NONPLANAR DOUBLET-POINT METHOD FOR SUPERSONIC

UNSTEADY AERODYNAMICS

Ashish Tewari* Nat iona l Aeronaut ica l Labora tory , Bangalore, I n d i a

Abs t r ac t

A new method i s devised f o r t h e c a l c u l a t i o n of p r e s s u r e s and aerodynamic i n f l u e n c e - c o e f f i c i e n t s on nonplanar 1 i f t ing- s u r f a c e c o n f i g u r a t i o n s o s c i l l a t i n g i n a supe r son ic f r e e s t r e a m . The method i s an ex t ens ion of t h e methodology in t roduced i n t h e p l ana r supe r son ic Doublet-Point scheme of Ueda and Dowell, which i s based upon t h e concept of concen t r a t ed l i f t f o r c e s and uses t h e a c c e l e r a t i o n p o t e n t i a l doub le t as an elementary s o l u t i o n of t h e wave e q u a t i o n . These f e a t u r e s make t h e method capable of be ing inco rpo ra t ed i n a u n i f i e d code f o r bo th subsonic and supe r son ic speeds , as w e l l as amenable t o r a p i d a e r o e l a s t i c c a l c u l a t i o n s . Resu l t s on va r ious l i f t i n g - s u r f a c e c o n f i g u r a t i o n s are i n agreement wi th o t h e r supe r son ic o s c i l l a t o r y methods, v a l i d a t i n g t h e Doublet-Point approximation f o r nonplanar supe r son ic case.

In t roduc t ion

I n modern a e r o n a u t i c a l development, t h e r e i s a renewed i n t e r e s t i n a i r c r a f t capable of s u s t a i n e d supe r son ic c r u i s e and maneuve rab i l i t y . The aerodynamic c h a r a c t e r i s t i c s of such a i r c r a f t d i c t a t e s t r u c t u r a l c o n f i g u r a t i o n s whose f l u t t e r speeds a r e i n t h e supe r son ic regime. In a d d i t i o n , t h e gus t- response and a e r o s e r v o e l a s t i c i n t e r a c t i o n s need t o be i n v e s t i g a t e d a t supe r son ic speeds . I n o r d e r t o add re s s t h e s e problems, an e f f i c i e n t method of p r e d i c t i n g t h e unsteady supe r son ic a e r o d yna m i c s on p r ac t i c a 1 1 i f t i rig- s u r f a c e c o n f i g u r a t i o n s i s r e q u i r e d . The unsteady aerodynamic loads due t o a gene ra l motion of t h e s t r u c t u r e can be de r ived from those a r i s i n g o u t of s imple harrtionic rootion, by us ing a n a l y t i c con t inua t ion i n t h e Laplace domain. Hence, an o s c i l l a t o r y s upe r s on i c 1 i f t i ng - $2 u r f a c e t h e o r y becomes neces sa ry .

For subsonic speeds , t h e

Double t- Lat t ice method i s w e l l e s t a b l i s h e d f o r i t s s i m p l i c i t y and amenab i l i t y t o a e r o e l a s t i c c a l c u l a t i o n s . However, t h e supe r son ic l i f t i n g - s u r f a c e schemes a r e much more complex by n a t u r e s i n c e t hey must account f o r t h e a d d i t i o n a l f e a t u r e s t h a t are t y p i c a l of supe r son ic f low, such as k e r n e l s i n g u l a r i t i e s on t h e Mach cone . The supe r s on i c count e r p a r t o f t h e s ubs on i c Double t- Lat t ice method has been sought i n

”. - -._- X S c i e n t i s t , S t r u c t u r e s D i v i s i o n . Member, A I A A .

Copyright 0 1993 American Institute of Aeronautics and Astronautics, Inc. All rights reserved. 2466

t h e past‘ wi thout much succes s . Neve r the l e s s , mot iva t ion f o r a supe r son ic scheme wi th enough commonality wi th t h e subsonic Double t- Lat t ice method i n o rde r t o p r e s e n t a u n i f i e d code f o r bo th t h e speed regimes, has p e r s i s t e d .

There a r e t h r e e c l a s s e s of methods p r e v a l e n t i n t h e s o l u t i o n of t h e supe r son ic three- dimens ional , o s c i l l a t o r y problem, namely those employing t h e v e l o c i t y p o t e n t i a l , t h e g r a d i e n t of t h e v e l o c i t y p o t e n t i a l , and t h e a c c e l e r a t i o n p o t e n t i a l , r e s p e c t i v e l y , as t h e

3 elementary s o l u t i o n . Ga r r i ck and Rubinow p re sen ted an i n t e g r a l equa t ion f o r t h e v e 1 o c i t y p o t en t i a 1 The most common v e l o c i t y p o t e n t i a l s o l u t i o n procedure i s t h e Mach Box method in t roduced by Pines e t a ~ ~ , and f u r t h e r

r e f i n e d by s e v e r a l D i f f i c u l t i e s wi th t h e Mach Box method inc lude t h e n e c e s s i t y t o e v a l u a t e t h e v e l o c i t y p o t e n t i a l i n diaphragm reg ions o f f t h e l i f t i n g - s u r f a c e and t h e dependence of t h e g r i d on Mach number. The re f inements sugges ted t o a l l e v i a t e t h e s e d i f f i c u l t i e s add complexi ty t o t h e method. Another v e l o c i t y p o t e n t i a l

13 approach i s t h e ex t ens ion of Evvard * s s t e a d y - s t a t e t heo ry t o t h e o s c i l l a t o r y

This scheme case by Burkhart . e l i m i n a t e s t h e need f o r diaphragm r e g i o n s , b u t i s l i m i t e d t o p l ana r a p p l i c a t i o n s .

