The Aerodynamic Derivatives with respect to Rate of Yaw for a Delta Wing with Small Dihedral at Supersonic Speeds

12 Juli 1950

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2 TECHNISCHE üir; T DELFT IXI

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Kluyverweg 1 - «lo-^ï?' rsö DELFT

THE COLLEGE OF AERONAUTICS

CRANFIELD

THE AERODYNAMIC DERIVATIVES WITH

RESPECT TO RATE OF YAW FOR A DELTA

WING WITH SMALL DIHEDRAL AT

SUPER-SONIC SPEEDS

by

REPORT No. 28 MARCH, 19U9

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

The Aerodynam.ic Derivatives with respect

to Rate of Yaw for a Delta Wing with Small Dihedral at Supersonic Speeds -hy-Sq.uadron L e a d e r J . H . H u n t e r - T o d R . A . P . , M . A . , D . C . A . , A . P . R . A e . S . — o O o — SUMMARY

Expressions are derived for the yawing deri-vatives on the assumptions of the linearised theory of flow for a delta wing with small dihedral flying at supersonic speeds at small incidence.

The non-dimensional derivatives are nunerically decreasing functions of Mach number. The

non-dimensional rolling and yawing derivatives are also numerically decreasing functions of aspect ratio.

When the wing lies entirely within the apex Mach cone there is a leading edge suction force

proportional to incidence v/hich makes a destabilising contribution to the yawing moment and side force which m^ay be of the same magnitude as that from the induced excess pressure distribution.

1. Introduction

A number of papers have been published during the past two years in which expressions are derived on the basis of the linearised theory for the various force coefficients acting on a delta wing flying at supersonic speeds. (C^, C^^ , C^^ in Refs. 1 and 2; L^,N^,Y^ in Ref .3; L ,M in Ref.U) The present paper^ in which the

aerodynamic derivatives with respect to yawing are calculated, completes the list of derivatives with respect to linear

and angular velocities relevant to the stability calculations for the delta wing.

It is clear that a steady rate of yav/ is not possible if small deviations from a neutral position are only to be considered. In consequence the interpretation of derivative with respect to yav^^ing is a matter of

convention. In this paper the forces are taken to be those that would result from a hypothetical steady motion with the same instantaneous velocity distribution at the boundaries; the pitching derivatives are calculated on the same basis in Ref.U. The hypothetical pressure distribution at the aerofoil differs from the true by

an amount of the same order in the frequency oarameter, r c/V (for notation see next section)• the errors in the

resultant force derivatives are reduced to second order

in r c/V by the addition of appropriate sideslip acceleration derivatives. The latter may be of the same magnitude as

the yawing derivatives in the form assumed here,

and a short acccn^.t of them will be given in a subsequent report.

The present investigation is confined to a wing of small dihedral at small incidence, of vi^hich the two halves are flat. The deviations from the neutral position are assumed small and in particular it is

assumed that, if both leading edges lie within the apex Mach cone v/hen in neutral position, they will remain so when disturbed, and vice versa.

When the leading edges lie within the apex Mach cone, a solution to the potential equation is obtained by extending the method of cone fields introduced by Stewart (Ref.1) to cover velocity distributions which are of the first degree in the space co-ordinates. V/hen the leading edges protrude through the Mach cone the problem reduces to the integration of a simple source distribution.

2. Notation

V = Free stream velocity

$ = Angle of dihedral 1^= Semi vertex angle 0^= Angle of incidence c = Max. chord

S = Wing area s = semi span

A = Aspect ratio (Us / S) L = Rolling moment

N = Yawing moment y = Side force

r = Angular rate of yav/ P = Air density

M = Free stream Mach No.

(i =

/M'CI

^ =

jï. tan (^

K,E = Complete elliptic integrals of 1st and 2nd

< kind of modulus k n = ^ L//jjVSs = n o n - d i m e n s i o n a l r o l l i n g moment ÓA derivative

n r ~ ^ N / Q V S S = non-dimensional yawing moment

^^ derivative

^r ~ ^ Y/QVSC = non-dimensional side force ^'^ derivative

3. Results

The non-dimensional aerodynamic derivatives quoted are referred to body axes and not to wind axes. The rolling axis is taken to be the axis of the aerofoil, that is the line common to the tv/o halves. It will be noted that the signs of n and y are reversed if the X and z axes are taken in directions opposite to the arrangement of Pig. Ua.

