The theoretical wave drag at zero lift of fully tapered swept wings of arbitrary section

r 26

m,m^

DELFT

HE COLLEGE OF AERONAUTICS

CRANFIELD

THE THEORETICAL WAVE DRAG AT ZERO LIFT OF FULLY

TAPERED SWEPT WINGS OF ARBITRARY SECTION

by

T. NONWEILER, B.Sc,

of the Department of Aerodynamics

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

VIJEGTUIGBOUWKUNDS Koiiaaisüaat 10 — D£LFT Report No. 7

7^

m.mh

Octoher 1953 T H E C O L L E G E O F A E R O N A U T I C S C R A N F I E L D

The Theoretical Wave Drag at Zero Lift of Fully-tapered Swept V/ings of Arbitrary Section

-by-T.Nonv/eiler s B. Sc.

/ /

-S U \M M A R Y

An expression is deduced for the wave drag of a fully tapered swept wing of arbitrary section in the convenient form of a double integral involving the variation of wing-surface slope. It is concluded,

in the general case, that the drag may best be computed by numerical integration, the method for which will be

the subject of a further report.

In certain cases, however, the drag may be deduced for simple sections by direct integration, notably for the condition that WM -1 cot Aj_ is small (where J\^

is the half-chord sweepback). Some results are quoted for this condition. Calculations suggest that the re_sults thus obtained are approximately valid for

V M -1 cot A i < 0.3J and A tanV^^^ ^ 24,. It is shown that, in this range, the relative drag of different wing sections, with the same root thickness and the same sweep on the maxim-urn thickness line, is roughly that for the infinite wing (which has been deduced elsewhere)*, except that there is some rise (or fall) in the drag with dec/aase in aspect ratio A, due to the presence of a convex (or concave) curvature aft of the maximum thickness.

A further result deduced is that the drag of a fully tapered wing tends to a lower limit than that of an

untapered v/ing, as the aspect ratio becomes infinitely large.

PDF

CONTENTS

Pago

List of Symbols 3 1. Introduction 5 2. The Expression for the Drag. 7

2,1 Both Leading -.nd Trailing Edges Subsonic 8 2. 2 Both Leoiding and Trailing Edges Supersonic 8 2.3 Leading Edge Subsonic and Trailing Edge

Supersonic 9 2. i|. Some particular conditions. "lO

3- The Application to the Section with Discontinuous

Surface Slope. 12

k' The Particular Conditions in which the Expression

for the Drag is Simplified. 13 5. The Drag of Slender Wings. 16

5.1 Limits of Applicability 16 5.2 Drag of Polygonal Sections "18 5.3 Drag of 'I'Jlr.g Sections Expressed as a

Fourier Series "'S 5.I; Drag of Sections Expressed as Polynomials 19

5.5 Drag of Reversed Sections 20 5-6 Discussion of the Results 21

6, Some Comments on Further Work 22

References 23 Appendix I I Derivation of the Expression for the Drag. 2i+

Appendix 71 : P r o o f of a n Integra] Identity. 35

Appendix III : The Edge Force. 36 Appendix IV : Integration of the Expression for the Drag

of a Double-¥/edge Section. 39 Appendix V '. Integration of the Expression for the Drag

of a Wing Section expressed as a Fourier

Series. k"^

3

-List of Symbols

a = 1^Q{Z' (^)-'^H ^ in Appendix III only.

a Coefficients of Fourier expansion for z'(k) - see equation (18).

c Wing root chord.

g( êsy?ni) defined by fc4.uation (3) of Appendix I. h(A,K,Ti) defined by equation (12) of Appendix I.

i defined by equation (1) of Appendix I. 3' defined by equation (12) of Appendix I. k = ^/c, whex'e g; is defined as in Appendix I. m cotangent of angle of sweep.

m(a) cotangent of angle of sweep of spanwise line which is placed at a fraction — of chord.

n maximum thickness position as fraction of chord. tip perturbation pressure.

Q. = ipU^

r r a d i u s of curvature cf wing leading edge a t

root s e c t i o n .

s wing semi-span.

t maximum wing thickness (of root section) w upwash perturbation velocity

x,y,z system of cartesian co-ordinates v/ith origin at wing apex, with the positive x-axis lying

down-stream of apex along wing root, and the plane of the wing being z = 0.

z(x) the wing serai-ordinate at the root section at a distance x from the apex.

A wing aspect ratio (= i|s/c) D wing wave drag at zero lift. H(a) = 0 if a < o; = 1 if a >^ 0.

I(A,K) defined by equation (I5) of Appendix I.

I^ (A,K),l2 (A,K) defined by equation (22) of Appendix I.

I,(A.,K) defined by equation (3) of main text. J(T]) defined by equation (16) of Appendix I. M free-stream Mach Number.

P(A,K') defined by equation (6) of main text. Q(X>H) defined by equation (9) of main text. J? I I 'the real part of ^ ^'.

S ?/ing area (= sc) U free-stream velocity

X defined by equation (12) of Appendix I. Z(k) = ^ z(kc)

a = Ps/c

e used variously for a small quantity, and also for source plane strength (Appendix I)

r\ = y/s 6 = cos"^ (2k - 1) K = o - k

A = a - f

M, = mB [i = 3m (ac) -1

g = cosh (20-1) in main text, and Appendix III*, also used as a variable of integration in

Appendix I, being the distance aft of the wing apex of ti\e elementary source plane.

p air density in free-stream

a = s/cm(0)

0 = cos"^ (2^-1).

Primes denote differentiations with respect to the argument of functions.

N. B. Accepted convention of signs and principle parts I

-Jx^ i\JTx\

j if X < 0,

tn(x+iy) = -^-fnlx +y | + i tan" ^^\ , where -K >^tan~'([|)>0 if y > , 0.

cosh" X = Tn(x +'v/x2-1). 0 <• cos X < 7C, if |x| < 1.

5

-Introduction.

At the present time the greatest need in assessing the theoretical wave drag of svifept Yifings at supersonic speeds, is for some yardstick by which we may assess the effect of wing-section upon drag. Numerical results are available from linearized theory for -.Tings of double-wedge section for a variety of planforms, and for untapered wings of biconvex section . However most of the swept wings envisaged for supersonic aircraft have round-nosed sections, and apart from a single solution existing for

2

v/mgs of infinite sp^n , no theoretical information is available for the drag of ¥/ings with such sections. This solution is of some value in suggesting the drag of untapered wings of finite aspect-ratio, because the bulk of the drag at low supersonic speeds derives from the flow near the root-, but if applied to tapered wings one cannot properly account for the interaction of the varying effective angle of sweepback over the section on the total drag.

It is known for instance that for fully-tapered wings of double-wedge section the drag is reduced at low supersonic speeds by putting the maximum thickness line forward, so that it is more highly swept . Whilst, of course, such a step would not be envisaged without taking into account the effects of such an alteration of the section upon the other characteristics of the wing, nevertheless it appears to be an important consideration in \i'lng design.

However there has as yeL been no method suggested for computing the drag of tapered v/ings of other sections than the double-wedge, except by the rather laborious process of approximating to the section shape by a polygon . Our first concern here is then to formulate the problem and find the expression for the drag of a tapered wing assuming a completely arbitrary section.

We do this here for a fullj'^-tapered viring, with a geometrically similar section at each chordwise station-along the span. For the purposes of the discussion the slope of the section is assumed continuous, although we shall afterwards discuss the validity of the results if the section slope is discontinuous.

The answer may be written down as a double Integral, involving of course an expression for the variation of the section slope, and including parameters which relate to the T/ing lolanform and free-stream Mach Number. The integrand is a lengthy expression, and even if the wing section could be expressed by a simple formula, it seems doubtful whether in tho general case the integrations could be performed to yield a value for the drag in terms

of known functions of the planform and Mach Number parameters. In view cf this consideration, and in view of the

additional fact that very fev/ ¥/ing sections can be

expressed adequately even by means of a complicated formula, it is desirable to consider a numerical method of integration', this will be the subject of a later report. In two

particular cases, however •> direct integration is possible for T/ings with simple section shapes (including some with bluff noses). One case is that of the wing of infinite

span, and this has already been studied', the other case is the one which will be considered in some detail here, and vvhich we shall call the application to 'slender wings' or, in other words, v/lngs of high sweepback.

To be more precise it is shown in para. 2. ij. that if A l s the wing aspect ratio and .Aj^ is the half chord

2

sweepback angle, then for small values of the parameter

\x ='VM2-1 cot J\^^i the exuression for the drag may be

'•- 2

developed as a power series in \x of the form f ^ (A tan yi 0 + f, (A tan A 1 )M-^ +

U 2' ' ' T

where the functions f , f... of (A tan A i ) are bounded provided the wing telling edge is sv/eptback \X»Q. the value

of A tan y\ jf exceeds 2). Thus for highly swept, or 'slender', wings (where cot J\ ^ — * 0 ) , the expression for the drag reaches a finite limiting value which is a fiinction of A tan-A j_> this limiting value is at least a good

approximation even if u- is not zero but merely small. The same limiting value apparently applies If M — ^ 1 , but of

course it must be borne in mind that the linear theory is not strictly applicable to this condition. In fact, the

same kind of restrictions apply to this approximation as are involved in the development of the well known 'low aspect ratio' or 'slender wing' theories of the lifting wing developed by R.T.Jones and others.

