The wave drag of highly-swept wings: A comparison of linear theory and slender body theory

.TECHNISCHE HOGESCHOOL

\n i^-ixuiGBOUWKUNDE

straat 10 - DILFT

THE COLLEGE OF A E R O N A U T I C S

CRANFIELD

THE WAVE DRAG OF HIGHLY SWEPT WINGS;

A COMPARISON OF LINEAR THEORY AND

SLENDER BODY THEORY

by

TECHNISCHE HOGESCHOOL

VLIEGTÜIGBOUWKUNDE Kanaalstraat 10 - DZUfT

N o t e No. Mj. O c t o h e r

19511-THE COLLEGE OF AERONAUTICS C R A N F I E L D

The Wave Drag of HighlySwept Wings

-A Comparison of Linear Theory and Slender Body Theory.

-hy-T. Nonweiler, B.Sc.

SUMlvlARY

This note comments on the comparison between tne answer obtained by linear theory for the wave drag of

Slender-wings (as interpreted by the limit Ju -1 cot y\-*0) and that value for the drag obtained by Slender-Body Theory. It is shovm that for fully tapered wings the agreement is exact, and that there is reason to suppose that the same

is true for all wing planforms, unless the trailing-edge is unsv/ept, or the wing section has a finite trailing-edge thickness. Some remarks are included concerning the drag of slender delta wings.

on the basis of the linearised theory of compressible flow, for the wave drag at zero incidence of a sweptback fully tapered wing of arbitrary section. As a x^articular example of the method developed therein, the drag of such Ti/ings of slender planform was considered as a particular case, and a few numerical results evaluated.

It has subsequently been pointed out to the author that these calculations are in close agreement with

numerical estimates of the drag of such wings calculated on the basis of the " Slender Body" Theory advanced by

2

Ward some years ago. The present note discusses the comparison of the tv/o theories on a formal basis.

This agreement noticed by other investigators is, in fact, an exact one; this is shown in the Appendix of this note by a straightforward mathematical argument. The

2

so-called " Slender Body" Theory yields the answer that the drag D is given by

•-1 •-• 1 1 ' ' ' '

« = ^ 1 j s"(OS"(ri)

(n'-L-1 27CL Jo j 0 ''^"^ dr drj

subject to certain' reservations (which we shall detail subsecuently), where S(e) is the cross-sectional area of the body (or wing) at a distance £'L do?/nstream of its nose

(or apex), - L being the overall length of the boèy. On the other hand the linear theory yields a relation of the kind :

2

-5 = s^ T^ f (A tan ,/A_{q o o ^^} J M ^ - I cot A . Z2 )

where s is the wing semi-span, T the thickness/chord ratio, A the wing aspect ratio and /j the angle of sweepback of the leading-edge. The parameter £] is used here to

describe the section shape. In reference 1, the drag of "slender" wings was stated as

D 2 2 _{• lim (f(A tanA,v^/i^-1 c o t j S , ^ ) )}

yi -1 cot/v-»o -'

= s T f (A tany^ , 0, £ ^ ).

The slender body theory gives, of course, a very similar form of answer :

2 2

I = s T 0 (p ^ ) s say,

where c is the wing chord,* and the two functions f and

0 are shown, in the Appendix, to be identical.

This deduction alludes, of course, only to the fully-tapered wings. It is strong support for the argument,

however, that the "Slender Body" Theory gives the appropriate limit of the linear theory for the general case of the

highly-swept viring, whatever its planform. The correspondence is by no means obvious from first principles, as the

"Slender Body" Theory uses the raomentmTi theorem, in con.j\inction with a quadratic approximation to the pressure, to evaluate

the drag force, - the potential of the flow, hoY/ever, being obtained from the linearised equation of motion,

In this sense, the "Slender Body" Theory may be said to be more exact than the linearised theory. Hovifever, a

scrutiny of the analysis of roference 2 shows that the quadratic terms in the pressure equation do not contribute to the expression for the drag given above, but reveal themselves in additional terras m^hich are in general zero for wings with closed sections. Thus, the agreement with linear theory is not surprising ; without the

quadratic terms, the momentum integral for the drag, as well as the flow potential, have both a "linear" form.

