Bound and trailing vortices in the linearised theory of supersonic flow, and the downwash in the wake of a delta wing

rECMNISCHE »'"'^'^«^^ïTEST D E L R

LUCHTyAI\[;T-?! Mimmj^ü Juli 1950

Kluyverweg 1 - 2629 HS DELR

REPORT NO.10 October, 1947

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

Bound and Trailing Vortices in the Linearised

Theory of Supersonic Flow, and the Downwash in the Wake of a Delta Wing

-by-A. Robinson, M.Sc,, -by-A.F.R.Ae.S., and

Squadron Leader J . H , Hunter-Tod, M.A,, A.F.R.Ae.Sc oOo—

SIMLfiRY

-The field of flow round a flat aerofoil at incidence can be regarded in linearised theory as the result of both botinu and trailing vortices for supersonic as well as for low speed flight. This leads to a convenient method, given the lift distribution over an aerofoil, for calcvilating the flow round it at s\ipersonic speeds.

As an application of the results the downwash is

calciilated in the wake of a delta vdng lying within the Mach cone emanating fran its apex. The downwash is found to be least just aft the trailing edge and is everywhere less than the downflow at the aerofoil. It increases steadily to a limiting value which is attained virtually within two chord lengths of the trailing edge, The ratio of the downwash at any point in the woke to the dovrnflov;-at the aerofoil decreases with increasing Mach number and apex angle,

2

-1. INTRODUCTION.

In the first paper on three dimensional supersonic aerofoil

theory, written by Schlichting in 1936 (ref.l), the idea of a supersonic

horseshoe vortex

\7a.s

used as an auxiliary concept. However, the

Prandtl-Lanchester vortex approach, which is of such fundemental

importance in low speed aerofoil theory, has been almost entirely

abandoned in the subsequent treatment of the matter. This, of course,

is no accident, for it appears that the alternative methods of the

supersonic theory lend themselves more readily to the solution of the

main problem of finding the pressure distribution over an aerofoil of

given shape and incidence; furthermore there exists no supersonic

counterpart to the lifting line theory to which is due the remarkable

success of Prandtl's approach. The purpose of this paper is to show

that once the lift distribution is known, the vortex appraoch oan still

be of use in determining the flov/ round an aerofoil.

The general linearised theory of a field of flow due to an

arbitrary distribution of vorticity under steady supersonic conditions

is developed in the College of Aeronautics Report No. 9, 1947, (ref.2)

and is applied in the present paper to aerofoil problems; in particular

the dowmrash along the continuation of the centre line of a delta wing

is calculated for the quasi-subsonic case (apex semi-angle smaller than

the Mach angle).

Other methods of determining the field of flow from the pressure

distribution, such as first deriving the "acceleration potential" due

to an equivalent doublet distribution, have been found, at least in this

particular case, to lead to considerably more complicated calculations

than those involved in the method adopted here.

2. RESULTS.

^he downv/ash, w, along the continuation of the centre line of

a delta vidng moving at a supersonic speed such that it lies entirely

within the Mach cone emanating from its apex is given

by:-When d <> .^Hc

- _o

w

V/*(, -rTE'(A)

Hh]-{^irS]Hh)]

When d ^ '•X c

w

/

Vot •TrE'(A)

E

K(k)-E(k) ,^ _^i K(k) - E ( k )

,^^\

,,,(i^i)

k + A J4- k^(l+kA)

(%r^" K(k) - E(k) ,^ ) ^^^(-^^,.)

^ J k + A

where K, E & E' are the well known complete elliptic integrals

V = the velocity

^

= the incidence

c = max. v/ing chord

d = distance aft the trailing edge /A - Maoh angle

y = wing apex s e m i - a n g l e , \ = cot Artj t a n Y"

The condition

d'C Xo

indicates that the point in question

is outside the Mach cones emanating from the wing tips, and

The corresponding spanwise lift distribution over the aerofoil

as given in R.A.E. Report No. Aero 2151 (A.R.C. 10222) (ref.3)

is:-:i

£{y)

=

^<^ "^ -^

/ Q 2 t a n ^ y - y^, where

[^ =

air density ...(2)

E' ('-^ ) V y = spam'vlsö coord,

As d tends to infinity in (l,ii),

'^.

tends to _ i _ _ _ , which

Y<JL E H > 0

is exactly the same result as obtained for the downwash in incompressible

flow far behind the trailing edge of an aerofoil with spanwise lift given

by (2), This is a special case of a more general result stated in

ref.5, according to which, for a given spanwise lift distribution, thó

trailing vortex field tends in regions far behind the aerofoil, where the

chordwise coordinate is large compa.red to the other coordinates, to the

same limit in supersonic as in subsonic flow.

