AIAA84-0425 Application of the Green’s Function Method for 2- and 3-Dimensional Steady Transonic Flows K. Tseng, ICARUS, Inc., Boston, MA

AlAA 22nd Aerospace Sciences Meeting January 9-1 2, 1984/Reno, Nevada

For permission to copy or republish. contact the American Institute Of Aeronautics and Astronautics 1633 Broadway, New York. NY 10019

U APPLICATION OF l I lE G R & ~ ' S FUNmION KE'JROD POE 2- AND 3-DlllhlSIoNAL STEADY TIWJSONIC PLOWS'

Kedin Tseng. ICrnS, 1nc.

Boston. Uassiohosetts

Abstract

A Time-Domain Green's function mothod for the nonlinear time-dependent three-dimensional aerodynamic potential equation is presented. The Green's theorem is beins used to transform the partial differential equation into an integro- differential-delay equation. Pinite-element and finite-difference methods are employed for the spatial and tire di.cretizatioms to a p p r o ~ i r a t e t h e i n t a s r a l e q u a t i o n b y a s y s t e m o f differential-delay equations. Solution may he o b t a i n e d h y s o l v i n g for this n o n l i n e a r simultaneous system of equations in time. T h i ~ paper discnsaes the applioaticn of the method to tho Transonio S m a l l Distorbsnco Eqnstion end nometical resnlts zor lifting and nonlifting sirfoiIs and wings in steady f lows are prerentod.

I Y, n "r N P P. P P. R t T

B

Q 0,

y. Y

~Omenclstnre

speed of sound aspeot ratio prosanre ooefficient chord domain rnnCtion. see Eq. (121 nondimenaionsl transonio parameter, V,lfl' reference length Yich nluber, UJa, norms1 ro 0

noma1 to point having coordinates ~ , y . r control point, (x.,y,.z,) point having ooordinates 1.Y.Z control point, (I.,Y,,Z,)

time nondimsnsionel time 0,tli velocity of nndistorbed flow space ooordinetes

E norma1 to 0"

IP-P,I

nondimension.IPrsndtl-Gl*n~~t

snrf;ce of body in 1.Y.Z space snrface Of wake in 1,Y.Z SPaCe ocnvective time delay, Eq. ( 8 ) perturbation velocity potential nondimensional pertorhation velocity

nonlinear terms normal wash in x,y,; space

Potential, 9lU,P

n,, a l i n 1.Y.Z spree +This wo:i?as suppotted by NASA Langley Rasearch Center under Grant NOR 22-004-030 to Boston University and Contract NASI-17317 to IcbpIIS, Inc.

Research Scisntfst, member AIM.

e. - transonic parameter. (T+i)u,/a: 1. nondimensional nonlinear tems,fllU., v

OB -H

SUZface snrronnding body end WL e snrfmce or body In x,y,x space surface 01 wake i n x.y.z IPSEO

,%biorints ~n~erscrints

B body TE trailing edge W wake 0 freestream oondltion

L. Introduction The work presented in this paper Is b81.d

upon the Green's Pnnctioq Yethod for unsteady potential 11ows of Yorino . T%is mlt\$d differs from other integral equation methods - In that theboondarp ocnditicn is satisfied on the true sorfece rather than on a zero thickness surface. PurthCrmOre. the theory I s applicable t o arbitrarily complax configuration nnderaoing onsteady motion.. The nnmerioal formulations for (linear) anbeonic and snpersonic potential f l o w s are presented in Refs. 11 en4,_13, Snbseqnently. Yorino and his oc-workers progressively implemented tbie linear formulation into a computer code SOWSSA (Steady. Oscillatory e n d U n s t a a d y , S u b s o n i c a n d S u p e r s o n i c Aerod~naoica). The Linearized velocity potential formulation is adequate in so far a s it is for the prediotion of tlows in which the entire flow field is well within either the sobsonic or snpersonic rinse. For tlows with local speed close to the speed of sound, the nonlinear terms Of the Original tu11 potential equation beoomes increasingly important end the linearized potential 110" assumption is no longer correct. More importantly. at thia high subsonic (01 low snpersonic) range, the :low field becomes mixed. In other words, a snpersonic (snbsonic) resfon may appo8r 1n an otherwise snhsonio (snpersonic) flow field. Thus. the n o n l i n e a r velocity Potential eqnation. snob as the Small Distnrhanca oqnstion. must be used in order to model this physical Ilow phenomenon.

T o w a r d s r h i s s o i l . Yorino and Tseng" de8onstrated the feasibility of a time-domain Green's rnnction fornolation t o subcritical as w e l l as (shock-free) suPerffitica1 flows. In 1982, Tieng and Motino presented the formulation for unsteady treqspic flows which is >asad upon the work of Tseng .

1

Presented h a r e i s a solution method, the detail of wnich may bo found in Ref. 21. for the nonlinear nnsteady t h r e e - d i m e n s i o n a l aerodynamic potential equation wherein the Green's fonction, the finite-element and the finite-difference method, are used to transform the differential sqoation into integro, differential-delay equations in time. These sqnations may then be discretired, in space and time, to yield a nonlinear algebraic system of equations In time. This system o f diffcrenos equations relates the valuns of the unknown velocity potential at time t to the prescribed normal wash and the values of the potential and its derivatives at preceding times. The veivcity potential can thus be obtained step by step in time. Shock n e v e , if present, i s 'captured' in the time marching procsas. Tho formulation h a s been implemented into a computer program. Nnmerical results obtained from this code indicates that the method i. numerically stable and it is capable of capturing shock w m e .