TECI-n>JISCMn 'HOGESCHOOL^

Vl,IEGTü!GBOUV7KUND2 Kcmaolstiaat 10 — DELFT

7

-The expression for the drag in this limiting condition is again a double integral, but is considerably simpler than the general expression to evaluate, so that a fairly v/ide numerical survey of the drag of various wing sections

can reasonably be undertaken. The results of this

'slender v/ing' theory can be used to reach certain conclusions about the relative dra^ of various sections over a range of aspect ratio, and provide a check on our results in relation to data already published^ as well as enabling subsequently an assessment of the accuracy of the processes of numerical integration to be made.

The Expression for the Drag.

The details of the mathematical argument will be found in Appendix I, Briefly the method is first to represent the wing by a superposition of source planes (representing one integration) the pressure distribution for which has then to be Integrated over the wing area to give the drag force (representing two further integrations). By suitable choice of oblique axes, one of these integrations (representing the drag of one elementary soiirce plane on a spanv/lse line

across the wing) may be conducted without knowledge of the v/ing ordlnates. The resulting expression for the drag is thereby suitable modification, expressed as a double integral, the kernel of which Includes the expression for the wing slope

z'(i^) =

¥

^ > k = 1 (^)

?/here c is the root chord, t the root maximum thickness, and z(x) is the viring semi-ordlnate at the root at a distance X aft of the wing apex. The remainder of the kernel is a complicated expression involving the parameters

a = J- A tan A ^ -- j - A tan A j ^ "^ 2" )

. ' > (2) and a = r-v/M^-l cot A \

4 o J which may best be expressed in different vmys according to

the oosition of the Mach lines from the apex on the wing. In the following paragraphs these various expressions are written down, and are extracted from the final results enunciated in the Appendix I, In some particular cases

these expressions are cai3able of considerable simulification, /and.

2.1 Both Leading and Trailing Edges Subsonic (CASE I) This is Case I of the Appendix I, and the relevant result Is obtained from equations (31) and (32) of that Appendix. We have from these that

D 2a qt 0QtJ\ „ % 2 r1

de

o

o

'(k)z'(-?)rP(A,K) + P(K,A)1 dk

^ f j Z'(k)7'(«e) fp(A,K) + P « , ^ ) ] dkd-«

Jo Jb

(3)

where P(a,b) is a function of a, as well as of a and bj which is given by

P ( a , b ) = 2 g 1

^ [

4^^Z^'^ (a+b)'

.(2a ^J-b^-a^)

2ab (a+b) 2b Ub , b - > 2 a b - a f " T " + 2 è a+b a - a JL

(a+b) (a~b)Jb -d"

, - 1 / a b - a \ „V,-1 / a i cosh /—:—-—\-cosh ƒ— VI ( a - b |

(a+b)V^^-o^lb^-a^

a+b

f

"0

(k)

and where and K = 0 - k (5)

The parameters c, and o have been defined in equation (2), and Z' (k) has been defined in equation (1). It should be noted in the present case that A and K are both greater than 0. over the range of integration.

2.2 Both Leading and Trailing Edges Supersonic (CASE II) This is Case II of Appendix I, and the relevant

result is obtained from equations (33) and (3k)- Thus we find that

D

fCz'mJ

-2

rr-— = o - \

7_-Ur. dAJ +

^~\

def Z'(k)Z'(f)Q(A,K)öJc

qt^cotV^o Jo'/a2-A^ % /where... 2n1 p«

-\ de Z'

Jo Jo

(6)

_ 9

-where % ( A + K ) - Ajd -A c o s

\ a

K^+2KX->>F

(a^-X^)(K+A)^ [/a^->?

A ^ + 2 A K - > ' V ^ (a2-K^)(K+;,)2

^ c o s - V ^ \ - 1

J:^J

c o s -1 a

';^i-i

y a

-(7)

is a function of a as well as of A and K 5 as before

and where

A = o -I ,

and |< = a - k.

i

2.3 Leading Edge Subsonic, and Trailing Edge Supersonic (CASE III) Both /\ and K" are in •uhis case less than a. This is Case III of the Appendix I, and the relevant

result is obtained from equations (35) and (36). It is the most complicated case to deal with, since we must divide the range of integration over the wing surface so as to

isolate the regions of 'subsonic' and 'supersonic' flow. We have from the A-o-oendix I that

D

qt^cot^A,

. n2 f fz' a)!^

.2 2a %

1'

'a-a

ilililde + •^^['^'d\i|' Z' ( O Z ' (k)[p(^,K)+P(^<,>5ldk .

^ -Jo Jo

z'ii!)

1

f a_a \ b-b-a Z' (k )Q (A, K)dk +\ Z' (k )R (.>», K ) dl<:| d'i 'o (8)

Here the functions P and Q, and the variables ?\ and K have been defined before in equations (ij.), (5) and (7) above," the ftmction:

R(;^,»<) = 1 rK(K + A) ^ _ 2 L K^-O-^ , 2 K A " | kic K:^+2Air-A^

'\+K

a - A 2 x2 cos .1 M^-'; _o,(»<-A),

£ 2 ^ . ^ _ U P \

(^

^».)'y^^WK\'^-aV uj

+ 9 ? K -2K;»>-A K + 2 X A - A ^ ( a 2 - A ^ ) « + A ) ^ (K^-a^)(A+K)^ +

(A + K ) ^ - A ^

K

UA ^ A^+2XC^

(^+K)VK^-a2 U+lC y<"-n,^

2. I|. Some Particular Conditions.

(9)

These general formulae are rather too complicated for Integration in terms of known functions, unless

(perhaps) the surface shape were expressable in a

particularly simple manner. However for a few particular types of planform» or ranges of Mach Nximber, it is possible to develope expressions which permit Integration even for complicated section shapes. These further cases, which are arrived at by assigning certain limiting values to the parameters a and a, will no¥/ be distinguished.

I

I

Case IV ! a ~i oo, 0, = M, o, Oi < 1).

=0—^—^^-o '—

This Is a particular example of Case I,* from (ij.). expanding P (./»,«) as a series of powers in

2 P ( A , K ) = C.+C_sgn(K-A) + — ^ — ^ ' ^ ^ " ^ t n ^ 20^^(1-LLf)^2

1 .

o ' 1

K-A

^ 20^(1-1.2)^2 ^ . ^

(2-iiai£rl< H. o / i 3 \

N o t i n g t h a t , s i n c e Z(0) = Z(1) = Os 1

I

1 ci

d^ I Z'(k)Z'(t).^k =

o J O Z ' ( f ) Z ( t ) d J l = 0 /we . . ,

4

11 -we h a v e i n t h e e q u a t i o n s (3) a f t e r s u b s t i t u t i n g from (10) t h a t ; 2 r1 r^ D 2-[X, + 2 +2 . q t c o t A %

( 1 ~ t f ) ^

1 2 - ^ L^Q._, 7C ( 1 - ^ i ^ ) - 2 d€ ^0 1 ^1

Z' (k)Z' {{) in

0 1

k-f

dk + 0

Ö,

QJ 0

/;' (k)Z' (t)tn| ldkd£ +

of-> (11) C a s e V : g-Hüco, a = 0 ( 1 ) > 1 . T h i s i s a p a r t i c u l a r e x a m p l e of C a s e I i ; a n d f r o m (7) e x p a n d i n g Q ( A , K ' ) a s a s e r i e s of —, we e a s i l y e s t a b l i s h t h a t Q(Pj,K) = O j - ^ j . Thus i n e q u a t i o n (8) s i n c e

(oot^7^„) t=^,V. = i(|)%= = i ( i ) \

w h e r e S i s t h e w i n g p l a n a r e a , i t f o l l o w s t h a t 3D

m

' 1 2

[2'(^)] d-f + o A

(12) Case VI : a -* 0, o > 1 T h i s i s a p a r t i c u l a r e x a m p l e of C a s e I , a g a i n ; e x p a n d i n g P ( A J K ) a b o u t 0. = 0 , we f i n d f r o m (i+) t h a t

P(,,,K)= ^i^-fna+4,fnK

K I 2K' A K-/^ so t h a t

P(A.*<)+P(K,A) - -^^nK+^enA+fV-Tf^!--^!- "^ + 0^°'^)

>^ x"/ IK-A» ^K

Thus i n ( 3 ) D qt^cot^^N^, 2 0

_{- 'f de [ Z ' (k)Z ' (of ^ n -J^l+ \in M}

rt

( 1 3 )

l l

~ - j d l c Y/e n o t e t h a t a s o - ^ - o o , i t f o l l o w s f r o m e q u a t i o n (11) t h a t t h e e r r o r i s n o t m e r e l y 0 ( a ) b u t 0 (M, ) = O j ^ j « A l s o / I f . . .