For the v^^ing entirely within the apex Mach cone

(A

<i):-(i)

-e^

= £ • (8 -

7M)E

- A'CU - 3/(^)K

6 (2 - A^)E - A^K

(ii) n

= - 1 f

cot Y (6 - 5A^)E - A^H - 2A^)K

+ l o C S c o t y cosec Y ( 2 E - / ( ^ K ) ( 1 - ^ ^ ) ^ ^ ^

E (2 -/('^)E - A^K

( i i i ) y ^ = - k ^hanY (6 - 5A^)E - k^(^ - 2^^)K

3ir (2 - A^)E - A^K

+ U_ o ^ ^ o s r (2E -X^KU1 - >?)^^^

^^ (2 - ;^2)E - K^K

Por the leading edges protruding through the apex Mach cone (X. ^

"1):-(i) -C = f/^-^

(ii) n^ = - Sfcoljrf ) 1 +

2)?

- 3

[2

(ili)yr

It can be shown that the derivatives are

continuous as the parameter A passes through unity. In Pig. 1 the quantity ^^/^ is plotted against Mach number for different aspect ratios.

2

In Pig. 2 the quantity n^/S ^ ° ^ ^ero incidence and the contribution to n^/ocS ^'^^ *o leading edge suction. are plotted similarly. In a like manner the variation of y with Mach number and aspect ratio is shovm in Pig. 3.

1+, Delta Wing Enclosed within the Apex Mach Cone Linearising the equation of continuity for steady supersonic flov/ gives the Prandtl-Glauert

equation:-~ A ^ + kï + ^ = "^ •

' > x 2»^ ^z

where u,v,w are the induced velocity components in the x,y,z directions in the cartesian co-ordinate system indicated at Pig. l+a.

When the flow is irrotational there exists an induced velocity potential,(O , and it and u,v,w all satisfy the

eqiiation-.-.2 \2^ \2^ \2

fr

0 1 + 0 1 + è_f = 0 (1)

^x2 ^y2 az2

Define G J = / w + i y = A ^^^?r + Iz

X +/

Prom the analo^^ue of Donkin' s general solution of degree zero of Laplace's equation in three dimensions

(Ref. 5) it follows that the real part of any analytic function of \jj is a. .solution of degree zero of equation

(1) also satisfying Laplace's equation inO^J, "^ ,

X X X /

Suppose u ,v ,w are functions of' degree zero in x,y,z, derived from a potential (0 » and satisfying equation (i); we can take them to be the real parts of functions U,V,W of u^. It was shown in Ref,1 (compare,

Ref.3) that in these

circumstances:-i/^dU = idV - = dW ( 2 ) 2UJ ^ ^\jf 1 - U)^

It can be shown by dimensional arguments in our present problem in v/hich the aerofoil has an angular velocity about the z-axis that the induced velocity components are functions of degree one in X, y, z in the region ahead of the trailing edge.

It is assumed that the motion is irrotational. Therefore, the first derivatives of the velocity components with

respect to x are of degree zero, are derivable from a potential function and satisfy equation (1). In consequence these derivatives are the real parts of functions U„, V„, W„ of (W > v/hich are connected by

X X X

a relation of the form (2); it follows likewise that the derivatives with respect to y and z are the real

parts of functions U„, V,, W„ and U„, V„, W„ respectively,

y y y z z' z ^ ' which are similarly connected.

For the boundary conditions at the aerofoil we make the usual assumptions of the linearised theory

of thin wings with small incidence and dihedral that the kinematic boundary conditions are fulfilled at the normal projection of the aerofoil on the x y - plane rather than at the aerofoil itself. In calculating the aerodynamic derivatives with respect to yawing, referred to body axes, we can, except for the purpose of assessing the suction forces at the leading edges, ignore the incidence v/ithout loss of generality. Therefore, the boundary

condition at the aerofoil reduces to v/ = - r x o > y ?" 0 and w = + r x ^ , y < 0, aj; z = 0.

The other relevant boundary is the shock wave emanating from the apex of the delta wing, which in accordance with the principles of the linearised theory is taken to be the Mach cone corresponding to undisturbed flow. It is further assumed in the present problem

of a wing v/ith small dihedral lying entirely within the Mach cone that the shock wave is infinitely vifeak. The boundary condition reduces to the requirement that the

induced velocity should vanish at the apex cone. Clearly a sufficient condition, Yi^hich will also be shown to be

necessary, is that the velocity components shall vanish at one point and their first derivatives at all points on the Mach cone.

Since the induced velocity vanishes at the Mach cone we can write a number of relations of the

form:-1 form:-1 form:-1 form:-1 form:-1 u (x.y.z) - u (x.y.z^ = y - y. u(x.y.z) - u(x,y.z) ,

1 X - X 1 X - X

T

.1 1 y - y

where (x,y,z) and(x|y^z) are points on the Mach cone, from which it can be shown that in any region on the Mach cone where one velocity derivative is finite the remainder are finite.

