Conclusion. Application of the differential analyzer to the construction of par

ticular solutions. As soon as a sufficient n u m b er of [Q , p ] stencils (i.e. q^„) h av e been prepared we can, by (3.1) and (3.2) and sta n d a rd m ethods of ev aluation of polynom ials on punch card m achines, d eterm in e th e values of p2,_i an d p2, for a suffi

ciently dense lattic e of points.

T h e second step, the d eterm in a tio n of th e coefficients a, in the expression (1.2) in ord er to o b tain a solution of (1.1) w hich assum es the prescribed b o u n d ary values, will be discussed in a su b seq u en t paper. T h e basic idea of th e procedure to be em ployed has alread y been indicated in [2]; we shall how ever discuss th is in m ore d etail from th e p o in t of view of num erical analysis.

Som etim es we need solutions of (1.1) in connection w ith qu estio ns o th e r th a n the b o u n d ary value problem , and it is convenient to ap p ly th e m eth od of p a rtic u la r solu

tions in th e following slightly modified form.

As w as proved in [ l] , §1,

U(z, z) = J ‘ E(z, 5, 0 / [ ( * / 2 ) ( l - n ]d t / ( 1 - r-y>\ (7. 1) w h e re/ is an a rb itra ry a n a ly tic function of a com plex variable, w hich is regu lar a t the origin, is a solution of (1.1).

T h ere are instances in w hich a large n u m b er of solutions of th e sam e eq u a tio n are required, and th e corresponding f u n c tio n s /a r e know n. (See (7.1).) (T his situ a tio n occurs, for exam ple, if an “a tla s ” of solutions has been p repared.) In these cases it is th en very convenient to em ploy graphical m ethods. In th e following we shall in d icate tw o graphical m eth o d s for the ev a lu a tio n of (7.1). B oth can be perform ed con- v ien tly by use of a differential analyzer.

I. One prepares once an d for all for a given eq uatio n (1.1) diag ram s in w hich th e curves

Y = E x(z„, s „ t ) = Re [E(z„ z „ t ) j, — 1 g t g 1, (s„ , z , ) fixed, Y = E 2(z,, z„ t) = Em [E(s„, z„ /)], — l i d 1, (z,, z„) fixed,

for a n u m b er of po in ts (x, y) = (x„, y„), v = l, 2, 3, • ■ • are draw n. F u rth e r one has to prepare tab les for the values -¿2,(1—/£), for fM= —1, — 1 + a , — l+ 2 o r, • - - , 1, w here a is a sufficiently sm all positive co n stan t. z„ — x , + i y , den o te th e coo rdin ates of p oints m entioned above. If now th e fun ctio n f(z ) = u(z, s)-Ht>(s, 2) is given, say in th e form of tw o d iagram s for curves m(z, z )= c o n s t. and i»(z, z) = co n st., we draw (using the tables m entioned above) th e curves

Y = « [z ,(l - /2), ¿ (1 - I2)], V = v [z,.( 1 - /-), z ,(l - /?)], - l i / i l . (7.3) Using these d iagram s an d those m entioned above an d em ploying a differential a n a lyzer (or sim ply an in te g ra to r) we com pute th e real p a rt of (7.1),

/ . <-i

[ E l { z „ Zy, t ) u [ j j Z y { 1 — t ' ) , 1 - / - ) ]

i — 1

— E 2(z„ Zy, / ) » [ ï Z , ( 1 — t'1) , i s i . ( 1 — t 2) \ \ d t , ( 7 . 4 )

and analogously its im aginary p art.

II. Som etim es it is not sufficient to d eterm ine the values of (7.1) a t a set of points (.v»t y,) which arc prescribed in advance. T hen one can ap p ly th e following procedure w hich was suggested to the a u th o r by M r. H an s K raft.

One prepares (once and for all) diagram s

E i { z , z , h ) = const., E->{z, z , t v) = c o n s t.,/„ const. t7 -5 ) for a set of values ly — — 1, — 1 + a , — 1 + 2 a , * - - 1.