2. B a s i c Eaaationa

The governing nonlinesr equation of the aerodynamic potential may be written a s unsteady

surface IS embedded in the flow field. Therefore. only the potential jump across the wake need be consxdored a s aireontiauitg. The boundary conditions aseociated w i t h the partial differential oquation, Ey.(l). are v

at - (4 )

where a is the outwardly directed unit normal to the m d y surface, eB, (see Figure 1). v i s the n o r m a 1 w r s h , A i s the difference of the function bstwaan the infinitesimally t h i n surface l8yers on the wake and r 1s the time necessary for the vortex-point to travel from the point, Pte (origin of the trailing edge, see Fig. I ) , to the point P on the w a k e .

With 9 governed by Eqs. ( 3 ) . through (8 ) . the pressura coefficient C may be evaluated through the Bernoulli theorem, P by

where $ is the perturbation potential w h i l e a, and U- a r e t h e speed of sound and fluid. respectively, of the undisturbed flow and

where sobscripts 1, y, I and t denote spatial and time derivatives and y i s the ratio of specific heat.

Eq. (1). w i t h 7 g i v e n b y Eq. ( 2 ) . represents tho exact nonlinear velocity potential equation. With t h e small disturbance assumption. this equation is reduced to the well known Transonic Small Disturbance Equation, i.e.

Aaaociated with ' 9 . ( 3 ) are two sources o f diacontinuitier. namely, the potentisl diacontinuity a o r o s s the w a k e and the discontinuity in the normal deriyativs of the potential acroaa the shock. Tseng showed that the intsiral formulation derived for a shock-free flow contains the shock implicitly. In other words, Reference 1 proved that the integral formulation, with the shock surfass treated a $ discontinuom sorfaee, is identical to the shock- free tormulation and therefore the latter 1 6 capable of crptnrins the ahock wave if a s c h

( 9 )

- 3. Transonic Integral Eauation

Introduce the gsnoraliesd Prandtl-Glauert transformation and nondimcniion.liration. 1.0.. set 4

,b - Wv,e

where 1 is a reference lenlth.

Appiying Green's Theorem to Equation 1, one obtains

where S m c e Discretization

As noted in Section 2. dnp,to tho shock- 1 if P. 11 outside LE captnring natura of the method the explicit

E(P*) - 112 if P* is on I (12) treatment of the .hock surface 1 s not needed for 0 Equation (11). The xollowing procedure 11 taken

to diacretire the apace integral operator over the body (aircraft) aurfaceEB, wake surface Zw and flow f i e l d V :

if P. is i n r i i o Z E

I % / a

I v

R - [(X-X.)'+(Y-X,)'+(Z-Z,) 1 (13)

(14)

while

0 - [Y,(X-X.)+RIY.I~ (151) and

Q [ ~ ( X ~ - X ) + R l M ~ l 8 (15b)

In aadition, alaN i s the normal derivative in the Prandlt-Glauert apace and

(16)

a) L is aivided into NE small quadrilateral psne8s. w i t h i n which Vand d, are t a k e n t o be constant.

b) As L oonsequence of Eq. (17). m a l o n g the r a k e 1s not an additional unknown. To facilitate the use of Equation (17). the wake rnrf*ceL is divided into strips. Theses atrips are define! by (ateaay-itate) vortex l i n e a emanatin: from the nodes on the (wake-generating) body trailing edge('). These strips a r e then subdivided into N elements with nodes along the vortex l i n e s . Wlth Eq. (17). the potential diacontinuity. say. of wake element En can then be expresaed. in terma of potential on the body trailing edge of the wake atrip in which Xn is located) as

Indioatea the component of the nondimensionil ve1ocity in che airection of the normal N to the anrfsce of the X.T.2 space (not in the direction of the normal n to the rurfsoe a of the physical apace ) and is known from the boundary conditions. The relationship between Y and" Is discnssed in Reference 21.

The potent#rl jump, Ad, (P,T). across the

u AWP,T)-A6(PTE,T-ID (17)

where n is the nondimensional time necessary for a particle on the wake to travel (along a vortox l ine , within the steady flow) from the point P cri:in or the vortex line at the trailing e d g e y to the point P.

wake IS :Ivan by

Note that rhe aircraft anrface,Z, can be thon:ht of a s composed of two branches. The first & a the. (cloaad) snrface of the hody,Is. Tho secon6 is the (open) surface of the w a k e , Lw. Note a 1 1 0 chat =he snrface of the wake is considered twice since it is a cloaed snrfacs. In other words, the two aids. of the wake a r e considered to he two independent surfaces, having the same eqnation but oppoaite outwardly-dir8cte.d normals, Nz and N.. respectively. Porthermore. AV-0 on the wake.

