NASA Technical Paper 1241

M. J. Queijo, William R. Wells, and Dinesh A. Keskar

AUGUST 1978

https://ntrs.nasa.gov/search.jsp?R=19780022111 2018-12-05T16:02:51+00:00Z

TECH LIBRARY KAFB, NM

NASA Technical Paper 1241

Approximate Indicia1 Lift Function for Tapered, Swept Wings in Incompressible Flow

M. J. Queijo Langley Research Center, Hampton, Virginia William R. Wells Wright State University, Dayton, Ohio Dinesh A. Keskar University of Cincinnati, Cincinnati, Ohio

NASA National Aeronautics and Space Administration

Scientific and Technical Information Office

1978

0234527

SUMMARY

An approximate i n d i c i a l l i f t f u n c t i o n a s s o c i a t e d wi th c i r c u l a t i o n has been developed f o r t ape red , swept wings i n incompress ib le flow. The f u n c t i o n is de r ived by r e p r e s e n t i n g t h e wings wi th a simple vo r t ex system. The r e s u l t s from t h e de r ived equa t ions compare w e l l w i th t h e l i m i t e d a v a i l a b l e resul ts from more r i g o r o u s and complex methods.

The equa t ions , as de r ived , are no t ve ry convenient f o r c a l c u l a t i n g t h e dynamic response of a i r c r a f t , parameter e x t r a c t i o n , or f o r de te rmining frequency- response cu rves f o r wings. Therefore , an expres s ion is developed to conve r t t h e i n d i c i a l response f u n c t i o n to an e x p o n e n t i a l form which is more convenient f o r t h e s e purposes. The e x p o n e n t i a l form is n e a r l y as accurate as t h e form de r ived from t h e s i m p l i f i e d vo r t ex system.

INTRODUCTION

The i n d i c i a l l i f t f u n c t i o n fo l lowing a u n i t step i n c r e a s e i n wing ang le of a t tack is fundamental to t h e c a l c u l a t i o n of a i r p l a n e t r a n s i e n t motion. The o r i g - i n a l d e r i v a t i o n of t h i s f u n c t i o n ( g e n e r a l l y des igna ted by k l (SI) was developed i n 1925 f o r two-dimensional incompress ib le f l o w by H. Wagner ( r e f . 1 ) . S ince t h a t time t h e i n d i c i a l response has been c a l c u l a t e d f o r a number of s p e c i f i c wing planforms ( r e f s . 2 to 4, €or example) and v o r t e x l a t t i ce methods are a v a i l - a b l e f o r c a l c u l a t i n g i n d i c i a l l i f t of a r b i t r a r y wing planform ( r e f . 5, f o r exam- p l e ) . I n a d d i t i o n , t echn iques are c u r r e n t l y a v a i l a b l e f o r e s t i m a t i n g t h e l i f t on wings undergoing o s c i l l a t o r y motions. (See r e f s . 6 to 9, for example.) Ref- e r ence 10 ' conta ins a review and b ib l iog raphy on work i n unsteady aerodynamics.

Although a v a i l a b l e methods appear to be adequate f o r de te rmining t r a n s i e n t l i f t loads, they are g e n e r a l l y complex and do no t appear to be s u i t a b l e f o r i n c l u s i o n i n a i r c r a f t motion s t u d i e s ( excep t f o r a few s p e c i f i c wing p lanforms) .

The purpose of t h i s paper is to develop a s i m p l i f i e d method f o r de te rmining t h e i n d i c i a l l i f t f u n c t i o n a s s o c i a t e d wi th c i r cu la t ion . The part a s s o c i a t e d wi th t h e "apparent a d d i t i o n a l mass" is not t r e a t e d (eq. (31) of ref. 2) . A second purpose is to p rov ide an e lementary approach to unsteady aerodynamics which, a l though it l a c k s t h e mathemat ica l e legance of more e x a c t t h e o r i e s , it does g i v e t h e novice a s t a r t i n g p o i n t for s tudy i n an impor tan t and i n t e r e s t i n g

~ area of aerodynamics.

SYMBOLS

I 1.