Using these d iag ram s and the tables (described in m ethod I, for every required value of 2 we can easily d eterm ine the curve (7.3) and ev alu ate th e real an d im aginary p a rt of (7.1).

Remark. T h e procedure I can be perform ed by the use of punch card m achines.

In this case instead of d iagram s (7.2) it is necessary to prepare m aster cards.

T h e a u th o r should like to th a n k Professor George E. H a y for his exceedingly help ful advice and friendly criticism .

Bi b l i o g r a p h y

1. Stefan Bergm an, Z u r Tlieorie der Funktionen, die eine lineare partielle Differentialgleichung bc- Jriedigen, Recueil M athém atiq ue (M at. Sbornik) N .S . 2 , 1169-1198 (1937).

2. Stefan Bergman, The approxim ation of Junctions satisfyin g a linear p a rtia l differential equation, D u ke M ath. Journal, 6, 537-561 (1940).

3. W . J. Eckert, Punched card methods in scientific computation, C olum bia U n iversity, 1940.

4. E verett K im ball, Jr., A method of technical com pulations by punched card equipment, Publication of the Bureau of the Census, W ashington, D . C.

5. R. Lorant, D igiting without sorting, I.B .M . Pointers N o. 461.

246

T H E L IF T O F A D E L T A W IN G A T S U P E R S O N I C S P E E D S *

M A C H C O N E AT T H E

H. J. STEW ART

California In stitu te of Technology

1. Introduction. T h e use of th e tw o d im e n sio n a l lin e a riz e d th e o ry of su p e rso n ic flows in th e s o lu tio n of airfo il p ro b le m s a s in tro d u c e d b y A c k e re t1 h a s b e e n e x tre m e ly successful in so lv in g th e se p ro b le m s a n d th e re s u lts h a v e g e n e ra lly been c o m p le te ly s a tis fa c to ry fo r e n g in e e rin g p u rp o ses. T h e g e n e ra liz a tio n of th e se re s u lts to th e th re e d im e n sio n a l fin ite s p a n p ro b le m s h a s, h o w ev er, p ro g re sse d r a th e r slo w ly d u e to m a t h e m a tic a l c o m p lic a tio n s. T h e flow n e a r th e tip of a r e c ta n g u la r w in g w as given (in c o rre c tly ) b y S c h lic h tin g.2 T h e d ra g of a “d e l t a ” w ing (a w in g h a v in g a n isosceles tria n g le fo r its p la n fo rm w ith th e s y m m e tric v e rte x p o in tin g in to th e o n c o m in g flow as in F ig. 1) h a s b een d e te rm in e d b y P u c k e tt.3 T h e se tw o flow p a tte r n s a n d m a n y o th e r te c h n ic a lly in te re s tin g finite s p a n flow p ro b le m s a re p a r tic u la r cases of conical flows. A co n ical flow is one fo r w h ich th e fluid p ro p e rtie s (p re ssu re , v e lo c ity , e tc .)

a re c o n s ta n t a lo n g each ra d ia l lin e e m a n a t

in g fro m th e g iv en o rig in . T h e c o n c e p t of a co n ic al flow w as given b y B usem ann* w ho d e v e lo p e d c e rta in gen eral te c h n iq u e s for tr e a tin g th e s e flows a n d w ho a p p lie d th e m e th o d to se v e ra l p ro b le m s in c lu d in g S c h lic h tin g ’s p ro b lem .