If o is finite and > 1, we may put 0 (a ) = 0 ( ^ j * thus in general we have that

;a^) = o ( ^ ) ;

Jo Jo ''I

_{'•^-^' \fo '^ Jl}

-• 04)

of

It is sometimes more convenient to refer to the half chord sweepback (A 1) rather than that of the leading edge

{AQ}» in general

cotA^ = °°*^o(^-|) ^^5^

so that in (lij.) we obtain an expression for D/qt cot A J L ^y

2

replacing o by (o--|).

The Case of the Section with Discontinuous Surface Slope. It is unnecessary to examine in detail each stage of the arguments Y\rhich lead eventually to the equations (3) to (lAj.) if the expression Z' (k) is not continuous as has been assumed, For the most part, it will be sufficient to show that for the kind of discontinuities envisaged the integrals Involved in

the equations remain convergent.

The integrals are all, in fact, absolutely convergent if Z' (k) is merely bounded over the region of integration , so that the only case in doubt is that in Y/hich Z' (k) is somewhere singular. The most coinmonly occurring case is that for v/hich Z' (k) has a singularity of half-order at the

/leading... Exceut in the case where a = 1, c, = 0, which corresponds to the delta wing at M = 1. Here the double integrals are

non-convergent unless Z-'(k) is zero at k = 1 (i.e. the trailing edge is cusped).

TECI-n^SCHE HOGESCH

yUEGTUIGBOUWKlJNDH KocaülaiiGGt 10 — DIIiT

13

-leading-edge *. this corresponds to a wing section with a

bluff, or radiused, nose such as is common to all conventional low speed aerofoils. It is easy to shovir that for such a

condition the integrals remain convergent, unless the line of the singularity on the wing plan form is 'supersonic',

(i.e. it lies ahead of the Mach lines); in this case the single integral in equations (6) or (8) is non-convergent.

One further point does arise in connection with the

evaluation of the drag of such bluff nosed sections *. R.T.Jones has pointed out-^ thar the drag of such sections has to be

estimated with care as there is, on the basis of the linearised theory, an edge force at the nose virhich is not included by

the usual methods of integrating normal pressures over the v/ing to yield the resultant drag. This edge force exactly cancels a term which occurs in the integrations and which has the form

\ \

f ziMz.'lil a<.

Jo Jo k - <

This re-|peated Integral is non-zero if, and only if, the function Z' (k) is singular in the range of integration; and the edge force exists if, and only if, the slope (given by the same function) is infinite at the nose. It v/ill be noted that in our arguments in Appendix I v/here Z' (k) is assumed continuous, and so bounded^ a repeated Integral of

the form of that given above has been isolated and shown to vanish; this occurs in the derivation of equation (25) based on the arguments advanced in Appendix II. If we are to

consider singularities in Z'(k) it follows that this repeated integral no longer vanishes '. hov/ever, as we show in Appendix III, it yields a value which is precisely cancelled by the inclusion of the appropriate edge force as obtained from ref.5, so that its neglect is quite justified whether or not the surface slope is any^-vhere infinite.

Of course, the usual reservations must be made if T/e are to consider the drag of round-nosed sections by the linear theorj'' '. but in so far as it has yet to be shown Y/hether the ansv/er obtained by it is at all reasonable, it is evidently desirable to have a method of estimating it.

k' The Particular Cases for Yfhich the Expressions for the Drag are Simplified.

In the discussion of para.2, we found that in the three /conditions. . .

conditions designated as Cases IV, V and VI, the expression for D was greatly simplified. We shall consider these one by one.

Case IV is that of the infinite swept wing with subsonic edges, or of the highly swept v/ing of finite aspect ratio. That is, it is the result for the condition

a — «) i.e. :-r.— -* «O, and n =p cot A . < 1.

C CL'TI , ^ O O

which may be interpreted either as

f--* «o (cot /\^ finite), 3 cot A < 1. or cotT^ -* 0 (!• not small), R < r-rx

cot 7\ o

Because it holds for the infinite Y/ing, we should expect the result to agree v/ith that already published for such wings. A comparison v/lth the result of ref. 2 reveals this agreement

p

v/lth one exception - that a factor (3-1^^) is replaced in (11) by the factor (2-u^).

This discrepancy is at first sight Irreconcilable '.

however it v/ill be noted that the results for ref. 2 were derived by considering the drag of the Infinite v/ing as the limit of that for an untapered v/ing of large span, whereas here it is the limit of that for a fully tapered v/ing. The fact emerges that the limiting values are different, which may be substantiated by examining the trends in numerical computations carried out for both types of v/ing

(tapered and untapered) for plan-forms of finite aspect ratio. Figure 1 shovrs some results extracted fromref.l, where it will be seen that for the sam.e root v/ing section, Mach Number, and half-chord-line sweepback, the drag of the fully tapered wing is considerably smaller at high aspect-ratio than that of the untapered wing. The fact that they are not the same is understandable, but that they do not tend to approach each other as the aspect ratio increases is at first sight a surprising result, for the wing plan forms in the limit would seem to coincide near the root, where most of the drag is developed.

Yet the span of the tapered wing is only a half that of the untapered wing with which it is compared, and Its

-

15

-area is only a quarter of that of the unta-oered v/ing, and of course this applies even as the aspect ratio increases to infinity. For this reason the smaller drag of the tapered v/ing does not seem unreasonable. There are, too, other properties of v/ings of infinite span which suggest that the value of tho- proper-uy depends on tbe manner in which one reaches the iiraiting answer. Consider, for example, the aerodynamic centre of a laminar delta v/ing! this is (theoretically) a'c the -^ c. point of the root-chord independent of asioect ratio, however large this may bet yet as v/e allov/ the asioect ratio of a straight (rectangular) v/ing to increase, on the other hand, its aerodynamic centre moves aft to the 4-chord point of the root-chord. In the limit both delta wing and rectangular v/ing appear to have

the same planform if the aspect ratio is infinite, and yet the aerodynamic centre of the wings is at a different

position.

Returning to our consideration of Case IV, since, apart from the slight modification mentioned - which only affects the variation of the drag v/lth .a , - the results of this case bave alread-y been discussed in ref. 2, v/e shall not drav/ any further conclusions here about the effect of

section shape on the drag of any of the planforms the case includes.

Nor is Case V of any great interest : this result (equation (12) ) merely states that if a is sufficiently large, to a good approximation the drag coefficient of the wing (based on its planarea) is the same as that for the

same v;ing section in two-dimensional flow at the free-stream Mach Number. The conditions for this to be true are that

a = iiË -^ c», a = ~r-:r- = 0 (1 ) > 1.

c c cot J\ "^ \ J -^

v/hich implies either that the Mach Niunber of flight is

very high, or that the aspect ratio is high but the sweepback is small', in either case the flov/ is two-dimensional outside the Mach Cone from the wing apex, and this latter affects only a negligible portion of the Y/ing. Strictly the two conditions may be expressed as '.

either (3 -* oo, \^ and cot,A not small

o

•)

or f "^'^y cot/\ = 0(|)? (l^ not small).

This deduction about the drag coefficient of such planforms has been noticed before, of course.

Of greatest interest to us here is Case VI, v/here a --* 0 and a > 1.

As i s shov/n in the d i s c u s s i o n of the formulae, i t i s

s u f f i c i e n t i n deriving these r e s u l t s if we assume

( J , ^ 0, a n d a > 1.

These conditions may be Interpreted as meaning that either the Mach Number is near unity, or that the leading-edge

sweepback is high (provided in either case the trailing-edge is sv/ept back.) The consideration of the ?imit M-* 1 is not strictly justifiable on the basis of the linear theory,

so that we shall restrict our interpretation to the condition that the wing sweep is high so that

cot /^^ -» 0. (8 finite)

Whereas in Case IV, ?.re could also consider such highly swept v/ings, but only those of finite aspect ratio, we may now consider also those of small aspect ratio as well. For

this reason v/e shall distinguish Case IV as that of the 'slender wing', since it conforms with the same kind of

restrictions as the 'slender-body' theory applied to lifting wing surfaces, developed by R.T.Jones .

5. The Drag of Slender Wings. 5.1 Limits of Applicability.

Equation (1i+) gives us the expression for the drag of slender v/ings, if v/e interpret these as satisfying the restrictions noticed above. Our first problem is to ascertain v/ithin v/hat limits practical wing shapes are likely to have approximately the same drag as that found by this slender-wing theory. The drag of fully-tapered wings has only been extensively computed for those v/ith double-wedge sections, and v/e shall therefore base our deductions on the results for this v/ing-sect ion.

