An extension of Multhopp's lifting surface theory

''''CoA Report^Nö.'Iiï^^^

THE COLLEGE OF AERONAUTICS

CRANFIELD

AN EXTENSION OF MULTHOPP'S LIFTING

SURFACE THEORY

by

KSPORT NO. 132 Mayj19éO T H E C O L L E & E O F A E R O N A U T I C S C R A N F I E L D An E x t e n s i o n of Multhopp' s L i f t i n g S u r f a c e Theozy t o I n c l u d e t h e E f f e c t of F l a p s , A i l e r o n s , e t c . b y -R o b e r t ¥ , Simpson, B . A . S c . Smyli^IARY

The subsonic lifting surface theory due to H. Multhopp (Ref. "l) has been extended to include a chordwise discontinuity in the slope of

the lifting surface, i.e. to include the effect of flaps, or ailerons. By representing the chordwise loading of a two-dimensional flapped flat plate in closed form, a new loading function •& is used in the

2

representation of tiae chordwise loading, and a new influence function k is defined. This function is dependent on the parameter d which describes the hinge position, and so the tabulation which has been done for the influence functions, i and j, in Multliopp's original method have to be repeated for values of <f>, .

The method is restricted to linearized, non-viscous flow about thin wings of moderate aspect ratio of any planform. Values of lift,

pitching moment and centre of pressijre can be obtained across the span for deflected flaps, and the theoretical effect of planform can be studied.

An exanple has been calculated for a straight rectangular v/ing of aspect ratio 6 v/ith a full span 20^ chord, trailing edge flap, A

comparison with available ejqoerimental results shows that viscous effects

are important in obtaining the correct magnitudes of lift and pitching moments,

Based on a thesis submitted in partial ftilfilment of the requirements for the Diploma of the College of Aeronautics.

Page S^aramary L i s t of Symbols 1 • I n t r o d u c t i o n 1 2. Theoretical Background 2 3. Theory 8 3.1 . General 8

3.2. Vorticity distribution on a flat

flapped plate 8 3.3. Representation in closed form 11 3.4. Construction of the loading functions 12

3.5. Choice of pivotal points 13 3.6. The influence functions 15 3.7. The calculation of k lé 3.8. The spanwise integration 18 3.9. Correction for the logarithmic singularity 19

4. Example Calculation 20

4.1 . Method 20 4.2. ResiJ-ts 21 4.3. Discussion of resiilts 22

5. General Discussion of Theory 23

6. Conclusions 24 7. References 25 Appendix I 27 Appendix II 32 Appendix III 33 Figures Tables 1 - 23

LIST OF SB-BOLS

x , y , z r e c t a n g u l a r c a r t e s i a n co-ordinate system (see F i g . 1)

X-» y . p o s i t i o n of inducing s t a t i o n on wing

X

^ = :r- non-dimensional co-ordinates r e l a t e d

72

V2

t o semispan

X - ' -' . 'sj—«—- \ Non-dimensional vmig co-ordinates r e l a t e d t o

Y =•

c^y^

T-X^

inducing wing section (suffix o)

X _ _ = p o s i t i o n of leading edge of

in.duojLng wing section

-1

<P

K cos" (1 - 2X) = angular chordwise co-ordinate

Q = cos

V

= angular spanwise co-ordinate

Note: 6 is also used in tvTO-dimensional derivation of flapped

plate vortici-ty distribution as an auxiliary co-ordinate

_1

0 = cos

E, ,

See Section 3.2

U velocity of undisturbed flow relative to the wing

co-ordinate system

u,vw perturbation velocities in x,yz directions

a -

-^

(x) local wing incidence

a* incidence at forvraxd pivotal point,

a'

-reaj: pivotal point

P pressure

p

density

F upper surface - P lov/er surface ,. • „T

. . rf :

•,••..,., ^.^».._.,^.^..,, i„i,.- - non-dimensxonal

•gpU^ T/ing loading

"h

wing span

O wing chord

V =

pivotal station

n = inducing station

6

- "o - Ji -

angular co-ordinate of pivotal stations

n, = cos Ü, = s m •.

V V

m + 1

a , b coefficients for approximate integration

y = "-pV •- = non-dimensional lift per unit span

\'

°

^ _ ' = non-dimensional moment per unit span

i,j,k influence functions for chordwise downv.'ash integrals

E

%

chord for flap

, I position of wing line of flap

M 1 >•

"l • Introduction

•\

The lifting surface theory developed by H. MixLthopp produces aooxirate spanwise distributions of lift CL and pitching moment G^. for a minimum of computing effort. The method consists of a computing scheme to replace the double integral equation of the lifting surface, using influence functions and carefully selected pivotal points to do the chordwise integration, and an approximate sparovise integration technique similar to Multhopp' s treatment of the lifting line problem, The influence functions are tabulated as functions of tvro variables, X and Y, for any wing, and simileir standard coeffioiants are given for the spanwise integration. The solution of the integral aquation ie thus reduced to an iterative solution of a set of linear equations.

Multhopp' 3 original paper, summarised brief3^r at the beginning of the present paper, dealt with steady lineaxised potential flow about an infinitely thin wing of moderate or large aspect aratio. Garner-^ has extended it to slow pitching oscillations, but oscillatory derivatives, etc, for flutter or high speed oscillations cannot bo estimated. The method suggested by Multhopp for dealing with discontinuities in the surface (due to ailerons, flaps, e t c ) in Appendix 2 of Ref, 1 is a

15

calculation of equivalent incidences for chordwise discontinuities, and a fairing or interpolation process for spanwise discontinuities.

The present paper deals with wings having discontinuities in the chordwise direction by developing a closed form representation of a flapped plate, and using this as one of the chordwise functional representations. This representation requires the calculation of a new influence function \*iich turns out to be implicitly tied up with the spanwise position of the discontinui-ly and it is necessaiy to calculate and tabulate the function foi? eveiy position of this dis-continiiity, unless interpolation should prove practical. An example of the calculation of the required coefficients for a 2C^ flap is included in this report to illustrate the mathematical treateient and to demonstrate that it provides a satisfactory method for the evaluation of the properties of flaps,

2 , Theoretical Background

1

The b a s i c theory of Multhopp i s given here t o introduce the

problem and h i s nonenclature has been c l o s e l y f oUovirad,

Consider a t h i n vdng i n i n v i s c i d p o t e n t i a l flovr. Let ( x , y , z )

be an orthogonal system of axes such t h a t the x - a x i s coincides with the

d i r e c t i o n of the undisturbed flow, and the z-axis i s almost perpendicular

upwards from the vdng a r e a , ( F i g , l ) ,

< i

The perturbations (u,v,w) in velocity due to the vdng are assumed small as a necessary condition for the linearisation of the \i±iole problem,

Using the lineaxised Euler' s equations and tlie continid.ty equation, and replacing the \Tdng by a discontinvdty sheet built from doublets, the relation between local downwash and local loading density for the lifting surface is expressed as :

YKX.Y.Z)

-1 ff^^^o^o)

^ ( ^ o ' ^ o ) =

{'•J

AP

X

(x

-

xj

+

(Y

- Y^)^

dx dy o ^ o

(1)

where x , y , are positions of doublets and x,y, aro general points on the wing,

The double integral equation for •C'(x Y ) has a strong singularity

o o

« 3

-To satisfy the Kutta-Joukowski condition for smooth flow from the trailing edge, •^(x 7 ) is restricted to functions v/hioh vanish at the trailing edge, i

Since this equation cannot be solved directHy, Multhopp constructs linear combinations of independent loading distributions which can be made to satisfy the integral equation at a certain number of "pivotal" points. The more points are taJcen on the wing, the more independent load distributions can be used, resulting in more accurate results, but the computing effort is proportionally greater,

The spanwise integration technique tised by Multhopp in his treat-ment of the lifting line problem is employed again, and the spanwise distribution of pivotal points is similar, i.e. equal angular increments

n = 0, ±1, ±2 +£L^-1

For the chordwise integration, Multhopp assumes that locally, two-dimensioneO. relationships still hold. He constructs his loading

functiQn&&( 0) from terms of the Fourier expansion for thin airfoil theory,

l{4>) = a^ cot -^ + 2 a sin n ^ (2)

NT-/ O 2 1 n '^ '

0 at L.E,

IT at T,E,

and chooses his two pivotal stations on the basis of matching expansions of CL and CL^ in terms of a", c£',the incidences at the pivotal points to the thin airfoil results :<