T h e m e th o d s o f a n a ly s is used b y B u se

m a n n h a v e , h o w ev er, p ro v e d to be r a th e r o b sc u re , a n d it h a s b een fo u n d d iffic u lt to follow th e se m e th o d s in th e so lu tio n of a d d itio n a l co n ical flow p ro b le m s, in p a r tic u la r th e c u r r e n tly v e ry in te re s tin g p ro b lem o f th e lift of a d e lta w ing. A new m e th o d of tr e a tin g th e se co n ical flow a ir foil p ro b le m s w h ich u ses th e w ell k n o w n th e o r y of co n fo rm a l tr a n s f o r m a tio n h a s been d e v ise d . I t is th e p u rp o se o f th e p re s

e n t p a p e r to d isc u ss th is m e th o d a n d to a p p ly th is m e th o d to th e p ro b le m of th e lift of a d e lta w in g . In th is a p p lic a tio n it is o n ly n e c e ssa ry to c o n sid e r th e ca se for w h ic h th e le a d in g edges of th e d e lta w in g a re w ith in th e M a c h co n e fro m th e v e rte x . T h e o th e r c ase fo r w h ich th e le a d in g e d g es a re o u ts id e th e M a c h co n e h a s a lre a d y been so lv e d b y P u c k e tt.

In th e p r e s e n t m e th o d no e sse n tia l m a th e m a tic a l d ifferen ce is fo u n d in th e so lu tio n of th e tw o cases.

Fig. 1. D elta wing in a supersonic flow.

* Received May 21, 1946.

1 J. Ackeret, Z.F.M ., 16, 72 (1925).

5 H. Schlichting, Luftfahrtforschung, 13, 320 (1936).

3 A. J. Puckett, Aero. Sei. (To be published shortly).

4 A. Busemann, Schriften der Deutschen Akademie für Luftfahrtforschung, 7B, 105 (1943). Also Luftfahrtforschung, 12, 210 (1935).

2. General theory of conical flows. I t is well k n o w n t h a t th e lin earize d th e o r y of

This result was first communicated to the author by Mr. W. D. Hayes.

248 H. J. STEW ART [Vol. IV, No. 3

'/* + E q . (5) b ecom es

d ( d P \ d2P

s — I s ) -t = 0 . ( 6 )

d s \ d s ) dd2

T h is is th e n o rm a l form of th e L a p la c e e q u a tio n in tw o d im e n sio n a l p o la r c o o rd in a te s . I t is seen t h a t s is a fu n c tio n o n ly of n a n d is th u s c o n s ta n t on a n y o ne of th e cones fo r w hich co is c o n s ta n t. T h e re la tio n s b e tw e e n 5 a n d co a rc as follow s:

\ / M 2 — 1 tan co

5 = ... 7= ^ -— ^ .-:.:r ... (7 )

1 + \ / l - (M2 - 1) ta n2co

y 25

\ 2M 2 — 1 tan co = --- (S)

1 + 5 2

I t m a y f u r th e r be n o te d t h a t 5 = 1 on th e M a c h co n e th ro u g h th e o rig in .

Since th e re d u c tio n to E q. (6) is possible, a n y of th e q u a n titie s w h ich P m a y re p re s e n t ca n b e w ritte n a s th e real (o r im a g in a ry ) p a r t of a n a n a ly tic fu n c tio n of th e co m p lex v a ria b le f w h ere

r = se'e- (9)

F u rth e rm o re , all th e m e th o d s of tr e a tm e n t of su c h fu n c tio n s, in p a r tic u la r th e m e th o d of co n fo rm a l tra n s fo rm a tio n , m a y be used in th e a n a ly sis of th e se q u a n titie s . If P is th e h a rm o n ic c o n ju g a te of P a n d

P + iP = P ( f ) , (10)

th e C a u c h y -R ie m a n n e q u a tio n s fo r th e se c o n ju g a te f u n c tio n s m a y be w r itte n

dP dP dP

s = --- = (¿u2 — 1) --->

ds dd dy.