For the double-wedge section v/ith a maximum thickness /at.,.

17

-a t -a f r -a c t i o n n of the chord, i t i s shown i n Ar)pendix IV t h -a t

I

D

^ 2afƒ injo-n) _ f n [ a ( l - n ) l _ fnt(a-1)n

q t ^ c o t ^ A ^ % (0(0-1 )n(1-n) (a-n) (a-1 )n a(a-n)(1-n)_

This expression has been evaluated for n = 0 . 2 ( 0 . 1 ) 0 . 6 , and

the r e s u l t s are shown i n f i g u r e 2, where from (15)

M^\

(16

(- iy

D

q t ^ c o t ^ A i \ 2a j qt^cot^y\

is shown plotted against (1/Atanj^O = 1/(1+0-2).

It will be seen that over the greater part of the range of aspect ratio (for Atanv^j_ < 7), the forv/ard position of the maximum thickness reduces the drag, as opposed to the

conditions existing for ver;;- high aspect-ratio v/ings. The variation in drag with

yM^TcoM, = .^(^°_)

for v/ings of double-v/edge s e c t i o n (v/ith maximum t h i c k n e s s

a t 0.5 chord) i s shov/n in f i g u r e 3* v/here the r e s u l t s are

taken from ref. 1 with some slight modification near (but

y

p '^ 2

M -1 c o t y i j = 0 to include the f a c t t h a t D oc jo,

near u^ = 0 . I t w i l l be seen t h a t provided AtariA 1 X- '^•5?

o - y-p"^ "* 2

the change in drag i s under 10/C for v a l u e s of vM -1 c o t A i ,

l y i n g between 0 and 0. 3>i so t h a t the slender-wing approximation

(|j,Q-• 0 ) , i s v a l i d over a reasonable range of Mach Number.

This does not apply t o the r e s u l t s of lovi^er aspect r a t i o ,

Moreover the r e l a t i v e drag of the v a i i o u s s e c t i o n s

does not vary g r e a t l y with Mach Number. This i s shown i n

figure 4> where the r e l a t i v e drag i s shov/n for various

M -1 c o t ^ j _ ,

2

/ f o r . . ,

" In other respects the agreement of the present calculations with those of ref.6, for M = 1, is excellent.

for a v/ing having Atanj'^_i= 6. These results are again obtained from ref.1. What change there is appears to emphasise the reduction in drag by putting the maximum thickness line forward - v/hich is, physically, v/hat one v/ould expect.

5.2 The Drag of Polygonal Sections.

The drag of polygonal sections may be evaluated quite easily using the method of Appendix IV. For instance, using the results of the Appendix, it may be shown that if

Z'(k) = n^, for k^^_^ < k < k^ f o r a l l V = 1,2, . . . p , where k = 0, and k = i ; t h e n 2 P v-1 q t c o t ^ A ^ r, v=1 n - 0 ^ '+^ M- M.+1 V n v/here S ( < , k ) = s ( k , € ) = - - - ^ L d ü i _ t n | k - q + M ^ { n ( a - k ) + f M ^ ( a . ^ ,

(a-i)(a-k) ' ' \aA) \g^k)

and n and n . ï^re taken as zero.

This formula is quoted for reference only and no numerical examples have been v/orked '. the drag of such sections may of course quite readily be estimated not only for the condition

[X = 0, but for other Mach Numbers, using standard methods. 5.3 Drag of Wing Sections Expressed as a Fourier Series.

If we suppose that the ordlnates of the v/ing-section are expressed as a Fourier Series, i.e.

Z(k) = H ) a^ sin no, where k = ^ ^ ^ (18) n=1 ^

the double integral of (1i+) is capable of evaluation in terms of the coefficients a : this is done in Appendix V.

Some of the simpler results may be noted here : for

ï l l i p s e , a =1, a, = a , • n 2 3 v/e may t h e n c a l c u l a t e t h a t j

t h e e l l i p s e , a =1, a,. = a , = . . . = 0. From Aupendix V, n ^ 3

/ _ D _

qf^cot A i

19

-D 2

— _ _ = 7^ coth g , where cosh E = 2a-1 qt cot A ±

2

qt cot A ^ k o(a-1 )

2

The Joukowskl section is given by

Z(k) = ~^[k^(1-k)'^]

373^^

i . e . a. = -%- , a^ = - —-- , a , = a, = . . . =o; we t h e n

^ 3B ^ 343 ^ ^•

calculate that

,-\ 2 _ D ^ 83_ q t ^ o t ^ A 1 27 (19) ^ ) ( - - ^ ^ ) ( - )

v/here '6, is defined in the previous equation.

The 'double-cusped' aerofoil section is given by

Z(k) = 8[k(^-k)J ^

3 1 i.e. a. = f^, a^ = 0, a, = - 7-, a, = a^ = . . . = o; then 1 4 ^ 3 L]. 14. ^ qt cot Aj_ 8 2

5.ii. Drag of Sections expressed as Polynomials

The drag of sections v/hose shape is given by polynomial expressions involves in its evaluation a great deal of

labour, and only the simplest examples are quoted. For the biconvex section, v/here

Z(k) = I|Ic(l-k)

evaluation of the Integral (1I|.) reveals that

D

qt^cot^A

_=: i(2o-1)^ ( " ( 2 0 - 1 ) ^ 1 / - ^ ) - 1 - a ( a - 1 ) { t n - ^ j I (22)

_{1, % \a-1/ V 0-I /}

In t h i s Integration, the ansv/er i s obtained in terms of

Euler's dilogarlthmic Integral Lp(x). However, these

terms may be expressed in terms of logarithms, if note is

made of the identities

L2(x) + L2(1-x) = L2(1) -"tnlxjfnjl-xf.

^S^ Mz:) -' -*

MTT-These relations are also used in estimating the drag

of a section such as that given by

Z(k) = ^ k ' ^ ( l - k )

2

which has the 'conventional' bluff leading edge and angled

trailing edge; calculation reveals that

"^cot A i

16% \ a J

qf^cotV^i 167C

z' 2

5,5 Drag of Reversed Sections,

^coth'^'a^

J

(23)

The v/ell-known reversed flov/ theorem a p p l i e s to the

drag of the slender wings I the drag i s the same if the

flow d i r e c t i o n i s reversed. This m.ay e a s i l y be v e r i f i e d

2 2

from equation (1i(.b) if the value of (D/qt cot .A±) i s w r i t t e n

dov/r., and (1-k), ( 1 - f ) , and (I-0) are s u b s t i t u t e d for k, "f,

and a r e s p e c t i v e l y : the r e s u l t i n g double I n t e g r a l i s then

i d e n t i c a l in both cases. The physical i n t e r p r e t a t i o n of

t h i s r e s u l t i s t h a t the drag of a sweptforward v/ing i s the

same as t h a t of the reversed (sweptback) wing v/ith a reversed

s e c t i o n shape. As there appears l i t t l e i n t e r e s t i n the

drag of sv/eptforward i/ings a t the moment, no data on the

drag of such wings with sections asymmetrical f o r e - a n d - a f t

i s presented here. For v/lng-sections with f o r e - a n d - a f t

/symmetry. , .

II I. .,1. .aw». -. • - » • m . I . . . I. n 'yii. . . • • . i. n ^ . n i . i ,i i , „ . . 1 . . 1 . . . 1 1 . . . , 1. » j. 1 « . , . n i l •

*It is not established, yet, that the reversed flov/

theorem may be extended to ixiclude the cases where there

are edge forces present (as in the application to the

bluff-nosed wing). The fact that the present work

indicates that the reversed flov/ theorem does apply in

this particular case, lends substance to the belief that

such an extension may be possible.

21

-symmetry, the drag is the same, of course, for the reversed (sweptforward) v/ing,

5.6 Discussion of the Results.

The variation of the dra^ v/ith aspect ratio and sweepback, of the five sections given by equations (19)

-(23), is illustrated in figure 5. Once again it v/ill be seen that a reduction of the aspect ratio tends to reduce the drag of those wings v/hich have a forward position of the maximum thickness, as is the case with double-wedge wings. Moreover (particularly as the aspect ratio is reduced so that the trailing-edge becomes nearly unswept) we see that the sections with cusped trailing-edges appear

to have least drag.

