Swept Wings _quot; in Supersonic Flight

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					                                                                                                                  R. & M. N o . 2818
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                                                    i                                                                      A.R.C. Technical Repor~
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                 MINISTRY                                                OF S U P P L Y

      AERONAUTICAL                                                  RESEARCH                COUNCIL
                REPORTS                                       AND            MEMORANDA
                                                                                                                           a        ..I     ..........

Swept Wings " Supersonic Flight
                                               S. B. GATES, M . A .

                                               Crown Copyrig/Jt Reserved
                                                                                                      •.,   !('

   LONDON:        HER                          MAJESTY'S STATIONERY                             OFFICE
                                THREE S H I L L I N G S                               NET
                         Swept Wings in Supersonic Flight
                                                           S. B. GATES, M . A .

                                         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 ~peither 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 cos2 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 . = s e c ~ 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 w a y to r e m o v e s o m e of t h e
 c o n f u s i o n s w h i c h exist in assessing t h e a e r o d y n a m i c m e r i t of s w e p t wings. Y e t this p a p e r still
 seems to leave o p e n t h e a r g u m e n t b e t w e e n t h o s e w h o t h i n k of t h e w i n g s e c t i o n along t h e direction
 of flight a n d those w h o t h i n k o f it as p e r p e n d i c u l a r to {he l e a d i n g e d g < N o r does it fully r e b u t
 t h e n o t u n c o m m o n o p i n i o n t h a t as sweep m e r e l y delays t h e critical Mach n u m b e r its usefulness
 at supersonic speeds is d o u b t f u l . As A c k e r e t ' s t h e o r y of w a v e r e a c t i o n at supersonic speeds
 can be s i m p l y e x t e n d e d to include sweep on t h e lines s u g g e s t e d b y M c K i n n o n ~vVood, I h a v e
 w o r k e d this o u t in t h e h o p e of throveing f u r t h e r light on t h e subject. T h e c a l c u l a t i o n given
 b e l o w follows T a y l o r ' s o u t l i n e of A c k e r e t ' s t h e o r y (R. & M. 1467 ~) w i t h s o m e c h a n g e s of n o t a t i o n .

      2. Ackemfs Theory Extended to Include Swe@.~Consider an infinite u n s w e p t w i n g of u n i t
• c h o r d m o v i n g at speed V (Mach n u m b e r M > 1) a n d incidence c~ referred {o t h e c h o r d joining
 t h e s h a r p l e a d i n g a n d trailing edges. If t h e wing is n o w y a w e d t h r o u g h an angle W in ;the p l a n e
  of its edges w i t h o u t altering its m o t i o n , its speed a n d incidence in a p l a n e p e r p e n d i c u l a r to its
  edges are r e s p e c t i v e l y V cos ~o a n d c~sec W (Fig. !). I t is t h e m o t i o n in this p l a n e : w h i c h p r o d u c e s
 t h e p l a n e s o u n d waves s p r i n g i n g f r o m t h e surface, a n d so we c a l c u l a t e t h e w a v e r e a c t i o n b y using
 t h e a p p r o p r i a t e Mach angle # in this p l a n e a n d s u b s t i t u t i n g V cos ~0 for V a n d c~ sec ~0 for ~ in
 T a y l o r ' s equations. T h e g e o m e t r y is s k e t c h e d in F i g . 2.
    T h e Mach angle ~, is given b y

                 sin      =   V cos        -- M     cos w               .   .   .   .   .   .   .   .   .   .   .   .   .   .                     (1)
  (60497)                                                                                                                                         A
   At the surface the c o m p o n e n t velocity of the air n o r m a l to t h e surface is equal to the velocity
of t h e surface n o r m a l to itself. The surface condition is therefore
     V cos ~./3     =    u c o s (~ - / 3 )

if u is t h e w a v e velocity and /3 the inclination of the t a n g e n t of the surface to the u n d i s t u r b e d
flow. /3 being small c o m p a r e d w i t h # we h a v e from this
                  u =    T7/3 c o s ~p s e c # .                .      .    .     .     .    .       .       .           .           .           .           .           .               .           .           .                           (2)