(11)

dP dP dP

= S = (m2_ ! )

---de ds dn

In th e d ire c t airfoil p ro b le m , th e airfo il g e o m e try is g iv en , a n d if th e 2 a x is is ta k e n n o rm al to th e a irfo il p la n e , th e b o u n d a ry c o n d itio n s for d e te rm in in g th e flow a re th u s given in te rm s of th e d is tu rb a n c e v e lo c ity c o m p o n e n t w. I t is d e sire d in th is case to c o m p u te th e p re ssu re d is trib u tio n w h ic h m a y be e asily ex p ressed in te rm s of th e a x ia l d is tu rb a n c e v e lo c ity c o m p o n e n t u. In th e in v e rse airfo il p ro b le m , a p re ssu re d is tr ib u tio n is d e fin e d , a n d it is d e sire d t h a t th e airfoil s h a p e b e c o m p u te d . In e ith e r c ase th e b o u n d a ry c o n d itio n is given in te rm s o f o ne v e lo c ity c o m p o n e n t a n d a n o th e r v e lo c ity c o m p o n e n t g iv es th e d e sire d re s u lt. F o r a co n ical flow th e re a re sim p le re la tio n s b e tw e e n th e co m p lex fu n c tio n s re p re s e n tin g th e v a rio u s C a rte s ia n v e lo c ity c o m p o n e n ts . T h e use of th e se re la tio n s is th e essence of th e p re s e n t m e th o d of t r e a tm e n t of co n ical flows. T h e s e re la tio n s b e tw e e n th e co m p le x fu n c tio n s c o rre s p o n d in g to th e C a rte s ia n d is tu r b a n c e v e lo c ity c o m p o n e n ts u, v, w w h ich will b e w ritte n

it + in = U (f) , i- + iv = V (<-), w + iiv - W (f), (12) are e sse n tia lly th e v o r tic ity re la tio n s.

T h e fu n d a m e n ta l lin e ariz ed re la tio n s g o v e rn in g th e s te a d y flow of a fluid a t s u p e r sonic sp e e d s are th e v o r tic ity re la tio n s

dv dw dll dw dll dv

— = --- , (13) — = — , (14) — = — , (IS)

dz dy ds dx dy dx

a n d th e lin e a riz e d e q u a tio n of c o n tin u ity

dv dw dll

— + ---= ( M 2 - 1 ) — ■ (16)

dy dz dx

If th e se a re solved s im u lta n e o u s ly , it is e a sily seen t h a t ea c h of th e v e lo c ity c o m p o n e n ts o b ey s th e P r a n d tl- G la u e r t e q u a tio n , E q . (1). F o r a n y conical flow each of th e v e lo c ity c o m p o n e n ts m u s t be a fu n c tio n o n ly of th e c o o rd in a te s n a n d 9. B y m e a n s of th is fa c t, E qs. (13) to (16), re sp e c tiv e ly , m a y be w ritte n as follow s:

, / dv d w \ ( dw dv\

— 1) I cos 0 sin 0---) = I cos 6--- |- sin d — ), (17)

\ d)i dii) \ dd dd)

(li2 — l ) 31'- dw du du

= ii(ii- — 1) cos 9 — sin 0 — > (18)

V/ M - - 1 du dii dd

(,ii- — 1) 3/'- dv dll du

— = ,u(ii2 — 1) sin 0 --- j- cos 0 — t (19)

\ / M ‘z - 1 dii d/i dO

dll ( dv dw

- \ / M 2 - 1{ii2 - 1)3/2 du

(

_sin₀_—■^dv_{+ cos}₀ ^{d w \}_I

d/i d ii)

/ dv d w \

F ( cos 6 sin 6 ■— - ). (20)

\ dd dd J

If E qs. (18) a n d (19) a re c o m b in e d , it is seen t h a t

du On2- l ) s / 2 / dv d w \

— = —...—.... ; I cos d --- sin d---). (21) dO \ / M 2 - 1 V d/i d / i )

dii (n2 — l)l/2 / dv d w \

— = — ( sin d--- f COS d --- ). (22)

dii \ / M 2 — 1 \ d/i d/i /

F u rth e r m o r e , E q s. (20) a n d (22) show t h a t

li1 — 1 / dv d w \ ( dv d w \

( sin d 1- cos d ) = — ( C(,s 9 sin 9---). (23)

H \ du d i i ) \ dd d d )