In Pig, 1 the downwash is plotted against the distance from the

trailing edge for various values of the p.arameter A . It will be noted

that for A = 0, that is for very small aspect ratios or at speeds very

near that of sound, the downwash becomes equal to the downflow at the'

aerofoil ( -••—•• = 1 ) .

Fig. 2, shovYs what the downwash would be if the entire lift

were regarded as being concentrated at the trailing edge for the given

value of 0,4 for

X

.

To assist in applying the results given in Fig.l to particular

cases, the values of

,\.

for specified values of aspect ratio and Mach

number can be found from Pig. 3.

3. VORTEX PLA])]E THEORY FOR SUPEESOMC COITOITIONS.

Consider a flat aerofoil placed approximately in the xy-plane

at a small incidence in an airstream of velocity V , greater than that

of sound, in the positive x-direction. Then according to linearised

theory we

have:-The equation of continuity - ... (3)

^ffi^.iÈ.^i$

= 0 where p"

\ ^x' ^i^ Ó? i

= M^ - 1

= velocity

.potential.

The Eulerian equations - . . . (4)

_ ' i h = y i a )

I • ^ )

where p = pressure

,i. M^ Uèv. ^'^'^ = velocity

0

X

"

"

'

v ' ^ "

^

components.

4

-It is assumed as in subsonic lifting plane theory that the kinematic boundary conditions must be fulfilled at the normal projection of the aerofoil on the xy-plane rather than at the aerofoil itself, and that (p is continuous everyv/here except across the wake. The latter is taken to be the strip lying in the xy-plane subtended downstream by the aerofoil. Finally it is assumed that the pressure is continuous across the vTake. The exact or approximate validity of these assumptions under supersonic conditions is, in the last resort, a matter for

experimental verification.

Since the flow is assumed to be irrotational, j^ v and •d w a x ;> X n-iay be replaced in equations (4) by <^^ and " ^ respectively.

è>y ès z

Integrating these v/e obtain the linearised foim of Bernouilli's Equation:-Vu + /» = const. .... (5)

f

where the constant is the same throughout space, that is both sides of the wake. It. follows that u, like the i^ressure p, is continuous across tb.e wake.

Furthermore the normal velocity, w, is continuous at the aerofoil, because it is assumed to be flat, and similarly across the

T/ake since a discontinuity would indicate the presence of sources contrary to the condition of continuity.

This also follows from the boundary conditions which require Cp - Vx to be ajiti-symmetrical vd.th respect to the xy-plane, so that

§ (x,y,*z) = " <J(x,y,-z). Hence 7ro

have:-At the aerofoil:- u(x,y,+0) - V = -[ufx,y,-0) - vj

v(x,y,+0) = -v(x,y,-0) ,...(6) w(x,y,+0) = +w(x,y,-0)

v(x,y,+0j = -v(x,y,-0) (7) w(x,y,+0) =

and At the wake:- u(x,y,+0) = +u(x,y,-0) =+V -v(x,y,-0^ +w(x,y,-0

These equations show that we may regard the area comprising the aerofoil txnd its wake as a vortex sheet with a surface distribution _^(x,y) of vortices given

by:-ÓJ = (-{v^ -v_),(u^ -u.),0j = (^-2v ,2(u-V),0) ....(8) and, in particular, in the

v/ake:-ÓJ = (-2v.^,0,0), where u^. = u(x,y,+0) &a. ....(9)

Now, since the flow i s i r r o t a t i o n a l , we h a v e :

-èv+ _ iut. a 0, . . . , ( 1 0 )

ex ^y

and

èv- . au- = 0. . . . . ( 1 1 )

èx èy

/ Hence

Hence, at the

aerofoil:-^^"^ -"-^ . i K j ^ O .0, .... (12)

è X è y

and at the wake, taking into account equation

(7):-^^^^ -^-) = a i l t =0. • . . . . (13)

<^ X Ö X

Equations (12) and (13) show that div C^ = 0, as required for a vorticity vector.