A aspect ratio, b2/S

aolal ,a2,bi ,b2 cons t a n ts

- . . .. . , , . . . .

b wing span

ACL(t) th ree-d imens iona l i n d i c i a l l i f t c o e f f i c i e n t , a f u n c t i o n o f t i m e

s t e a d y - s t a t e l i f t curve s l o p e ("".),, C wing chord

cz (0 ) s e c t i o n l i f t c o e f f i c i e n t a t t = 0

Acl (t) two-dimensional i n d i c i a l l i f t c o e f f i c i e n t , a f u n c t i o n of t ime

C r wing root chord

C t wing t i p chord

i =p K nondimensional c o n s t a n t

k l (SI Wagner f u n c t i o n

L wing l i f t

1 s e c t i o n l i f t , per u n i t span

P,Q,R terms de f ined i n e q u a t i o n (12)

S wing area

S Laplace v a r i a b l e

t t i m e

U f ree-stream v e l o c i t y

W downwash v e l o c i t y due to v o r t i c e s

XO d i s t a n c e from th ree -qua r t e r chord p o i n t of root chord to s t a r t i n g p o i n t of shed v o r t e x a long root-chord l i n e

Y,= c o n s t a n t s i n e x p o n e n t i a l form of i n d i c i a l response f u n c t i o n

a a n g l e of attack

r t o t a l s t r e n g t h of v o r t e x

r i vo r t ex s t r e n g t h of i t h v o r t e x (i = 1 , 2, . . ., n)

2 4

A

A S

x P

T

w

sweep of wing quarter-chord l i n e

sweep of shed-vortex l i n e

wing taper ra t io , ct/cr

a i r d e n s i t y

dummy v a r i a b l e

angular frequency

ANALYSIS

The b a s i c concep t s used i n this s t u d y invo lve t h e u s e of l i f t i n g - l i n e t h e o r y to model t h e change i n l i f t fo l lowing a u n i t s tep i n c r e a s e i n a n g l e o f attack. The p r i n c i p l e s are de r ived for a two-dimensional wing and then extended to swept, t ape red wings.

Two-Dimens i o n a l Wing

The two-dimensional wing is r e p r e s e n t e d by a bound v o r t e x l o c a t e d a t t h e quarter-chord l i n e . (See f i g . 1 (a) .) The boundary c o n d i t i o n of no f l o w through t h e wing is s a t i s f i e d o n l y a t t h e wing th ree -qua r t e r chord l i n e . I f t h e wing a n g l e of a t tack a is suddenly g iven a u n i t i n c r e a s e , t h e bound ( l i f t i n g ) vor- t ex s t r e n g t h i n c r e a s e s by a v a l u e r. S i n c e t h e t o t a l s t r e n g t h (or c i r c u l a t i o n ) i n t h e f l o w f i e l d must remain a t t h e v a l u e it had b e f o r e t h e u n i t i n c r e a s e i n CL, a counter v o r t e x o f s t r e n g t h (-r) is formed near t h e wing t r a i l i n g edge. The vor t e x s t r e n g t h is such t h a t t h e normal v e l o c i t y component a t t h e wing th ree -qua r t e r chord l i n e , induced by t h e t w o v o r t i c e s , is e q u a l and opposite to t h e component of t h e free-stream v e l o c i t y normal to t h e wing s u r f a c e a t t h e same p o i n t . As t i m e i n c r e a s e s , t h e shed v o r t e x moves downstream, b u t r e t a i n s i t s o r i g i n a l s t r e n g t h . T h i s movement would cause a d e c r e a s e i n downwash a t t h e th ree -qua r t e r chord p o i n t , b u t t h e dec rease is o f f s e t by t h e shedding of a d d i t i o n a l v o r t i c e s . (See f i g . 1 (b) .) T h i s means, i n t u r n , t h a t the bound l i f t i n g - l i n e v o r t e x s t r e n g t h i n c r e a s e s i n o r d e r to keep t h e t o t a l c i r c u l a t i o n unchanged. Conse- q u e n t l y , t h e r e is a d i s t r i b u t i o n of v o r t i c i t y behind t h e wing. The v o r t e x s h e e t is extending downstream a t a ra te e q u a l to t h e free-stream v e l o c i t y .