Thus, G^ and C ^ x^alues are estimated as accurately as would be possible with an arbitrary choice of three stations, and the inclusion

of another term of the series, The linear combination for

is taken by Multhopp as

^(V^o) =

2 C^ B C., — i ( ï ) e ' ( x ) + - — ^ ( y ) ^ ' ( x ) TT o' o ^ o*^ IT ^''o^ r o ' •vrfiere ^' c o t ~ , •&' = c o t -^ » 2 s i n 0 a ( y ) and a ( y ) a r e v / e i g h t i n g f u n c t i o n s v/hose v a r i a t i o n v d t h y i s t o be d e t e r m i n e d . Here t h e y t u r n o u t t o be e x p r e s s i b l e a s t h e q u a n t i t i e s C^ and G,, xising e q u a t i o n s ( 3 ) , ( 4 ) , ( 5 ) » a c c i r c a t e l y r e p r e s e n t e d b e c a u s e of t h e c h o i c e of p i v o t a l p o i n t s , and whose v a r i a t i o n i s of p r a c t i c a l i n t e r e s t . T h i s i s t h e r e s u l t of c o n s t r u c t i n g ( a c c o r d i n g t o •fchin a i r f o i l t h e o r y ) a combination of a " n o n - l i f t i n g moment" d i s t r i b u t i o n ' c o t ^ '1 and a " n o n - l i f t i n g " camber d i s t r i b u t i o n

c o t p - 2 s i n (p which g i v e s t h e moment,

The i n t e g r a t i o n i n t h e chordyri.se d i r e c t i o n can now be t a k e n t o form t h e i n f l u e n c e f i m c t i o n s , b / 2 T^,XYZ) _ ^ i QTT C T ( Y ) . c L o ' '-b/2 ^ ^ - ^ o ^ '

2e (x )

o^ o' TT 1__ BTT b/2 C.. (Y ) . c _{M ^ o^} - b / 2 ^y - ^ o ) ' 8 TT X - X 1 +

( x - x j % ( y - y j '

dx <ay.

(6)

^/x^) I 1 +

X - X (xv-x^) +{y''y^) dx _<3y. Vi*iere Multhopp d e f i n e s

cot I

1 + s i n (p d (f> ( 7 ) TT

a(x,Y) =

= it

_{vr./ ^ ° ° ^ 2} - 2 sini^) 1 +, M 2 x_ LJ::PPAÉ]\J^ sini^ diji (8)

and t h e g e n e r a l i s e d v a r i a b l e s X,Y 8LS : X = Y - Y V o -1 0 = C O S " (1 - 2 X ) o Combining e q u a t i o n ( 6 ) w i t h ( 7 ) and ( 8 )

1 1"^^ G^c(Y^) i(Xï) + ^ c U J J W

- b / 2 « (X,y) = - ^ ^ (y - y^)'

ay^ (9)

or in non-dimensional variables as defined in the nomenclature

1

Ul (n-n^)'

72

Si!

2b

n'

72

2b

The spanv/ise i n t e g r a t i o n i s done by t h e t e c h n i q u e u s e d i n Multhopp' s t r e a t m e n t of t h e l i f t i n g l i n e problem a s d e s c r i b e d i n S e c t i o n 4 of t h i s r e p o r t . The imknown f u n c t i o n s y ( n ) , /^(n) a r e r e p r e s e n t e d by p o l y n o m i a l s i n terras of t h e i r v a l u e s a t t h e p i v o t a l p o i n t s i n t h e u s u a l manner «,. (x) = K,, ( y i + Mj), -vv where ^vv vn. m + 1 4 c o s m.-1 2 V Z ~ v n t , ^ ( y i + /^ö) _n (11) nH-1 » ^ v n V = n n o t i n c l u d e d = 0 f o r j y - u ] = 2 , 4 , 6 ( 1 2 ) c o s nTT mfl mfl ( s i n — r - sirr—7) ^ im-1 m+1'

for|y-^ I « 1,3,5 (13)

The speed of corvorgence is dependent on the number of spanjfri.se

stations selected; when account is taken of a logarithmic singularity

in the integral, the rate of convergence is increased.

2

Multhopp' s method was improved b y Mangier and Spencor . T h i s took t h e form of an a d d i t i o n t o t h e i y y , Q^i; t e r m s , and c o r r e c t e d v a l u e s a r e denoted 3^^ , "j^^ . The c o r r e c t e d boundary c o n d i t i o n s a r e

t h e n e x p r e s s e d a s .

(X ( x ) = b ( Ï y + 0 /i ) - 2 b ( i y + d M ) (14) _{v^ ' vv^ vv ' v ^vv ^v^ ^ i^n t'n n t'n n ' ^ '}

This is satisfied at tv70 points at each pivotal station, denoted by

x^ = x^ L.E. + 0,9045

%

x"

V

= Xj^ L.E. + 0.3455 Cy

(15)

Prom the two conditions at each pivotal station, the unknovms y^ , /^

are separated by elimination. Thus, the 2m equations

axe

expressed

in a more convenient form for solution as follows

:-m-1

^ = a U' a' - e" a") + VV^ V V V v^

2

2

m-1

" 2

m-1_

2

a (m"

a"

- m' a') + S.

vv^ V V V v' (S^\ '^ 2 '

a

U'±' U"

i" )y

+U'n'

-C"V' )/i

vn Y/' '^ '^^ 1^ vn' n ^ V^vn f'^W n . vn

(16)

•vAiere V vv I" V vv

m'

m.

'vv V 'I 'vv 1 n _{vv "fV} vv '^VV

The equations are solved by an iterative process T.iiGreby even and odd

values of n define two sots of equations. This is possible since

a = 0 for |y-n = 2,4,6 .,, Thus, for

v

odd, i need be calculated

vn ' ' ' ' * ' vxi

• • ' . ƒ • *

The aerodynamic forces and moments, t h e i r d e r i v a t i v e s , and the c a l c u l a t e d p o s i t i o n of the c e n t r e of p r e s s u r e are obtained from the values of v„j jLi„ quite r e a d i l y as suggested by Multhopp,

TOfA ^ n ' '^n HL m+1 - ( ^ y^ cos nir _m+1 (17) C^ = a . c . TT A IQttI 1 Jiiï m+1 m>«1 2 - ( ^ ) y * n m-1 2 E y n s i n 2ngr m+1 n ;<^ 0 n ^ n b

72

(18) (19) - y n

W E ,

V2

(20) about the y axis,

This completes a brief sumnary of Multhopp' s paxx3r ( R & M 2884). The chordwise pressure distribution has been composed of two special distributions in arbitrary proportions, and the speinwise distribution

calculated. The spanvdse distribution may be symmetrical, or antisymmetrical (•vAiich gives some simplification to calciiLations) or quite asymmetrical, The only quality the distribution must have spanvd.se is smoothness, The interpolation functions cannot be expected to work if there is an irregular behaviour between two interpolation stations, Miilthopp recomnends that the distance between spanwise stations should be less than 0,4 or 0,5 wing chord.

3.

3.1 • G-encral

We choose a functional representation for the chordx'dse loading distribution in Multhopp' s method \*Lich represents more closely the section loading with a flap deflected. In linearised subsonic theory, the wing loadings can be considered as independent for a flat plate

wing at zero incidence with a control flap deflected. The flap represents a discontinuity chordwise in the slope of the wing, and may take any

chordwise position, e,g. a nose flap, or a trailing odge flap. As Multhopp has done, we resort to thin airfoil theory' to construct this functional representation,

3-,2, Vorticity distribution on a flat flapped plate

T-t-T-T»:-;-rg-'

y.

Consider a flat airfoil • of chord 2c with z axis as shown in Fig. 2, A trailing edge flap is hinged at X and deflected through an angle 6.

Thin airfoil theory gives the downwash relation c

w(x,0) = - 4

Zir

. d?

(21)

X - C

-o

•vAere y (S) = circulation/unit distance, or vorticity distribution to represent airfoil,

Making the variables dimensionless by putting

X = —

_{S =}

we get

w(x,0) = - ^ ƒ

-1 ya

(g)

X -

.

^ï

(22)

9

This equation can be solved f o r our case by inversion. I t i s proved

by SdE.'iVi'-'n' tliat for aany tvro f-jnctions f and g which are continuous

except f o r a f r n i t e number of s i n g u l a r i t i e s , a unique s o l u t i o n t o the

i n t e g r a l e qua-lion

•v\ftiere the v a r i a b l e s are dimensionless andi^(l) i s f i n i t e or z e r o , i s

given by

2

ll^^ rif^ t t ^ - ^

-1

i£

j

f(?)

^ = 0,

-1

Applying this to equation (21),

1

y,(5) = I

f^

i

M .

^^

. al (25)

-1

^ f ,0) - u, = u ^

Therefore,

In our case (Fig, 2)

^ a

dx

1

^1

= 0

= 5

+

X X

1

for

for

"l -5

X < 5^

da

a

dx

X - 1

<a<"

1 . * .

y.(x) =

a 2U 1 - X -1

SS LnJL / IL+J ±.i„

^ ^rTs J ^'1 -f' 5 - f

. <aè

I f we make the s i i b s t i t u t i o n = - c o s 6 X = - c o s 0 x ^ = - cos (l>^

when X goes from -1 t o 1 , ^ goes from 0 t o TT

df = s i n 6 d e t h e i n t e g r a l r e d u c e s t o 'TT 1 - c o s

<f>.

cos 6 - 00s <P do

The value of this integral as obtained in Appendix I is

/ V (1 - c o s ^ -,„ ^ ^h' s m 56 sin 0^ + ^ sin ^j^ « 0 (26) (27) Therefore, y (x) = y (0) a

y (0) = 2U6 jUl^ogè

'^a^^' TT \/ 1 - COS^ 2U6 s m -?^h^?i r ^ ^ 4. ^ 2U6 ., (."" - 01,) cot -^ - - — In •w h' 2 •^ ÏT 5^1, +i> sin

?^v, - 0

sin ^ h - ^ (28)

This is the vorticity distribution of a flapped flat plate from 2-D airfoil theory.