If u is to be zero over the Mach cone then at the Mach cone in a region where its derivatives are

defined the following conditions

hold:-X ^u + y ^ + z ^ = 0 èx ay ^z z \a - y Vu = 0 so that xz ^ + (y + z )^u = 0

(3)

Since ' ^ = ^ everywhere and x = /J (y + z ) on the Mach cone, the last equation can be rewritten as

:-i -'^X ^ X = 0

"b^

ih)

Put

^ = l ^ ^ ^ F ( U ) )

dUJ U ) ^

so that

^ = li p(u:))

dk> ^uJ

and equation (k) reduces

to:-LJT. tJx. 2iPduJ +

U) P d t O

_{= 0}

u>

u>i

U)'

/where ... /

whereu). ,iJ2 are points in the same region in which

'bxi. and \w are defined J>x ^x

Since on the Mach cone fu) I = 1 , we may substitute g ^ for ^ in the last equation and

obtain:-r .02 .02 ^

R J. Sin Ö 2 I P<^Ö - \ P s i n £ ) d Ö / = 0

whence A 2 ^

R cosöi I P d'-f I cLö = 0

from which it follows that P is pure imaginary on l iJƒ = so that ^ 2

= 2R f p d ê = 0

/lu\

'

(^\

= 2

Therefore \u and similarly Vi, ^ are bx ^y ^z

constant and clearly from equations (3) zero, in the

region on the Mach cone where they are all three defined. In a like manner it may be shovm that the other derivatives vanish in the same regions. Since these functions are continuous at points just inside the Mach cone it follows that they cannot be infinite in some regions and zero in the remainder and therefore either zero or infinite everywhere on the cone.

Now for thin aerofoils which are approximately in the xy - plane and v/hich are symmetrical with respect to y the induced velocity potential is anti symmetrical with respect to z, so that \n, Vv, 'bv, \^ vanish at all

^x ^x >y ^z

points in the xy - plane not on the aerofoil. Therefore all the derivatives vanish everywhere on the Mach cone.

Consider the

transformation:-cn(r,k) = cn(P+i(5-,k) = 2i»J

where cn(Xk) is the Jacobian elliptic function of

modulus k in Glaisher's notation. The interior of the Mach cone is represented in the X•- plane by the interior of the rectangle with vertices + 2iK',K + 2iX', the imaginary 'ixis between + 2iK' corrcs-oondin,^ to the I/iach cone and the

parallel line between K ± 2iK' to the aerofoil. (See Pig.

Since the flov/ is irrotational it is clear that V - W = W - U = U - V = 0 . and it will be

z y X z y X seen that in the ^ - plane

^ d U ^ = idUy = dU^ = idV^ (5) cn'^ sn't sn't

= - dV^ = idVg, = dW^ = id'Vy =^ dW^ /jsnTscr y^scy /j s c T ^JncY

I t now remains t o choose one f u n c t i o n , say X , s a t i s f y i n g the boundary conditioïis i n such a way d ^

t h a t t h e v e l o c i t y p o t e n t i a l i s single valued and the aerodynamic f o r c e s a r e f i n i t e c

B y reasons of symmetry ^ = 0 o n y = 0 ^x

and therefore, referring to pig. he, d'.v m.ust be pure imaginary on the lines OC,AB and A'B'. On integrating

"x along OCB dw must jump in value from 0 to -^ r ^

d Y ^x

at the point C and on integrating-a long OCB' to - r o • in order that the boundary conditions at the aerofoil may be met: hence dW must have a simple pole with a residue

X

dr

of imaginary part 2r^/jf at c('T = K) and similarly poles

with residues of opposite sign at B,B' i^= K * 2iK'). In addition v/e require qjN to be constant and ^ to vanish

^x ^ y over the tvro halves of the aerofoil, which need is met by

choosing dW to be real on BB'.

The boundary condition at the Mach cone that the first derivatives of the velocity components should vanish requires dW„ to be real on the im.aginary axis, AA*, to have

X

dr"

no singularities on AA' that contribute to the real part of its integral and to have at least a simple zero at p and

P'(r= ± iK').

For the velocity components to be single

valued dW must have no branch points or poles with

dr

residues inside the rectangle AA'B'B. The aerodynamic forces will be finite provided dU and dU do not

Tf

dr

have any singularities of too high an order on the aerofoil, BE'.