Wagner (ref. 1) o b t a i n e d a s o l u t i o n for t h e l i f t fo l lowing a u n i t s tep i n c r e a s e i n a by applying a conformal t r ans fo rma t ion and s a t i s f y i n g t h e physi- ca l principle that t h e v e l o c i t y a t t h e w i n g t r a i l i n g edge is f i n i t e a t a l l t i m e (Kutta c o n d i t i o n ) . H e d e r i v e d for t h e l i f t , as a f u n c t i o n of nondimensional d i s t a n c e S = Ut / (c /2 ) , t h e e x p r e s s i o n

2 = 2TTPUwkl (E)

Wagner d i d no t d e r i v e an e x p l i c i t a n a l y t i c a l e x p r e s s i o n for b u t gave o n l y numerical va lues . Some y e a r s la ter (1936) , Kussner ( r e f . 11) d e r i v e d a long, slowly convergent series f o r k l (E).

k l (Si)

3

The mathematical complex i t i e s of o b t a i n i n g v a l u e s or e x p r e s s i o n s f o r arise because of t h e p h y s i c a l model d e p i c t e d i n f i g u r e 1 (b), or its e q u i v a l e n t i n t h e transformed-plane method of r e f e r e n c e 1. The p r e s e n t report develops a simpler model and a simple, y e t reasonably accurate, a n a l y t i c a l e x p r e s s i o n f o r k l (E) . same ( t i m e va ry ing ) s t r e n g t h as t h e bound vor t ex and moving downstream a t a v e l o c i t y Ku. The downwash v e l o c i t y induced a t t h e th ree -qua r t e r chord p o i n t by t h e two v o r t i c e s (bound and shed) is e a s i l y found by use of t h e Bio t -Savar t l a w (or see ref. 12, p. 12'7), and is

k l ( Z )

The shed v o r t e x s h e e t is rep laced by a s i n g l e shed v o r t e x having t h e

1

Solv ing f o r r results i n

The boundary c o n d i t i o n is m e t by

w = U s i n c1

or , f o r sma l l a n g l e s

w = ua ( 4 )

Equat ions (3) and (4 ) can be used wi th t h e Kutta-Joukowski equa t ion f o r s e c t i o n l i f t

to o b t a i n an e x p r e s s i o n f o r t h e s e c t i o n l i f t c o e f f i c i e n t due to a u n i t s t e p i n c r e a s e i n a n g l e of a t tack

1 A C , ( t ) = 2~ 1 (6) (

Under s t e a d y - s t a t e c o n d i t i o n s ( t -f a), The d i s t a n c e xo is r e l a t e d to t h e s t a r t i n g l i f t ; t h a t is, xo is r e l a t e d to c l ( 0 ) . From equa t ion (61,

A c z = 2n, which is t h e c o r r e c t value.

4

Using Wagner's conc lus ion t h a t t h e s t a r t i n g l i f t is one-half of t h e s teady- s t a t e l i f t l e a d s , f o r t h e two-dimensional wing, to

Equat ion (6), t h e r e f o r e , becomes

The term i n pa ren theses is t h e c o u n t e r p a r t of t h e Wagner f u n c t i o n k l ( 5 ) . T h i s term is found to approximate t h e Wagner f u n c t i o n ve ry c l o s e l y i f K equals 1/2. Equat ion (9) then becomes

1

G a r r i c k ( r e f . 13) a r r i v e d a t e x a c t l y t h i s same expres s ion by c u r v e - f i t t i n g t h e Wagner func t ion .

The i n d i c i a 1 response is approximated c l o s e l y by u s e of t h e s i n g l e bound- v o r t e x l i n e of f i g u r e 1 (c) and t h e shed v o r t e x (which r e p r e s e n t s t h e shed dis t r i - bu t ion of v o r t i c i t y ) s t a r t i n g a t a d i s t a n c e c/2 behind t h e wing th ree -qua r t e r chord l i n e and moving downstream a t one-half t h e f ree-s t ream v e l o c i t y .

Three-Dimensional Wing

The simple v o r t e x model of t h e two-dimensional wing compares w e l l w i th Wagner's e x a c t a n a l y s i s ; t h e r e f o r e , t h e same i d e a s have been t r i e d on three- dimensional t ape red , swept wings. There a r e s e v e r a l c h o i c e s to be made rela- t i v e to t h e mathematical model, and t h e s e cho ices inc lude :

(1) A t what p o i n t should t h e boundary c o n d i t i o n be s a t i s f i e d ?