The mathematical explanation of this latter effect lies in the fact that the pressure distribution near the trailing-edge becomes singular as AtanAi-» 2, Thus, f or

2

slender wings v/ith a f i n i t e t r a i l i n g - e d g e angle there i s

a logarithTiiic s i n g u l a r i t y in drag a t AtanAi = 2 (and of

/'"2 ' 2

coursevM -1 coty\j_= o) v/hilst the drag remains f i n i t e for

2

s e c t i o n s with a cusped edge; a b l u f f edge, as on the

e l l i p t i c section, produces a simple s i n g u l a r i t y a t AtanAj_= 2,

2

Hov/ever i t i s doubtful whether t h i s has much physical

meaning! the t h e o r e t i c a l l y high drag of d e l t a - l i k e wings

near M = 1 has not been recorded experimentally.

In f i g u r e 6 the drag of the v a r i o u s s e c t i o n s I n v e s t i g a t e d

i s shov/n in r e l a t i o n to t h a t of the double-v/edge v/ing (v/lth

maximum thickness at h a l f - c h o r d ) . The b a s i s of t h i s

comparison i s t h a t the root maximum t h i c k n e s s and h a l f c h o r d

-l i n e sT/eep i s the same for both s e c t i o n s , and the r e -l a t i v e

drag i s shov/n as varying v/ith A t a n A i ' the comparison serves

2"

f u r t h e r to emphasise the remarks made previously about the

effect of maximum t h i c k n e s s p o s i t i o n and. the t r a i l i n g - e d g e

shape.

A particularly apt way of accounting for the effect

of the maximum thickness position, is to quote the comparison as above except that not only the maximum thickness but also the sweepback of the maximum thickness line (instead of the half-chord line) is the same for both sections. This is done in figure 7? where the relative drag on this basis is

p l o t t e d a g a i n s t Atanj^i as before '. hov/ever, v/e nov/ note

2'

t h a t since j/\j_ i s not the same for b o t h v/ings, the aspect

2

ratio A will also differ. The effect of this artifice is to eliminate to a large extent the variation of the relative drag with the parameter Atan/i ^_. The only broad tendency

2

to be observed from figure 7 is that for sections with convex curvature aft cf the maximum thickness there is an

increase In drag as Atan/\i falls, v/hllst the opposite is

2"

true if there is a concave curvature I the effect is more pronounced if the maximum thickness is back at the half-chord - that is, if +he curvature near the rear of the section is also more pronounced. For sections without

any curvature (i.e. the double wedge sections), the relative drag, on this basis, is not greatly affected by change in aspect-ratio.

Some Comments on Future Yifork.

It will be evident that even in the case of the slender v/ing, and where the section shape may be expressed in a

closed form, considerable effort is required to obtain an expression for the drag. There seems no doubt that for other wing shapes or Mach Number conditions, where the

variation of a must also be taken into acco\int (in which case the kernel of the double Integral is far more complicated), and where the wing section (if described even by an

approximating closed form of expression) is more involved, the evaluation and computation of the double Integral for the drag presents a tedious problem,

Bearing this In nlnd, some v/ork has been done to find the best method of performing a numerical integration of

the drpg double-integral. This in itself presents a problem as proper account has to be taken of the singularities in the integrand *. if the v/ing edges are subsonic, v/hich is the case of main interest, the range of integration is over the unit rectangle, with a logarithmic singularity across a diagonal (k = t ) and (if the wing is bluff nosed) with a singularity of half-or der along each of the tv/o adjacent sides. Hov/ever, the formula for nvimerical integration, once set dovvn, is applicable v/hether or not an expression for the v/ing shape is known in a closed form, and the value of the kernel of the double integral (independent of the expressions for the surface slope) may quite easily be asslmilcited v/lthin the weighting factors of the formulae.

- 23

This method of computation therefore promises to be easy to apply, once the fundamental calculations are complete.

The method has something in common v/ith Beane's approach to the problem of the drag of a tapered biconvex wing^ by using an approximating polygon to the section, but is likely

to be easier to apply, and to be a more accurate -".nd more flexible method.

As the problems of the method are peculiar to the subject of N-umerical Integration, it has been thought desirable to discuss these in a separate report, which it is hoped v/ill shortly be published.

References 1. T. Lawrence 2. T. Nonweiler

3.

4. A. E, Puck e t and H . J . S t e v / a r t , B. Beane 5. R. T. Jonei 6. R. T. J _ones

C h a r t s of t h e Y/ave Drag of Wings a t Zero L i f t .

A.R.C. Current Paper 116, Nov. 1952 Theoretical Supersonic Drag of Non-Lifting Infinite Span Wings

sv/ept behind the Mach lines. A.R.C. R. and M, 2795 (1953)

Aerodynamic Performance of Delta Y/ings at Supersonic Speeds.

J.Ae.Sci.,Vol. 14.No. 10 Oct. 19ii.7. pp.567 - 578.

The Characteristics of Y/ings having Biconvex Sections.

J.Ae.Sci., Vol.18,No. 1. Jan. 1951. pp. 7 - 19.

Leading Edge S i n g u l a r i t i e s i n Thin A e r o f o i l Theory.

J . A e . S c i . , V o l . 17»No.5. May 1950. p p . 3 0 7 - 310.

P r o p e r t i e s of Low Aspect r a t i o p o i n t e d Wings belov/ and above t h e

speed of sound.

Appendix I

Mathematical Formulation of the Expression for the Drag.

A uniform plane distribution of sources in the plane z = 0, bounded upstream by the lines ^ = + (x-g) for x X ^,

m

w i l l produce (in the plane of the sources) a pressure

2

d i s t r i b u t i o n given by

2e q

^ ( ^ , ( = o . - |

( x - g ) - M . ( y / m )

ia|x-£-(y/m)|

+ cosh'

- 1 (x-g;)+M. ( y / m )

H^-^+ml J)

l l H ( D } l H ( D •

v/here H(j) = 1 i f j > 0 and H(3) = 0 if j < 0> and where

j = (x-5) - 3|y) i f ^i < 1 ^

= (x-^) - (|y|/m) if n > 1 b

0)

This expression may be virritten more conveniently as

7c3 A P 2s q iL _^.^

_{osh-V'-^ .M^-^^-M^}

2 ^2 2

yn;^^'"" L u^(x-^)2-pVij

= g(x-ê, |y.|, m), say.

• H ( ó ) ( 2 )

The quantity e v/ill be termed the soiirce plane strength, and is, in fact, related to the angular deflection of the flow normal to the source nlane '. we have that

lim

z->0

w

(sgn z)Ue H[m(x-e)-|yl] (3)

Consider now the following s u p e r p o s i t i o n of source-planes I

(I) a source-plane bconded by y = +m(0)x of s t r e n g t h z' (O)

( I I ) a d i s t r i b u t i o n of source-planes bounded by y = +m(^)(x-E)

of I n f i n i t e s i m a l s t r e n g t h z"(f;)d£, for a l l 0 ^ g <. c.

( i l i ) a source-plane bounded by y = +m(c)(x-c) of s t r e n g t h

z ' ( c ) .

Further, v/e suppose t h a t , i f s > 0,

Then from (3)y using (k), v/e f i n d t h a t for t h i s f i e l d of

flow :

ih)

L)

(sgn z ) ^

_{= z}

z=0

m(g)(x-g)-ïylj dg

25

-'(0)H[m(0)x-jyjl+f z«(g)H

-z' (c)H[m(c)(x-c)-|y|]

- • ( ° ) H [ ( - ^ - 9 ( ^ - ^ ) ]

Performing the integration, we find that, for

1

( yj ^ s

(sgn .)ü ..[(x- J^)/(1- M ) ] If J ^ < x ^ i ^ . =

a^

z=0

IcT

(5)

Hence the superposition of source-planes described above

s a t i s f i e s the boundary conditions for an arrowhead wing of

semi-span s, with root chord c, and v/ith swept-back leading

and t r a i l i n g edges. The leading edge i s the l i n e s |y| =m(0)x,

and t h e v/ing s e c t i o n i s g e o m e t r i c a l l y similar a t a l l spanwise

p o s i t i o n s , the surface a t the r o o t being given by | z ^ z ( x ) , and

the surface slope being everywhere continuous.

From (2), the p r e s s u r e d i s t r i b u t i o n on t h i s v/ing i s

given by

^ ^ = z ' ( 0 ) g [ x , | y l , m ( 0 ) ] + ] ° z " ( £ ) g [ x - e , ly|,m(.g)] dg

- z ' (c)g[x-c, | y | , m ( c ) J

(6)

The drag force on the wing is found to be

D = Uq|%yn^-{(^)z'[(x-^)/(l-|)]

ra(o)

dx

(7)

For convenience v/e s h a l l now change t h e n o t a t i o n and

v a r i a b l e s , as d e t a i l e d below. We put

a = 3 s / c , o = s/cm(0), Z ' ( a ) = ( ^ ) z ' ( | )

-. = or] + (1-ri;

Z

_s

ri f •- = k _c

3

where t is chosen as the maximum wing thickness (at the root). After some calculation we find that

pm(E,) = • — and a > 1, from {k)

and V ^ W - = *^'

'^

^7).