L Numerical Pormnlations Upnation (11) describe. the problem of

n o n l i n e a r u n s t e a d y p o t e n t i a l transonic aerodynamics around complex confi'nrationa. In ora81 to aolvethia problem. it is necessary to introdnce a nnmerical approximation for Equation (11). Soon a nnmerical spproximation inclndea the introduction o f a :enoral finite-element repreaentation for the space dircretization and the time discrstization is accomplished through finite-difference. The spatial dstir.tivss of 6 in the nonlinear term i n spproximsted by mired csntrs~l/backward aiffersncing of Mornan and Cole.. .

where u h-l (S*--l), if n tdentities the upper- ride (?ower-aidn) panel L h on the body correaponding to the nth element Z n o n the wake , and Snh - 0 otherwise. c ) Shilsrly, the flow field i 8 divided into N voltme elements. Within each of these elementr. 3 1% a i i o approximated by ita valne at the centroid of the element.

With the spatial discretizaticn procedurea ( a ) , (b) and (c), Eq. (11) becomes

N"

whore 6 (TI- d,(P ,T) w h i i e P is a Control point whioh m i y eithoi bo On the dody or In the flow field. In addition,

3

These integrals m r y either be oalcolated analytic$+ly or b y a mixed analytical-nnmcrical proceaore . Time Discretization

Bqnation (19) indicates the nature of the aerodynamic operator relatin& potential and normalwash aa obtainod by n t i n g finite-element rspresenr.tion to disorotire tho spatial problem. In this section the time derivatives term8 sn Equation (19) will be discretized. Set

t- (m+a)At (m-integer and OLa<1) ( 2 1 )

Next, consider a continuoar function F(t) snch that, by using L rorward finite-difference approximation. the first dsrivrtivo of F may be written as

( 1 2 ) dF(t) F(t) - P(t-At') + __ - __

dt At,

where At' is an arbitrary time inorement not neoessari ly the same as At.

With the forward diffsronoing, Eqeation (22). P(t) may then be expressed s t

F(t) - F[(n+a)Atl z(l-a)F(nAt) +aF[(n+l)Atl (23)

Yakins nse of Eq. (21). the delry times of Eq. (191 m.7 be written .I

ejh - ( m j h + ajh)At

ejh +At' = (mjh + ajh)At

e. + n n - (mjn + ajn)At ( 1 4 ) Jn

ejn + nn + At' = (mjn + ajn)At

ejq - ( m l q + ajq)At

and

N e x t . with Eq. (24 ) . define

uf = B. (1-a ) J h Jh l b

~3~ - B. Jh a. ~h

Co = ( C + D lAt*)(l - a],,) Jh lh lh - 0

DSh = - D lh IAt'(1 - ajh) = o

D. = 0 i h

E - D l a IAt'(l-ajh)

Df = - D IAt'a' i b lh l h

if mjh - 0

if mjh # 0

if mjh - 0 if mlh # 0

( 2 5 )

if mlh - 0 if # 0

if mjh - 0 if mlh # 0

F l n = (ejn + G J ,,lAt')(l - aIn) if mjn = 0 u = o if mjn # 0

if mln - 0 if mJn f 0

(26)

G9n - QjnlAtl(l - ajn) if mln - 0

= o if mJn # 0

Ff = 0 Jn

= (Fjn + GlnlAt')(l-ajn)

Fin = (Flu + GjnlAt')a In

if mjn - 0 Gf - 0 J n

= - 13jn/At'(l-a;m) if mjI # 0

4

(28)

In Eq. (281 . At has been omitted for notationaI simplicity, Lo., WMt) is written as Q(k). Furthermore. the last two terms on the riiht band side O X Eq. (29, w e r e obtained by makine use of 8q. (18).

Note that the non1ina.r t e r m at delay- time k-m becomes the current time%f mJP-O. A simple p&ictor-corrector (relaxation) scheme is employed here. Specifically.

u

t ( k ) - A " t ( k ) + 1 1 - A ) :-'(k) ( 3 0 )

where the superscript. n and n-1 indicate iteration count w h i l e A 1 s an arbitrary relaxation coefficient valued between 0 8nd 1. In addition, waile '(k) i s governed by Equation

P ( 3 0 ) . :;(k) is approiimated by takin& finite- difference of the flow field potential. Note that in order to account for the local behavior in the flow field ( i . 8 . . subsonic or supersonic), a mired centtal/backward, ginita-difference scheme due to Yurman and Cole is introduced for the eV8lUStiOn Of .

Thus, iterate on Equstion (28) yields the solution. 4, . at curront tins step k. With the exception or the Isat term. the right bsnd side of Equation (28) i s oompletely determined and may be represented simply ne

lb:ln = 17.B;hVh(k-mjh) + i B;h@l'h(k-m j h -1) h h

with J - 1.2.3 ....., NE (31)

where subscript '1' indicetea that the Ng control points a r e on the body surface while the superscript N indic8tss iteration counter. Note that Equation (28) includes the nonlinear term, 1 ,which is not known st current time step k and hsnoe an iterative prooesa is necessary S O that @ on the aorfacs may be solved. The current time step index k has been omitted in Eqmation (31). Unless otherwise noted. a11 erptsasions discussed hereafter w i l l ha understood a i at a fixed time step k. Nsrt. write

lbxln = IA*l-' l h i l n (32)

IB,1 - I A ~ I - ' I d I (33)

where [AI-' is the inverse of ,the NE I N matrix detinod 1n L(q. (291 while 1E11 is t h e a n x N matrix obtained by applying Eq.(27) to NB controY points on tha body sorfaoe.