Jones and Appa proposed t h e P o t e n t i a l Grad ien t method. They used a series expansion of t h e k e r n e l f u n c t i o n , which l o s t v a l i d i t y a t h igh reduced- frequencies and low supersonic Mach numbers. Hounjet avoided t h e series expansion by us ing an i n t e g r a t i o n scheme s i m i l a r t o t h a t of t h e subsonic Doublet La t t i ce method f o r d i r e c t l y downstream r e c e i v i n g p o i n t s . Chen and

Liu17 app l i ed ano the r approach t o avoid t h e series expansion by us ing a pa rabo l i c c u r v e - f i t f o r t h e exponen t i a l p a r t of t h e i n t e g r a n d , i n o r d e r t o i n t e g r a t e t h e d i p o l e spanwise s i n g u l a r i t y of t h e p l ana r k e r n e l . The schemes of Refs . 15-17 needed t o cons ide r t h e wake r eg ion between l i f t i n g s u r f a q e s . Also, t h e computation of p r e s s u r e i n f 1 uence- coe f f i c i en t s r e q u i r e d a d d i t i o n a l s teps i n t h e s e schemes s i n c e t hey d i d n o t formula te a d i r e c t r e l a t i o n s h i p between pressure and

normalwash. Appa18re-derived t h e i n t e g r a l equa t ion of t h e b a s i c Po ten t i a l- Grad ien t method and a r r i v e d a t a d i r e c t ’

s our c e - s t r e rig t h .

au tho r s 5-12

1 4

r e l a t i o n s h i p between p r e s s u r e and normalwash, which i s a f e a t u r e common wi th t h e subsonic Double t- Lat t ice method. Ref . 18 also showed t h a t t h e k e r n e l could be expressed i n a form a n a l y t i c on t h e Mach cone , and eva lua t ed t h e nonplanar i n t e r f e r e n c e by u s i n g a f i n i t e - d i f f e r e n c e approximation. There was no need t o i n t e g r a t e on t h e wake r eg ion l y i n g between t h e l i f t i n g s u r f a c e s . The r e c e n t l y pub l i shed Harmonic-Gradient

ZONA51C scherne of Liu e t a l . l9 uses t h e same i n t e g r a l formula t ion as of Appa 18 . Ref. 19 claims t o handle all ke rne l i n t e g r a t i o n s a n a l y t i c a l l y , b u t an absence of d e t a i l s p r even t s meaningful d i s c u s s i o n . I t i s a l s o impl ied i n Ref. 19 t h a t t h e method uses a Double t- Lat t ice t ype curve- f it approximation, which appears t o c o n t r a d i c t t h e claim of e x a c t i n t e g r a t i o n s .

The t h i r d ca t ego ry of supe r son ic l i f t i n g - s u r f a c e methods adopts t h e a c c e l e r a t i o n p o t e n t i a l doub le t as t h e elementary s o l u t i o n of t h e wave equa t ion . The advantage of t h i s approach l i e s i n u s ing t h e same i n t e g r a l equa t ion as t h a t f o r subsonic f low. I t should be po in t ed o u t t h a t t h i s i n t e g r a l equat ion can also be obta ined by us ing a r e- de r ived

Po t e n t i a l YGradi e n t Watkins and Berman" der ived t h e i n t e g r a l equa t ion from t h e a c c e l e r a t i o n p o t e n t i a l

18 ,19 f o rmu 1 a t i on

approach. Harder and Rodden"' provided t h e nonplanar k e r n e l f u n c t i o n f o r t h e i n t e g r a l e q u a t i o n . One s o l u t i o n procedure f o r t h e i n t e g r a l equa t ion i s t h e Kernel

Funct ion method of Cunningham , which uses assumed p r e s s u r e polynomials whose c o e f f i c i e n t s a r e determined by s a t i s f y i n g t h e normalwash boundary cond i t i on at a number of c o l l o c a t i o n p o i n t s . N i s s i m and

L o t t a t i 2 ' ? developed a v a r i a t i o n of t h i s ap2roach by employing box- l ike d i v i s i o n s wi th a cont inuous p r e s s u r e polynomial i n each box. The l i m i t a t i o n of t h e supe r son ic Kernel Funct ion methods l i e s i n t h e i r extreme s e n s i t i v i t y t o t h e cho ice of t h e assumed polynomials . Ueda

and Dowe11Z4 proposed ano the r type of a c c e l e r a t i o n p o t e n t i a l scheme, c a l l e d t h e Supersonic Doublet-Point method, l i m i t e d t o planar a p p l i c a t i o n s . I t uses concen t r a t ed l i f t f o r c e s (or p o i n t d o u b l e t s ) and an averaged normalwash f o r d i s c r e t i z i n g t h e i n t e g r a l equa t ion . The most a t t r a c t i v e f e a t u r e of t h i s scheme i s t h a t it a f f o r d s t h e g r e a t e s t degree of s i m i 1 a r i t y and supersonic: c a l c u l a t i o n s . TJeda and Ilowell had ear l ier devised a subsonic