To find the field of flow due to this vorticity distribution we apply formula (60) of ref.2. which states that the velocity vector

(u,v,w) due to vorticity ("? ,»-« , f ) is given by:-(u,v,w) = curlh ;^ + (V, 0,0)

nr ""

(^^^

i^(x,y,z) = ^ l ( J , » ^ , | ) d x j | ^ ,

.^1

where (a) s =J ( X - X Q ) -j'5 [ (y-yo) • +(z-Zo) J (b) R' is the subdomain of the region

concerned for which s is real and XQ < X .

(c) curlh is the hyperbolic curl and

"f

1 Hadamsrd's "finite part of an

infinite integral" as defined in ref.2. In our case the vortex layer becomes infinitely thin, while the product (?',n , t)^Zo remains finite and equal to £0. We

obtain:-•f' = -—-' \ ^(^o>yo) ^SitiZo^ ^here now Z Q = 0 in a (15)

2 "^

JR'

^

The vorticity distribution over the aerofoil will be called the bound vorticity and that in the wake the trailing vorticity. The latter consists of straight vortex lines of constant strength extending from the trailing edge to infinity in the negative x-direction.

If we write u** = u - V and v* = v_^, equation (15)

becomes:-X =-^ \ ("^^^^o) ''^° ^y° .... (16)

JR

By equations (7) and (13) we have in the

wake:-u* = 0, .... (17) , . ^ = 0 . .... (18)

è X

Also, by equation (5), u*^ is connected mth.the pressure difference, ^-p = ^^ - p_, between the top and bottom aerofoil surfaces by the relation:

-u« = ^ . .... (19)

é

-Hence, if the lift distribution is known, so is u* and^ vice versa. Equation (l8) shows that v* is independent of x in the wake, so that to evaluate the integral (l6) over the wake it is sufficient to know the variation of v^"*' along the trailing edge only. It will be

shown that at the trailing edge, as vroll as elsewhere on the aerofoil, V = JlL i u' dx, where the integral is taken along a chord from the

'>y j

leading edge.

Consider a circuit ABCDD'C'B'A'A, where AB,CD are parallel to the X-axis and AD,BC to the y-axis, so that AB and CD are separated by a small distance Sy; A,B,C,D are just above the jry-plane and A',B',0',D' form their image just belov.' it; A,A',D,D' are points ahead of the

aerofoil. (See Fig.4). Applying Stokes' theorem to the flow round this circuit we

obtain:-rB '"^ '"D rC' r.B' f A ' , . I % d x + I v^dy + 'v \x,^dx + \ u_dx + I v_dy + i u-dx = 0. ...(20)

JA JB

ic

JD' JC' JB'

This may be

written:-r (u^-u_)dx -

{

' (u^-u_)dx + (v^-vj^y = 0. ,...(2l)

J A .J D

Hence, as ö y tends to zero:-\ ('B

(v^-v ) = -£. \ (u^-u )dx, since u^ = u , as far as the aerofoil. (22)

-^y

J A

Now u.-u = 2u'^) ' •'• " at the aerofoil, so that

v^-v_= 2v^0

v'^s-2- ( u'^dx, as asserted, ....(23)

«y

J

This relation might have been derived directly from equation (12) , which can be v/ritten as èy^ - èu^ = Q^ but for the possible

.^x èy

irregularities of èü and H Z at the leading edge and at the envelope- . ^y ÓX

of the Mach cones emanating from it,

Define ü = j u*dx with the condition u = 0 at the leading edge. Then v^ = aü . It will be seen from equation (19) that u

^y

is- proportional to the excess pressure integral from the leading edge to the point in question.

Divide R' into two subdomains S' and W' , belonging to the normal projection on to the xy-plane of the aerofoil,S, and the wake, W, respectively.

Then:-vT. ^ J:- r I - èi , ^ , 0 I ^o^yo + J L r /-/^ ) , 0,0\ dx„dyo

X -n-

Jg.l^ ey ax

)

- ^ — IT Jw.\^Uy)t j ^ T "

where | '1" j is the value of I H at the trailing

Wh

^y

edge for a given y .

Now r ^ ^Xpdyo ^ f ÏÏ ^ _ ^ f Ü (x-Xo)dxodyo

JS' ''''

^

Jc'

^ "

Js'

s^

by i n t e g r a t i o n by p a r t s , where G' i s t h a t segment of the t r a i l i n g edge included i n E ' .