Finally, with Eq. (31 ) to ( 3 3 ) . Eq. (28) may be written in matrix form 8s

141,1n = Ib,ln + IB.ll=Jn 134)

where IID1ln i a the vector of the potentials st N control points on the body surface st the ntf itoration.

Siailarly, the oorrespondine potentials in the flow field are obtained by applying Equation (28) to Nv control points in the flow field. Once these field potentials are computed, may be approximated Dy taking finite-difference of these potentiall. With subscript '2' to represent flow field control points, these field potentisls can compactly bo written as

(4bIn = lbsln + IB,1 1 5 1 ' (35)

Unlike 89. ( 3 4 ) . where the inversion of matrix A. is necessary, Eq. 0 5 ) re resenti a

included in (b,ln, i a known from Eq. ( 3 4 ) . Consequently,

aimple algebraic equation, since lea) B , which i s

Smmarv of a Nnmerical Algorithm The numerical aI(orithm described ahore may

be slumarired in the following :

(1) For xtime step k, mako initial auoss for (I 1 , whioh i s taken in thia report as the last iterated value of 1-71 of prosedine, or (k-l)th t i m e step.

5

(2) Calcula te ( , P a l n 8ecording t o Bq. (34).

( 3 ) With lc41n, c a l c u l e t s (Ubln a c c o r d i n g t o Rq.

( 4 ) Compute I ?TIn by m i r e d c s n t r a l / b a c k v a r d

(35)

d i f f e r e n s e .

(5 ) Compute (:In according t o Mq. ( 3 0 ) .

( 6 ) R e p e a t ( 2 ) . (3) . ( 4 ) and ( 5 ) u n t i l d e s i r e d convergenos c r i t e r i a i s met.

( 7 ) Repoat (1) t o (6 ) f o r ( k t l ) t h t i m e s t e p .

z ti- I C & -u

A nonl ine8r Green'a func t ion method f o r the t i m e - d e p e n d e n t t h r e e - d i m e n s i o n 8 1 t r 8 n s o n i c . e r o d y n 8 m i c p o t e n t i a l e q u t t i o n h a 8 b e e n p r e s e n t e d . Tho numerical f o r m u l a t i o n o f t h i s method has b e e n implemented i n t o a computer code.

N u m e r i c a l r e s u l t s a8ve b e e n o b t a i n e d f o r s e v e r a l c a s e s u n d e r d i f f e r e n t f l o w c o n d i t i o n s . The &low c a s e s p r e s e n t e d i n t h i s p a p e r may be s e p a r a t e d i n t o two c a t e 8 o r i e r . The q u a s i two- d i m e n s i o n 1 1 o a s e s w i l l bo p r e s e n t e d f i r s t , fol lowed by the three-dimensional r e s u l t s . Most of t h e r e s u l t s ubtained w i t h t h e p r e s e n t method a r e compared w i t h e x p e r i m e n t a l d a t a a n d l o r f i n i t e - d i f f e r e n c e so lu t ions .

For hiph a s p e c t r a t i o wings, the p o t e n t i a l , a n d h e n c e t h e p r e a s u t e c o e f f i c i e n t Cp. i s cons tan t a long the spanwise d i r e c t i o n so t h 8 t the number uf unLnowns, p o t e n t i a l , st c e n t r o i d s ox f i n i t e - e l e m e n t p a n e l s , m a y b e r e d u c e d s i g n i f i c a n t l y . C o n s e q u e n t l y , t h e o o s t f o r c o m p u t i n g t h e r e a u c e d a l g e b r a i c s y a t e m is much lower than rhe f u l l system. Thus, t h e ques t ions of sccu racy , s t a b i l i t y and c o n v e r g e n t - r 8 t e of t h i s m e t h o d m a y b e a n s w e r e d r e l a t i v e l y iuexpensively through ana lyses of two-dimensional problems. A s long as no swept shock, which is not c o n s i d e r e d in t h i s paper , is i n v o l v e d , t h e c r i t e r i o n e s t a b l i s a e d f o r two-dimensional f lows s h o u l d be v a l i d f o r t h r e e - d i m e n s i o n a l f l o w s a s w e l l . This is ~ a i n g substantiated by t h e nnmerical r e s u l t s o f two r e c t a n g u l a r wings a t f l i s h t maon number of 0.82 and 0.908. I t s h o u l d he p o i n t e d o u t here t h a t a l l t w o - d i m e n s i o n a l r e s u l t s a o t a i n e d in t h i s paper w e r e computed throurh a quasi two-dimensions1 manner. I n o ther words, h i g h a s p a c t r a t i o wings were osod t o achieva tha t-o-dimensional s f f a c t of t h e flows.

