantage of sweep in reducing the incidence necessary for any flight conditions is obvious.

Lift/Drag.--In Fig. 7 LID is plotted against c~ for several values of M for ~o = 0 and 45 deg, at thickness t = 0.1. This shows clearly the merit of the yawed wing, but is chiefly of interest when studied in relation to the incidence survey of Fig. 6. At a given value of w/r the (L/D)max available can only be utflised if the incidence range in which this occurs is exceeded by the emax given by Fig. 6 or equation (16). If w/ris of the order 100, supersonic flight is confined to incidences of less than 0.01 radians and L/D's less than 1. Even at w/r = 1000 the L/D reached is con- siderably less than that available, and it is only when w/r exceeds 1500 that (L/D)max is reached m some part of the M range. The wedge-shaped curves sketched in the figure show the operating conditions for several values of w/r. Thus in the case considered maximum efficiency can only be realised in the troposphere at enormous wing loadings ; with loadings of less than 100 it is necessary for maximum efficiency to fly high in the stratosphere. Conditions become easier with thinner wings than the 10 per cent illustrated, since c~for (L/D)ma.~ decreases with thickness. I t is nevertheless generally true that efficient supersonic flight puts a premium on high altitude.

Specific Drag D/Sr.--It is clear from equation (15) that ~ affects both the induced drag (arising through w/r) and the form drag (arising through thickness t). To separate these effects, the specific drag is plotted in Fig. 8 for the ideal case of zero thickness, and also for t -- 0.1".

The effect of 45 deg yaw on the induced drag is shown in the lower group of curves; i t is favourable but small except near the critical M = V'2.

The yaw effect on form drag is much more important, as is seen by noting the difference between a curve in the upper group and its corresponding one in the lower group. For instance at w/r = 500 and M = 2 the form drag of the unyawed wing is AC; this is reduced to BC by yawing through 45 deg. The yaw effect increases with M, and at large M the form drag i~s approximately halved by yawing through 45 deg.

I t should perhaps be noted that L/D values can be quickly obtained from a specific drag curve; we have only to divide w/r by its ordinates. If the diagram is examined in this way it

be found to conform to the L/D discussion already given.

* A rather extreme thickness has been chosen for illustration. At the more usual thicknesses of 7 per cent and 5 per cent the form drags shown would be respectively halved and quartered.

5 ( 6 0 4 9 7 ) A *

6. Yawed and Sheared Wings.--The discussion so far has compared a straight wing with one of the same thickness which isyawed. It is evident, however, that the drag equations (18) and (15) admit another interpretation, for t cos ~ is merely tF the thickness/chord ratio of the yawed wing measured in tile direction of flight. If we Use tF instead of t cos ~ in (15) we are clearly comparing the drag of a straight wing of thickness/chord ratio tF with that of the wing sheared through an angle ~p. The result of this at t~ = 0.1 is shown in Fig. 9, to be compared with Fig. 8. The sheared wing has more drag than the straight wing above the critical M because its thickness/chord ratio in a section perpendicular to its edges has been increased, and the comparison with the yawed wing is very striking. The lift and incidence comparisons of Figs. 5 and 6 remain unaltered.

This distinction between the yawed and the sheared wing seems very pertinent to supersonic design. We have seen that supersonic form drag Js particularly serious because it increases as t 2. The minimum thickness which can be tolerated is usually dictated by non-aerodynamic considera- tion, and should be settled in relation to the straight wing. The critical M can now be increased from 1 to see ~0 by sweeping the wing through ~, however this is effected. If the wing is yawed the

( M~ -- I ~ 1'~ which form drag is multiplied, relative to the straight wing, by the factor cos ~ ~p \ M ~ _. sec ~ ~o/

becomes unity shortly after M = see ~ and ~ cos ~ ~ as M -+ co. If the wing is sheared the

( M ~ - 1 "~*J~ which is always > 1 and -+ 1 as M - + co. corresponding factor is \ M ~ _ sec ~ ~P /

It seems then that in designing for a range of supersonic speeds beginning at M = sec ~o the correct course is to turn the thickness which has to be tolerated through an angle ~o away from the direction of flight by yawing the wing through VJ. The application of this simple rule to plan forms of small aspect ratio and high taper, such as the delta wing, is of course very doubtful. But the above analysis seems to resolve the argument between the ' fore-and-af t ' and the ' normal to leading edge ' schools of thought, in so far as effects of sweep can be isolated from the other parameters of a finite plan form.

No. Author

1 R. McKinnon Wood

2 G . I . Taylor . . . .

REFERENCES

Tit le, etc.

Notes on Swept-Back Wings ~or High Speeds. A.R.C• 8806. September, 1945• (UnpuNished.)

Application to Aeronautics of Ackeret 's Theory of Aerofoils moving at Speeds Greater than that of Sound. R. & M. 1467. 1932.

6

FIG. l ,

V,~

I '0

J

C~

CM

Vcos J~

-Upper suPFac¢

0"8

0"6

Trop~p~ R¢l~blw. ~ D r ¢ ~ J S u r ¢

O,,~

0.~

800o0 4 0 0 0 0 60O00

FIG. S.

6 0 0 0 0 I O 0 0 0 0 "

v ~ ~ " / p ~

FIG. 2.

Low.c- 5urr- a C~¢

O0

0.5

0,~

D.3

Ct.

0-2

0.1

\ ~=ooo

IO0

M

FIG. 4.

4

G

~c L-

4

2 M

FIG. 5.

0'09

0-0.

¢D

0"03

0"02

0"01

I

~ " 4.¢o

/! / \. ".. r,,.

I / ~ " " '%) '~

~/ / - ~ . . -~¢= 7oo -

' " Ill I

FIG. 6.

4

't.lo

~ /~ L ~ , ~ / / / / -='°°°

Z' ,i/,,

, ,i'!/,Y

tg °,oo o o.o~ o,to o-15

o~ r '~d ians

FIG. 7.

~ "- I ,'~ / ' X I

OqO Oq5

= 4 5 °

Comparison of Yawed and Unyawed Biconvex Wings ; 10 per cent thickness.

,600

50(

t~

I01

I "

I I

. ~ f - - ~ ' S ~

I ,j ~ 2 5

M

U~ewepb

b g'r = &~q p~r N I:B p~r rel~biv¢ pressure

= liFB p ~ r ~ Fb p e r r¢l~Biv¢ p r¢e~ur¢

~IG. 8. Performance Comparison of Yawed and Unyawed Biconvex Wings.

E ,e t~

O

th

c

a~

Nt

~ 0

500

ae N D

~,001

200F ---~

t00

o

FIG. 9.

I I I I

~ I ~/~,

. 1

Fligbb

~'y 2 M ~ 4

Performance Comparison of Sheared and Unsheared Biconvex Wings.

Ro & Mo Moo 281

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