1 1

-3»3. Representation in closed form

To obtain a load distribution vfe note from linearised theory

P - P ^ u L

u

4 S ( W"** 0T ) J 1 C . . . ^ ( ^ ) = ^ c o t ^ + ^ m TT 2 TT

1

c i n - ^

s m 2 (29) (30) (9)(lOl

This loading function can a l s o be represented py a F o u r i e r s e r i e s

of the form _{J CO} ^(<f) = a cot ~ + Z a s i n n^i ^ ' o 2 J n (31) \iiiere ir 8 s i n n^, h n ir (32)

and the u s u a l t h i n a i r f o i l r e l a t i o n s apply :

^L = f ( % - 4 )

%

='TZ

(^

-

^2^

(33)

Thus we see that we have obtained the series summation in closed form, The cot ^ term may be talcen as the flat plate distribution due to an induced angle of attack caused by the flap deflection. The logarithmic term represents the distribution due to camber line shape, and has a singulsLrity at the hinge position,

Using relations (33)

Oj^ = 26 (TT - (^j^ + sin^j^)

°^ Gj^j = - 2 ^'' "* °°^ ^h^ sin0j^

Now, if E denotes the % chord of a trailing edge flap (Figs. 2

and 3) ( 1 - E) =. l l ° ü A

2

/ /

4M.-y

*- r

'\ • > .>... In terms of a,, Fig. 3

= --g (1 -E)a^

(36)

O-T, 3L t b , etc., are calculated in Appendix III.

3.4. Constaruction of the__^]^ajl3jn^J\jnc_tiqns

Following Multhopp, vre construct two chordwise loading distributions: one giving lift and no moment; the other a moment but no lift. This is done in order that the spanvdse distribution of C- and G.. -vidll be a direct result of the solution of the equations,

The functions vre choose eire the cot ^ term, and the logarithmic function. Since this logarithmic loading has a lifting contribution, •we ad.d a negative cot ^ contribution to get a zero lift representation with the same C., value,

Considering a loading of the general form

u oo

•& (0) = a cot -r + S a sin no

^ ' o 2 J n ^

1 then Gj. will always be zero if

^ =1 K*r) = 0

i,e, a = - 1 = — ~

1 3

-Therefore, the new functional representation chosen is

V*) = --2 ( - * ! - s r r . ^

s m s i n

K

\ + -2. 2 9i

0

(38)

This loading r e q u i r e s the calcxiLation of a new influence function denoted by "k" (ilefined on pages 1 5 £>nd1 6 ) ,

The other loading function i s i d e n t i c a l with Multliopp's,

tt = A c o t i

o o 2

°L =

•nA,

A is quite general. They theory can be applied to flapped wings at incidence, (incidence being measured as 'T" ) . Wo can use the influence function "i", as tabulated by Mtilthopp for this loading,

Thios, we have replaced the continuous loading function by a linear combination of the form:

2 Or ("^ ) h G , X Y )

«(.„, y,) = - V ^ . « (x„) . ^ f e )

\

(^) (¥»

"ViAicre (,' ^ t,' are yuxoly functional representations,

3»5. Choice^ of Pivota.1 Points

We arc taking two independent load distributions, and therefore need t\7o pivotal points at each chordwise station, Multhopp selected the position of these points on the basis of using the first two terms of the Fourier expansion to give accurate estimations of the values of CL and 0^^, and the same positions have been chosen here; namely,

x' = x ^ ^ ^ + 0.9045c , ^; = - ^

•" "" \ . E , ^ 0.3455c , 0^' = f 2w

where c is the local chord. It should be noted that they are well away from the hinge position (Pig, 4) .

PIVOTAL POINTS

2 0 /o SPLIT FLAP

TAPER 2 . 1

F I G . 4 . WING OF EXAMPLE CALCULATION N A G A T.N. 1759

SCALE : l " = 2 0 "

1 5

-3.6, The influence functions

The chordTdse integration is carried out through influence

functions of the form ;

,T.E,

e(0) 1 +

L,E,

X - X

(x - x^)''+ (y « y^)'

dx

Using the co-ordinates

X - X X = —

oL.E.

_{Y = ° (f> = oo3"Hl - 2 X )}

Y - Y

c ' o'

- 1 ,

this becomes c 2 IT 2X - 1 + cos 0 ^ ^ ) 1 + ° ^ (ZK - 1 + cos 55)S 2,2^ sin 0, d(f)

The influence function i(X,Y) is defined as

fir

1

_o

_{J. 6 C A}

_{2 X - 1 + cos}

_i>

0, c / 2 • % °°* 2 ^

-^

r-'^.^^^^-^^-^^^—r-^ o <- J (2X - 1 + cos9i)'+ 4ïr-'^.^^^^-^^-^^^—r-^

s i n

<p , d <l)

|(2X - 1 + cos 0 ) % lif-di> (41)

d ^

(2X - 1 + cos <!>)'+ 4 X '

Multhopp suggests that computation cf this integral be done by graphical means, and he gives tabulated results.

Our new influence function k is defined as

k(X,Y,^) = ^ c a

₁

c .^ 4

cot

t

1

2 sin 56,

In

s m

h

s m •=-^

^j^ + 0 « 0 1 + 2X - 1 + cos

(2X - 1 + cos i>y+ iiY^

TT

cot I

_ J

sin 0,

. In

sin

^h-^^

. ^ - ^

sin

r -I -^J:; -1,-+__QP-S .^ | j,

J (2X - 1 + cos0)^+ 2,Y^

This is reduced to the form (Seo Appendix II)

sin i> d<t)

(^)

i(X,Y) 1

-where

TTsin 0

s m

h

2X - 1 + cos

(2X - 1 +

aos^pf^- Uf'

(43)

Jd5i

(W

The value of K would seem to be best eveiluated by graphical means.

The logarithmic function has a singularity at the hinge position, but

graphical integration is possible (see Figs. 11 - 1 6 ) ,

It is important to notice that

^,

adds a nev/ parameter to the

calciilation of the influence function, and it cannot vary in its

chord-wise position unless a complete set of k functions are calculated,

The value o± 8 has hovrever been taken outside the calculation of the

influence function, and lies explicitly in the values of C_ , G^ at

each station. An induced flap ejigle S is presumed to exist to allow

for the spanwise variation of these quantities,

3.7. The calculation of k

From k =

2

1 - E

K

i(XY) - 1 - — - ^ - r

^

^ TT

Sin 9

h

i(X,Y) can be found from tabulated results in Ref, 1 ; K can be evaluated

by graphical integration,

K =

IT

In

. ^ h ^ ^

s m ^ '

sm ——g—

r 2X - 1 + cos (!) ~]

^ |(2X - 1 + cos ^'Y-^ l^ -^

sin0 d^ (45)

-17

-The braxsketted term (or the induction term) is independent of 0, and can be calculated for values of X and Y. The logarithiiu-c term must be calculated for a particular value of ^p. (see Table l ) . This

expression can be sin5)lified for purposes of calculation to the form

In

A cos •* + B sin -^ A cos ^ - B sin ^

Tfjhere

^ h ^ h

A = sin ~ and B = cos —r

The product is plotted against 0 as shown typicallj' in Pigs, 5 - 10 to give an idea of the rarir:;. of values and then integrated by a planimeter, A check of grapnjcal results with exact integration reveals an accuracy

of the order of 1vo,

The calculation of k^^ requires special treatment. In this case y = 0, and the bracketted term becomes either 2 or 0.

A + 2X - 1 + cos ^ i (2X - 1 + cos^) = 2 X < X o = 0 X > X o

If we associate ^ with X, i.e. cos ?!i = 1 - 2X then tlie term has the value 2 for ^ < 9i 1 ,'. k(X,0) = ~ T r ' - ^ N /(cot-^ - - r ~ - , In ^ ' ' ' 7 r ( 1 - E ; j \ ^ 2 sind ^1 sin 0j^+0

K ^

s m

'^h

s±n<l> d(f)

and this, using Appendix I reduces to

^ 1 " " ^ /COS0, - cos (f>\ sin 0. - ( "- • •.• • •;- •-—— ) In 1 V s m 0j^ Bin s i n 2 2

(46)

Thus k may be calculated analytically if the positions of the pivotal points ^ , and hinge position d are known.