S t e 8 d ~ s t a t s Nonl i f t ing Plor

P r e s e n t e d zn F i g u r e s 2 t h r o n g h 8 a r e t h e n u m e r i c a l r e s u l t s o b t a i n e d w i t h tho p r e s e n t mathod t o r L 6% t h i c k c i r c u l a r a i c o n v e x a i r f o i l in a t e a d y - s t a t c f l o w w l t h Y, = 0.908 and a s p e c t r a t i o AR-20. I n Figure 1, t h e p r e r e n t r e s u l t is compersd w i t h t h a t O t t h e f i n i t e - d i f f e r e n c e method of R e f e r e n c e 23, Filmre 3 shows t h e t i m e h i s t o r y o t rhe pressure c o e f f i o i c n t Cp a t a t i r e d poin t on the a i r f o i l . The data shown 8re f o r two and three i t e r a t i o n s (rs l .xat ion c o e f f i c i e n t 2 =

0 . s e e Bq. ( 3 0 ) ) p e r t i m e * t o p , w i t h t i m e i n c r e m e n t AT e q u a l t o 80 p e r c e n t chord . Shown i n Figure 4 is the comparison between t h e r e s u l t s of t w o a i f f e r e n t r e l a x a t i o n c o e f f i c i e n t s while t h e number o f i t e r a t i o n s is f i x e d a t two. The phenomenon o t o s c i l l a t i n g convergence disappeared f o r I = 0.5 . Note t h 8 t . with 1-0.5, t h e same smoothing t r e n d of oonvergence is observed for 3 i t e r a t i o n s p e r t i m e s t e p . I n c l u d e d in F i g u r e 5 8 r e t h e t i m e h i s t o r i e s o f t h e p r e s s u r e c o e f f i c i e n t a t a f i x e d p o i n t on t h e a i r f o i l b u t w i t h various t i m e i n f r e m e n t s . Figures 6 and I s h o w t h e p l o t s o t cb v s x a n d @ v s z r s s p e o t i r e l y . l?mse two f i g u r e s d e m o n s t r a t e t h e ex ten t LO which the nonl inear term i a importnnt. Accord ing t o Figure 6, t e n d s t o a c O n s t 8 n t ( a l o n g t h e x - d i r e c t i o n ) h a l f a chord l e n g t h in f r o n t o f t h e a i r f o i l l e a d i a g edge and h a l f a chord l e n g t h Behind t h e 8 i r f o i l t r a i l i n g edgo. F i g u r e 7 i n d i c a t e s , on t h e o t h e r hand, t h a t v a r i e s r e l a t i v e l y s l o w l y in t h e v e r t i o a l (2 -1 d i r e c t i o n . O u t s i d e o f t h e a i r f o i l region (1.0.. o u t s i d e YI 0 < I < 1 ) . is p r 8 c t i c a 1 1 y l i n e i s in I (end t e n d a l s o t o zero). Summariz ing . b a s e d on F i g u r e s 6 end 1. r a t h e r th8U a f l o w f i e l d which extends t o i n f i n i t y . a computat ional domain t h a t is oonfined t o the immediate neighborhood Of t h e a i r f o i l is s u f f i c i e n t f o r t h e e v a l u a t i o n Of 0 ( a n d h e n c e pressure c o e f f i c i e n t C,,)

Computed r e s u l t s *or a NACA 0012 a i r f o i l . w i t h M, and zero d e g r e e 8 n p l e of a t t a c k a r e p r e s e n t e d i n F i g u r e r 8 8nd 9 . Figure 8 shows t h e o o m p a r i s o n of t h e p r e s s u r e o o e f f i o i e n t b o t w e e n t h e p y p s s n t m e t h o d a n d f i n i t e - d i f f e r e n c e

f i e l d .

L/

accura te ly .

mothod . Figure 9 is a p l o t Of a /in Of the flow ~f

d Based on t h e r e s n l t s of F i g u r e 5 , i t may be

concluded t h a t the p r e s e n t method is n w e r i o a l l y s t a b l e and t h e sccuracy o f t b a s o l u t i o n is r e l a t i v e l y i n s e n s i t i v e t o the t ime increment ( t h e Cp c o e f f i c i e n t of s e v e r a l d i f f e r e n t r i m e i n c r e m e n t s c o n v e r g e s t o w i t h i n 0.25 o f e l o h o t h e r ) . F u r t h e r m o r e . f o r s t e a d y - s t a t e t l o w s . la rge t ime increment is more d e s i r a b l e since l e a s t i m e s t e p s would be r e q u i r e d f o r the t r a n s i e n t 6 0 l u t i o n t o resch s teady s t a t 8 (see Figurs 5).