C - Ï)

Using the notation of equations (8) and (9)? we find from (7) that

Again, from (6), in the new notation,

^ A£ ^ a m L , (o)h(A,a,r,) + f Z"(k)h(A,K,^)

2 q 2 U / C "^0

ok - Z ' ( 1 ) h ( A , a - 1 , r i ) (9) ( 1 0 ) ( 1 1 )

'J

where, from e q u a t i o n s (2), (8) and (9)

Z"(k) = M I M

dk

A = a - f

K = a - k

(A,K,rO JUy-^2^ (cosh-^X)H(d')]

" ' J _1 L(i.,)Ai2 - nV]/|[K-(i-.)>J WK^

and j ' = (K-A) + i^-o^h i ^ K > a

= (K-A) i f K < a„

Substituting (11) in (10), and taking the integration v/ith respect to x] first

(12)

D

q.

(- ^ 1 * ^ j ^^ J Z' (f)(1-r,)5^Z' (0)hC^,a,Ti)+] Z"(k)h(A,K,r))dl<:

-Z' (1)h(A,o-1,r,)| dr., (13) Consider the repeated integral:

\

27 -fi ri

dr) S Z"(k)(1-n)h(A,K,T..)dl-.. ^0 Jo

The function Z "(k) (1-r|)h(A> «"SITI) is continuous within the

range of integration except along certain lines, k = constant, and the integral of it v/ith respect to k between k = 0 and 1 / is convergent for all r. between 0 and 1, provided that Z' (k)

is nov/here singular (which is a necessary condition for the applicability of the linear theory). Thus v/e may Interchange the order of integration, and (13) may be written as

-p-S-r-g-? = ^ ^ f [Z'(0)l(A,ö)+ j Z"(k)l(A,<)dk q[m(0)J f ^ 7C Jo"- ^ 0 -^' (1)l(A,a-l)]z'(t)d^ (1i|) where I(A,K) = I (1--i)h(A,K,Ti)d-n. (15) ^ 0 If we put . J(r]) = f n ^ f ^ (1-ri)dri (16) then, from (12) in (15), v/e have that

I (>!*•<) = S l { j ( ^ ) - J(0)] if >>> < K

=5l(j(^) * 4 ^ ~ ) | ^^ A > K:* a ^ (17)

= 0 if A > K, K < a.

But, in (16), i n t e g r a t i n g by ü a r t s

In t h i s expression v/e may c a l c u l a t e t h a t , from (12)j

9X ^ U(x^-a^')K%CK-(1-ri)Al an a2(1-r))[(K-A)+(K + A ) r j ] | [ K - ( 1 - r i ) A ] % V l

and a l s o ,

T x T T = ^JIZ^J^- (1 -n ).N]^-a%^] (K- (1 -q)), I / 1[K- (1 -ri) Af-Ti^., 2|

Thus, since [K-(1-n)A] > 0 f o r the range of values of TI,KS

and A i n v/hich we a r e i n t e r e s t e d (see e q u a t i o n (17) )i s u b s t i t u t i n g t h e s e r e l a t i o n s i t follov/s t h a t

2ir;2-a2 J

[(K-A)+(K+^O^]

- f

Where ^ = (A^-a^)Ti^+2A(K'-A)Tl + ( K - A ) ^ = a\i + 2bT] + c s a y .

To effect an integration we first resolve into partial fractions

^ ( 1 - ^ ) 2 < ( A - K )

[(K-A)+(K+A),jj . K+A (^ + K)^ (A + K) [Ci-A)+(K+A)Ti]

= Ar; '• B +

Ti--n,

, say, v/here ri >>-K

A+K

Upon I n t e g r a t i o n v/e t h e n have t h a t )

2yK^-a.'^

-•[yf-I^)ii--7^

+b_^ a c j K+A ( ^ cosh -1 Ti^n, ( a r ] ^ + b ) T ] + ( b t i ^ + c ) (Tl-i1^)^/b"2-ac

S u b s t i t u t i n g f o r A, B, C, a , b , c,X,'>l'', and ri_, Y/e t h e n f i n d t h a t

K

(K+ A) (>f^

j{[K-i1-r,)Xf-a%']

.2K. , > ^ ^ e o s h - i (A--)rA{^->^) (^+:<) A"-a J a | K - A |

(A+o/^2_;^

^ ' ^ ' ^ ^ g f e ^ L cosh-fsgn(K-A)-^^^'-^^^^^"^^^-^^^ (A+K)VK -o-^ \^ a[(kf-A) + (K+A)Ti]

J

I n p a r t i c u l a r , i n (17)> v/e r e q u i r e t h e v a l u e s of J ( 1 ) = -(K+A)(A^-a2) (A-^ _ .2K (A-^ AiiL-iiilcosh-(A-^

+><^)/A^-a2'[A+K A^ -a^i

^ 2 K . ^ ^ i ^ L . eosh -1 _{sgn((^-A) g +K} 2 2 2a>« (18) A K -rj (19) / J ( 0 ) , . .

29 -J(0) = -

J^jzA

«+A)(A -o, ) (A + K)/x^-a: 2 ••""2"

2K ^(A-K) ~ 2 2 A+K A - a , cosh"^ ^ a — 2-^.<2 (A+K)'

;osh"Y^jfor K

_>^

(20) and

IA-J

= 0 for A > K > a. (21)

Substitution of these expressions in (17) v/ill yield the appropriate form of l(A,K)j which is in general discontinuous ^t K = A' where it has a logarithmic singularitjj and at

K= a < As v/here there is a simple discontinuity in its derivative with respect to K , Let us nov/ put

— I(A,K) = I. (A,K) for K > A

aK

(22)

9K

I(/^,K) = l2(A, K ) for K < A (KT/ a )

Then b o t h I. ( A > K ) and Ip(A? K ) have.simple singularities at |<=^and a discontinuity at K = a. Using these definitions in

(1I|.), we have on integrating b y parts v/ith respect to K> and taking the principle part of the improper integrals, that!

2 '1

?. , _ 2 a _ i ^ , /.>x 1 1 m

q[m(0)]'^t

- Z ' ( a - a ) l ( A j O L - s

% "0

Z ' (t-e ) I (^,A+e )-Z ' «+6 ) l (?y,A-s )

i +Z' (a-a)l(A, o+e| +\

iMa lA>a (

u

-a)l(A,o+e1 +\ Z ' ( k ) l ^ (A,K)dk lA^a 0

•j;

+ 1 Z'(k)l„(,VK)dk +s d-^ ₍₂₃₎

Now, from (17), (19), and (20), if A > g, for s u f f i c i e n t l y small s,

l(A»A±e) = - - J-.-,^.. + ^

>y?':^2^ 27^2:

fn

2(A^-a^) ae

^ J n / ^ . 0 ( e )

I (A, a+s )| = 0(e), and I (As a-s )j = 0. If A < a, again for sufficiently small e, I(A,A+e) = — 7 = ™ ^ + 0(s)

2 A ^ - A l(A,A-s) = 0.

- 30 -

1

Thus (23) becomes

D

q[m(0)] ^t^

= o'

+ ^ ^ \ ^'(^llo f ~^Z.(k)l (A,K)dk+f Z'(k)l2(A,K)dlc\ (2^+) Excluding, for the moment, the range of a such that

a ^ a :;^ 0-1 (so that 1^ is discontinuous v/ithin the range of integration), the second (repeated) integral in this expression is capable of simplification. We note that at K = A s "both I.^ (A,K) and I2(A>K) have the same singularity and that Z'(k) is continuous, so let us put:

z' (k)i^ (A,K) =

I*(A,K:)

+ ^ ^ ^ ; z' (k)ip(A,K) = ip(A,K) + ^ M

A-K 2 2 ^ _ ^ where I^ and I2 are now everywhere continuous except for a

logarithmic singularity, and C is some function of A which is also continuous. Then

lim s^O

if:

^ ^ a . +

A-i^

>i +e

A-k

V

t

dk

dt

Also we have that, on first changing the order of integration and then the symbols

^Z'(n eiS C l2(^sK)dk]d^t = fz'(|)lfl2(A,K)dlc d^

f1 ' fk s \ Z'(t)lp(A,K)d?

Jo[ "^0

'^

^1\rt X Z' (k)lp(K,A)dk JQ ^0 '^ % X

Thus, substituting I^ and I2 for I.^ and 1^^ using the above Identity, and then returning to the original notation*.

\2'(t)eiS

\'^

Z'(k)I. (A,K)dk + [^ Z'(k)l2(A.K)dk df.