We can nov? r e p r e s e n t U-' as the sum of two v e c t o r s , *4 = (VK , '^ , M-- ) and 4 ^ = ( i f ^i,^l),

where:-=- •>• ^ ^ a; 1 2 3

"^ ig.'^y

. . (25)

and U^. = - J: \ ih\^^l9 )

, . . (26)

It will bo soen that ^* coincides with ^ ' if we imagine that the whole bound vorticity, for any given span position, is concentrated at the trailing edge.

It will be observed that, if the aerofoil is assiimed to be symmetrical vd.th respect to the zx-plane and to have a straight trailing edge, yx may be regarded as being duo to the sum of a set of horse shoe vortices of strength -2 /•iH] whose spanwise segments extend from.

"• "-y / t

-yo to + yo (see ref.2, sect.6(64)). ^o find the velocity components (u",v",7/') due to this combination it is necessary to integrate the expression given at equation (68) of ref.2 from the midpoint of the trailing edge to the positive endpoint with respect to y^^ thus

:-2 f (•' , - (y-yo)z.dyo

u" = - £ • ; I

n I ] a y , [{x-x,f-fz^]Jix^^,f-pHjy-yo)^ ^]

( èu (y+yo)z.'iyo \

"^yo r . N 2 « 2 2 ] ^ ' ; ~T . 2 , " 2 2 ] / • • • • ( 2 7 ) [ ( X - X Q ) - | 3 Z j^(x-Xo) - (^ [ y^y^) + z J / where . . .

8

-where the integrals extend over those segments of the trailing edge for

which y^ is positive and the integrands are real. Prom the above

assumption it follows that v^= SÜ is anti-symmetrical with respect to

hy

the zx-plane, so that the second integral in (27) is equal to but

opposite in sign to the first taken over the remainder of the path of

integration, C'.

Hence u" =

-Similarly

i ( ^ü (y-yo)z.dyo , _ ,

iu

'- .

.... (28)

^yo l(x-Xo) - /-i z J . s

c

and

W"

- - I

_ (x-Xo)z.dyo .... (29)

>-yo [(y-yo)^+ ^ ^ .s

— ^ (x-Xo)(y-yo) f (x-Xp) -g f(y-yo) '+2g j) dye

-" ^,,Vp[(x-x„)2.p2,2J[-(;.y^).^ ^2j,3 r ^

.... (30)

In calculating (u",v",w") x v.dll take the value of x at the

trailing edge,

To calculate the field of flow due to

^

behind the aerofoil

on the assumption that it is symmetrical with respect to the zx-plane

and that it has a straight trailing edge, or more precisely one that is

not so curved that it meets a line parallel to the y-axis in more than

two points, we revre-ite equation (25) in the

form:-^ 1 = l \ ( form:-^ o ) form:-^ o ; V g " i Vform:-^o)<lform:-^o ; y form:-^ = 0 .... (31)

where:-X. = - J . ( è : f ï o ; X^ = - ± T ^•(fZo)££o ...(32)

It will be observed that, for a given x,,, X-, is obtained from

Ux" by differentiating it with respect to x and putting i^ièi - ^ ,since

^ 1 \<^yit *^o

.X - 17 ^x j ^ ^ ^ é y j , B r. \^A -^

r

= - _i

- C '

c>u j ^yp , [ 4 H I being independent of XQ.

-^yL s Uy/t

Similarly we may derive X from

^^-

I* ^'vill be noted that, though the

limits of integration may be dependent on x, we are justified in

differentiating under the integral sign of the finite part of an infinite

integral as demonstrated in para,2, of ref.2.

Thus (u',v',w') is obtained from (u",v",w") by differentiation with respect to x and sulDsequent integration \7ith respect to x across

the aerofoil.

Hence:-.2 Sf

u' = - £

èu ^ I (y-y^z ,^^^^y^

, ^ o ^^ l[(x-Xp)^-/3^z'^j ,;

....(33) V' = - _ 1

AÏÏ è

(x-Xo)z ^^yo^x U"(y-yo)^+ z 2 ] . s • ^x„dy, o^J'o W' - ' i J ' ^ E i . f(x-xo)(y-yo) [ ( X - X Q ) ^ - a H{y'yo)K2z^]]

r>

Jg.'^yo'^^ !

[ ( X - X , ) 2 .