S t e a d y s t a t e L i f t i n i Plows

P r e a e n t e d in F i g u r e 10 a r e t h e computed r e s u l t s of a NACA 0012 airfoil a t M,-0.65 w i t h angle O t a t t a c k d-2 degrees. A comparison between t h e computed resv,l,ta of t h e p r e s e n t method and experimental d a t a is presented. The computed r c s n l t r a r e g e n e r a l l y in good agreement w i t h the e x p e r i m e n t a l d i t s . I n c l u d e d also in t h i s f i g u r e are the computed Pressure o o s f f i c i s n t 8 . 8 r e s u l t o f n e g l e c t i n g t h e sr.11 d i s t u r b a n c e n o n l i n e a r te rm i n t h e V e l o c i t y P o t e n t i a l Equat ion . Note t h a t r h i s c a s e is only m i l d l y t r a n s o n i c -- the l o c a l Mach number exceeded u n i t y (&e., e x c ~ ~ d e d Sonic s p e e d ) only on t h e u p p e r s i d e o f t h e a i r f o i l and t h e r e f o r e a s shown in t h e f i g u r e , t h e r e is no d i s c s m 8 b l e d i f f e r e n c e between l i n e a r and nonl inear computed r e s u l t s on t h e lower s i d e of the a i r f o i l .

d I n F i g u r e 11, t h e computed r e s u l t s of t h e a i m 0 a i r f q i ) are compared w i t h those of experimental d a t a . The d i s c r e p a n c y o b s e r v e d b e t w e e n t h e numerical r e s u l t s and experimental d a t a is dns t o

t h e f a o t t h a t t h e r e s u l t s o f t h e p r e s e n t method are obta ined through i n v i a o i d o a l o u l a t i o n w h i l e e x p e r i m e n t a l d a t a a r e s t r o n g l y a f f e o t s d by t h e viaoon. e f fec t . .

$ t e a d r s t a t e Three-dimensional Plorr W

Inoluded in tho fo l lowing are the nomarical r e s u l t s uf t h e p z e s e t l t method a s a p p l i e d t o three-dimensional ( i . 0 . . l o r a s p e c t r a t i o ) a t e a a y - a t a t e n o n l i f t i n g flow. The r e s u l t s p r e s e n t e d a r e f o r an a s p e c t r a t i o 4 wing w i t h zero angle of a t t a o k , oons tan t 6% t h i c k c ircular biconvex a i r f o i l s e o t i o n s ( w h i c h is t a p e r e d o f f t o zero a t t h e t i p ) and t h e f r e e s t r e a m Macb number 8- - 0.908, F igure 12 oomparss tho r e s u l t s of the p r e s e n t method w i t h t h e f i n i t e - d i f f e r e n c e method o f Ref. 27 a t t h r e e a p a n w i s e l o o a t i o n r . The d i s c r e p a n c y b e t w e e n t h e two methods i s p o s s i b l y due t o t h e f a o t t h a t sinoe t h e a c t u a l r i n g surfaoe is being used i n the p r e s e n t method. t h e r a p s r i n g of the a i r f o i l t h i c k n e s s near t h e wing t i p 800.lsr.teS t h o flow i n t h e s p a n w i s e d i r e o t i o n . Ref. 27 on t h e o t h e r hand, t r e a t s t h e r i n g as a zero th ioknea . surfaos ( a l t h o u g h t h e a i r f o i l s l o p e i s r e t a i n e d i n t h e b o u n d 8 r y c o n d i t i o n ) .

WNCLUDING pEIuRI[s

A s o l u t i o n msthod f o r t h e n o n l i m a r t i m s - d e p e n d e n t t h r e e - d i m e n s i o n a l v e l o c i t y p o t e n t i a l e q u a t i o n using t h e Green ' s f n n c t i o n method has been out l ined . The theory has been implemented i n a oomputer code and numerical r e s u l t s , f o r s t e a d y - s t a t e a i r f o i l s and w i n g s , have b e e n o b t a i n e d . T h e r e r e s u l t s showed t h e metbod i s c a p a b l e o f ' o a p t u r i n g ' s h o c k w a v s ( r ) , a s demonstrated mathematical ly i n Ref. 21, a s .harp c h a n ' e s i n p r e a s m r e r a t h e r t h a n a s d i s o o n t i n u i t i s s . It s h o u l d be emphas ized h e r e t h a t t h e present method is s time-domain ansteady method. The s t e 8 d y - + t a t e r e s u l t s were o b t a i n e d through marching In t ime (not pseudo-time).

Based on the p r e l i m i n a r y n u m e r i c a l r e s u l t s computed f o r v a r i o u s r e p r c a e n t a t i v c c a s e s , t b e fol lowing Eonclnrions may be drawn :

1. A s s v i d o n o e d in t h e c o m p u t e d r e s u l t s d i scussed in Sect ion 5, the captured shock i s being smeared over rcver.1 computational g r i d p o i n t s . T h a t t h i s is S O i s due t o t h e f 8 E t t h a t i n s t e a d o f t r e a t i n g # b X x as s D i r a c d e l t a funct ion. i t is t r e s t e d as a continoous one which r e s u l t e d in t h o smooth ing of t h e p o t e n t i a l f u n c t i o n and t h u s t h e s m e a r i n g of t h e s h o o k . A p p r o x i m a t i n g i t b y backw.rdlscntr .1 d i f f e r e n c e smears i t e v e n f u r t h e r .

2 . The soorce and donblet integral c o e f f i c i e n t s a r e r a t b e r i n v o l v e d i n t e g r a 1 6 a n d consequent ly expensive t o compute.