(2) Where should t h e shed vo r t ex be s t a r t e d r e l a t i v e to t h e wing?

(3) How should the shed v o r t e x be o r i e n t e d r e l a t i v e to t h e wing?

I n order to keep t h e model a s simple as p o s s i b l e , it was dec ided to s a t i s f y t h e boundary c o n d i t i o n o n l y a t t h e th ree -qua r t e r chord p o i n t of t h e root chord. The o t h e r t w o c h o i c e s a r e made i n a l a te r s e c t i o n of t h i s report.

5

The vortex model for the three-dimensional wing is indicated in figure 2. The downwash at the control point can be derived by using the Biot-Savart law (or see p. 127 of ref. 12) and is of the form

-l

b/2 cr Cr - - _ sin A - sin A cos A 2 2

cos A

1 P =

r w = -(P + Q + R)

TT

sin A,

[(xo +-; ut) cos As]i

-~ - . .

where

> (12)

I

b 1 b

.2 2 2 2 - - _ xo + - Ut + - tan A,

Q=i[ d(x0 + 1 ut + - b tan As

2

b/2 + (.. + 2 1 ut)

1 cos As

(xo + ut) sin As - /m+ [to + ut) cos As]’I

6

... . . . .. .. . . . . -. . ..

The P,Q,R expressions are associated with the bound vortex, the wing-tip trailing-vortex pair, and the shed vortex, respectively. The geometric rela- tionship b/cr = A ( 1 + X)/2 can be used to put equations (12) into more con- venient forms

bP =

A ( 1 + 1) secZ A - t a n A

A ( 1 + A )

sec A - s i n A + cos2 A I’ + t a n A 1

The lift of the wing is determined fran the Kutta-Joukowski equation

Nondimensionalizing and using equation (11) results in

2nwb

US(P + Q + R) CL(t) =

The boundary condition requires that w/U = 01; hence, equation (15) becomes

2Trm

b(P + Q + R) CL(t) =

7

Under s t e a d y - s t a t e c o n d i t i o n s (t -+ m), equa t ion (16 ) , with proper s u b s t i t u - t i o n of equa t ions ( 1 3 ) , results i n

Solving f o r 2lTA and s u b s t i t u t i n g t h e r e s u l t i n t o equa t ion (16) resul ts i n , f o r a u n i t s tep i n a,

f

+

A ( l + A ) 1 - t a n A

2 . . t a n A]' + [ A ( l + A ) 1'

A s t becomes ve ry l a r g e , equat ion (18) reduces to

However, equat ion (17 ) is no t a c c u r a t e because of t h e simple vortex-system rep- r e s e n t a t i o n of t h e wing. A concept proposed is to u s e t h e b e s t a v a i l a b l e source

r a t h e r than equa t ion (17 ) . I n t h i s manner, a t t a i n i n g correct va l - for (CLa),,

u e s f o r equa t ion (18) is insured , a t l ea s t f o r l a r g e va lues of time.

Before equa t ion (18) can be used (with eqs. ( 1 3 ) ) , t h e terms xo/(cr/2) m u s t be determined, and a cho ice made f o r t h e sweep ang le of t h e shed vor tex . As noted p rev ious ly , f o r a two-dimensional wing, a va lue of %/(cr/2) = 1.0 is appropriate. Th i s va lue is r e t a i n e d for s i m p l i c i t y so t h a t t h e inboard ends of t h e shed v o r t e x s t a r t a t a d i s t a n c e %/(cr/2) = 1 behind t h e th ree -qua r t e r chord p o i n t of t h e root chord.

S e l e c t i o n of t he sweep of t h e shed vo r t ex was made by t r y i n g r easonab le va lues ; t h a t is, (1) paral le l to t h e wing t r a i l i n g edge, and ( 2 ) p a r a l l e l to t h e wing quar te r -chord l i n e . Comparison wi th results from more e x a c t methods showed t h a t t h e shed vo r t ex sweep ang le equal to t h a t of t h e wing quar te r -chord l i n e was t h e better choice.