The necessary f u n c t i o n i s found to b e :

-^ = i s c T n d -^ i ; (A d n -^ r + Bk'-^) (6) d r

v/here A and F a r e r e a l c o n s t a n t s such t h a t \ji ^x

and ^ w have the correct values on the tv/o halves of the

aerofoil. Any other function of this form would lead to an inadmissible singularity of one or more of the functions such as d?/^ at either ^T = + iK' or T = = K + iK'

d ^ , This function has a residue - i(A + B ) A

at the point C, ^ = K, and therefore A + B = - 2 r S k ' ^ TT

Prom equations (5) and (6) we have:-K = /^/?^sc^'rnd^'r(Adn^r + Bk'^)da:

f

= l_/^^^f'^(A + UB)K(T) - (2A + ^(7H-k^)B)E(r) k ^ V, ^

-»fk^(A + i(U+k^)B)snrdcr+ (A+B)snTcdr + ^ Bk^k'^snTcnrnd^r j

= L^fi (k'^(A + 1±B)K - (2A + ^(7+k^)B)E ]

^' I 3 )

forf = K + iCr

Therefore in order that ^ w may vanish on the

^ y aerofoil:-B = 6k'^r S . 2E - k'^K 2 *7~ k (K+E) - (K-E) /Also .. .

Also from equations (5) and (6) we h a v e :

-.r

^ u = 1 R i s n T n d ^ r (Adn^T + B k ' ^ ) d ' ^ SfK ^ 'Ó '0 - 1 R i (A + B ) i c d T - 1 B k ^ i c d ^ T f k ' 2 4 ( 3 -^ Now c n T = 2 i t o = i / ^ y on z = 0, so

1

'i/

/x^ l^^^y^'

that icdT = y/^ x^tan^ V - y^ for z = +0 and is of opposite sign for z = -0

Henco

^ = - j i (A + B)y ^ + • Bk^y^

^x k'^/^ hx^tanV -y^)^ 3(x2tan^r - y^)^/^

on the upper surface of the aerofoil with opposite sign on the lower siirface.

Again from equations (5) and (6)

rX

Vu = R sn^Tnc'Xnd^'^(Adn^T+ Bk'^)d T

by J

= R J A + B

!

ch'^'dcT' " k'sdr/ - iBk^sd^T

/ = ~ A 4- B 1 ch" xtan V - x tan X _ „ /

k'3 1 I y| /I^tan^r^J

• Bk^x^

3/?(x2tanV- y2)3/2

on the upper surface and with opposite sign on the lower surface.

Hence u = - A + B „ ch" x tan V + Bk xy tan T

k'5 (y| 3k'3/x^tan^rly^

(7)

on the upper surface of the aerofoil.

The excess pressure is approximately - ^uV

and therefore the force derivatives at zero incidence

may be readily calculated from expression (7). However,

when the aerofoil is at incidence the interaction between

the two fields give rise to a leading edge suction which

contributes to the yawing moment and sideforce.

It is shown in Appendix IV of Ref.2. that, if

the total induced velocity perpendicular to the leading

edge is of the form

\c%''^

+ bounded terms

I ,

where "^ is

the distance in from the leading edge, then the suction

force is

h SrC

k cos T per unit length.

The velocity along the leading edge y = x tanV^

induced by the yawing is (u cosV^ + v sin

Y )

and

^ (u COS Ö + V s i n V ) = JL cos Y" R

^ x fi"

i

k*snT+ icrfl

c n T ö n ^ T

•Bnr{Adn^'t + B^k'VdT

which vanishes at the leading edge

{'^r

= K - iK'),

and <i (u cos V + v sin V) vanishes in like manner.

>^y

Hence the velocity induced by the yav/ing perpendicular

to the leading edge is (u cosecY'+ bounded terms).

The velocity 'ootential induced by incidence alone is by

Ref. 2 V»<./x tan f - y/E

Therefore the total induced velocity perpendicular

to the leading edge

is;-\

VoC + Bk y i x sec V"^ . + bounded terms

{ E 3k'-^j /x^tan'^r-y'^

~ J "^g^, + Bk XQ / /xntanV^ sec

Y +

bounded terms

E 3k'3 J / 2 ^

where x = XQ + "^ sinYand y = Xotan

Y -'"z

°os

Y\

Neglecting second order quantities the suction

force resulting from yawing when at incidence is

therefore:-I T t

'^

VBk\^tan Jï^/3Ek ' ^ (8)

Integration of the pressure distribution obtained from expression (7) and of the suction forces at expression (8) yields the values of the non-dimensional derivatives ^ , n , y given in Section 3. for A ^ 1 •