Doublet-Point methodz5 us ing t h e same concept of d i s c r e t e a c c e l e r a t i o n p o t e n t i a l doub le t s and d i s t r i b u t e d normalwash, r a t h e r t han t h e

d i s t r i b u t e d Double t- Lat t ice method's doub le t s t r e n g t h and t h e normalwash eva lua t ed a t c o n t r o l p o i n t s . Ques t ions

22

between s u bs on i c

1

a rosez6 about t h e "phys i ca l exp lana t ion " of t h e Doublet P o i n t approximation s ince i t reve r sed t h e t r ad i ti ona 1 d i s c r e t i z a t i o n procedure . However, Ueda and Dowell" Q 4 25 demonstrated t h e concep t ' s v a l i d i t y f o r p l a n a r cases i n bo th subs on i c and supe r son ic c a l c u l a t i o n s . The subsonic and supe r son ic Doublet-Point methods can be combined i n t o one code s i n c e they d i f f e r on ly i n t h e k e r n e l i n t e g r a t i o n . T h i s , and t h e concen t r a t ed l i f t f o r c e assumption of t h e Doublet-Point scheme, are inva luab le f e a t u r e s when r epea t ed a e r o e l a s t i c c a l c u l a t i o n s a r e t o be c a r r i e d o u t e f f i c i e n t l y . Tewari 27 ' " extended t h e supe r son ic Doublet Point, method t o nonplanar a p p l i c a t i o n s . When compared t o t h e p l a n a r k e r n e l f u n c t i o n , t h e nonplanar k e r n e l i s of a very complicated c h a r a c t e r and r e q u i r e s s p e c i a l t r e a t m e n t . While t h e p l a n a r k e r n e l ha s an i n t e g r a b l e 1 / 2 power s i n g u l a r i t y a long Mach l i n e s , i t s nonplanar c o u n t e r p a r t h a s , i n a d d i t i o n , s i n g u l a r i t i e s of u n i t power and t h e power 3 /2 on t h e Mach cone. The k e r n e l averaging procedure has t o account f o r t h e Mach boundary cu rva tu re i n lionplanar i n t e r f e r e n c e , l e a d i n g t o many more averaging cases than a p l ana r c o n f i g u r a t i o n . A s a consequence, t h e gene ra l nonplanar supe r son ic Doublet-Point method i s a much more complex unde r t ak ing than t h e scheme of

Ueda and Dowe1lZ4, which was l i m i t e d t o p l a n a r c a s e s .

The I n t e g r a l Equat ion

The i n t e g r a l equa t ion r e l a t i n g p r e s s u r e d i f f e r e n c e ampl i tude , ACp I a c r o s s t h e l i f t i n g s u r f a c e a t p o i n t ( x , Y , Z ) and t h e normalwash ampl i tude , w, a t a p o i n t ( r , n , T ) on a nonplanar l i f t i n g c o n f i g u r a t i o n p laced i n a uniform supersonic f r ee s t r eam of v e l o c i t y u and

2 7 , 2 8 o s c i l l a t i n g a t a frequency o, i s

W(C* v , 6) = - 8', 1 ~ ~ ( x , y ~ z ) K ( ~ -x, tl - Y, f -Z, w)dXdY

(1)

Uprlream Mach cane

1 i

Figure 1. Co- ordinate geo e t r y and i n t e g r a t i o n n r e a .

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The i n t e g r a t i o n i n equat ion (1) is performed over t h a t a r e a W of t h e l i f t i n g - s u r f a c e which i s conta ined w i t h i n t h e upstream Mach cone emanating from t h e normalwash p o i n t (F igu re 1 ) . The k e r n e l f u n c t i o n i n equa t ion ( 1 ) can be expressed a s

K ( x , y , z , k ) = Kl(x ,Y,z ,k)T1

f K 2 ( x , ~ ~ , z , k ) T 2 ( 2 )

T = C O S ( ) . ~ - F , ) 1 where,

T 2 = ( ZOcos). -YOsinr ) ( z0cosy -Yosinys) r r s

Z = z cosy f y s i n y 0 S S

Y = y cosy - z s inys 0 s

M2e-ikx K l ( x , y , z , k ) =

R x+x2

1 - ikx

x+x2 r

x+xl

- ikX2

3

X2”

( X + X 2 1

- ikv

4 r e 1 ( 4 )

dv J - 3 r

2 2 ( r +v 15/2 x1

k=ab/U i s t h e non-dimensional reduced f requency , b i s a r e f e r e n c e l e n g t h , Y , ~

and yr a r e t h e d i h e d r a l a n g l e s a t t h e

sending ( p r e s s u r e or l o a d ) and r e c e i v i n g (normalwash) p o i n t s , (X,Y,Z) and ( t ’ , v , C ) , r e s p e c t i v e l y , (F igu re 21, and a i s t h e speed of sound f a r upstream.