3 . 8 . The spanwise inte_gra;^qri

With the d e f i n i t i o n s of our chordv'dse l o a d i n g s , equation (9) becomes

V2

, , 1 [ ^ ° ( ^ o ) ' i ( ^ . y o ^ + CLT <-C)' k ( 2 i y . y A )

"" J^/2 (y - yo^'

(47)

Introducing non-dimensional terms t o proceed T/itli the i n t e g r a t i o n t e c h n i q u e ,

r °L-°

^ ~ bu " 2b » Y Then (47) becomes .1

%'°

. =

"Ir

X \ _•/ ,- \ f

J

(h3)

«(^'^) = - 2 ^ i ^ ^ . - _ ^ ^ ^ _ - ^ - _ » ^ . ^ . . , = „ ^ ™ . ar,^ (49)

with the techniques described in Section 5 of Ref. 1, Multhopp reduces the solution of this integral equation to the solution of a set of simultaneo\is linear equations of the form,

m-1 2 a = b

V vv

\lv \ -"^.v^^- ^,\n^\n^n'-Kn''n^

^^^^

I.. J / m - 1 \ - ( ^ )

whei^ Ï and Ic are values corrected for the logarithmic singularity in the expansion of tlae influence functions with respect to Y. These equations may be transformed into equations for y and /i and the solution done by iteration as originally done in Multhopp' s paper.

Generally, the number of points vdll be fifteen or more for good accuracy, so that discontinuities in flap in this direction can be handled more easily. Tliis discontinuity vdll appear in a', the rearward pivotal station slope, and this value can be faired by the method

suggested by Multhopp for interpolation purposes. However, if the

discontinui-ty is inboard, an excessive nijmber of points may be necessary. The loading spanwise may be symmetrical or antisymmetrical, so that flaps or aileron,cases can be calculated:- there is a slight difference in the solutions,

- 19

3.9. Correction for the lop;arithmic singularity

Multhopp has pointed out that in expanding the inflijence functions for small values of Y, an additional term should be added due to the logartihraic singulari-ty arising from the expansion of the induction term, (See Appendix 13» Ref. l ) . He used distributions of vorticity which are regular across the chord, and included an additional term as a correction to the d^^ term. Mangier and Spencer^ have given a better form of this correction as

^\v

=

^ \

\

p.(e)

"vrtxere F ( e ) i s g i v e n f o r a number of spanwise p i v o t a l s t a t i o n s ,

I n t h i s c a s e , we have a l o a d i n g d i s t r i b u t i o n which i s l o g a r i t h m i c , and h a s an i s o l a t e d s i n g u l a r i t y a t the p o i n t 0 = ^ ; On s i n s m 2 E 1 2 s i n n 9^, TTn s i n n 0 T h i s f u n c t i o n c a n be r e p r e s e n t e d b y a F o \ r r i e r s e r i e s , and i s q u i t e r e g u l a r e x c e p t a t t h e p o i n t ^ = 0, , I t i s r e a s o n a b l e t h e r e f o r e t h a t we may p r o c e e d a s Multhopp h a s d o n e , u n l e s s one of t h e p i v o t a l s t a t i o n s happens t o c o i n c i d e w i t h the f l a p h i n g e p o i n t . The p i v o t a l p o i n t i s p l a c e d on t h e f l a p t o a v o i d t h i s c i r c u m s t a n c e , Then, ws can e v a l u a t e t h e k t e r m

d e (x)

"dfx

1 ^

J

sin?!» d I

_iO

72

B u t Oj, = - | ( 1 - E ) a , o o t | -• -• ^ , 7r(1 - E; s m ^

wTTTÊTsïn^

j o a e ^ * cosec s i n ^ _h 2 sin? On s i n •^.

••i.

c o s ^, - 0030 cos d - COSI0 See Appendix 1.4 9i = 0,

4. Example Galcula.tion 4.1. Method

To ensure that there axe no unexpected difficulties in using a logarithmic function for chordiTise loading, a simple calculation has been attempted for a single planform for which experimental results axe available , The wing selected is rectangular, i:ins%vept, with full span, split flaps of 2C^ wing chord (see Fig, 4 ) .

TvTO chordwise points, and seven stations across the span have been used. Since the flaps are full span, we can expect no appreciable irregular spanwise distributions due to the small number of stations,

The calculation of k is made for 20^ chord, or 0, = 126,9 ,

First, the values of X, Y are calculated and tabxolated in Tables 3 and 4. For eexch pivotal point, the value of K is obtained by graphical

integration which is illustrated in this report by a typical set of graphs drawn in Figs, 5 - 1 0 based on calculations such cm TaiblG 2

and tabulated in Table 8. The values of i(X,Y) are given in Table 6 from Ref, 1, and the forms of calculation follow that suggested there,

The values of k!(x', O ) , andlc'(x", O) may be evaluated (see Table 5 ) . 'COS 126.9 - cos ^ .0

^(^'°) = ^rrfTEj

s i n - ^ „ i — -^-.^ ^^-^ ^ ••. •^. . Jin. ^ ^ eia 126.9° sin 8.5 = .286 Similarly, S::(X",0) = 2,88

The values of k , k'. can also be calciiLated for use in the correction

3 ' 3

to k for the logarithmic singularily.

The original thesis contained a complete set of tables and figures for the whole range of values of k, samples onl;/ being reproduced here,

21

-k =

_{7r(l - E) s i n 0} Co see

i -

_{c o s ^i, — «»os9i}

4>

^<l>.

k^

3

A.

,87r s m • ? - _{s i n} 2ïr - , 6 - C03

'hs

= -22.7

<

A_

2'W .87r s i n •?-5 - , 6 - c o s 27r

=

8.50

Thus t h e v a l u e s of 2^^^ can be e s t i m a t e d . The s o - c a l l e d c o r r e c t i o n t e r m s a r e r e a l l y a d d i t i o n a l t e r m s , and the v a l u e of Ak o b t a i n e d h e r e i s f i v e t i m e s t h e v a l u e of fc . The c o r r e c t i o n s a r o made i n T a b l e 7 .

Form I and I I ( T a b l e s 10 - 12) d e a l v d t h t h e c a l c u l a t i o n of t h e

values of B

vn' vn' D and E given in Tables 13 and 14. The problem vn vn then reduces to the iterative solution of equations of the form

y = a 'v vv

= a

vv m-1

2

(A' a' - I" a") + V

^ V V V v' 't m-1 (m(', af_ - n/. a';) + |

B

_vn V V V V'

D

vn ^n^

y +

'n

m-i

2

I C

0

m-1

2

vn y E

n

n

A shortened form of this solution is given for a'= 2, a"= 1 in Tables 16 and 17» aiid summarised and checked in Table 15. For a'= 1 , a"- 0, the solution is given in Tables 19 axid 20 and summarised and checked in Table 18.

The actual solution of these equations is fairly slnrple in this

case compared to the amount of work involved in caclulating the coefficients B , etc. Thus, many calcxilations can be more readily made once the

coefficients are evalixated for a given wing. 4.2. Results

The r e s u l t s of the c a l c u l a t i o n a r e shown i n P i g s . 12 t o 16 and T a b l e s 21 - 2 3 .

The spanvdse d i s t r i b u t i o n s of G! a r e shown i n F i g . 13 f o r z e r o i n c i d e n c e and cc = 6 , 7 . T h i s l a t t e r r e s u l t h a s b e e n o b t a i n e d b y l i n e a r l y i n t e r p o l a t i n g t h e d i f f e r e n c e i n t h e z e r o and u n i t i n c i d e n c e c a l c u l a t i o n s . S i m i l a r l y 80 i s e s t i m a t e d a s 3 . 8 9 / r a d i a n .

The distribution for equal integrated C^ and C^ values is shown in Pig. 14. A similar model comparison is made in Fig. 12 v/here the centreline ordinate has been taken as imity. Experimental results are conipared vdth a. lifting line solution by Multhopp* s method, and those calculated by the theorj'' of this paper,

The spanwise distribution of C^^ is shown in Pig. 15 and there are no connparable experimental data,

The centres of pressure are plotted in Pig. 16, An experimental check is not possible, and the comparison is made vdth thin airfoil theory estimations,

4*3. Discussion of results

The resiolts of the calculations do not agree qtmntitatively vdth the results of Ref, 6. This woiiLd be expected since the effects of

viscosity are neglected, and a split flap v/as vsed in that investigation. Gompaxing the zero incidence, flap deflected, results, vre see that

experimental results are approximately 60^ of the theoretical estimations. It is v/ell knovni that results from tv70-dimensional thin airfoil theory for flapped controls is generally 20^ too high.

Making the comparison at 6.7 » or an experimental 0^. of 1,60, •there is again a considerable difference, but that it is almost the

same as at zero incidence. This suggests that the experiinental difference is due mainly to viscous effects on the flap. Thus, it would seem that a bettor estimation could be made if flap effectiveness could be

aooounted for by empirical or experimental means. The information input to the equations could be modified by using an effective flap angle r6 instead of 6, so that

a'' = a" + T 6

The distribution of lift across the span agrees moic closely

vdth lifting line theory than the experimental resiolts as seen in Pig. 12. However, it must be remembered that rather few spanvdse points havo been

taken in both calculations, and that the experimental values have not been corrected for vd.nd tunnel interference.