If th e d e riv a tiv e s w ith re sp e c t to ju a re e lim in a te d from E q s. (17) a n d (23) b y m e a n s of th e C a u c h y -R ie m a n n e q u a tio n s [cf. E q . (1 1 )] fo r v, w, v a n d w, th e se e q u a tio n s m a y be w r itte n a s follow s:

250 H. J. STEW ART [Vol. IV, No. 3

T h is m a p s th e in te rio r of th e u n it circle in th e f p la n e in to th e e n tire f i p la n e w ith th e region R e f >0 c o rre sp o n d in g to th e region J m J i >0. T h is tra n s fo rm e d p la n e is

im£,

JÂT^{w » 0}

• A ^-I

«iC, A

Fig. 3. Boundary conditions in the plane.

show n in F ig. 3. T h e p o in ts a t th e w in g tip s, f = ± iso, a re tra n s fo rm e d in to th e p o in ts

<Ti = ± 1 / k w h ere

2s q ,---—

k = = \ i M - — 1 tan o)0. (32)

1 + S eco n d , a p p ly th e tra n s fo rm a tio n

i.e.,

_ ___

v a - r i ) ( i - m

i i — sn(£i),

(33)

(34) w h ere th e e llip tic fu n c tio n h a s th e m o d u lu s k. T h e n th e region I w f i > 0 is m a p p e d in to th e re c ta n g le h a v in g its c o rn e rs a t fa — ± K , i K ’ ± K w h ere K a n d K ' a re th e c o m p le te e llip tic in te g ra ls of th e first k in d h a v in g a m o d u lu s of k a n d k ' w here

k> = ; v i - k2 = (35)

B y in te g ra tin g a ro u n d th e s lit from —1 to 1 in th e p lan e, it is seen t h a t th e region m a p s in to th e re c ta n g le h a v in g c o rn e rs a t £ « ~ 2 K ± K , 2K ± K - \ - i K ' .

N o w , th e tr a n s fo rm a tio n given b y E q . (31) is d o u b le v a lu e d , i.e., tw o p o in ts in th e f p la n e c o rre sp o n d to each p o in t in th e f i p lan e. T h e f i p la n e m u s t th u s b e c o n sid e red as a tw o sh e e te d R ie m a n n su rfa c e w ith one sh e e t c o rre sp o n d in g to th e in te rio r of th e u n it circle in th e if p la n e a n d th e o th e r sh e e t c o rre sp o n d in g to th e e x te rio r of th e u n it circle in th e f p lan e. F u rth e rm o re , th e v a lu e of th e d o w n w ash v e lo c ity w m u s t b e e q u a l a n d o p p o site a t in v erse p o in ts in th e f plane. T h is p e rm its th e a n a ly tic c o n tin u a tio n of w th r o u g h o u t th e e n tire f p la n e ; in p a r tic u la r i t is seen t h a t w — —wo on th e e x te rio r p o in ts c o rre sp o n d in g to th e airfoil. T h e tw o sh e e ts in th e f i p la n e a re c o n n e c te d th ro u g h th e slit from —1 to + 1 . A c o n to u r c u ttin g th is line p asse s from th e u p p e r to th e low er sh e e t or v ice-v ersa. T h e sh e e t w h ich c o rre sp o n d s to th e e x te rio r region of th e f p la n e is th u s seen to b e m a p p e d in to th e re c ta n g le h a v in g c o rn e rs a t £2 = K ± 2 K , K ± 2 K —i K ' . T h e e n tire p la n e is m a p p e d in to a b a sic re c

252 H. J. STEW ART [Vol. IV, No. 3

ta n g le in th e if2 p la n e a s show n in Fig. 4. A s h a s p e rio d s of 4 K , 2i K ' [see E q . (34)]^

in j"o, th is p a tte r n is re p e a te d th r o u g h o u t th e ifs p lan e.