Z

5 171 ECElVlNG POINT

Y

Figure 2. Dihedral Angles at sending and receiving points.

The D i s c r e t i z a t i o n Procedure

A numerical g r i d i s formed by d i v i d i n g t h e l i f t i n g- s u r f ace c o n f i g u r a t i o n i n t o a number of boxes. TWO

d i s t i n c t approaches are a v a i l a b l e f o r 18,19

d i s c r e t i z i n g equa t ion ( 1 ) . The f i r s t t h

assumes a c o n s t a n t p r e s s u r e i n t h e j box, and e v a l u a t e s t h e normalwash a t t h e ith c o n t r o l p o i n t . The assumption of c o n s t a n t p r e s s u r e a l lows AC P t o be taken o u t s i d e t h e i n t e g r a t i o n s i g n i n equat ion (1) , which i s c a r r i e d o u t over t h e area of t h e p r e s s u r e box i n t e r s e c t e d by t h e

The o t h e r upstream Mach cone . t h e d i s c r e t i z a t i o n procedure i s

Doublet-Point methodz4, which assumes a concen t r a t ed l i f t f o r c e AjCCp i n each

g i v i n g t h e non-dimensional p r e s s u r e d i f f e r e n c e i n t h e jth box as

box, whose non-dimensional a r e a i s A j 9

AjACp(X,Y) 5 ( X - X . ) J d ( Y - Y J , ) .

In a d d i t i o n , it i s assumed t h a t t h e normalwash i n t h e i th box can be r ep re sen ted by an average va lue wi such t h a t

where AW i s t h e p a r t of an averaging r eg ion l y i n g i n s i d e t h e downstream Mach cone emanating from t h e load p o i n t ( X j , Y j ) . The d i s c r e t i z e d i n t e g r a l equa t ion then becomes

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. A .

Ueda and D o w e d 4 s e l e c t e d a r e c t a n g u l a r averaging r eg ion f o r t h e normalwash, wi th t h e a r e a and width same as t h a t of t h e r e c e i v i n g box, and d i sp l aced downstream such t h a t i t s leading-edge pas se s through t h e box-center ( F i g u r e 3 ) . The load p o i n t i s assumed t o be l o c a t e d a t t h e c e n t e r of t h e sending ( p r e s s u r e ) box. These choices of t h e normalwash averaging reg ion and t h e load p o i n t a r e found t o be numer ica l ly j u s t i f i e d , a l though no a n a l y t i c a l reason i s o f f e r e d f o r them i n

Ref . 2 4 . F u r t h e r , it was found2' t h a t t h i s cho ice of ave rag ing reg ion works when a l l t h e boxes are of a uniforro wid th . Trapezoida l averaging r eg ions are a l s o

p o s s i b l e , b u t t hey i n c r e a s e the number of c a s e s of Mach cone i n t e r s e c t i o n d r a m a t i c a l l y . A l l boxes have s i d e edges p a r a l l e l t o t h e f r ee s t r eam f o r s i m p l i c i t y .

28

Y

SENDING BOX (LOAD POINT)

X r = =-F

4* BOX CENTER RECEIVING BOX (NORMALWASH POIM)

Figure 3. Discretization and averaging [eo

Equation ( 6 ) r e q u i r e s i n t e g r a t i o n of t h e k e r n e l f u n c t i o n . However, t h e ke rne l f u n c t i o n has s i n g u l a r i t i e s on t h e Mach cone where R + O . I n o r d e r t o i n t e g r a t e t h e s e s t r o n g s i n g u l a r i t i e s , t h e k e r n e l f u n c t i o n i s s i m p l i f i e d t o a form given i n t h e Appendix. S ince a d i r e c t i n t e g r a t i o n of t h e s i n g u l a r o s c i l l a t o r y k e r n e l i s n o t p o s s i b l e , it i s f u r t h e r s epa ra t ed i n t o s t e a d y and unsteady f a c t o r s . The s t eady f a c t o r i s ob ta ined by l e t t i n g k=O, and is esp re s sed as

The uns teady f a c t o r i s obta ined by d i v i d i n g t h e k e r n e l f u n c t i o n by Ks:

The unsteady f a c t o r K U , c o n t a i n i n g t h e

o s c i l l a t o r y f a c t o r s , is a n a l y t i c on t h e Mach cone, and i t can be shown t h a t

-ikM2 r /P ( 9 )

l i r n R + o K U ( x , y , z , k ) = - e

T2 S ince t h e unsteady f a c t o r i s va ry ing slowly i n s i d e t h e averaging r eg ion , it i s eva lua t ed a t t h e midpoint of t h e averaging r eg ion and taken o u t s i d e t h e double i n t e g r a l s i g n . Hence, equat ion ( 6 ) becomes

where s u b s c r i p t m r e f e r s t o t h e c e n t r o i d of t h e r e c t a n g u l a r averaging r e g i o n . A s equa t ion ( 1 0 ) i n d i c a t e s , on ly t h e s t eady k e r n e l must be i n t e g r a t e d . The s t eady k e r n e l r e t a i n s t h e 3/2 power s i n g u l a r i t y on t h e Mach cone , as well a s t h e 1/2 power s i n g u l a r i t y . The reg ion of i n t e g r a t i o n AI4 i s obta ined by t h e i n t e r s e c t i o n of t h e Mach cone wi th t h e ave rag ing r e g i o n . The equa t ion d e s c r i b i n g