The matrix. A,,,, on the left-hand-aide of the a l ~ e b r a i s . $ s t e m , Eq. ( 2 9 ) . is d e n s e l y p o p n l a t s d (and f o r l a r g e t i m e i n c r e m e n t s i t is f u l l y p o p u l a t e d ) a n d h c n s s is t i m e

3 .

u oonsuming t o 0.rry omt ." inverrion. 4. The e b o v e s h o r t - o o m i n g s h o w e v e r . i s

compensa ted f o r by t h e fmct t h o t i n t e g r . 1

methods, inc luding t h e preaent one, oonvsries mseb f a s t e r t h a n , f o r i n a t a n o e t h e f i n i t e - d i f f e r e n c e method. (PDM). T y p i c a l l y , t h e Green ' s i n n o t i o n method requires 25 t o 40 t i m e s t e p s Ior 8 a t e a d y - a t a t e s o l u t i o n t o c o n v s r & c t o w i t h i n 0.1 % d e p e n d i n g on t h e Mneh number a n d l o r wing a i r f o i l S e c t i o n c o n s i d e r e d . T h i s i s o o a a i s t e n t w i t h tb: 30 i t e r a t i o n s r e p o r t e d by Nixon and B8UOOOk .

5 . Where.. FDM g e n e r a l l y r e q u i r e s t h a t sm.11 t i m e i n c r e m e n t s be used i n o r d e r t o a v o i d numetioal i n s t a b i l i t y , f rom t h e t e a t o a s e s s t u d i e d . t b s p r e s e n t method s p p e a r e d t o be i n s e n s i t i v e t o t i m e i n c r e m e n t s i z e a n d t h e r e f o r e i a numer ica l ly very a tab le .

6 . Since f in i te -e lement r e p r e s e n t a t i o n is used, t h e present method requires no s o p h i s t i c a t e d a r i d - g e n e r a t i o n t e o h n i q u e a , as d o f i n i t e - d i f f e r e n c e methods.

7 . Al though source and d o u b l e t i n t e l r a l a 8re e x p e n s i v e t o c o m p u t e . t h e y a r e t i m e - independent q u a n t i t i e s and need be c a l c n l a t s d o n l y onto a n d s t o r e d . F o r u n s t e a d y c a l e v l a t i o n s , t y p i s a l l y severa l oyoles would be needed t o reach a converged s o l u t i o n . I t is expected t h a t tho present method would be more oompetitivw as a computat ional t o o l f o r rout ine t ransonic ana lys i s .

z n e p w

1. Morino, L., 'A General Theory o f U n s t e a d y C o i p r s s s i b l e P o t en t i a1 Aerodynam i o s' , NASA CR-2464. 1974.

2 . Oswatitrch,K., 'Die Gstchrindi~keitavorteilung ab S y m m e t r i s c h e P r o f i l e n be im a u f t r e t s n l a k s l e r U b o r s c h 8 l l g s b i c t e ' . Acta P h y a i o a A u r t r i a o a , Vol . 4. pp. 228- 271, 1950.

3. S p r e i t e r , J.R. and Alksne. A.Y., ' T h e o r e t i c a l P r e d i c t i o n of Preasurc D i s t r i b u t i o n s OD Non- L i f t i n g A i r f o i l s a t H i g h S u b s o n i c Speeds ' , NACA Rapt . 1217, 1955.

4 . Crown, J.C., 'C8lcu l . t ion of T r a n s o n i c Plow a v e r Thick A i r f o i l s by I n t e g r a l Method''. AIAAJoornal . Vol. 6. pp.413-423. 1968.

5 . N o r a t r o d . 8.. 'High Speed Plow p a s t Wings', NASA CR-2246, 1973.

6 . Nix0n.D.. 'Transonic Flow Around S y m m e t r i c A e r o f o i l r a t Zero I n c i d e n c e ' , J o u r n a l o f A i r o r 8 f t . Val. 11. pp. 122-124. 1914.

7. Nixon. D., 8 n d l lancock. G.J. , ' l n t o i r a l E q u a t i o n M o t h o d s - a P o a p p r a i r a l ' , i n S y ~ e o r i u ~ I a r r e r i n r ~ 11, E d i t o r s : I(. Oswat i t sch and D. Rues, S p r i n j s r Verlag. New York. 1976. pp.174-182.

8. Nixon. D., ' P e r t u r b a t i o n of a D i s o o n t i n o u s Transonic Flow, ' A I A A P i p e r No. 7 7 - 2 0 6 , J a n u a r y 19r7.

9 . N i x o n , D., ' C i l c u l a t i o n of U n s t e n d j

E q u a t i o n Method.' AXCC Loosgil. Vol. 1 6 , No. 9. S o p t . 1978, pp. 916-983.

Transonic Flows Using t b e I . t e*ra l

7

10. Piers, W.J.. and Sloorf. J.W., 'Calculation of Trrnronio Flow by Means of a Shock- Capturing Field Panel Method', AIM Paper 79- 1459, July 1919.

11. Morino, L., 'A PinLts-Element Formulation tor Subsonic Plow Around Complex Configurations.' BOstOD University, Department OK Aeroapace Engineering, 111-73-05. 1973.

12. Yorino, L., 'A Pinite-Element Formulation for Supersonic Flow Around Complex Canrigurations.' B o s t o n U n i v e r s i t y , Department of Aerospace Engineerin#. TR-74- 01, 1974.