I|.. To enlarge on this effect of the second-order terms, it may be mentioned that the pressure equation is used in "Slender Body" Theory to find the momentum flux through a finite plane area, perpendicular to the flow direction, and embracing the body at the furthest downstream cross-section of the body (or wing). The pressure there depends on the local value of the cross-flov/ potential 0 , and its change over this plane (i.e. the derivatives of 0 with respect to r and 0, say, the polar co-ordinates in this plane ^=1); and in "Slender-Body" theory, the appro-oriate second-order terms in velocities included in the expression for pressure involve [tf°j and -2 (If*^) • Nov/, provided the

trailing-edge is swept and the wing section is a closed one, the value of 0 over this plane (E=1) is a constant

(given by b in the analysis of reference 2)j this may easily be shown, as the disturbance to the cross-floy^ vanishes at this plane. The boundary conditions at the v/ing surface normally require a line distribution of

h

-sources and sinks to obtain the appropriate cross-flow

in the transverse planes E = constant, and this distribution tends to vanish as B --»1. Since 0 is a constant, it will be evident that both |^° and -i-o must be zero, in the

dr do

plane f = 1, and so there is no contribution to the pressure at that plane from the quadratic velocity terms. That is why there is, for the general case of the wing, nothing

incompatible between "Slender Body" theory, and one in which the linearised form is used throughout. Consequently, agreement betv/een the two theories is to be expected where they meet - that is, for slender wings.

5. The exceptions noted above bear on important cases. The wave drag of wings with a finite trailing-edge thickness has not received a great deal of attention, but the linearised theory may not give the same answer for the slender wing as the theory of Ward j in this case, the latter theory could be said to be more exact. The other exception, which includes all wings having a straight trailing-edge but a highly swept leading-edge - that is, all wings v/hich have a slender

"cropped" (or "uncropped") delta planform - is of great interest, but as we shall see is a case which does raise

some difficulty in interpretation. In both exceptions there are terms in the expression for the drag given by "slender-body" theory which are additional to that given in para.2

and which are quoted at the beginning of the Appendix.

6. For the delta wing (with a finite trailing-edge angle) it will be found that the variation of S"(g), in the double integral for the drag quoted earlier, has a logarithmic

singularity at E = 1, but nevertheless the integral

remains convergent, and so yields a finite contribution

to the drag. Of the remaining tv/o terms (as given in

equation (1) of the Appendix) the term involving S'(1)

remains finite, as S' (f;)has a finite limit at E = 1 (the

trailing-i dge), but the last term, though again finite,

involves the expression in(1/vM -1 cot-A ) and the

complete expression for the drag is of the forms

D

_{s T^ 0 (ZI) + constants}

i i m r fn/

JF^ cotJ\

_{o /}

The question arises as to the compatibility of this result

with the answer obtained by the same theory for "near-delta"

wings - i.e. those with values of o = T- A tan

J^_ = T-

which

are a little greater than unity.

f A tan y^

_k

_{o L}

= r

7. For such wings, "Slender Body" theory has been shown to

be in agreement with linearised Lheory, from which it is

found in reference 1 that.

D q. 2 2 S T

H^)

a s 0 = r-j r-j 1 from above,

i.e. the drag becomes infinite. How is this result

reconcilable with the finite drag of the slender delta

wing?

This question is easily resolved if the well-known

result of linearised theory is recalled, that for a delta

wing (or its reverse)

=• r^ s T in ,~rr—, as 7 M -1 cot /\

q. VM'^-I cotTs^ "

- this result being well-known in the sense that it will be readily agreed that the drag of delta wing is (by this

theory) infinite "at M = 1", It also implies that (D/qs^) is infinitely large for a slender wing. Now the results of linearised theory for the slender and the "near-delta" wing, are both of course compatible v/lth one another, and

only differ because of the reversal of the order of taking the limits in the general expression for the drag,

-allowing first, either the wing to become slender, or the trailing-edge sweep to vanish. Viewed in the same light, there is therefore nothing incompatible about the answer given by "Slender-Body'' theory for a delta wing, despite the infinite drag of a v/ing with 7- = 1+j since we see that it is at least qualitatively similar to that of linearised

2

theory. The value of (D/qs ) is certainly infinitely large for a slender delta \?ing, and the finite expression derived for it merely shows the nature of this singularity.

8. Conclusions

(i) That the drag of fully-tapered swept T/ings calculated by linear theory in the limiting

fo—' *

condition ^/M -1 cot/\ -• 0 is identical v/ith the result of "Slender Body" Theory for such wings.

(ii) That any difference betv/een the drag calculated by "Slender Bodj''" theory and linear theory could

approximation to the pressure employed in the former.

(iii) That the quadratic terms in the pressure equation do not contribute :o the drag of swept wings, in general, so that the agreement noted above for

fully-tapered vifings should extend to all planforms. (iv) That the exceptions to this rule are those ?/ings

which have a section with a finite trailing-edge thickness, or a planform with an unswept trailing-edge (i.e. a delta wing). For these v/ings, the expression for the drag given by Slender Body theory involves a contribution from the quadratic terms, and so y/ould be expected to differ from the ansTifer found by linearised theory.