a Mach cone is x2-P2(y2+z2)= 0 , while an averaging r eg ion can be expressed as

z = y t a n ( y - y ) + h where h is t h e v e r t i c a l s e p a r a t i o n between t h e sending and r e c e i v i n g boxes. Th i s g ives t h e equa t ion f o r t h e reg ion of i n t e g r a t i o n AW as

r s

which i s t h e equa t ion of a hyperbola . AW i s t h e r eg ion bound downstream of t h i s Mach hyperbola . F igure 4 shows t h e v a r i o u s p o s s i b l e c a s e s i n which t h e Mach cone i n t e r s e c t s an averaging r eg ion .

The Kernel I n t e g r a t i o n

I n o r d e r t o i n t e g r a t e t h e s t eady k e r n e l , t h e i n t e g r a l

Xb 'b

a a I = S, Jy ~ ~ ( x , r ) dy ax

i s d iv ided i n t o t h r e e p a r t s , I = I +I +I where,

1 2 3

a Rr

dy dx a R r

dy dx

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/ I I

AW d

CASE I L

CASE VI El

Figure 4 . Avergaing cases of Mach cone intersection.

xa9xb>Ya>Yb denote the limits of integration corresponding to the area AW. The integrals I1 and I2 are seen to have a 1/2 power singularity as R-0. This singularity is removed by using the relation

- X

8, R - _ -

When this relation is incorporated into the integrands, the two integrals become

Note that for the general case of both receiving and sending boxes having dihedral angles, T2 is only a function of y and can be expressed as

Hence the above indicated operations are valid. The integrals I1 and I2 are non-singular and can be carried out analytically. Integral I3 contains a 3/2 power Mach cone singularity, whose treatment requires the consideration of the following integral :

where

and

Yb = - 4 G K 2

Since f(y,,,y) is continuous and integrable in the range of integration and has the following property:

r 1

then Hadamard ' s finite -part 0 1 ex i s t s for the integral in expression (13) and is defined as follows:

PP .

"Fp. ' I denotes the finite-part of the singular integral. The definition of the finite-part2' is such that, instead of taking the integral up to its singular limit where it would diverge, the integrand is expanded in a polynomial series near the point of singularity and only those terms of the series are retained that remain finite after integration. It is interesting to note that, as shown by Schwartz3', Hadamard's finite-part for any integral with first order singularity exactly corresponds to the Cauchy's principal-value for the same integral. Heaslet and Lomax3' showed that equation (14) is also valid for a higher order of differentiation with respect to the integration limit, By using the relation

a - 2 'b d

ayb P --

x ax ~-

in equation (14), the following is obtained:

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The integral on the left hand side of equation (15) is recognized as the inner integral of 13, whose finite-part can now be written as

A half order singularity remains in the integrand and can be integrated in Hadamard's sense, to produce

'b "b 'a a

Ig -PCCg(x,y)l Ix

-2 In the coplanar limit ( z = O ) , an r singularity of the kernel appears in boxes directly downstream of the sending point, and is integrated in the Mangler's3' sense, as shown in Ref. 24.

Numerical Results

In order to validate the nonplanar Doublet-Point method (DPM), comparisons are presented with other methods for selected test-cases.

I. Non-coplanar Rectangular Wing-Tail

This test-case examines the nonplanar interference between a rectangular wing and tail (Figure 5). The tail is vertically separated from the wing by 0.4 units. Figures 6-8 show the chordwise pressure distributions at 5%, 45%, and 95% span locations, respectively, when the wing undergoes uniform plunging oscillations at kZ0.2 and M=1.2. Good agreement is observed in these figures with the Mach Box method . Results are obtained with 200 boxes in the DPM and 300 boxes on surfaces (plus additional boxes in the diaphragm region 1 in the Mach Box method.

10

11. Rectangular Wing with Folded Tips

Here the nonplanar DPM is applied to

-I 0 1 1 1 X

Figure 5. Ron-coplanar rectangular wing tail (test-case I ) .

Doublet - Poi nt Met hod Mach-Box Method

o'80 1

-0.40 ' I I I I v I t I I I I I 8 I I I I t I I 0 I 0.60 0.60 1.60 1.50 2.00 2.50

X

+ L

0 - a -0.00 ' .OO 1 -2.00 I I I I I I I I I I I I I I I I * ' " I

0.60 0.60 1.60 1.60 2.60 2.50 x

Figure 6. Chordaise pressure distribution for test-case I at 5% span.

a rectangular wing of aspect-ratio 4.0, with tip dihedral beginning at the mid-span location. The wing is at a steady angle of attack in a stream of

M = F . Figure 9 shows the lift-curve slope, CLa , as a function of the tip dihedral angle, y . Results from Ref.33 obtained with a Mach Box method and analytical theory, are used €or comparison. The three results are seen to be in good agreement, although the DPM slightly underpredicts the exact values at high fold angles.