B u t the w a v e pressure p is given b y
                  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 respectively the slopes of t h e u p p e r and lower surfaces referred to the chord,
# is (-- c~ sec W + #,) along the u p p e r surface and (c~ sec w -¢- f12) along the ]ower Surface.
  H e n c e the u p p e r 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 C O S = ~o
                P2 =     V(M       ~ cos 2 ~ _             1)       (~ s e c     v' +       #2) .        .           .           .           .       .           .               .           .           .           .                       (5)

  W e can now integrate along the u n i t chord to get t h e section coefficients as follows
                             1      ,
                c ~ - - ~p v 2 fo (p~ -                #1) d ~

                    =    v~(M~ _            seo2 ,p) ( ~ -                 ~o),                     •.           .           .           .       .       .           .               .           .           .           .                   (G)
                  ~0 =   ~ cos ~
                                        f'  0
                                                (#~ - # 2 )           dx              . . . . . . . . . . . . . .                                                                                                                            (7)

                c~ -     ½pV 2 o A(/31 -                      ~ s e c ~o) + P2(/3~ +                         ~ s e c ~)                              &

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

                                                4cz                    ~
                         + V(M ~ _                    s e c 2 ~) fo (#2 -               #~) dx .             .           .           .           .       .           .               .           .           .           .       .           (8)

CM about leading edge

                         ~pV 1 flo
                    =    V(M
                                   2 - - s e c 2 ~o)
                                                           {    -      2~ +       2
                                                                                      ;   o (/3~ -           #~)~ d x
                     =   -       ½c~ +          x / ( M 2 - - s e c 2 v,)                    o.                                                                                              --                                              (9)

Thus the aerodynamic centre is at half the chordl and

                          cos~         {2/f 1                     fl                }
               C.0 = ~v/(M~ _ sec 2 ~)   ~ o ( ~ - - ~2)x d x - - o (t~ - - t~2) d x . . . . .                                                  (10)
In t h e above Co and CMO are referred to the yawed chord and t h e y a w e d span. We are more
interested in their components C~ cos 9, CM o cos ~oreferred to the direction of flight, and denoting
these b y t h e suffix F we have

              C~ F ---- ~ / ( M ~ _ sec ~ ~o) 4c~23- 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')

             c~o~      =   V ( M
                                   9   -     s e e 2 9)    l;  2
                                                                       (~ -- ~)x dx
                                                                                          ;   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 t h e wave reaction produced b y an infinite
y a w e d 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 t h a t

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

   If the thickness/ch0rd ratio is t and the camber of the centre-line is ~ we h a v e

It follows t h a t
                                   (1 (~1 -         ~9) dx = 0
                                   d 0


                                                                       = ~(~2 + 4~)

                                   J 0

  The results for a biconvex section are therefore
                 C~ = V ( M ~ - - s e c ~ ~)

                                   2 _ s e c 9~0) 4 ~ + - -
                                                           {             ,(t9 + 4~9) cos 9v }
                 D         ~9 + {(t2 3- 4 v~) c os9 ~o                                    t
                                              C O S ~ ~v
              CMOt 7 __        8
                               3 ~/(M        2 _    sec 9 9)

  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 K C 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
               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) -~ ),iboM2 .            .        .       .       .       .       .       .        .   .   .   .   .   .   (12)

              Cvr. = V ( M ~ __ sec . ~f) (z3 + ,~t~ cos" ~) + KCI                                                                                       (13)
                                                                                    K                                       . . . . . . . . .            (14)
                D --        +   . t2 cos       v +         _   sec         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 )  M2      + ~t~ cos~ v • 7Po • V ( M ~ _ sec ~ v)
                       + ~:,poM 2          •
                                               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 CI depends on the Reynolds number,
                                     w h i c h varies between 107 and 108 in the range considered. In this
                                      range CI is of the order 0. 0020 to 0. 0025, and so the value of K C 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.
                       t            0 and 0.1.
                       ~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
 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 :