T h e fu n c tio n d~W/d^2 (b u t n o t W itself) m u s t be d o u b ly p erio d ic in th e p lan e w ith p erio d s 4 A a n d 2 i K ' , th e first c o rre sp o n d in g to a loop a ro u n d th e p o in ts f i = ± 1 a n d th e second c o rre sp o n d in g to a loop a ro u n d th e p o in ts f i = l , \ / k o r —1, — l / k . T h e o n ly s in g u la ritie s of W o r d W / dfs m u s t be a t th e p o in ts c o rre sp o n d in g to th e a ir foil le a d in g edges, i.e., a t th e p o in ts c o n ju g a te to i K ' ± K . F in a lly d W / d f a m u s t be p u re im a g in a ry on th e lines I m { t = n K ' a n d Re£2 = K + 2 n K (n b ein g a n y in te g e r).

All of th e se c o n d itio n s a re satisfied b y th e J a c o b ia n e llip tic fu n c tio n d W

— = iDcd2n(fr), d{°

(36)

w h e re n is a n y p o sitiv e in te g e r a n d D is a real c o n s ta n t. If th is is in te g ra te d i t is seen t h a t fo r « > 0 , W h a s a pole of o rd e r 2 n — 1 a t th e w in g tip s. T h e cases fo r w > l c a n th e n be d isc a rd e d a s th e s in g u la rity a t th e w in g tip s is seen to c o rre sp o n d to a so u rc e- sin k co m p lex w hich h a s a n in fin ite to ta l lift. F u rth e rm o r e , th e case fo r n = 0 m a y be d isc a rd e d a s [see E q . (2 9 )] it re q u ire s t h a t U (f) h a v e a lo g a rith m ic s in g u la rity on th e

M ach cone. T h e a p p r o p r ia te s o lu tio n is th u s ,/W

= i D c d ^ 2).

T h e c o n s ta n t D m a y b e e v a lu a te d fro m th e fa c t t h a t

' i K ' d W

•wo

If th is in te g ra tio n is c a rrie d o u t, it is seen t h a t D — — k-wo

E ( k 0

(37)

(38)

(39)

w h ere E ( k ') is th e c o m p le te e llip tic in te g ra l of th e second k in d h a v in g a m o d u lu s k' a s g iv en b y E q . (35).

If th e v a ria b le is e lim in a te d from E q . (37) b y m e a n s of E q. (31) a n d (33), it is seen t h a t

d W

~dF

T h u s , from E q . (29), d ü

2 w<i ( i + n

-kE(k')

(40)

4 Wo

m + n

₍₄₁₎

S ince U = 0 a t f = l , th e in te g ra l of Eq... (41) is

kvio f 1

U =

( r2 +

1 / 2 (42)

On th e to p side of th e airfoil f = hj w h ere — .Vo < 17 < .''0, so

fewo 1 + V'

11 — —

E ( k ' ) \ / A n - 1 ^11/2

[M - (1 -

(43)

T h is re s u lt m a y b e c o n s id e ra b ly sim p lified if we in tro d u c e [from E q . (8) a n d (3 0 )]

k = \ / M 2 — 1 ta n w0 (44) Wo = — Ua

a n d

E q u a tio n (43) th e n b ecom es

ta n w ta n wo

4 ta n wo

(45) E ( k' ) \ / l -

t-T h e slope of th e lift c u rv e dC i./da is given b y th e m ean v a lu e of A / a ( u / U ) o v er th e su rfa c e of th e w in g ; th u s

dCi.

= —

_{R( k}4

⁷

₎

-

_J^fr₀ “° sec2 w(/w a n d , b y E q . (44),

da E (k') J 0 V l - Ï-

dCj. ²^t ta n wo

da E{V)

(46)

(47)

In th e lim it fo r w hich w0 o r 50—>0, k' —> \; so E {k')^> \. F o r th is case w hich w as given b y J o n e s5

* R. T. Jones, N .A.C.A., Technical N ote 1032 (1945).

254 H. J. STEW ART

k • \|m2 - I Ian co.

Fig. 5. Lift of a delta wing.

F ig. 6. dC iJda. vs. M for a delta wing with oi =10°.

dCL da ^{= 2}

^r t a n t o 0 .