- ]%i. j ^ Z'(Ol"(AsK) + Z'(k)l2(rs,A)'dk +fz'(f)C(A)^nj^^\df

= f d^ f Z' (k)z' (Oil, (A,K)+l2(K.A)]dk+fj^ef-[^-'(^-)^(^)-^'(^)^-^l'

'^o "^0 L '

-

> -» [ 0 -^oi t- k

(25)

31

-The expression contained within the curly brackets in the

last expression may be shown (see Appendix II) to be identically

zero, so that on substituting from (25) in (2/Lj.),

D

,[m(0)J

2 - ^ = 0

VMl^'^'f''

^2 ri r* r

1

•2-.J dC Z'(^)Z'(k) I. (A,K) + I2(<,A) dk

7^ ''0 ^0 • i '

. '^

J

(26)

v/here t h e I n t e g r a n d of t h e s e c o n d I n t e g r a l h a s nov/ o n l y a l o g a r i t h m i c s i n g u l a r i t y a t K = A ' To f i n d t h e v a l u e s of I , (AsK^) a n d I g (As »<) we r e q u i r e t h e v a l u e s of 1 9 J ( 1 ) = _ „ 9K (A+K)7k^-a^ 2K K(t^+A) ^ 2lc P p + —p ^ ^ - a « - A A•^«^ A - a A+K 2 2 A K - g •"2 2 •sgn(K--a) k\<' ^ K^+2Ai<-A^ (A+r.) A^-a.^ " ( A + K ) V A 2 r ^

+ -iK^Hni^l^/22, _ -/^)eosh-^r

aU-Al

c o s h " sgn(K-A) 2 ' 2 g +K 2 g K

,nd 9 J M = .

aK K^+2kA-A^ A^+2KA-V;^ + i(A^-g2)(vC+A)^ ( K ^ - a 2 ) ( A + k ) ^ '

^ (A + i^)W^l>^+*^

" i i i l + K^+2XK-> v2 2 A - a c o s h — K ( A + K ) V K 2 - a ^ 'Jib- + X^+2>>K-K^~' A+K- (K^-a^) c o s h — g (27) (28) T h e n from e q u a t i o n s (22)^ (17}> a n d ( 2 1 ) ! l l 3 K J H ( K - g ) (K < A) (29)

Using (27) and (28) :'n (29) to find I^ (A.

K )

and

I 2 ( A > K ) P

and

substituting for I. (AsK) and l2(»^sX) in (26) v/e may then

write down our general expression for the drag. If o ^ g ^ o - 1 ,

then equation (2i|) must be used in place of (26). We may

conveniently consider the precise form of this expression in

certain particular cases, one by one.

CASE I Both Wing Leaddng and Trailing Edges Subsonic, (g^ a-1 ) In equation (29), v/e then have that K= a-k > a-1 > g, and similarly Ay > a, so that

I2(A,K)

•m

_\

We also note that 9J(0)/9K niay be expressed in the form

where from (28), we may infer that \<^+2<A-A^ (30) A ['.LK , K^+2M->£

5

'

~ (>2_c,2) (K+ >») 2 ~ (X + K)

V}f^

L^+r; X^ -g^

cosh" — . a Thus in (26), putting m(0) = cotA s we have that

^"^ J d ? | Z'(k)Z' (<[j[p(A,l<)+?(l<,A)]dk

D q t ^ c o t ^ A , % "0 •^0 (31a) where P(X,K) = ^ ^ ^ ^ +I,(AsK) . aK ^ ( 3 2 )

I t follov/s immediately, since the i n t e g r a n d i s symmetrical i n k and t t h a t an a l t e r n a t i v e e x p r e s s i o n i s

2 2 qt cot A o

— f ( Z ' ( k ) Z ' ( { ) r p ( A , K ) + P ( K , A ) l d k d t (31b)

7C ^0 '^O I J

In (32), we may calculate from equations (27) and (30) that

P(AsK) = 2X^ 21<A

>?-g^ ~ (A^-a^)(A+K)^ "

{>?-<i^)JiF-'"c? ~

(A + k)^(y^-A)/K2rg2'

A./;rc^

(A+<)^(A^-a^) v,-1 -^ - c o s h —

A

(A+IÖW^

UK k^+2K.X-X^'

A+K

A -a \2 2 ^eosh-^/^<---''' a a 2K

'(x^K^y^"^'^'^'''

a \K

i-4-ö - ^Kn(KA)

-g A+K,

(32)

CASE II Leading and trailing edges supersonic. (a > a) In (29), since K = a - k < g and similarly X < a, v/e have

33

-that I 2 ( A , K ) = I 2 ( K , A ) = 0. Thus in (26), putting

I., ( A S K ) = Q ( A , K ) say, D — ^ ^ —

qt^cot^yi

2 n1 A = a

I ^-;(1IJ d f + ^ l dd Z'(f)z'(k)Q(X,<)dIc. (33)

''0 Var-X 7C Jo J o To f i n d Q ( A S H ) we must f i n d t h e v a l u e of I^ (A>lk) f o r \ < K < » ; i n doing t h i s we n o t e t h a t i n (27)? ' . . , „ t , - 1 / A K - g ^ \ ' ^•^-^ < - 1 , so t h a t IJ c o s h ' CX|K-A| ( VajtC-AJ

J

and also I

Lsh-^rsgn(K-A)(2^]

_0.

Thus w e m a y calculate from the equations (27) and (28) in (29) that Q(A,IC) = - ^ (A + K ) ^

';*p •"•' (- 9

_Va^-K""

cos

-1/K

+ X +2KV-A"

(g -A ) (K+A)' . X ^ 2 A K - R ^ + « p - ! (g'^-K'^)(3<: + A ) '

A -l/A\ .

rr, C O S ƒ - \ -'

/o^-\^

la)

^--"^e -

(31+)

CASE III Leading edge subsonic, and trailing edge supersonic.

(a-1 < g < a )

In this condition we use equation

{2\\)

in place of (26)

and since 0 < a-a < 1, so that (a-A) is positive if

1 > 't > o-g, the first Integral in this equation becomes

rai|4[z.(t)]2dt = r J ^ ^ * ^

-Jo /c. -A^ •- ^ J a - g

Vg'^-X-In the second (double) Vg'^-X-Integral, v/e note that the integrand is (within the range of integration) continuous except for a discontinuity at K = g. It should b e noted that the individual terms of the expressions for I. (A,K) and l2(A»vC) have simple singularities at A = g or K = g though these

singularities do not occur in I. or I2 regarded as a whole. Nov/ for K < A and also k < a, it follows from (29)

I>'(«-s[J

«+s

Z'(k)l^(A,K)dk

dt

= [''•%.'({)df 11^ r"%.'(k)i2(AsK)dk

«o J f+s

where I2(A, K ) is now continuous over the range of integration Hence, as in deriving (25) of this Appendix, changing first the order of integration and then the symbols, it follows on using the result of Appendix II that this expression may be

shown to equal

.{

f>o-g pl+s

z ' ( ^ ) d n z ' ( k ) i (k,A)dk.

Jo Jo

Thus i n (21;) t h e second I n t e g r a l becomes

£ 2 ' (£)3^^g f^"^Z' ( k ) i | (A,k)ak + r ^ z ' (k)i2(AsK)dk df

= [''"'^Z'(.^)df[ Z ' ( k ) r i . (A,K) + Ir^iKX)

Jo Jo L

+ f Z'(f)df I Z ' ( k ) l . (A,K)dk

J a - g Jo

^{[''"""^Z' ('f)dej Z'(k)[l^(A,K) + l2(KsA)]dk

+ r Z' (.^)df f r " \ { ]Z. (k)i^ (AsOdkf.

^'O-g \fo Ja-ctJ I

I n t h e f i r s t i n t e g r a l over the r e g i o n 0 .^ f ^ o-g,

0 <• k ^ " t , v/e have t h a t fC > A > ctj so t h a t as i n e q u a t i o n (Ij.) abo-^-e, we f i n d t h a t

I-, (AsK) -.- IgC^^sA) - P(A,t() + P(K,X).

S i m i l a r l y i n t h e r e g i o n a-g ^ f ^ 1, a-g 5 k g "Is v/e f i n d t h a t g > < > X : t h u s as i n (33) above, v/e xnay put

I., ( A , K ) = Q(A,K).

Finally, in the intermediate region, where o-g ^ f ^ 1 and 0 * k -é a-g, v/e have A < g < K? then v/e put

I^ (A,K) = R ( A , K ) , say.