                                                         C~                          %= (deg)

                                               M = 1.5        2.5             9=0              45 deg

                           ground       lg       0.008        0.003           0.24         0" 17
                                        Sg       0.040        0.015           1 "20        0"85
                           52,000       lg       0.08         0-029            2.4         1.7
                             ft.        5g       0.40         0.145           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)maxis 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 m a x i m u m 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. It 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; it 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. T h e yaw effect increases with M, and at large M the form drag i~s
approximately halved by yawing through 45 deg.
  It 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.

 (60497)                                                                                                                        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 ~0by sweeping the wing through ~, however this is effected. If the wing is yawed the
                                                                             ( M~ -- I ~1'~
form drag is multiplied, relative to the straight wing, by the factor cos~ ~p\ M ~ _. sec ~~o/ which

becomes unity shortly after M = see ~ and ~          cos ~ ~ as M - + co.      If the wing is sheared the
                        ( M ~ - 1 "~*J~
corresponding factor is \ M ~ _ sec ~ ~P
                                       / which is always > 1 and -+ 1 as M - + co.

   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 ~oaway 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 ' f o r e - a n d - a f 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                                         Title, etc.
  1   R. McKinnon Wood                  Notes on Swept-Back Wings ~or High Speeds. A.R.C• 8806. September,
                                          1945• (UnpuNished.)
 2    G . I . Taylor . . . .            Application to Aeronautics of Ackeret's Theory of Aerofoils moving at
                                          Speeds Greater than that of Sound. R. & M. 1467. 1932.



                              FIG. l,


                                                               R¢l~blw. Trop~p~


                                                                              800o0   40000      60O00   60000   IO0000"

                                                                                       FIG. S.
                                         -Upper suPFac¢

Vcos J~

                                         Low.c- 5urr- a C~¢

v~ ~                " / p ~

                               FIG. 2.



                                ~c L-


            \ ~=ooo
                                        2         M

                                            F I G . 5.


                      M     4

                  FIG. 4.

                                                    ~ " 4.¢o

                                                                                                          ~                  "-
                                                                                        I ,'~ / ' X I


            /!         /   \.                ".. r,,.                 't.lo
                                                                                                                                                  = 45 °

¢D               I /        ~    "                       "'%)'~
                                                                          ~ //~ L~,~
                                                                              ///                       -='°°°
                 ~/ /                - ~ . . -~¢= 7oo             -
                                                                                   Z' ,i/,,
                                                                           , ,i'!/,Y

     0"02                                                                     tg°,oo
                                                                          o              o.o~           o,to             o-15         OqO   Oq5
                                                                                                                 o~ r ' ~ d i a n s
     0"01   '"   Ill                            I                     F I G . 7.    Comparison of Yawed and Unyawed Biconvex Wings ;
                                                                                               10 per cent thickness.

                                F I G . 6.

                  I                "
                                                                                                                        I                 I                          .   1

                  I                                                                        O

                                                                                           th         D
                                                                                                ~,001             ~I   ~/~,


                .~f--~'S ~
                                                                                                200F ---~

           ,j     ~                          2                       5                          t00

                b                                                                                           ~'y        2         M        ~                 4
                g'r = &~q p~r N                      I:B p~r rel~biv¢     pressure
                          = liFB       p~r       ~   Fb   per   r¢l~Biv¢ p r ¢ e ~ u r ¢
                                                                                                     FIG. 9. Performance Comparison of Sheared and Unsheared Biconvex
                Performance Comparison of Yawed and Unyawed Biconvex                                                              Wings.
 ~IG. 8.
                                                                                                           Ro & Mo Moo 281

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