8

Based on these choices, equations (1 3) become

r \

A ( l + A) 1 ut A(1 + A) tan A tan A + - - 1 -~ 1 +

2 2 c,/2 2 ~~ +

A ( l + A) 2 tan ,,I2 + - [ 1 ( 1 2 + ,/I A ( 1 2 + A) tan A]’

1 Ut A ( l +x) -~

A ( l + A) 2

sec2 A - tan A

sec A - sin A + cos2 A 2 1’

A ( l + A) bP = ~

2

(19) ’

I, - tan A

A ( l + A) 2

bR =

A ( l + 1)

2

1 ut 1 ut

Equations (18) and (19), with suitable selection of an expression for (Cb) ss (use of ref. 1 4 or 15, for example), are the basic equations for the indicial response for tapered, swept wings in incompressible flow. Although the equa- tions are somewhat lengthy, several factors appear repeatedly and are constant during the lift buildup.

LIFT VARIATION FOR ARBITRARY CHANGE IN ANGLE OF ATTACK

The lift of a wing undergoing an arbitrary change in angle of attack can be determined by use of the indicial response function in DuhameP’s integral; so that

Frequency response characteristics can be obtained easily by using the Laplace transform of equation (20) so that

9

from which

Because of the manner in which time enters into equations (18) and ( 1 9 ) , closed- form Laplace transforms are not known to exist for the general indicial lift function kL(t). Therefore, a form for which closed-form Laplace transforms do exist, which would also be convenient, has been sought. indicial response for several specific elliptic wings were given in exponential series of the form

In reference 2 the

In the present study, equation (18) was approximated by just one exponential term, namely,

that leads to

A(1 + A)

2 1 - tan A

{WT Y[A(l;+ 1 - - k L (t)

bP + bQ + bR

The parameter y was determined by evaluating equation (24) at Ut/(cr/2) = 0 and solving for y. The resulting equation is

I A(l + A ) 1 - tan A

2 b P + l +

tan + [~(l; ')I2 (25)

1 y = l -

[bP + bQ + bRJUt/(cr/2)=0

10

where

A ( 1 + A ) 2

1 + tan A

(bQ) ut/ (cr/2) =O = A ( l + A ) 2

tan A I 2 + [ ] A ( l + A )

A ( l + A ) 2

tan”]+[ A ( l + A )

1 - tan A +

] A ( l + A )

2 -

1

A ( l + A )

2 sec2 A + tan A

- tan A

sec A + sin A + cos2 A I1 d

A ( l + A ) 2

A ( l + A )

2 - (bR) ut/ (cr/2) =o -

Determination of the parameter z is more difficult. If equation (23) is solved for z , the result is

1 1

Evaluation of equation (27) at Ut/(c,/2) = 0 yields the indeterminant form

O/O, since y is precisely equal to [' - can be used with equation (27) and the resulting expression evaluated for z . By using L'Hospital's rule

However L'Hospital's rule

Using equation (24) results in

1 I r

A ( l + A) 1 - tan A

2

r d 1

Ut/ (cr/2) =O

1 2

or

z = - ! [ + l Y +

-

A(l + A ) 2

1 -

A.(1 + A )

2 tan A]

tan A

ut/ (cr/2) =o

1

The d e r i v a t i v e s appearing i n equat ion (28) were eva luated , and the r e s u l t s a r e

1 k( l + x‘) 2

- 2 [ T I

1 A (1 + A) tan tl A ( 1 + = - ;(bR)Ut/(cr/2)=o + - --

4

sec A + s i n A s i n A + cos2 1 A ( l + A) 2

X

312 sec A + s i n A]’ + cos2 .>

13

Equat ions (25) and (26) were used to c a l c u l a t e y for a wide range of wing sweep a n g l e s , aspect ratio, and taper ratio. Equat ions (28), (26), and (29) were used to c a l c u l a t e z for t h e same range of geometric v a r i a b l e s . R e s u l t s f r m t h e c a l c u l a t i o n s are p r e s e n t e d i n f i g u r e 3 and can be used i n e q u a t i o n (23) f o r c a l c u l a t i n g i n d i c i a l response.

COMPARISON OF RESULTS W I T H OTHER METHODS

T h i s s e c t i o n compares results from t h e methods developed i n t h i s study w i t h o t h e r methods f o r which numerical r e s u l t s are publ ished.