247%

Doublet-Point Method Mach- Box Method

- 0 . 3 O 1 , , t , I , + ( I , I I 0.60 0.60 1.do 1.60 2.60 2.50

X

+ L

0 - -0.00 '.OO 1 -2.00 I I , I , I I , I , I I , 1 , , , I , , , I

0.00 0.50 1.60 1.60 2.d0 2.50 X

Figure 7. Chordwise pressure distribution for test-case I at 15% span.

111. Highly Swept Nan-coplanar Wing-Tail

Th i s example tests t h e a p p l i c a b i l i t y of t h e nonplanar DPM t o a h i g h l y swept w i n g- t a i l i n t e r f e r e n c e problem. The planform geometry i s shown i n F igure 1 0 , wi th t h e t a i l s epa ra t ed 0 . 2 u n i t s v e r t i c a l l y from t h e wing. The leading- edge sweep ang le s of t h e wing and

t a i l a r e 63 .44" and 50.2", r e s p e c t i v e l y . Resu l t s from t h e Piecewise Continuous

Kernel Funct ion methodz3( PCKFM) are s e l e c t e d f o r comparison. The PCKFM d i v i d e s a l i f t i n g s u r f a c e i n t o s e v e r a l boxes whose boundar ies are Mach l i n e s . A cont inuous pressure polynomial i s assumed i n each box, and c o n t r o l p o i n t s a r e c o l l o c a t e d f o r t h e s o l u t i o n of t h e polynomial c o e f f i c i e n t s . In t h e p r e s e n t example, PCKFM employs 38 unknowns i n t h e pres .sure polynomial . The chordwise p r e s s u r e c a l c u l a t i o n s f o r M = F and a s t eady ang le of a t t a c k a r e shown i n F igures 11 and 1 2 f o r 55% and 80% span l o c a t i o n s , r e s p e c t i v e l y . The PCKFM is seen t o produce jumps i n t h e p re s su re d i s t r i b u t i o n , which a r e n o t p o s s i b l e i n a l i n e a r i z e d f low, and appear t o be due t o t h e PCKFM's a l lowing d i s c o n t i n u i t i e s i n t h e p r e s s u r e polynomial a c r o s s Mach l i n e s z 3 . In c o n t r a s t , t h e DPM r e s u l t s ,

0'40 1 Doublet- Point Met hod Mach- Box Method

-0.20 ~ , , , , , , , , ~ i , , , , i , , , , i , , , , 0.00 0.50 1.00 1.50 2.00 2.60

X

ri

-1.50 ~ ~ ~ ~ ( ~ n ~ , , , , , , ~ , , , , , , , , 0.00 0.50 1.00 1.50 2.60 2.20

X Figure 8. Chordwise pressure distribution for test-case I at 95% span.

ob ta ined by t a k i n g 5 and 1 0 spanwise g r i d d i v i s i o n s ( s t r i p s ) , r e s p e c t i v e l y , a r e seen t o be w e l l behaved. I n a d d i t i o n , a l a r g e disagreement i s seen on t h e t a i l p r e s s u r e s , which appears t o be l a r g e l y due t o an assumption i n t h e PCKFM t h a t t h e Mach cone boundar ies i n nonplanar i n t e r f e r e n c e can be taken a s s t r a i g h t l i n e s , where a s t hey a r e a c t u a l l y hyperbolae .

The v a r i a t i o n of t h e o v e r a l l l i f t - c u r v e s l o p e , CLcv , with t h e nondimensiunal v e r t i c a l spac ing , h /b , i s presen ted i n F igure 13 . Both PCKFM and DPM p r e d i c t t h a t t h e s t eady CLe i n c r e a s e s s h a r p l y as t h e v e r t i c a l spac ing assumes a nori-zero v a l u e . With f u r t h e r i n c r e a s e i n t h e v e r t i c a l spac ing , t h e increment i n CLcr becomes less g r a d u a l . The d iscrepancy between t h e PCKFM and DPM r e s u l t s is seen t o i n c r e a s e wi th h/b. This is aga in probably due t o t h e f a c t t h a t while PCKFM ignores t h e Mach boundary c u r v a t u r e , which becomes l a r g e r wi th i n c r e a s i n g h/b, t h e DPM accounts f o r t h i s cu rva tu re c o r r e c t l y ( e q u a t i o n ( l 1 ) ) .

I V . F-18 Wing wi th Leading-Edge Flap

Ref.19 p re sen ted r e s u l t s of t h e r e c e n t l y developed ZONA51C Harmonic

2472

a Exact Theory A Oonalo L Huhn 0 Nonplanar Doublot Polnl Method '1

0 M to w Y

Fold ,Anale (Oapnea)

Figure 9 . V a r i a t i o n o f l i f t c u r v e- s l o p e w i t h f o l d a n g l e ( t e s t - c a s e 11)