(v) That the expression for the drag of slender, cropped (or uncropped) delta wings^ is (if not quantitatively) at least qualitatively identical in both theories.

References

T. Nonweiler The Theoretical Wave Drag at Zero Lift of Fully Tapered Sv/ept Wings of Arbitrary Planform.

C. of A. Report No. 76 (1933) G. N. Ward Supersonic Flow past Slender Pointed

Bodies.

Quart. J. Mech. and App. Maths, Vol. II Pt. 1 (iSkS)

8 -Appendix

The fundamental result of the Slender Body Theory 2

of Ward in relation to the wave drag at zero incidence

(D, say) i s t h a t

(tHÜ)"<^^>^"(->^"

g---.

d.d,- s i t u

r1

S«(g)fn

1-J dE

^ o l ^ ^ -

0)

E=1

where S(^) is the cross-sectional area of the body at a distance E,L downstream of its nose or apex (L being the body overall length), 0' is the limiting form of the perturbation potential of the flow on or near the body, and V and T are orthogonal co-ordinates in cross-sectional planes, v being measured normal to the body and T around itj the contour C is described around the cross-sectional perimeter of the body. The potential 0 is shown in

reference 2 to be derived from that for the cross-flow in planes ^ = constant, which satisfies the boundary condition

1^0 9v

dv

dE on (2)

i.e. on the body surface.

We suppose now that the body in question is a fully-tapered swept wing, with a swept-back trailing edge, and with a leading edge angle of sï/eepback given by cot" m ,

say. The cross-sectional area of such a body is given by

S(5) = k

-m EL

o

Konaalitragt 10 - DELFT

9

-where z is the wing semi-ordinate, and y is a co-ordinate o

measured spanwise from the wing root chord line, so that y = +m EL describes the position of the wing leading edge.

fie further suppose that the wing section is similar over the entire span and is given by

y=o

^I'E.L^

'Vc-)

(k)

along the root chord line. We define Z(k) as ^ero for k < 0 , and k > 1. Then it follows from simple geometry

that, in general, if s is the viring semi-span (so that s = m L ) ,

., = | ( i - | ) z ( . )

where k =

_{s i - S .}

1 . |lc

(5)

Substituting from (5) in (3)» it follows that

S(S)

rV^

2t S L - ^ m, Q. ^ s-^ J dy

This integral can be simplified b y changing the variable of integration from y to k» as defined in (5): for then

(^-|) = , s and d^

10

-Further, if we put

i A tan A ^ = j ^ = ^ = a, say ... (6) o

noting that o > 1 is the condition that the trailing-edge is sv/eptback (as already assumed), our expression for the cross-sectional area simplifies to

S(g) = 2sta^ (-1-^)^1 - ^ - ^ ^ dk ... (7) Jo (a-k)-^

Evidently S(g) i s continuous and vanishes a t g = 1. F u r t h e r Yire find by d i f f e r e n t i a t i o n t h a t

S-(E) = 2st f i 2 a . 4sta2{1-0 [ " ^ - ^ aic

^^'-^ 0 O ( a - k ) ^

or a f t e r i n t e g r a t i n g by p a r t s

S ' ( ^ ) = 2stö2(1-E) - ^ ^ - ^ - dk . . . (8) '^ 0 (a-k)

from which it appears that S'(C) is also continuous, and vanishes at E=0, and at E=1 (provided that a i». 1), Again differentiating, if follows that

S,.(g) . 2BtaZ'(a£) _ 2sta2 \ ' ^ - ^ ^ dk ... (9a)

1-ê '^o

io-ky

11

-parts, this has the alternative expression

J o S"(E) = 2st0 j-k 2sta ^ 0 for 0 <^E<^ ... .y

(9b)

for 3- <5<1

We see that S"(E) villi be generally discontinuous since Z'(k) will be discontinuous in general for 0 < k < o.

Even if Z'(k) is continuous in the range 0 < k < 1, if the wing section has a finite leading-edge angle, Z'(k) and

so S " ( E ) has at least one discontinuity at k = 1 and ^ = —, Now referring back to equation (1) we see that the second term on the right-hand side vanishes, since by (8), S' (1) = 0.

the furthest dovi/nstream cross-section E='l

The third term involves a knowledge of 0 at In general at any plane E = constant, 0 must be found so that, from

(2),

Mo

dz _z=0 dv dE z=0 / \ CIS „ JjU ,-7 f (sgn z ) — o = — Z ' dE 2c fL-i a c(1-^) ^ S'' ^ .(sgn z) dz

where we have used equation (5) to interpret ^ ^ . Such a boundary condition is met by a distribution of sources along z = 0, It is seen that on E=1,

94

9z"

Lt z=0

- 12

and consequently the sources vanish a t ^ = I, and 0 i s

as a r e s u l t independent of y and z i n t h i s plane. Thus

the t h i r d term on the right-hand s i d e of (1) a l s o

vanishes. Evidently then

r1 r1

D

2% ₀

S"(EjS"(rO f n j i ^ l dEdTi

. . . (10)

where S " ( E ) is given by equation (9) in terras of Z'.