Two-Dimensional Wing

The two-dimensional wing is a r a t h e r special case having A + a, A = 1 , The i n d i c i a l - l i f t f u n c t i o n given by t h e e q u a t i o n is r e l a t i v e l y simple A = 0.

(eq. ( 1 0 ) ) and does possess a closed-form Laplace t ransform. There is no need, t h e r e f o r e , to use t h e e x p o n e n t i a l form for t h e i n d i c i a l l i f t . R e s u l t s from e q u a t i o n (1 0) are compared wi th those of r e f e r e n c e s 1 , 2, 4 , 5, and 11 i n f ig - ure 4. The results from equa t ion ( 1 0 ) are i n e x c e l l e n t agreement wi th t h o s e of r e f e r e n c e s 1 , 2, 4, and 11, which are e x a c t s o l u t i o n s or numerical approxima- t i o n s of e x a c t s o l u t i o n s . The success of e q u a t i o n ( 1 0 ) i n matching t h e s e exact s o l u t i o n s is a t t r i b u t a b l e to t h e s e l e c t i o n of ~0 ( t h e s t a r t i n g p o i n t of t h e shed vo r t ex ) and t h e v e l o c i t y o f t h e shed v o r t e x to match t h e Wagner s o l u t i o n .

Three-Dimensional Wings

Vortex systems and doublet d i s t r i b u t i o n s used to model three-dimensional wings are g e n e r a l l y r a t h e r complex and lead to involved c a l c u l a t i o n s f o r deter- mining i n d i c i a l l i f t . Numerical r e s u l t s , t h e r e f o r e , are a v a i l a b l e f o r o n l y a few specific wings for comparison wi th e q u a t i o n (23) .

Rectangular wings.- Rectangular wings have A = 1.0 and A = 0. R e s u l t s f rm u s e of equa t ion (23 ) , w i th v a l u e s of y and z f rm f i g u r e ( 3 ) , are com- pa red wi th r e s u l t s from o t h e r more exact (and more complex) methods i n f i g u r e 5 for wings of aspect ra t io 6 and 4. The f i g u r e shows t h a t t h e approximate method of t h e p r e s e n t paper is i n good agreement w i t h t h e r e s u l t s of r e f e r e n c e s 4 and 5.

T r i a n g u l a r wings.- T r i a n g u l a r wings have A = 0 and, i n a d d i t i o n , t h e wing geometry leads to a simple r e l a t i o n s h i p between aspect r a t io and t h e a n g l e of t h e quarter-chord l i n e , t h a t is,

3

A t a n A = - (30)

R e s u l t s frm use of e q u a t i o n (23) and v a l u e s of y and z from f i g u r e 3 are compared wi th r e s u l t s of r e f e r e n c e 16 i n f i g u r e 6 for t r i a n g u l a r wings o f aspect ra t io 4 and 2. I n t h e s e cases t h e agreement between t h e r e s u l t s of t h e p r e s e n t approximate t h e o r y wi th r e s u l t s of r e f e r e n c e 16 is e x c e l l e n t .

1 4

LimLting case with A + 0.- The parameter y for wings having aspect ratio of zero can be determined readily from equations (25) and (26) and is

The indicial response function, therefore, is

=

This result agrees with the results of reference 16.

CONCLUDING REMARKS

An approximate indicial lift function associated with circulation has been developed for tapered, swept wings in incompressible flow. The function is derived based on representing the wings by a simple vortex system. Comparison of results from the derived equations compare well. with the limited available results fran more rigorous and complex methods.

The equations, as derived, are not very convenient for calculating the dynamic response of aircraft, parameter extraction, or for determining frequency- response curves for wings. An expression, therefore, is developed to convert the indicial response function to an exponential form which is more convenient for these purposes. The exponential form is nearly as accurate as the form derived fran the simplified vortex system.

Langley Research Center National Aeronautics and Space Administration Hampton, VA 23665 May 5, 1978

15

I

REFERENCES

1. Wagner, Herber t : Uber D i e Enstehung des dynamische A u f t r i e b e s von Tragflugeln,.Z.a.M.M. Bd. 5, Hef t 1, Feb. 1925, pp. 17-35.