O n th e o th e r lim it fo r w h ich k ' —>0; so E ( k ’)^nr'/2. F o r th is case dC i.

da ⁼ 4 t a n w 0 =

\ / M i - 1

(48)

( 4 9 )

1 h is lim it, th e sa m e a s th e tw o d im e n sio n a l so lu tio n , h ad p re v io u s ly been o b ta in e d b y P u c k e tt.

I t m a y f u r th e r be n o te d t h a t th e q u a n tity \ \ / A P —1 d C i / d a is a fu n c tio n o n ly of th e p a r a m e te r k = \ / A P —1 ta n w0- T h is re s u lt is sh o w n g ra p h ic a lly in Fig. 5, a n d th e slope of th e lift c u rv e fo r a p a r tic u la r case, w o =10°, is show n a s a fu n c tio n of M a c h n u m b e r in Fig. 6.

L IN E A R IZ E D S U P E R S O N IC F L O W S W IT H AX IAL S Y M M E T R Y *

WALLACE D. HAYES**

California Institu te of Technology

1. Introduction. T h e s tu d y of s p a tia l lin earized su p e rso n ic flow m a y he a id e d by th e s tu d y of som e sim p le fu n d a m e n ta l flows w ith axial s y m m e try . T h ro u g h th e p rin ciple of su p e rp o s itio n , th e se flows m a y be c o m b in ed to give m ore general flows a b o u t v a rio u s o b je c ts a n d a b o u t liftin g sy ste m s. I t is th e p u rp o se of th is p a p e r to ex p ress th e e q u a tio n s of lin earized su p e rso n ic flow in a sy ste m of co n ical c o o rd in a te s, to d e v elo p a th e o ry fo r fu n d a m e n ta l flows w ith axial s y m m e try , a n d to d escrib e e x a m p le s of su ch flows a n d of th e ir c o m b in a tio n b y su p e rp o sitio n .

V a rio u s e x a m p le s of th e fu n d a m e n ta l e q u a tio n s a n d so lu tio n s h e re d e scrib e d will be given in la te r p a p e rs, to g e th e r w ith th e d e v e lo p m e n t of so m e c o n c e p ts useful in th is field.

2. The velocity potential. S te a d y - s ta te co m p ressib le ir ro ta tio n a l flow ca n b e d e scrib ed b y a v e lo c ity p o te n tia l <p w hose g ra d ie n t is th e v e lo c ity v ecto r. U n d e r th e a s s u m p tio n t h a t th e v e lo c ity d e v ia tio n s from a u n ifo rm su p e rso n ic flow of th e M ac h n u m b e r M a re sm all, th e d iffe re n tia l e q u a tio n fo r th is p o te n tia l ta k e s th e lin e a r fo rm1'2

4>rr ---4>r-H--- - <t>M — { M ‘Z — = 0 ( 1 )

r r2

in c y lin d ric a l c o o rd in a te s.

T h e fu n d a m e n ta l u n ifo rm flow is given b y th e p o te n tia l <t>o= Vz w h ere V is th e v e lo c ity c o rre sp o n d in g to th e M a c h n u m b e r M . E q u a tio n (1) will be co n sid ered as y ie ld in g v e lo c ity d e v ia tio n s w hich m u s t b e a d d e d to th e v e lo c ity of th e fu n d a m e n ta l flow to d e sc rib e th e n e t flow.

A new c o o rd in a te is in tro d u c e d to rep la ce th e c o o rd in a te r:

t = (r/z ) (2)

T h is q u a n t i t y is th e r a tio of th e ta n g e n ts of th e p o la r angle a n d of th e M a c h angle.

E q u a tio n (1) w ith r e lim in a te d a n d t in tro d u c e d b ecom es

(1 — t2)<f>tt H (1 — < -)---4>ee + 2tz<j>iz — Z'<plz = 0. (3)

t /2

B y s e p a ra tio n of v a ria b le s a so lu tio n of th e form

* Received March 25, 1946.