Thus in {2I4.) we have on summarizing all the above results:

D q t c o t 7 \ 35 -,2 f' [ Z ' {^]^ ^i J a - g Vg2-.X^

+ ^ P ^ ' z ' ( O d H ' Z'(k)rp(A,K) +P(KsA)l

7\; / J o "Jo •- J

t

dk + n Z ' ( C ) d n Z'(k)Q(AsK)dk ' a - g '^a-g 01 r o - a ' ' a - g J o Z' ( k ) R ( A , K ) ^ ) •

I

(35) N o t e t h a t t h e s e e x p r e s s i o n s a r e n o t d e f i n e d f o r g = a - 1 o r g = a, a l t h o u g h t h e v a l u e of D f o r e i t h e r of t h e s e c o n d i t i o n s may b e c o m p u t e d a s t h e l i m i t i n g c a s e s , f o r a - » a-1 o r a. I n e q u a t i o n (35) we h a v e t h a t P ( A , K ) a n d Q ( A , K ) a r e g i v e n b y t h e e q u a t i o n s ( 3 2 ) a n d (3I4.) a b o v e , a n d R ( A , < ) i s t h e v a l u e of I ^ ( A , K ) f o r A < g < K : i . e . from ( 2 7 ) , ( 2 8 ) a n d ( 2 9 ) R ( A s K ) = 1 (A + K ) ^ ^ ^ * ' K(K + A) 2>^A _^ _A_ K^-g^ a 2 . ^ ' ^2^^2 ^ ^ ^ ^ 2 _ J ^ 2 J A _ >c^+2Aig-X^ - 2—"2 A+«* g - A

" (A+K)Va^-A2

, 2 K ^ (22^ . -^\U^\

( A + K ) V K 2 - ^ (^A+K

K^-g^^ U J

- 1 M-a a ( K - A ) K ^ + 2 K A - A ^ .+ k^-2K)>-X^ ( g ^ - K ^ ) ( K + A ) ^ ( k 2 - g 2 ) ( A + K ) ^ A r Uly K^+2X><-A^

(A+K)Va^Ix2^L^+'^ "

g - A 2 v2 (A

K _ _ r uA

^K)

V K ^ - ^ I U K

UA X ^ + 2 A K - V < P + 2 2 K - a

--'8

(36) A p p e n d i x I I P r o o f of a n I n t e g r a l I d e n t i t y We s h a l l h e r e p r o v e t h e i d e n t i t y >io Jo { - k ^ o W / (1)

Consider the integral

1(a)

= ]%(M.n^f)d

/Since.

Since

>m ^^ = °

(3)

it follows that if f ( 0 is continuous at <=

1(a)

=\y(i)

-.f(a)]^n^S^)d< .

a

ik)

Then differentiating

d l M =

da

q,m^.„,>t„(^)

d( (5)

a sufficient condition to allov/ the differentiation under the sign of integration being that f(t) is continuous at f= a, and integrable over the range (0,a). From (3) in (5)'.

dl(a) da

f"ff(f)

-Jo[ a -i

- f (a) d^ and so (6) (7)

1(a) - 1(0) = r I I M ak = i dk r LLÜ::1M at .

Jo dk Jo Jo k-^ But, using (3)

If f(f) is continuous at f= 0 it is easy then to show in (7) that 1(0) = 0, so that from (6) and (1), the identity (1) is proved: subject to the sufficient conditions that f(<) is continuous at f = 0 and a, and Integrable over (0,a).

Appendix III The Edge Force

The argument which leads to the neglect of the terms in the curly brackets in equation (25) of Appendix I is

given in Appendix II, v/here it is shown that this neglect is certainly justified if the function f(£) of the Appendix II is continuous at the end points of the interval of integration

(i.e. at 'C = 0 and a). Now in equation (25) of Appendix I we identify this function f(f) with Z'(-jJ)C(X), v/hich from its definition and the use of equations (27), (28) and (29) of

37 -Appendix I, may be shown to equal

2

[z'(f)jy2/^

This is, in general, continuous at the end points "^ = 0 and t = 1 of the range of integration, provided that the slope of the aerofoil section is continuous at the nose and the trailing edge. This is the assumption in Appendix I, and it is this fact which justifies the use of the results of Appendix II. However if the slope at either of these end points is infinite, then the arguments we have used must break down.

Let us suppose, for example, that Z'(k)~ —i as k-*0,

k2

which describes the usual type of singularity associated with a bluff nosed aerofoil. We shall attempt to deduce equation (26) of Appendix I from equation (21;) for this case, where the previous argximents used no longer apply. Similar reasoning

to the following may be used to discuss what happens if Z'(k) is singular at k = 1 (i.e. if the wing section has a bluff trailing edge).

Suppose we subdivide the range of integration of the repeated Integral in equation (21;) of Appendix I:

Z' (k)l„(A,K)dk

i y ( ^ ) l l o

J^"%'(k)I,(A,K)dk + j^

^ Jiojf^'^^) |\'(k)l2(A,K)dk d?+J%'(-e) |^Z'(k)I^(A,K)dk

df

Jis[l/ (C)^'s[f'2' (^)^ ^^^^)^ ^]\

5 "o Ö Z' (k)lp(A,K)dk dt V (1)

+e

Using arg\aments similar to those employed in deducing equation (26) from (2i|.) of Appendix I, v/e may show that

lim

Ö-V0

Z' (k)l^ (A,K)dk+r Z' (k)l2(A,K)dk

l^e

^. Jdft Z'(f)Z'(k)ri^(A,K) +l2(KsA)]dkl

dt

(2)

These arguments, including the results of Appendix II, are justifiable since Z' (k) and Z' (f) have no singularities in the range of integration.

Now the second limit in the right-hand side of (1) may

be shown to vanish as ö --• 0} also the limit of the integral

on the right hand side of (2) may be shov/n to e:èist, so that

taking this limit and substituting in (1),

J z ' l O e i o j ' ' ' z ' ( k ) I . , ( A s » ^ ) d k + J z ' ( k ) l 2 ( A s « < ) d k

=J d/j Z'(k)z'(^)k(A,K) + l2(K,A)

d«

dk + F

v/here /•

[ '"z'(k)I. (A,K)dk + r z'(k)lp(AsK)dk

Jo Jl+e

(3)

dik)

But in this repeated integral (on the right hand side of

ik))*

the region of Integration is confined to the immediate

neighbourhood of the point i = k = 0. Thus it is permissible

a.

to replace Z ' (k) and Z ' « ) by i^f and

j±

respectively; and

further we may replace

Il(A,

2 2

K)by|c. r^^. + —^^T^J^ ^^

l^

27^r^(K-A) 2o(a^-g^)^

1

K-A

2 2

l2(AsK)-byfc - •-L + - 2 2 ^ r £ ^ y ^ n | - ^ | l

-^

) 2 2yo2-a2^(K-A) 2a(o'^-g^) 2

J K - A I J

where c. and c^ are finite constants, as may be established

on examination of equations (2"') (28) and (29) of Appendix I,

Thus we find that in (4)j

>/,2 2

i-/a - g

J(

-F =

-s f 5

a

I.e. F =

-ö^0|j^ ^^2e..o|4 h.+ek^ !-k (^"0{Jo-t

-^ ^

2a^ f i

/'2

—2* J

*/a -a^ •'o X

.•««> . . 2 2

c o t h ~ \ dx = - a_7c__

_^Ja^Za^

(5)

Hence in equation (26) of Appendix I, the expression for the

drag D remains unaltered, except that from (1+) and (5) above

there Is an additional drag

AD = - ^

q.[m{0)}^a\^t^/y<K^

(6)

Suppose that the radius of curvature of the wing leading-edge

at the centre-section is r: then we may calculate that

2^2 /

r = a t / 2c

(7)

so that in (6) we have, after substituting for a and g, that

/AD...

39

-AD - - ^q.m(0)sr ^

/ l -3'^m2 (o)

•rcqs cosyAnr

THM^cosl^i''

(8)

However, the corrected equation (25) of Appendix I is no longer the complete expression for the drag, since it omits the edge force acting on the v/ing, which is not included in the integrations of the pressure over the wing surface. This edge-force is given in reference 5 as a force normal to the leading edge of amount

'J

%q, cos A ^ r ^ •/l-M^cos^y\_

per unit length

where r is the nose radius measured normal to the leading edge. But from the geometry of the wing section,

^n = a sec

A^f)--AC-Ö

so that the contribution of this edge force to the drag is

AD^ = 2 cosA^

VtllT^-° J o v : ^ ^ M ^

•y^oTn s2yr

M,

cos A , rqs COsAnT / Ü M ^ C O S 2 7 ^ (9)

Comparing (9) with (8) v/e see that the contribution of this edge force exactly cancels the additional term arising from the Integrations v/hen the leading edge is rounded. Hence, Yi'-e deduce that equation (25) of Appendix I, in the form given In that Appendix, is the correct expression for the total drag whether or not the leading edge of the wing section is roundedj if the leading edge Isroimd, we have seen that the derivation of this equation leaves out a term which hov/ever is equal and opposite to that obtained from the edge force, v/hich should also properly be Included.

Appendix IV

Integration of the Expression for the Drag of a Double-Wedge Section

We have to evaluate the Integral of equation (39) of the Appendix 11 -dkdf (1)

q[m(0

•^2"^ = ^ [ f z ' ( k ) z . ( e ) [ ^ y ^ l

-1

(a-k)(a-0

/for...