^2,2jj-(y.y^)2, ,2] . 3

<iXodyo

It -will be seen that those latter equations involve the

representation of a vortex sheet by a system of line vortioes. Hence, in accordance with a remark at the end of ref.2, they are not valid

everywhere, but can be shovm to be so inside the envelope of the Mach cones emanating from the trailing edge. In particular, the formulae are valid in the region of the wake. Thus for the do-wnwash,

w = w' + v/', we have in the wake, where a = 0, by equations (30) and

(33):-r \-

èü j ( X - X Q ) ^ - p^(y-yo)^

^ 57o (x-Xo)(y-yo) 'dye +

e\u è

èyo èx

S' dxo<ayo

..-.__..! J(x-xo) 'ji- ^-{y-y^S]

(x-Xo)(y-yo)

.... (34)

I.e. w =

-| - H . » , T , «.-a •••I. I I •gÉ1..I.É...i,,yiM.Ki.. ^j.>._.-_.j,.-|-n, fm^

^ 't^^-^o) -P^(y-yo) Hy H. (èü g (y-yo)

-rrl 1 ^3yo (x-xj(y-yj

'C' hy^ ( X - X Q ) V ( X - X O ) 2-fi2(y-y,,) 2 i» r S dxodyo _{.... (35)} Before applying our results to calculating the downvrash in the wake of a delta wing, it is- instructive to consider the case of two-dimensional flow.

X

In two dimensional flow parallel to the zx-plane v = 0, so by equation (I6) lu is given by U' = ^^ = 0

and:-Jr. ' 1 3

"Yo = JL ( u* ^^o^yp ^

IT

.... (36)

. S'

which we can integrate directly -with respect to y^ since u is independent of y,.

Hence;-u -1 - 1 fi(y-yo) 1 "^2! - _ _ S m 11 II ' \i _ ; I I ^ J(x-Xo)^-(3 2z^J dx, ....(37)

where y and y are the roots of 3^= 0.

Therefore:-- 1 _{u*^ dx.}

where the integral is taken from the leading edge to x^ to the trailing edge, -whichever is the smaller,

.... (38)

X - /3 I z I o r

1

l o

-in particular, if u'^ is constant and the lead-ing edge co-incides with

= 0 and the trailing edge with

x^ - a,

then;-If^ =>i, u^ (x -.;:*Jz| ) or

1 5*

~ u c

•... (39)

2 /3 i

i

t I

The components, u-V and w, are given by the hyperbolic curl

of 4" , which in this case is /- AÜlL.^ o, -A ^ ^ ^ 2 ^ .

Hence:-V r» z

èx I

u = V + u" )

: 3j ) if X - /3|z| ^. c

.... (40)

and

_{^ " y^}

_I

_{if X - /3 jzj > c, which is in agreement}

with Ackerot's theory. On the other hand formulae

(33),

while

providing the right ans-wer for x - j5 [z/ > c, fail for x - 6 izl < c

for the reasons mentioned previously,

4. TIffi DOViOPJASH IK THE WAKE OF A DELTA Y/IMG.

Consider a delta wing at a small angle of incidence at , in

a uniforn'i airstream of supersonic velocity so that the apex semi-angle,

Y )

is less than the Mach angle. The apex is at the origin, and

X = 0 at the trailing edge, such that the wing is approximately in the

xy-plano m t h its axis along the x-direction.

Under these conditions accordino; to ref.3. we

have:-2

u- =

JL£

E'(,^)^x2tan2r-y^

E'(A)

^"^^^ X , . , -vThere A = |3 tan ^ .... (41)

hence u = _ I ± v/^tan^ Jf " yo

and

dy

_{S'(;\) ,yx2tan2 y _ y2}

Here u is in fact identical with the induced velocity

potential of the aerofoil and can be obtained directly,

The downwash, w = w' + w", at the centre line of the wake can

now be found directly from equation (34) by substitution and integration.

We Irnve: v/' = - V ^ ' T r E ' ( A ) ; x 2 t a n 2 ^--yg è^[ / S '

è f J(x-Xo)2-|3 2y2

x-x. <ix_dy, o «^o -V 0^. ' p E ' ( l )

1 rrzüTf

js' l^-'^o;

£Zc

x-x^ dxodyo (x-Xo)s/x2tan2 ^ - y o ...{k2) / I n

In S' the limits of integration with respect to y have to be such that the integrand is real; they are + x^tan )^' or + _J; ( X - X Q ) ,

whichever is numerically the less. The limits are + x tan ^ i^ - — 2 < 1, which is always the case if d = x-c \ A c.