2. Jones, Robert T.: The Unsteady L i f t of a Wing of F i n i t e Aspect R a t i o . NACA Rep. 681, 1940.

3. Lehr ian , Doris E.: I n i t i a l L i f t of F i n i t e Aspect-Ratio Wings Due to a Sudden Change of Inc idence . R. C M. No . 3023, B r i t i s h A.R.C., 1957.

4. Jones, W. P r ichard : Aerodynamic Forces on Wings i n Non-Uniform Motion. R. C M. No . 2117, B r i t i s h A.R.C., Aug. 1945.

5. P i s z k i n , S. T.; and Levinsky, E. S.: Nonlinear L i f t i n g Line Theory f o r P r e d i c t i n g S t a l l i n g I n s t a b i l i t i e s on Wings of Moderate A s p e c t Ra t io . CASD-NSC-76-001 , Gen. Dyn. Convair Div. , J u n e 15, 1976. (Ava i l ab le from NTIS as AD-A027645.)

6. Theodorsen, Theodore: Genera l Theory of Aerodynamic I n s t a b i l i t y and t h e Mechanism of F l u t t e r . NACA Rep. 496, 1935.

7. Watk ins , Cha r l e s E.: Woolston, Donald S.; and Cunningham, Herbe r t J.: A Sys temat ic K e r n e l Funct ion Procedure f o r Determining Aerodynamic Forces on O s c i l l a t i n g or Steady F i n i t e Wings a t Subsonic Speeds. NASA TR R-48, 1959.

8. Albano, Edward; and Rodden, W i l l i a m P.: A Double t -Lat t ice Method f o r Calcu- l a t i n g L i f t D i s t r i b u t i o n s on O s c i l l a t i n g Sur faces i n Subsonic Flows. AIAA J., vol. 7, no. 2, Feb. 1969, pp. 279-285; Errata, vo l . 7, no. 11, Nov. 1969, p. 2192.

9. Morino, Luig i : A Genera l Theory of Unsteady Canpres s ib l e P o t e n t i a l Aerody- namics. NASA CR-2464, 1974.

10. Edwards, John W i l l i a m : Unsteady Aerodynamic Modeling and Act ive Aeroelastic C o n t r o l . NASA (33-148019, 1977.

11, Kussner, H. G.: Zusammenfassender B e r i c h t bber den i n s t a t i o n a r e n A u f t r i e b von Flugeln. Luf tahr t forschung, Bd. 13, N r . 12, D e c . 20, 1936, pp. 410-424.

12. G l a u e r t , H.: The Elements of A e r o f o i l and A i r s c r e w Theory. Second ed. Cambridge Un ive r s i ty P r e s s , 1948.

13. G a r r i c k , I. E.: On Some Reciprocal R e l a t i o n s i n t h e Theory of Nons ta t ionary Flows. NACA Rep . 629, 1938.

14. Queijo, M. J.: Theory f o r Computing Span Loads and S t a b i l i t y Derivat ives Due to S i d e s l i p , Yawing, and Ro l l ing f o r Wings i n Subsonic C m p r e s s i b l e Flow. NASA TN D-4929, 1968.

16

15 . USAF S t a b i l i t y and Control D a t c o m . Contracts AF 33(616)-6460 and F33615-7542-3067, McDonne11 Douglas Corp. , O c t . 1960. (Revised Apr. 1976 . )

16. Dr i sch ler , Joseph A.: Approximate I n d i c i a 1 L i f t Functions f o r S e v e r a l Wings of F i n i t e Span i n Incompressible Flow as Obtained from Oscillatory L i f t C o e f f i c i e n t s . NACA TN 3639, 1956.

17

n n w n w x h Y V.. I I ..I< I I

Bound vortex Shed vortex r

Ua

(a) vortices a t i n s t a n t a f t e r a is inc reased by a s t e p .

Dow nw as h r l+ r2+. . . +rn

n t

vortex t

r4 r3 9 r1

3 x 0 Shed vort ices

U

Ua

(b) vortices some f i n i t e time af te r a step i n c r e a s e i n a.

Downwash

I Total

P + >

Total

Ut 3 +T--I

t Ua

(c) S impl i f i ed v o r t e x system used i n t h i s study.