1s. Morino. L.. and Kuo. C.C.. 'Subaonic P o t e n t i a l A e r o d y n a m i c s f o r C o m p l e x Coniigurations: A General Theory', AIAA Journal, Vol. 12. Fob. 1974, pp. 191-197.

14. Yorino, L., Chen. L.T., and Yuc%u, E.O., 'Steady snd Oscillatory Subsonic and Supersonic Aerodynamics Around Complex Configurations'. A I M Journal, Vol. 13. March 1945. pp. 368-314.

15. Tseng, K.. and Yorino, L.. 'A New Unified Approach for Analyzins Wins-Body-Tail Confisnrations with Control Surfaoes', A I M Paper No. 16-418, July 1916.

16. Yorino. L.. and Tseng. K. , 'Steady, Oacillatory and Unsteady. Subsonic and Supersonic Aerodynamics (SOUSSA) for Complex Aircraft Coufigurstiona', in AGABD-CP-227. Unsteady Aerodynamics. Ottawa, Canada. September 1971, pp. 3-1 t o 3-14.

17. Yorino, L.. 'Steady, Osoillatory, and U n s t e a d y Subsonic and Supersonic Aerodynamics - Produotion Version 1.1 (SOUSSA-F 1.1). VOI. 1. ThSOretiCai Manual'. NASA a-159130. 1980.

18. Smolkr, S.A.. Preuss. R.D.. Taena, K.. and Yorino, L.. 'Steady. Oscillatory. and U n s t e a d y S u b s o n i c a n d S u p e r a o n i s Aerodynamics- Production Version 1.1 (SOUSSA- P 1.1). v01. 2. UserlPro#rammer Manual', NASA Cp159131, 1980.

19. Morino, L.. 8nd Trona, K.. 'Time-Domain Green's Function Method for Three-Dimensional Nonlinear Subsonic Flowr', AIAA Psper 78- 1204, 1978.

20. Taenp. I. and Morino, L.. 'Nonlinear Green's Punction Method for Unsteady Transonic Flows,' Transonic Aerodynamics, edited by D a v i d Nixon, Vol. 81 of P r o g r e s s in Astronautics and Aeronauticr. 1982.

21. Tseng, L, 'Nonlinear Green's Function Method f o r Tr.nsonie Potential Flow.' Ph. D. Oisserration. Math. Dspt.. Boston University, Boston. ~8sas0husetts. Jan. 1983.

22. Murman, B.Y., and Cole, 1.0.. 'Calcul8tian O K P l a n e S t e a d y Tr8naomic Flora'. AIAA Journal, Vol. 9. Jann.ry 1971. pp. 114-121.

23. Khasla, P.K. and Rubin, 9.0.. 'Tranronic Flow Calculations in T w o - a n d T h r e e - Dimensiona.' Polyreohnlc Inatituta of Brooklyn, Department of Aerospace Engineering and Applied Mechanic., Report No. 13-13, June 1913.

v

24. Lee. K.D.. Dickaon, L.J., Chon, A.K., and Rubbert, P.B., #An Improved Matching Method for Tranaonio Compntaticna,' AIAA Paper No. 78-1116, July 1978.

25. L o c k . R . C . , 'Test Cases for Numerical Methods In Two-dImsnriona1 Transonic Flora,' MARU Report NO. 575, Novsmbor 1970.

26. Knechtsl, E.D.. 'Bxperimental Investi(ation a t T r a n s o n i c S p e e d s o f P r e s s u r e Distributions o v e r Wedge and Circular Arc A i r f o i l Section. a n d Evaluation of Perforated Wall Interference.' NASA TN D-15, Auguat 1959.

27. Bailey, F.R.. and Steper, J.L.. 'Relaxation Techniques for Three Dimensional Trmnaonic Flow About K i n p a , ' A I A A Paper No. 72-189. Janlrary 1911.

'J

8

6

*.I t..

A x

x/t

FIgwe 7. Flow fiekl potentid vorlatim h the z direction, M= 0.908, a =: 0 dag

'1

-1 \

F igm 8. Cp Disfribution for NACA ooiz M = 0.82, dt = 4, a = 0 &g.

airfoil

"'1

J I (I.. I L1 $

Figure 10. The effect of nonlinear term M = 0.63, a = 2 dag, ne = 20

NACA 0012 urfoii

b W E "

Y L O W "

0 ".( . l*Y

m 1.(.11-*

II

wc Figure 11 Cp Distributbn for 6 7. biconvex oirfdi

result compared with exp. data M = 0.857, a = 1 deg

0.8

0..

0.2

4

~

J

*.I-- , ,. 0.9 0.4 0 . 0..

Figure 12. Rectmgh- W h g h Steady Fbw with M = 0,908 dt = 2. nx = 20. ny = 5. 67. Bicwlvax sactiom AR = 4

Resrlts cmppsd wlfh fhii-differancs method OF Ref. 2 7

Fg~m 9. U variation h streomwise direction of a 2D steady fbw over a NACA 0012 airfoil

M = 0.82, a = 0 dag. dt = 4

d