X - X Q Consider J 2 2 ^ / 1 , i.e. x / x , X-XQ 1+ T P u t y^ = t x ^ t a n J^ and k = "'^ ^o , so t h a t x-x^ w' = - V o 4 T f E ' ( A ) -2Vü6 i r E ' ( A ) -2Vo^. TTE'(A )

\h-

k2t2 - 1 ^ K(k) - E(k) ^^ X - X b» J K(k) - E(k) A + k dk Consider now

A:

x-x. > 1, i.e. x^;, dxpdt

ix-x^)Jl-t^

l+> ....(43) ....(44)

Put « y ^ = t(x-XQ) and k = .^I^o , so that

I A X-w' = -Vo/,

-E'(A)

'1

-2Vo^

' T;E'(;I) r+1 |v/i-t2

-A - t

-1 dx^dt , A x p / i - k ^ K(k) - E(k)-(l-k2)K(k) I dxo •2 I iCp ....(45) = + 2Vo^ T7E'(A) K(ky - E(k) .-^ ^ k2(l+Ak) ....(46)

Now the range of integration with respect to x is 0 to c, so that when d <_ A c we must split the range into two parts -0 to

-JE and -iL_ to c. For the first part integral (42) reduces to

1+A 1+A

(1+1+) for which the range of integration with respect to k is 0 to 1, and for the aecond it reduces to (46) for which the range of integration •with respect to k is 1 to d/A c.

Therefore for d ^ A c we

have:-w.= -2Vc^ T r E ' ( A ) \. ^ - ^ dk + A + k 0 d Ac K - E k 2 ( l + A k ) dk • (47,i) / When

12

-When d > A c the full range of x = 0 to c is covered by k = 0 to Ac/d in the integral (44),

Therefore for d > '^ c -we

have:-w' = -2Vo^ d K - E

T . E ' ( A )

rr

dk

. . . . ( 4 7 , i i )

.. 0

for w" :

-We also derive from equation (34) the following expression

Vo/.

w " x *

/ ( x - c ) - p y^ dy^

- r ' E ' ( A ) \ (x-c) x/c2tan2 ^ -y^

. . . . (48)

As before, the limits of integration have to be such that the integrand is real; they are + c t a n ^ if d > A c , otherwise + Ê. .

Consider d > A c and put y^ = tctan Y and k = ii£« = ü£ , so

° ^

x-o d

that:--+1

w" = - _l£L_ 1

/^i.

at

1TE'(A) y 1 - t2

-Jl^E(k) ....(49)

Consider now d<, rl o and put R y = t(x-c) and k = -^"^ = d__ ^ ' ° (^ c ,:\o

so t h a t :

-w" =

r^n

i r E ' ( A )

V

J

- 1

/ i ^^' Idt =

-^ l-k2,2

2V E(k)-(l-k2)j^(j^)

7rE'(^) 1^

. . . . (50)

Therefore for d ^ A c we

have;-^. = - 2Vhave;-^. Ac I g( d )

T^E'(A ) <i I

'^°

And for d > A c

- [

1 -(^)^IK(I.)

Ac -' .^c I W" =

-irE'(A) d^ *

.... (51,i) .... (51,ii)

It -will be noted that w' and w" are continuous at d = A c. The gradient of w, hovrover, has a logarithmic singularity at this point.

The component w*' represents the downwash that would be

obtained if the entire lift were concentrated at the trailing edge for the same span-wise lift distribution; ^ is plotted in fig. 2. versus d/o

A "^^

for n = 0.4.

The t o t a l downwash, w = w' + v/', i s therefore given b y :

-i?here d < A o

w = ._ Vo( T T E ' ( A ) And where d > ^ c 2 w

Ac

d

M ü h Ë i i d k J ' K(k)-E(k)J

k + A j ^ k2(l+Ak)

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No.

Author Title

1.

H. Schlichting Tragflugeltheorie bei Uberschall-geschwindigkeit, Luftfahrtforschung, vol. 13, 1936.

2.

A. Robinson On source and vortex distributions in the linearised theory of supersonic flow. College of Aeronautics Report No. 9, 1 9 4 ^

3.

A. Robinson Aerofoil theory of a flat delta wing at supersonic speeds. R.A.E. Report No. Aero.2151 (A.R.C. 10222), 1946.

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