5. Delta Wing with Leading Edges outside the Apex Mach cone

Since the region of influence of a disturbance at a -ooint is contained within the Mach cone emanating from that point , the flov/ at the upper surface of the aerofoil is independent of the conditions at the lower

surface, and vice versa, when the leading edges protrude through the apex Mach cone. The flow at the uoper surface is represented, therefore, by a potential function which satisfies the boundary conditions at the top surfaces v/ithout regard to the lower surface, and such a function

is obtained by integrating a distribution of elementary solutions or sources that give the correct values of the normal velocity locally. The required source distribution is of strength - r^x for y y^ 0 and + rdx for y ZL 0

for the upper surface. '' The induced potential at the upper surface is

therefore:-X

,x tan y^ f f ^ V ri

(P(x,y) = - r i

^o^^o^^o

r-?: :2

0 o y(x-x^) -^j(y-yQ) o ^o^^o^^^o °-XQtan V^

In Pig. Ud P is the point (x,jr), OL-i and OLp the leading edges, and ?L^ and PLp are the boundaries where (x-XQ)^-^^(y-y^) ^ = 0

2

Put X = X - / q i±i^ , y = y-q^^p

The value of q and t vary as

follows:-When

(^Q,7Q)

is on (i) PL^, t = - 1

(ii) ^L2, t = + 1 , .

(iii)OP, t = t = X

-jx^-fy^

(iv) OX, q = q = • 1 ^

2t ^

(v) OL^, q = q^-- (x tanr-y) (1-t^)

X(Ut^)

- 2t

(vi) 0L2, q = ^2^ (x tany'+y)(1-t^)

J(1+t^)+ 2t

When p i s inside the Mach cone so t h a t x J>iy > 0,

v/e have

(

"^l ^o ^o ^^ ^2 ''

1

0 = 2A_I ( [A i^C" rx(1-t^)-/^q(1^^dq dt /

"^ i ii 0 t^ I i (^-*')' J

{ o ^o o y '

Having integrated with respect to q

we

can differentiate

iinder the i n t e g r a l signs v/ith respect to x, since

q^ = q^ = q„ v/hen t = t , a n d : -

_{0 1 2 o}

u = 2riA

IT

/ ' O /

:^i

X tan K* - 2At(x tanY^-y) dt

(Xt-1) 2+^2-1 |r[,lt-1)2+i-1 ƒ

- 2r<$^ 1 j X tan Y + 2At(x tanT+y) j dt

'^ tJUt+O^+A^-i £ (/{t+1) ^tA^-i ƒ j

1

+ 2r_£ j jrdt (9)

'O

I T "t *

= 2r£ y tan""*/ x t a n V / A ^ -1 I

^ TjJ^W^ I /x^tan2r-.^?y2j

+ 2i^ (A^-2)x tan)r tan""* j y / A^— 1 (+ 2r.Sych"''x

"r(>FIT)V^ ( /x^tanV-A^yM ^ /3|y(

the same r e s u l t being obtained for x > -/iy;?-0

/ F o r a . . .

Por a point outside the Mach cone,/^ y > x, ^1 .+''

(P = 2ri r \ x(1-t^)-i^^q(1+t^) dq dt

^ ^

V - 1

(^

-*')'

By putting t = 1 in equation (9), we obtain u = ref X tanV'(I^-2) -f y ,

and similarly for A y <^ -x

u = - r i X tanV( A^- 2) - y

By integrating the pressure distribution given by these expressions for u, the values of the derivatives quoted in Section 3. for A J^ 1 are obtained.

REFERENCES

2.

H.J. Stewart

A. Robinson

The Lift of a Delta Wing at

Supersonic Speeds. Quarterly of Applied Mathematics. vol.U.

October, I9U6.

Aerofoil Theory of a Plat Delta

Wing at Stxper<?onic Sneeds. May,19i|.6 to be nublished in A.R.C. R & M

Series.

3.

U.

5.

A. Robinson J.H. Hunter-Tod A. Robinson

E.W. Hob son

The Aerodynamic. Derivatives with Respect to Sideslio for a Delta Wing with Small Dihedral at

Supersonic Speeds. College of Aeronautics Report No. 12, 19U-7. Rotary Derivatives of a Delta Wing

at Supersonic Speeds. J. Roy. Ae, Soc. vol 52. November, I9U8.

The Theory of Spherical and

Ellipsoidal Harmonics (Section I06) Cambridge, 193I.

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T H E AEROFOIL IN T H E (V..<J.Z) FIELD

Flö.

4OL T H E U3 - PLANE FIG. 4 b .

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rZiK t K UI z o Ü o < -cK^

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T H E T- P L ^ N E THE AEROFOIL FOR

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F I G . 4 C . F I 6 . 4-dL.