The distribution of moment shov/s a change of shape for the two incidences, but not much change in overall magnitude. If the lift

and moment distributions v/ere independent vre would expect that the shape vrould be the same in both calculations, but it is obvious from the

equations "v

=

^ n ^n %v a vv m-1 m-1 2 2

H -I '•K^v)^l ^n ^n ^ I

m-1 2

(K'K^^-%vK^ + z D^^

E Ai vn n m-1 2

^''^^o

23

-where a" = a, a' = a + 6

that tho distribution of moment depends on the incidence a"and the lift distribution y . The main contribution to u comes from the

'^n n ayy(my ^i'^ - m^a') term, and so vre see the variation in A^ is mainly

dependent on the valioe od m!' - m' which modifies tlie a value. For a variation of a of one radian, the change of shape is apparent, but the

orcrall magnitude has not changed very much. This variation in CL^ vdth ir.'jidence is due to the fact that the aerodynamic centre no longer

coincides with the quarter chord point v/hen the flap is deflected. The theoretical centre of pressure shovre fair agreement vdth tvro-dimensional values, and some slight variation from a straight line for the zero incidence case,

5. ^General^Dj^sjsussioji .of^hBca^

i

To satisfy the integral equation at a limited number of pivotal stations would only be justifiable if the boundary conditions, i,e,

x,y), Eire fairly continuous, To overcone this, YJB have chosen an

xy) distribution vAich is discontinuous at the hinge point, and constant over each portion of chord, i,e, a flat flapped plate. The fijnctional representation of this has been obtained in closed form so that all terms of the thin airfoil series expansion are included, but only two pivotal statiens axe required, one on each portion of the vdng,

Since the points chosen by Multhopp vrould be used presumably for a

vdng vdthout flaps, those have been retained to reduce extra calculations, Generally, these points will satisfy this condition for trailing edge flaps since the rearv/ard point lies at 9C^ chord. Other choices may be necessary for leading edge flaps,

The method describes the flow over vdngs of lev/' or moderate aspect ratio, and any given planform. The effects of leading edge, trailing edge flaps, or ailerons on lev/ aspect ratio straight vdngs, sv/ept v/ings, or delta wings, may be indicated. Part span controls could also be

treated if enough spanvdse points -vTcre taken to describe the irregularities in the spanvdse distributions,

In estimating the effects of these controls on actual ivings, the effect of viscosity vrould be rather marked, and limits the usefulness of the calcvilation. The quantities 3Gj^ , 9C wotild be tlie only

valuable result expected from the theory, unless experimental evidence can suggest an empirical modification to the flap angle used. The trend towards laminarised flew vdngs with simple controls may reduce the

magnitude of this error in future. Flap blov^ing v/ovild. also reduce the size of this error,

There is a cotcpling betvreen the distributions obtained from the equations, v/hich suggests that the loadings are not independent. A cambered, flapped vdng may therefore require a calculation involving three pivotal points. Further theoretical calculations vdth available experimental checks should be done to investigate tliis matter, and to verify the use of an effective flap angle,

6, Gonclusiqns

By representing the flapped flat plate distribution of thin airfoil theory in closed form, Multhopp*s method has been extended to account for a discontinud.ty in siJrface slope in the chordwise direction. This permits effects of leading edge flaps, trailing edge flaps, and

etilerons on wings of any planform to be calculated by vrhat is considered to be a more accurate method than that suggested by Multhopp,

However, for quantitative agreement, empirical or experimental data must be used to allow for the effects of viscosity in the real case. In viev/ of the large amount of necessary calculation, this fact Ti^akens the argument for using such a calculation, Tlie effects of flap

or aileron deflection on different planforms will be indicated in a

qualitative fashion, and the values of dG^ , 9G may be quite representative,

L i/l 3a 90^

The computing e f f o r t involved would be reduced i f the influence function k vrere t a b u l a t e d . Unfortunately, i t s dependisnce on the parameter 0, , the p o s i t i o n of the v/ing l i n e of the f l a p , renders t h i s i m p r a c t i c a l , u n l e s s i n t e r p o l a t i o n i s p o s s i b l e , or r e p r e s e n t a t i v e values u s e d . The metliod o u t l i n e d i n t h i s paper can be u s e f u l i n tlie e v a l u a t i o n of the p r o p e r t i e s of f l a p s ,

7» Acknowledgements

The author ackno\7ledges vdth g r a t i t u d e the a s s i s t a n c e of Mr, W, R a i n b i r d , who supervised the vrork, and of l.'tr, J . L, Naylor vdio e d i t e d the paper for p u b l i c a t i o n ,

25 -7. References 1 . Multhopp, H, 2» Mangier, K.W,, Spencer, B.F.R. 3. G-amer, H.C., 4* G-amer, H.C, 5. Mangier, K.W, 6, West, F.E. Jr., Hallissy, J,M. Jr, 7. Palkner, V.M., Watson, E.J. 8. Vandrey, F, 9. G-lauert, H. 10, Perring, W.G.A.

Methods for Calcvilating the Lift Distribution of Wings (Subsonic Lifting Surface Theory),

R & M 2884

Some Remarks on Multhopp' s Subsonic Lifting Surface Theory,

R.A.E. TN. Aero 2181.

Multhopp's Subsonic Lifting Surface Theory of Wings in Slot Pitching Oscillations.

R & M 2885.

Swept Wing Loading: A Critical Coniparison of Pour Subsonic Vortex Sheet Theories.

A.R.C. 14138 C,P,102,

Improper Integrals in Tlxeoretical Aerodynamics,

A R.C. 14394 G.P,94.

Effects on Compressibility on Normal

Force Pressiore, and Load C h a r a c t e r i s t i c s

of a Tapered Wing of NAGA 66 - s e r i e s

A i r f o i l Sections witli S p l i t F l a p s ,

NACA TN,1759

Tables of Multhopp and other Functions

for i:ise i n L i f t i n g Line and L i f t i n g

Plane Theory.

R& M 2593.

Graphical Solution of Multhopp's

Equations for L i f t D i s t r i b u t i o n of

Wings,

A.R.C, 12(238 C.P.96,

T h e o r e t i c a l Relationships for an A i r f o i l

vdth a multiply Hinged Flap System,

R & M 1095.

The Theoretical Relationships for an

A i r f o i l vdth a Multiply Hinged Flap

System.

R e f e r e n c e s ( C o n t i n u e d ) 1 1 , B r e b n e r , ^.G-,, L e m a i r e , D,A, 1 2 . B i s p l i n g h o f f , A s h l e y , Halfman,

The c a l c u l a t i o n of t h e Spanwise Loading of Svreptback Wings,

R , A . E . R e p o r t Aero 2 5 5 3 . A e r o e l a s t i c i - t y .

Addison-Wesley P u b l i s h i n g Co. 1 9 5 5 .

13.

Pope, A. Wing and Airfoil Theory, MoGraw Hill, 1951.

14. Durand, W.P, Aerodynamic Theory, Vol, II.

15. Palkner, V.M.

16, Curtis, A.R.

17. SSlingen, H,

The Use of Equivalent Slopes in Vortex Lattice Theory,

R & M 2293

Tables of Multhopp* s Influence Functions. NPL Math, Div, Report Ma/21/(5)5 May,1952, Die Losungen der Integral gleichijng und deren Anwedung in der Traflugel theorie, Math, Bond 45 - 1939.

27

-APPENDIX I

(1) The Evaluation of the Integral

•TT

1 - COS

A COSÖ - c o s ^ , dQ

= ƒ

d a / , „ / COS0 . dQ

cosG - cosci J, J, cos© - o o s p

= - (TT - ^ ) + ( l - COS 9!) ) »r

d e

COSÖ - o o s ^ TT

de

The i n t e g r a l / —' "Q " ~ ' ' 'X i s s i m i l a r t o Glauert* s i n t e g r a l , and ^ J, COSÖ - OOS9

Tl

is evaluated in the following manner,

1

Noting that

, can be put in the form

1

2 sin^i

cot

(_LiJL

and that

^±ï±) ^

<3(1^^

2 " 2

dQ ^ 1

cos Q - eos 0 sin <

— cot

dQ_

2

2

In

sin

sin

1+i

2

_{+ c}

Since the denominator passes through zero when ö = ^,> ws integrate

from Ö, to

4> -^ t

and ^ + e to ir, and take the limit as e ^ 0.