Using the first form of equation (9) (a), it follows then in (10) that 2 2 ^1 ^^ D ^2öfsf qt^ %L^ JO''

f/.

OE r? I

0 [ 1-C

-^0

(a-k)

^61,

^a^

1

1-n Jo(o-f)2 I

Edri

or changing the variables of integration from g and r\ to k = OEJ and \ = OTJ, we have on expansion that

rt

D

2.2 qm t

2o'

% papar rk Z'(k)Z'(^) _ Zliil -'(P)dp,iliki

oJo ( a - k ) ( a - 0 (ö-f) J0(o-p)2 (a-k) 0(a-q)2

dq

,p_LLMap( -A^dq

->o (o-p)"^ ' Jo

(a-qy

fn

_k-e

dkd?. ... (11)

W e now attempt to cast the integrand into the form

-^ pi

Z' (k) Z' {{) f (k,f) dk d£ D

^7?

qm t

integrand is discontinuous along the lines k = 1, and £ = 1 . Taking the component terms in the expansion of the integral in (11) term by term, we note that

JOw o

Z'(k)Z'(^) fn

o(o-k)(o-e)

.1 ,A

k-f

dkdt = Z'(k)Z'(€)^ ^Ov)o(o-k)(a-'r) |k-C dkdf ... (12)

since Z'(k) = 0 for k > 1. Again

yoro

1 ^Jiil

JoJo(o-f)

"•'(-P' O(o-p)' rdp n k-<! idkd-t

Jo(a-t) J

f n 1-^

0 k-1 dk 0(a-p)^ .1

= 1 21ÜI

Jo(a-p) Jp •"0 a-*^ ^''o(a-p) k-dk "1 ;-1 rO in Jk " P-X dlcdt' v/here w e h a v e f i r s t c h a n g e d t h e o r d e r o f i n t e g r a t i o n b e t w e e n p a n d k, a n d t h e n i n t e r c h a n g e d t h e s e s y m b o l s . E v a l u a t i n g t h e i n t e g r a l w i t h r e s p e c t t o p , w e f i n d t h a t rOnO , f k r1 (-1 ^0^0(0-1!) ^•'O(o-p) (o-p)'

'in'

£-k

-' ' i. I i I a-k k-l 'k-t ^

hh(a-()

(a-k)

+ — = ^ in ' + 1 dlcdt

o-k ja-e.

... (13) and similarly

1i+ -rO (O 'O^'O Z'(k) a-k

Uo-ciy

dq

_m

- ^ dkdt' k - f i

'

•

'

' '^'^z'(k)z'(-:)

a-k o-f,

^ L k : : i i n ! ^

rn - ^ ^ 1 + ==—^;n|-^^ +1 Idkdf .

J

... (14) It will be seen that equation {M\.) enables the

repeated integrals of the second and third terms inside the square brackets in equation (11) to be represented in the form required. To put the last term in the brackets in a similar form, we proceed with a similar

series of manipulations to those used in deriving equation (Ii;), and consequently find that

>o o j Ak t H • V 0 ^ 0 T o n^

^ d p f - ^ H ^ d q E n

(o-p) r-l n1 '0 (ö-q)'

_J

k-?

dkd£.

dt

' ( k ) ^ ' ( g )

•'o .0^^-^) (^-^)

[

•,0 ,-o , • . dpi { n j - 2 - | d q

Jt.

ip-qi dk ,<1 •^1 - 0 '

0

io-t){o-k)

^

f n - ^

2(a-k) !a-l a-k

2(a-e)

•fn

a-k

i k r i l i _ fn - ^ 1 + ^

2 (a-0 (a-k) k-t: dkdtl ... (15) C o l l e c t i n g t h e r e s u l t s of e q u a t i o n s (12) t o ( 1 5 ) , and u s i n g them i n ( 1 1 ) , we have D 2 2 a % .1„1 0^0 Z ' ( k ) Z ' ( t )

(a-e)^ |k-f

1

(a-k)

rXn b - ^

2 -

f-k ( a - t ) ( o - k ) . . . (16) dkdt

This is the form of the expression deduced in reference 1, with a = r A t a n A .