G r a d i e n t program f o r an F-18 wing w i t h o s c i l l a t i n g l e a d i n g- e d g e f l a p . F i g u r e 1 4 shows t h e p lanform geomet ry . The f l a p i s o s c i l l a t i n g a t k=4.0 i n a s t r e a m o f Mzl.1. Chordwise p r e s s u r e ( r e a l p a r t ) d i s t r i b u t . i o n o f R e f . 1 9 a t 5 8 . 8 % span is compared w i t h t h a t o f t h e DPM i n F i g u r e 15. The f i g u r e shows a good agreement between t h e two methods on t h e p r e s s u r e jump a c r o s s t h e h i n g e - l i n e . However, l o c a l f l u c t u a t i o n s a r e obse rved i n t h e p r e s s u r e d i s t r i b u t i o n c a l c u l a t e d by t h e DPM n e a r t h e l e a d i n g and t r a i l i n g e d g e s . S i n c e t h i s i s a p l a n a r c a s e , t h e c a u s e of s u c h f l u c t u a t i o n s d o e s n o t l i e i n t h e n o n p l a n a r e x t e n s i o n p r e s e n t e d i n t h i s p a p e r . S i m i l a r chordwise p r e s s u r e f l u c t u a t i o n s were r e p o r t e d by Ueda and

Dowe1lZ4 w i t h t h e i r p l a n a r DPM on swept wings , when boxes o f low a s p e c t - r a t i o were u s e d . Al though an obv ious s o l u t i o n t o t h i s problem would seem t o be t h e u s e o f h i g h a s p e c t - r a t i o e l e m e n t s , t h e spanwise p r e s s u r e d i s t r i b u t i o n would n o t be a d e q u a t e l y r e p r e s e n t e d by such e l e m e n t s on a s u r f a c e such as t h e F-18

wing. Ueda and Dowe1lZ4 a t t r i b u t e d t h e f l u c t u a t i o n s t o h i g h l e a d i n g edge sweep a n g l e s , a r g u i n g t h a t t h e DPM's assumpt ion o f a c o n c e n t r a t e d l o a d w i t h i n a box makes it i m p o s s i b l e f o r a s i n g l e box t o a c c o u n t f o r t h e e f f ec t o f t h e sweep, s i n c e t h e i n f l u e n c e o f a box upon i t s e l f i s always e v a l u a t e d o v e r a r e c t a n g u l a r a v e r a g i n g r e g i o n , which i s i n d e p e n d e n t o f t h e sweep

2.0

1 0.0 1.0 2. 0 a. 0 4 . 0 1.0

x F i g u r e 10. lion-coplanar swept wing t a i l ( t e s t - c a s e 111)

2.0 3. 0 4 . 0 -0. I

1.0

f i g u r e 11. Chordwise p r e s s u r e d i s t r i b u t i o n a t 55% span ( t e s t - c a s e 111)

F i g u r e 12. Chordxise p r e s s u r e d i s t r i b u t i o n a t 80% span ( t e s t - c a s e 111)

0. 2 0 , 4 1.0 I 0 . 0 t I

F i g u r e 13. V a r i a t i o n of the l i f t curve- s lope wi th v e r t i c a l s e p a r a t i o n ( t e s t - c a s e 111).

2473

a n g l e . However, s u b s e q u e n t exper iments28 w i t h t h e shape o f t h e a v e f a g i n g r e g i o n r e v e a l e d t h a t t h e f l u c t u a t i o n s p e r s i s t e d ( t h o u g h o c c u r i n g a t o t h e r chordwise l o c a t i o n s ) when t h e a v e r a g i n g r e g i o n was a l l o w e d t o have t h e same s h a p e as t h e r e c e i v i n g box. I t now appears t h a t t h e problero i s c o n n e c t e d t o t h e l o c a t i o n o f t h e p o i n t d o u b l e t w i t h i n a box, r a t h e r t h a n t h e s h a p e o f t h e a v e r a g i n g r e g i o n .

The i d e a l s o l u t i o n t o t h i s problem would be t o f i n d a n a n a l y t i c a l j u s t i f i c a t i o n f o r t h e c o n c e p t o f normalwash a v e r a g i n g , which w i l l l e a d t o a un ique shape o f t h e a v e r a g i n g r e g i o n a s w e l l as t h e f i n a l l o c a t i o n o f t h e doublet . p o i n t i n a box . The a r b i t r a r y c h o i c e o f t h e s e p a r a m e t e r s made i n Ref. 24 a l s o r e s t r i c t s t h e DPM t o boxes w i t h e q u a l

widths": 1M .

m -

is .

im . >.

8 0 -

-40 I,, , , , * , , , , , , , , , , , , ~

-40 0 w 80 im 1s Dl

X

Figure 14. F-18 wing with leading-edge flap (test-case IV).