Figure 1 .- Vortex r e p r e s e n t a t i o n of a two-dimensional wing fo l lowing a step i n c r e a s e i n ang le of attack.

18

Downwash

Figure 2.- Vortex system for swept tapered wings.

Y

Aspect ratio

Z

Aspect ratio

(a) A = Oo.

Figure 3.- Values of c o n s t a n t s y and z f o r use i n t h e i n d i c i a 1 response -.[ut/ (cr/2II f u n c t i o n ACL C /( = ' - ye

20

.08E

0 0 E 1 . O E

E L

L

- A

I.. C

0.5

0.2

0

-0.7

-

I I I I

Aspect ratio

LLL! LLL! LLU

LLLL

Aspect ratio

(b) A = 15O.

Figure 3 .- Continued.

21

F--

-06E 0 0 I

Aspect ratio i

Z

( c ) A = 30°.

Figure 3.- Continued.

22

9 /

I I I I

-

I I I I

Aspect ratio I a

Aspect ratio

(d ) A = 45O.

Figure 3 .- Continued.

23

Y

1

Z

1 u.1

e ===

1 1 1 1

Aspect ratio

Aspect ratio

A = 60°.

-

1 1 1 1

8 9 I

F i g u r e 3 .- C o n c l u d e d .

24

1.0

. E

.4

.2

0

~ q . (IO), refs. 1, 2, 4, 11 Ref, 5 - _ _ _ _ _ _

4 I . . I I I I 2 4 6 8 10 12 14 16 m

W c J 2

Figure 4.- Indicia1 lift for two-dimensional wings.

1.0

.a

e 4

e 2

0

I . I 1 .. . I-- _ _ .-J 2 4 6 8 10

(a) Aspect r a t io 6.0.

F igure 5.- I n d i c i a 1 l i f t f o r unswept, untapered wings.

26

1.0

.a c

(b) Aspect ra t io 4 . 0 .

Figure 5.- Concluded.

27

1.0

0 2 4 6 8 l o

(a) Aspect r a t io 4.0.

Figure 6.- I n d i c i a 1 l i f t for d e l t a ( t r i a n g u l a r ) wings.

28

ss

0 . L 1 I I I 2 4 6 8 10

(b) Aspect r a t i o 2.0.

Figure 6 .- Concluded.

29

1. Report No. NASA TP-1241

-~

7. Author(s)

M. J. Queijo, William R. Wells, and Dinesh A. Keskar

9. Performing Organization Name and Address

NASA Langley Research Center Hampton, VA 23665

~

2. Government Accession No.

__ - _

12. Sponsoring Agency Name and Address

National Aeronautics and Space Administration Washington, DC 20546

3. Recipient's Catalog No

~ -. - - . - - . - - . 5. Report Date

August 1978 A .- ~-

6. Performing Organization Code

____ .~

8. Performing Organization Report No.

L-12110

10. Work Unit No. 505-06-63-02

11. Contract or Grant No.

13. Type of Report and Period Covered

Technical Paper

14. Sponsoring Agency Code

15. Supplementary Notes M. J. Queijo: Langley Research Center, Hampton, Virginia. William R. Wells: Wright State University, Dayton, Ohio. Dinesh A. Keskar: University of Cincinnati, Cincinnati, Ohio.

16. Abstract An approximate indicial lift function associated with circulation has been

developed for tapered, swept wings in incompressible flow. The function is derived by representing the wings with a simple vortex system. The results from the derived equations compare well with the limited available results from more rigorous and complex methods. The equations, as derived, are not very convenient for calculating the dynamic response of aircraft, parameter extraction, or for determining frequency- response curves for wings. Therefore, an expression is developed to convert the indicial response function to an exponential form which is more convenient for these

- .~ ~- 18. Distribution Statement

Unclassified - Unlimited

Subject Category 02 . . .... ;':,%. ;+;io -~

_ - -__I -~ . -

NASA-Langl ey, 1978

I- 17. Key Words (Suggested by Author(s))

Unsteady lift Indicia1 lift Frequency response Aerodynamics

-. . . .~ . - -. -

20. Security Classif. (of this page)

Unclassified 1 19. Security Classif. (of this report)

Unclassified ~ . . . . ~~ ~

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