Now 9 i - e dO <p, cos 6 - c o s <p s i n <p I n ( s i n — ^ - I n s m —^-^ ; 1 s i n 0 I n s i n ( ^~ ) - I n s m - ^

• ( r l

- I n + I n s i n 2 s i n ir d 6 ^ 1 and j c o s 6 - cos 95 sin9!i

(p+e Adding t h e s e i n t e g r a l s , vre g e t I n s m -e

(•hLt)

- I n s i n 9!) + — 1 s i n 0 I n

s3£Jè„-l/24 I + In

SÜ1 T^"+ e/2) As e -. 0 , t h i s t e n d s t o sxn ( - — 2 ~ ) s m ( — " 2 ° ^ ^ (2) s i n 0 I n s m ( — I " ) s m ( g ) TT

de

c o s 6 - cos 0 s i n <P I n s i n ( ' ^ -) s m ( — - " ^ J / c o t "^ s i n 95d9!i= / ( 1 + 00s!^ . d(p = TT . 9 ^ e (3) / °°"t 2 ( ° ° s ^ h ~ cos^i), sin^i , d(p TT IT

( COS 'Py^ - c o s ^i) d ?!> + j (COS ï^ t^ - COS ^ ) cos <P d<t>

'h

= COS A ( T T - Ó ) + s i n 0, + / ( c o ^ - 0 0 ^ ) d ( s i n ^

IT

'h . _{s i n 2 ^ h}

29

-(4) D i f f e r e n t i a l of In

s i n ( g )

s m ( 2 J

s m ( 2 )

s l n ( ~ Y - )

s m • • • — cos

-g + s m - - ^ — oos 2 s m

2 ^ h - ^

d9i

v^iich, on reduction, equals

s i n

A

cos 0 - cos 10,

d^i

(5) Evaluation of Integral

s i n 0 d0

d( cos 0)

= - In

. ^ - ^ s i n . ^ - * s m

cos 0

0.

,'' oos 0 s i n 0

cos 0 - cos é

d 0

This second i n t e g r a l i s Glauert' s i n t e g r a l i f 0 = w,

For general case

1/ cos 0

1 +

cos

- _ ^ . . ^ ] d 0 = 0^ + cos 0,

0 - COS0 y 1 '^h

d 0

As i n S e c t i o n 1 t h i s i n t e g r a l can be e v a l u a t e d a s s i n 0 . I n . ^h + ^ s m — — ^ ' s m - — g — I n s m s m 0 j ^ + 0 h

r?

s i n 0 d 0 = 0 s i n 0. + ( c o s 0. - cos 0^) I n 0 m

01,+0

s m " Y " T?hen 0 = i r , t h i s r e d u c e s t o ir s i n 0, ( 6 ) ir I n 5^ s i n

0 ^ . 0

s m 0 j , - 0 ( c o s 0 , - COS0) s i n 0 d 0 IT ( c o s 0, - COS0) d 0 s i n ^1^ + ( c o s 0 - c o s ^ I n s m s m

0^^

I n t e g r a t i n g t h i s b y p a r t s ( c o s 0, - COS0) 0 s i n 0 , + ( c o s 0, - cos 0) I n s m -^ • ^ s i n

0 ^ - 0

IT IT 0 s i n 0, + ( c o s 0. - c o s 0 ) l n sm=" \ * ^ s m \ - ^

4

s i n 0 d 0

31

-(cos 0, - COS 0) s i n 4>^ 0 + ( c o s 0^^ - cos 0 ) ^ l n

. \ *^ ' "

s m

s m

—-;r-Ü0.

ir ir

-/ 0 sin 0, sin0d0 -/ (cos 0^^ - cos 0)ln

A

<!>

Bin

sin

2

_{sin0 d0} ir . • . 2 / (cos 0j^ - cos 0) In

^h

sin

* h * *

sin

^^^ sin0 d0 = (cos 0j^ - GOS 0) In

sin

01,+0

sin

0j,-0 ir

- sin0 sin 0i^ + ^ sin 0, cos 0,

When evaluated over the range 0. - TT a question of convergence of the logarithmic term arises at 0, , The logarithmic term vanishes at 0 = TT, and also 0= 0, . ir (cos 0j^ - cos 0) In

sin

sin

i^*t

o^-i

sin0 a0

vr- 0; sin'' 0j^

AEÏENDIX I I The E v a l u a t i o n of k Prom e q u a t i o n (42)

k =

ir

T*^)

ir c o t ^ - —r-2 s i n I n h s m —"jf— s m

\ - ^

A 2X-1 + oos 0 ^ , ^ K2X-1+cos0)+4r'^- d0 2 TT ( T - ^ j ƒ c o t | s m 0 f l + . S L = L A . t . ^ q s 4 . _ l i n ^ a 0 o I (2X-1 + oos 07+2^^ TT I n s m s m 2^ A + 2^ - ,1 + c o s _i_ |(2X-1 + oos 0 ) % 4 r ' s i n 0 ,2 \ d 0 T ^ ^ ^ ^ ^ " 7;-(l - ^ E ) s i n 0. ^ ^ ^ ^ ^ ) ^'^^ Appendix I 7r(1 - E ) s i n 0, s i n 0 Z K - 1 + c o s 0 2 . . . 2 I(2K-1 + c o s 0 ) +4r d 0 .1 - E) ^^-^^ *" 1 -E " 7T<1-E)sin 0. Yiihere k ir I n s i n 0 h + ^ s m 0^_-0 s i n 0 2X - 1 _+ c o s ^ ^ _ _ _

|I;2X-1 + cos 0)% id

d0 k i s d e t e r m i n e d g r a p h i c a l l y .

33

-A i r a g ) K ^ i i

Two-dimensional Properties of Flapped P l a t e s

I t i s interesting to note that the logarithmic distribution has

a G- contribution,

n r, -

L = 2 0 s m 0,

C^c - - ^ h

2 •

\°

1 l**

This implies that the centre of pressure is

"^^^

behind the quarter

chord point, or

X =: ,25 + •!]•'• • for camber line distribution, (Pig. 3 ) .

cp

d

•i-c

1*'.w .-i»'^

Mk

Prom this vre see that the logarithmic distribution hsLS its centre of area,

or centre of pressure for\7ard of the hinge line. Also, the centre of

pressure for the v/hole flapped plate distribution may be estimated for

zero geometric incidence,

\

o.p.

i ^ H

-1 ^-1 (^ " ^ )

4 "• 4r^~rr^)

= T +

^ / 2 sin 0j^ (1 - E) X

~ 4 \^ "^

T^Ty^ + s±n<p^ J

Note: The centre of pressure of flapped a e r o f o i l does not change

vdth 5 , b u t i s a function of 0, , I t i s alv/ays behind the qxaarter

chord p o i n t , and ahead of mid-chord for t r a i l i n g edge f l a p s ,

The hinge moment, H, can be c a l c u l a t e d , using the logarithmic

function,

'>a • V 'V '. . f I I'll, g g

k

1 ^ I

L- 4.- •'•^^ n p' V*(2b)2 E

where P(x) is the pressure normal to the chord at the point x,

,', Q^ = ^ J I p(x) (x-5j^)d5 = - ^ e.(0)(cOS0^ - COS0)

\ in0 d0

This integral is evaluated in Appendix I to give

ir-• " ^ sin^ 0,

°H =

^

V " T " ^ ^ ^ °°^ % + — 2 ^

1

Ê

^o^2+ T2S

35

-Including the hinge moment due to the induced «ot •§ distribution

Hi

»7r

= "g ( ^ ^ - ^ ) J c o t I (cos 0j^ - COS0) s i n 0 d0

"vidiich from Appendix I becomes

E ^ ir ^ ( ïT 0 ) ( c o s 0. i ) + s i n 0 .

-s i n 2 0,

The total C„ is given by n

H

6,

TTE

Sljf 0^

( TT- 0j^)' («OS 0i, - -1) + - — ^ + (^ -0i,)s3n 0^

The usual derivatives may then be evaluated

^^H _^^ W ~ W E . 2 sin" 0, ("• - 9^1^)^(008 01^ - •§•) + • — ^ — + (7r-^j^)sin 0j^ m = ₃₆

^°M

sin 0, - ^ (1 - cos 0j^) 90,

= a r

= 2(- - 01, .

4 sln0 h ir

E:3q)erimental values of these derivatives are roughly 20^ lower than those calculated from the above formulae,

^ 1 ' ^ > . \ o < - Fi X O . 0 ^ -\ \ \ ^ ^ \

J

ha • \ \ o o X , - 0 iC ^ - ^ — - " " ^ (i X 0 a ^ - ' ^ " \ v ^ ^ " \ \ / / \

1

z o

s

— K \, -' ^ -•BS X ^ , = 1-255 ^

y

\ ^^ 3 ' ' ' ' ^

y

1 - ' ' ^ \ c \

V

2 0 4 0 I 2 0 ^ h ' ' ^ ^ ' ^ ° f DECREES

FIG. 9. METHOD OF GRAPHICAL INTEGRATION

K Y ^ , - " » 5 X ^ | = - 2 1 2 / • ^ Q /

V

11

' \ •) \ \

X

2 0 4 0 6 0 8 0 lOO I 2 0 9 K ' * ° ' * ° ^ DECREES

FIG. lO. METHOD OF GRAPHICAL INTEGRATION.

en sin sin 2 3 sir • ^< * , =i 126 • 9 ' - ^

A

y

2TT / ^ -/ / / ƒ

1

0 \

v

\ \ 4 T r 5

V

1 2 0 ^Y^ I 4 0 1^ DECREES

F I G . n . MF.THOD OF GRAPHICAL INTEGRATION.