5.00

4.00

3.00 A

5 a

0.00

- 1 .oo 5

(3 ZONA51C 0 Doublet-Point Method

3 1oo:oo 150100 200100 X

C o n c l u s i o n s

The c o n c e p t s o f d i s c r e t e a c c e l e r a t i o n p o t e n t i a l d o u b l e t s and a v e r a g e d normalwash a r e deve loped i n t o a g e n e r a 1 Do ub 1 e t P o i n t method, w i t h o u t i n t r o d u c i n g f u r t h e r a p p r o x i m a t i o n s . The p r e s e n t method o f f e r s t h e greatest d e g r e e of commonality w i t h s u b s on i c c a1 c u 1 at. i e n s , and i t s i nco r p o r a t. i on i n t o a un i f i e d s u b s o n i c / s u p e r s o n i c code i s t h e e a s i e s t compared w i t h o t h e r p o s s i b l e a p p r o a c h e s , d i f f e r i n g from t h e s u b s o n i c Double t- Poin t method o n l y i n t h e k e r n e l i n t e g r a t i o n . Al though t h e c o n c e p t o f normalwash a v e r a g i ng i s an a 1 y t i c a 1 1 y e s t a b l i s h e d , it i s s e e n t o be v a l i d n u m e r i c a l l y f o r some c h o i c e s o f t h e a v e r a g i n g r e g i o n . The scheme c a n be made more g e n e r a l i f a n a n a l y t i c a l j u s t i f i c a t i o n c a n be o b t a i n e d f o r t h e a v e r a g i n g c o n c e p t , which can l e a d t o a un ique s e l e c t i o n o f t h e a v e r a g i n g p a r a m e t e r s . I t may a l s o s o l v e t h e problem of s m a l l chordwise p r e s s u r e f l u c t u a t i o n s on swept g e o m e t r i e s , which i s i n h e r e n t i n t h e p l a n a r D o u b l e t- P o i n t a p p r o x i m a t i o n . Such l o c a l p r e s s u r e f l u c t u a t i o n s a r e , however , se l f -coropensat .ory i n a s e n s e t h a t t h e o v e r a l l p r e s s u r e d i s t r i b i i t i o n , which i s of i n t e r e s t i n a e r o e l a s t i c a p p l i c a t i o n s , i s n o t a f f e c t e d . With t h e p r e s e n t c h o i c e o f t h e a v e r ag i rig p a r axle t e r s , Doub 1 e t - Po i n t method i s l i m i t e d t o g r i d s w i t h uniform spanwise d i v i s i o n s , l i k e t h e Mach Box method. However, u n l i k e t h e l a t t e r , i t s g r i d geometry i s i n d e p e n d e n t o f Mach number. C o n s i d e r i n g t h e o t h e r a d v a n t a g e s of t h e n o n p l a n a r Double t- Poin t method, s u c h a s p o i n t l o a d s and commonality w i t h t h e s u b s o n i c case, it is a n a t t r a c t i v e :;theme f o r e f f i c i e n t a e r o e l a s t i c a p p l i c a t i o n s on g e n e r a l c o n f i g u r a t i o n s .

n onp 1 an a r s u p e r s on i c

n o t

t h e

Appendix

The k e r n e l f u n c t i o n g i v e n by e q u a t i o n s ( 2 ) - ( 4 ) c a n be s i m p l i f i e d by

34 u s i n g t h e series e x p a n s i o n s of Laschka

and Cunningham"" f o r t h e n o n - r a t i o n a l terms i n t h e i n t e g r a n d s , and e x p r e s s e d

or)

as :

For X , 1 0 :

K ( x , y , z , k ) = e r

-ikX2

2

e (x/R - 1) - i k A 2 / r + i k A , / r + - r

r

-ikX1 (x/R + 1)

4 I+-- R r r -+ T 2 e

Figure 15. Chordaise pressure distribution on F-18 Ring at 58.8% span.

2474

- ikM3 ] + T2e -ikX2 1% 2 - 24 -(x/R - 1) r 2 R (x+X1)

+ ikM3 + T2 (3ik A2/r3 - 3ik A1/r 3 R 2 (x+X2)

3 1 I - ik B2/r3 + ik B1/r )

For X1 < 0:

-ikX, e " + - ) + 2/r2 + ik A3/r - iB A2/r

2 r 9 r

2 1~ - p(x/R - 1) R r r

ikM3 -(x/R - 1)

-ikX1 {?Zx + 2k2 A4/ + T2 e

-iBX2 p2x

R2 ( x+X1 1 [;;;;- r 4 1

+ ikM3 + T2(-4/r4 - 6k 2 A4/r 2 2 R (x+X2)

-3ik A 3 / r 3 + 3ik A2/r3 + 2k2 B4/r2

where

3

I - ik B2/r3 + ik B3/r )

-(nc+ikr)Xl/r a e n

nc + ikr A1 = c A2 = 1

A3 c

n= 1 -(nc+ikr)X2/r

a e n nc + ikr n= 1

(nc-ikr)Xl/r a e n

nc - ikr n= 1

a n 11

(ncI2 + (kr)2 A4 = z n=l 11 -(2nc+ikr)Xl/r

- 7 bne B, -

11 -(hc+ikr)X 2 /r bne 2

n=l 2nc + ikr

11 (2nc-ikr)Xl/r bn e

2nc - ikr B.3 = 2

n= 1

The coefficients c, an and bn are given in Table 1.

r I Table I. THE SERIES CONSTANTS OF LASCHKA AN0 CUNNINCHAM

n

c = 0.372

4

I 1 -3.509407

2 2.7968027 57.17 120

3 -24.99 1079 -624.7548

4 1 1 1.591 96 3830.151

-0.24186198

5 -271.43549 -14538.51

6 305.75288 35718.32

7 4 1.183630 -57824.14

a -545,98537 6 1303.92

9 644.78155 -4W69.58

10 -328.72755 15660.04

11 64.27951 t -2610.093

Acknowledgement

A part of this work was carried out at the University of Missouri-Rolla, under contract with McDonnell Aircraft Company.

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