** This paper was prepared while the writer was employed by the Lockheed Aircraft Corporation.

1 R. Sauer, Theoretische E infiih ru ng in die G asdynam ik, Springer, Berlin, 1943. Reprinted by Ed

wards Bros., Ann Arbor, 1945.

5 G. I. Taylor, and J. W. Maccoll, The mechanics o f compressible flu ids, in Durand, A erodynam ic theory, vol. 3, Berlin, 1935.

<(> = zn <P(f, 6) (4a) o r

4> = z" sin (md + 0) T(l) (4b) is fo u n d . T h e fu n c tio n satisfies th e e q u a tio n

1 1

(1 - f2)4>(( + ^— (1 + ^{2 ( n -} l)/'-)T, - ^{n( n} - l)<f> + — 4>99 = 0 (5)

a n d m a y be called th e v e lo c ity p o te n tia l fo r g en eralized co n ical flow. If w = l , th e fu n c tio n <ï> d esc rib e s conical flow. T h e fu n c tio n T satisfies th e e q u a tio n

1 1

(1 - r-)T lt + — (1 + 2.(» - l )r - ) T t - — (rn* + n{n - 1 )t-)T = 0. (6)

t I“

S u p e rp o sitio n of s o lu tio n s of th e ty p e of (4 a), (4b) will give a g e n e ra l so lu tio n . T h e v e lo c ity c o m p o n e n ts are

25f> WALLACE 1). HAYES [Vol. IV, No. 3

V & P - 1

u = ---<f>i (7a)

in th e ra d ia l d ire c tio n ,

in th e a z im u th a l d ire c tio n , a n d

\ / M 2 - 1

v - ---4>i (7b)

w = <f>c i (¡>i (7c)

in th e axial d ire c tio n . T h e p re ssu re in lin e ariz ed su p e rso n ic flow is given in te rm s of th e v e lo c ity c o m p o n e n ts b y

/ u - + v- \ P = - p U 7«' H--- — )-an d th e p ressu re coefficient b y

/ w u- + a2 \ C P = - 2 ( — + --- ).

\ V 2 V ! )

(8a)

(8b)

T h e p a r t of E q s. (8a ), (8b ) in u a n d v is n o t n e c e ssa ry if w is of th e sa m e m a g n itu d e a s « a n d a. In m a n y im p o r ta n t cases, h o w ev er, w2+ a2 is of th e sam e m a g n itu d e a s Vw a n d E q s. (8a ), (8b ) m u s t be used in its c o m p le te fo rm . In th e se cases th e v a lid ity of th e so lu tio n sh o u ld b e c h eck e d .

T h e s in g u la rity of (5) o r (6) a t / = + 1 c o rre sp o n d s to th e tw o M a c h co n es e x te n d in g from th e o rig in in th e th re e d im e n sio n a l flow. V a rio u s ra n g e s of t co rre sp o n d to v a rio u s reg io n s of flow, a s sh o w n in th e follow ing ta b le .

R aille of t Region of Flow

T h re e sp ecial cases a re d istin g u ish e d a c c o rd in g to th e re la tiv e v a lu e s of n a n d m:

case A : — a> < n 5= — m —1, case B : — m g n ^ m — 1, case C : m £ n < »

T h e d is tr ib u tio n of th e se cases fo r sm all v a lu e s of m a n d n is show n in th e ta b le :

258 WALLACE D. HAYES [Vol. IV, No. 3

" n

m - 3 - 2 - 1 0 1 2

0 A A A C C C

1 A A B B C C

2 A B B B B C

3 11 11 B B B B

F ro m a c o n sid e ra tio n of E q s. (9) to (12) th e fo rm s of th e tw o ty p e s of so lu tio n s in th e v a rio u s cases m a y b e fo u n d . F o r all so lu tio n s e x c e p t so lu tio n s I-A (i.e., so lu tio n s o f ty p e I in case A) a n d s o lu tio n s I I-C , th e form is e x p lic it in te rm s of a p o ly n o m ial

W dokumencie The Quarterly of Applied Mathematics. Vol. 4, Nr 3 (Stron 46-70)