4 STATION

FIG. 13. SPANWISE VARIATION OF L I F T ,

THEORETICAL

Y sniN

FIG. 14. SRftNWISE LIFT DISTRIBUTION FOR EQUAL WING Ci_.

4 STATION

F I G . 15. SPANWISE MOMENT VARIATION.

PRESENT THEORY 2 - D ESTIMATE PRESENT THEORY 2 - 0 ESTIMATE UNIT INCIDENCE ZERO INCIDENCE

TABLE .1 <!> 20 40 60 80 100 110 120 125 130 1.1+0 128 150 160 170 <t>/2 10 20 30 40 50 55 60 6 2 . 5 65 70 64 75 80 85 I n A s i n A s i n s i n 9/2 .1736 .3it20 ,500 .&i28 .7660 .8190 .8660 .8870 .905 .9397 .900 .966 ,9848 ,9962

1 ^ B

1

-cos 9/2 .9848 .9397 .8660 .7660 .6it28 .5730 .500 .462 .422 .342 .439 .259 .1736 .0872 cos "I COS 1 s i n <t> A s i n 0 / 2 B c o s # / 2 .07c .152 . 2 2 ' ,881 .8J^0 .775 .287 .685 . 3 4 : .363 .33é .396 ..'+0; .415 .401 .431 .43S .44f ! .575 .513 .447 .413 .378 .306 .393 .232 .156 .073 2um .959 .992 .998 ,972 ,917 .878 .833 ,809 .781 .72Z.. .794 .663 .595 .523 ^h A Diff .003 .688 .552 .398 .237 M-fi ,061 .017 .025 .112 ,008 ,199 ,283 .367 = 130 -= oos 6 3 . 5 = s i n 6 3 . 5 Quot, 1,19 ^.ku 1,80 2 . 4 4 3,86 5 . 9 4 1 3 , 6 5 4 7 . 5 0 . 31.20 6.45 9 9 , 5 3.33 2 . 1 0 1.43 53.1 = 126.9 = .41|i> = .895 Itlg iQUDt 1 .1739 .3ó.'i.6 .5377 .8919 1.3506 1.7817 2 . 6 1 4 3.860 3 . 4 4 1.3640 4 . 6 0 0 1.2029 .7419 .3577 s i n 0 .3^*20 .6428 .8660 .9848 .984B .9400 .8660 .8190 .7660 .6428 .788 .500 .3420 ,1736 l n | - | 3 ^ j ,0594 .2340 .509 .876 1.33 1.670 2,260 3,160 2,640 1 ,20 3.62 .601 .253 .062 l n 0 TA3LH 2 i» = 2 , n = 1 Y = ,906 X ' = 1,015 't-20 40 60 80 100 110 120 130 140 1.'42 145 150 125 160 170 siXKJ) .342 .643 .866 .985 . 9 8 5 .866 .766 .62^3 ,616 .574 .500 .3^*2 .174 c o s ^ .91^ .766 .500 .174 - . 1 7 4 -.Jtó - . 5 0 0 - . 6 J ^ 3 - . 7 6 6 - . 7 3 8 - . 8 1 9 - . 8 6 6 -.573 - . 9 4 0 - . 9 0 5 I n | | s i j i 0 .02^03 .156 .337 .564 .828 1.14 1.38 2 . 0 2 2 . 5 3 2 . 0 3 1.13 .41 .094 ( 1 ) 2X+oos^ 2 . 0 3 2 , 9 7 0 2,796 2 , 5 3 0 2 . 2 0 4 1.856 1.69 1.530 1.387 1,264 1.242 1.211 1.164 1.46 1.090 1.045 .-12). _ . ( 1 ) - 1 1.97 1.796 1.53 1.204 0 , 8 5 6 .(^9 .530 .387 .264 .?/|2 ,211 .164 .'J> .090 .045 (i)

i^r

3 . 8 9 3.22 2 . 3 4 1.45 ,73 .476 .28 .15 .07 .053 .045 ,027 .212 .008 ,002

_Lü

( 3 ) + 4 r " 3.28 7 . 1 7 6.50 5 . 6 2 4 . 7 3 4.01 3.76 3.56 3.43 3.35 3 . 3 4 3.33 3.31 3.49 3.29 3.28 . . . . C 5 l _ 2 . 6 3 2.55 2 . 3 7 2.17 2 . 0 0 1.94 1.89 1.85 1.83 1.63 1.82 1.82 1.87 1.81 1.81 _Xé) . ..

^'V(5)

.735 .702 .645 .555 .428 .356 .280 .209 .I.W .132 .116 .090 .246 .050 .025 (6)1^11 .0594 .234 .509 .876 1.27 1.67 2 . 2 6 2 . 6 4 1,20 .601 3.16 .253 .062 s i n (f) .0296 .109 .217 .313 .354 .319 .289 .291 .33Jf .236 .106 .020 .002(5) (6)In II sin .0436 .165 .323 .486 .569 .595 .633 .552 .173 .054 .778 .0125 .0015

/i'A. 1 2 3 4 n = 1 2 3 4 ":. .9239 .7071 .3327 0 1 5 . 9 4 5 . 4 4 . 5 8 3.65

° .

1 7 . 2 2 0 , 7 25.9 32 2 7 . 6 9 7 . 1 5 6 . 3 3 5 . 4 ^ . 1?3 2 1 0 1 ^ 2 ^ ^ -0 3 10,29 9.75 8 . 9 3 8 . 0 ^ . S . - 8.6 - 1 0 . 3 5 - 1 2 . 9 5 -16 4 1 3 . 3 4 12,8 11.98 11.05 OALOULATION OP X AID Y

K

-2.66 - 3 . 2 -4.02 - 4 . 9 ;

n

+ 6.95 + 8 . 8 5 +10.45 +12.95 V n = 1 2 3 4 4 . 1 8 5.hB 2 . 7 3 2.25 1 15.55 1 7 . 4 5 19.05 2 1 , 5 5 \ \ - \ \ 0 ,2168 .5412 .9239 2 17.30 19.20 20,8 2 3 . 3 >, l ^ t V 2

| . , - . J

.2168 0 .31^4 .7071 3 1 9 . 9 2 0 . 8 2 3 . 4 2 5 . 9 t - = 72"

h,-.J

.5412 .3144 0 .3827 4 22.95 2 4 . 8 5 26,45 23.95

K-\\

.9239 .7071 . 3 8 2 7 0 TABLE 4 Y . X VALUES X' STA. V 1 2 3 4 .345 .314 .267 .212 .371 .345 .306 .261 .397 .376 .345 .309 .417 .40 .373 .345 .905 1 .015 1.11 1,255 .835 .925 1.0 1.125 .767 .801 .905 1.0 .716 .775 .325 .905 \ n 1 2 3 4 n 0 1 0 .906 2.26 3.85 2 .754 0 1.095 2.46 3.79 3 1.5 .875 0 1.065 4 2.065 1..59 .861 0 .861 6.80 3.03 -1 5.69 2.07

STA u 1 2 3 4 -3 1 -.842 -.414 -.281 2 -.802 -.700 -.387 -.203 kCx») 3 -.415 -.694 -.706 TABLE

_-5

TABULATION 01? k 4 -.230 -.390 -.725 -.725 1 .870 .490 .385 2 .525 .721 .470 .227 3 .185 .381 .741 4 .0655 .195 .455 .455 -.120 - . 1 6 9 . 1 3 4 . 1 2 3 - . 1 2 0 - . 2 3 0 .0815 .0655 TABLE 6

IWPLUÏÏICS FUNCTIONS i C x / T ) i ( X , ' Y ) , I^^

A ( ^ ï ) i(gY) . STATION n 1 2 3 4 1 2 3 4 V 1 1.575 1.15 1.20 1.15 2,081 1.58 1,30 1,25 2 1.075 1.913 1.14 1.12 1.61 2.289 1.50 1,35 3 1.0 1.06 1.978 1.13 1.35 1,54 2.330 1,52 4 1.0 1.0 1.06 1,844 1.25 1.35 1,56 2.247 - 3 1.05 1.13 1.20 1.52 •^ 1 . 0 0 1 . 1 0 1 . 1 2 1 . 2 0 -1 1.05 1.15 1.10 1.25

\v ~ "^i ^2c„

LCGARTTK.tlC COïffiSOTION Ai = ijk ( | ) ' ' P ( 9 ) , e t c .

1 2 3 4 1 2 3 4 V2c^ 4 . 1 8 3.48 2.78 2 . 2 5 (W2cJ 1 7 . 5 12.1 7.71 5.01 P(6) .00125 .00542 .00958 .01130 J W c " 1 7.76 7.76 7 . 7 6 7.76 4 k ; 34 34 34 34 k'(XO) A i " _l/v .170 .508 .573 .i:-39 A k " .743 2.23 2.51 1 .92 = 2.88

Kv

1.575 1.913 1.978 1.844 E " _UI/ 3.62 5,11 5.39 4 . 8 0 k ' = k " = 1 4 k ' i 4 . 8 0 4 . 8 0 4 . 8 0 4 . 8 0 4 k ; - 9 0 , 8 - 9 0 . 8 - 9 0 . 8 - 9 0 . 8 k' 1.20 1,94 (xo) k ' = k " = 3 A i ' _vv .106 .314 .355 .272 A k ' vv - 1 , 9 8 - 5 , 9 5 - 6 , 7 0 - 5 . 1 4 = ,236 - 2 2 . 7 +8.50 ï ' vu 2,081 2,289 2,330 2,247 E ' vv - 1 , 7 0 - 5 , 6 6 - 6 . 4 1 - 4 . 8 5 TABLE 8

IIWLO!a«KS FUNCTIONS k(JfY) k(X'Y) Z^^

„ktO;} k(,X'Y) STA V 1 2 3 4 -2 .121 .418 ,180 .375 -1 ,247 .ifiO ,180 ,560 1 3.62 1.02 .413 .280 2 1.17 5.11 .845 .385 .328 3 .912 1.04 5.39 .850 4 .430 .687 1.05 4 . 8 0 1.05 1 - 1 . 7 0 .660 ,388 .242 2 .925 - 5 . 6 6 .633 .408 .262 3 .565 .870 - 6 . 4 1 0,662 4 .560 ,630 ,.598 -V.85 .598

l'ïü-iL2 <t> V A L U E S 1 2 3 4 -3 -1 .1913 .3599 0 .023 0 .0064 0 .3599 .3536 .3879 0 .03;-i4 0 ,0064 0 .3879 .4619 .3943 0 .03V4-0 .028 0 .3943 0 , 5 .3943 0 .023 TAtlLS 10 POaM I y o r n -'vv -'vv ^'vv *^ vv I ' IE" vv w vv vv I' £" - T" vv vv vv V

K

<. E' vv ±1 2.081 1.575 - 1 . 7 0 3.62 7 . 5 3 - 2 . 6 8 10,21 .354 - . 1 6 5 .154 .204 + 2 2.239 1.913 - 5 . 6 6 5.11 1 i .70 - 1 0 , 8 5 22,55 ,227 -.251 ,085 .102 t 3 2.330 1.978 - 6 . 4 1 5.39 12,55 - 1 2 , 7 0 25.25 ,213 - . 2 5 4 ,078 .092 i ... 2.247 1.341+ -V.85 4 . 8 0 1 0 , 8 0 - 8 . 9 4 1 9 . 7 4 .243 - , 2 4 6 .093 .114 •niiere

K =

E " '^vv T" £' - T' E" _{vv w vv vv} 1 vv l" k ' - T' iE" vv vv vv uv -v'v •Vv .'V •\'v'^^v

"Ml -vn a i vn vn a k " vn vn a k ' vn vn a I' i ' vn V vn a «" i ' wi y un a (<' i ' -vn V -vn a « ' y i; n V vn a * " k " vn V vn a ( t ' k ' -vn V -vn %n "I K,n t .1 a m i vn V vn % n ^ % ' ^ -V % ><ai ^•^•"I. ^'vn n ODD u *".^;:n)

*>:„)

-'. ^.„)

( 1 ) 2 . 5 7 8 . 3 8 2 .367 .237 .132 - . 0 9 6 .228 •-.0538 - . 0 9 2 .1458 .0390 .0452 - . 0 1 0 2 .0375 .0201 4 . 0 3 5 . 0 2 8 .0078 ,0068 .0085 - . 0 0 6 9 . 0 1 5 4 ,00165 - , 0 0 1 9 .0036 .00320 .00326 - . 0 0 0 0 6 ,00089 , 0 0 0 6 4 2 . 5 8 5 .i.42 ,2,.J04 .338 .132 - . 1 1 1 , 2 4 3 .0768 - . 1 0 1 ,178 ,0450 .02,95 - , 0 0 4 5 .0i|.15 .0287 ( 3 ) 4 . 6 1 5 .418 . 3 3 5 .261 .149 - . 1 0 3 .252 .0635 -,082i^ .1459 .0477 .0571 - . 0 1 0 6 .0385 .0242 EVflNl ( 2 ) 1 .568 . 4 1 4 .420 . 5 3 3 ,201 - . 0 6 8 3 .269 .118 - . 0 6 9 2 .187 .0845 .0875 - . 0 0 3 0 .0856 .0513 5 .597 .411 .528 .246 .127 - . 1 0 4 .231 .0525 - . 0 8 3 4 .1359 .0578 .0467 . £ 0 8 9 .0502 .0192 (i-) 1 . 0 3 5 ,032 ,0135 .0155 . 0 1 2 4 . 0 0 5 2 8 -. 0 1 7 7 . 0 0 5 5 - . 0 0 2 2 2 . .0077 . 0 0 6 5 3 .0054 .00113 -.00275 .00258 3 , 6 0 0 ,445 .415 ,256 .128 -.113 .241 .0502 -.105 .1552 .0410 .0469 .0059 .O5S2 . 0 1 8 4 a (m'l!^ - m' k ' ) . 0 1 7 4 vn ^ vn V vn' .00025 ,0126 .0143 . 0 3 4 5 ,0112 .0004 . 0 1 9 8 TAT-LE 12 POK.: I I %n vn %n -vn a k " vn vn a k ' vn vn a t' i ' vn V vn V . *; Vn %n^'v Vn -a « ' k ' Vn V T / n a, < " k " °Vn V Vn ^n^K'^U.-a m" i " _{vn V vn} a m' i ' vn V vn a (m" i " -vn V -vn a m" k " _{vn V vn} a m' k ' vn V vn a ( m ' ' k " -vn V -vn n ODD V <" i " ) V vn' I" k " ) V vn m ' i ' ) V vn' m' k ' ) V vn' „ ^LD. .... - 2 ,00715 , 0 0 6 4 ,000775 .00115 .00162 - . 0 0 1 6 ,00522 .00026 - . 0 0 0 1 5 5 ,000ii£ ,000654 ,000608 ,oooai.6 ,00008 ,0000975 - , 0 0 0 0 1 8 - 2 .0413 .0578 .01i,4 .0129 . 0 0 9 4 - . 0 0 9 5 .0189 ,00292 - . 0 0 3 6 2 ,00654 .00386 .00350 ,00036 ,00147 ,00110 .00057 n irmi X2l -5 .0413 .0361 . 0 1 1 5 ,0090 ,0068 - . 0 0 9 1 6 ,0180 .00191 - . 0 0 2 8 7 .00470 ,00532 .00325 .00009 . 0 0 1 0 4 .00070 . 0 0 0 5 4 -1 .00705 .0067 .00158 .00115 .0025 - . 0 0 1 1 0 .0036 ,000407 - . 0 0 0 2 6 0 ,000667 .00136 .00109 .00025 ,000322 .000177 ,000145 _ ( 4 ) -3 , 6 0 0 . 4 4 5 . 4 1 4 . 2 3 6 , 1 2 8 - . 1 1 3 ,241 ,0502 - . 1 0 5 .1552 .0409 .0469 - . 0 0 6 0 , 0 3 8 . 0 1 8 4 . 0 1 9 6 -1 .0349 .0522 . 0 1 5 4 .0157 . 0 1 2 4 - . 0 0 5 3 1 .0177 .00555 - . 0 0 2 5 9 .00814 .00651 .00538 .00119 .00274 .00242 .00032

V n .0177 T/vJ:Li 13 E D !/n vn 1 269 0036 273 2 B f n 5 .231 .018 .249 .241 .228 .0032 .2312 .243 .0189 .2619 .0154 .0154 .0308 .252 .252 .504. V n 2 1 - . 0 0 3 0 .00025 - . 0 0 2 7 vn 2 3 - . 0 0 8 9 .00009 - . 0 0 8 8 .00113 .0060 .,0102 .OOCO4 ,00006 ,00006 -.0102 - . 0 0 4 5 .00036 .00iv1 -.00012 - . 0 1 0 6 - . 0 1 0 6 - . 0 2 1 2 TiiTiLE 1^4 i^n Vn n 2 V 1 .1870 .00i,S .1822 2 3 .1359 .OO/jÊ .1407 4 1 .O^i^Ö .00015 .0>!,3 2 3 4 .0112 .00034 .0115 .0077 .1552 .00035 .0198 .1456 ,0005 .1463 ,0036 .0072 .0174 -,00001 .0174 .00025 .0005 ,178 ,0065 ,1845 ,1459 .2918 ,0126 .0004 .0130 .0143 .0286