The Wave Drag of a Supersonic Biplane of Finite Span

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The Wave Drag of a S upersonic B iplane o f F in i te Span

A T hesis P rese n ted to th e F a c u lty o f th e G raduate School o f C o rn e ll U n iv e rs ity f o r th e d eg ree o f D octor o f Philosophy

Hao Sung Tan

F eb ru ary , 1950

ProQuest N um ber: 10834678

All rights reserved INFORMATION TO ALL USERS The q u a lity of this re p ro d u c tio n is d e p e n d e n t u p o n the q u a lity of the co p y su b m itte d . In the unlikely e v e n t that the a u th o r did not send a c o m p le te m a n u scrip t and there are missing p a g e s, these will be n o te d . Also, if m a te ria l had to be re m o v e d , a n o te will in d ic a te the d e le tio n .

uest P roQ uest 10834678 Published by ProQuest LLC(2018). C o p y rig h t of the Dissertation is held by the A uthor. All rights reserved. This work is p ro te cte d a g a in s t u n a u th o rize d co p yin g under Title 17, United States C o d e M icroform Edition © ProQuest LLC. ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 4 8 1 0 6 - 1346

Ackncwledgement The a u th o r w ishes to ex p ress Ills most S in c e re g r a titu d e t o P ro fe sso r V/0 R0 S ears f o r M s s u g g e s tio n of th e study* and h is u n f a ilin g guidance and encouragement th ro u g h o u t th e work® He a ls o ta k e s t h i s o p p o rtu n ity to th a n k P ro fe ss o r W© F e lle r* P ro fe ss o r H.

Conway and P ro fe ss o r J . N. Moodier f o r t h e i r

most v a lu a b le in s tr u c tio n s *

Biography The a u th o r was born i n Soochow* Kiangsu* China* on December 1 , 19160

He a tte n d e d p u b lic sch o o l i n Soochow*

was g ra d u a ted in 1935 from Soochow P r o v in c ia l High School* and th e n re c e iv e d in Jhine 1939 th e degree B. S c. from th e School of M echanical E n g in eerin g a t Chiao-Tung U n iv e rsity * Shanghai* China© A fte r g ra d u a tio n he fo llo w ed h is government t o Szechuen* where he con tin u ed g ra d u a te stu d y f o r a y e a r i n th e A ir Force T e c h n ic a l School* Chengtu* China©

From O ctober 19l*0

t o Jan u ary 19l*l* he se rv e d i n th e Bureau o f A e ro n a u tic a l Research* Chengtu, a s a ju n io r re s e a rc h fello w * engaged i n th e t h e o r e t i c a l stu d y of e l a s t i c i t y and aerodynamics© From Jan u ary 191*1* to June 191*5 he jo in e d th e te a c h in g s t a f f o f th e A ir Force T ech n ical School* in s t r u c t i n g m echanics and m athem atics. A fte r th e second w orld war* he came to th e U nited S ta te s f o r advanced s tu d y .

He e n te re d th e G raduate S chool a t

C o rn e ll i n O ctober 191*6* m ajoring i n a e ro n a u tic a l engineering©

C ontents

Part i .

1.

A L in ea rized Theory o f M u ltiplan e Supersonic $ave I n te r a c t io n ,

L in e a riz e d th e o ry o f su p erso n ic flo w .

2. . Theory o f su p e rso n ic so u rce d i s t r i b u t i o n . 3.

E q u iv alen t d is tu rb a n c e s h e e t.

L.

D eterm ination of so u rce d i s t r i b u t i o n .

5.

S o lu tio n of i n t e g r a l e q u a tio n .

6.

V e lo c ity p o t e n t i a l .

P a rt I I .

A*

Formal s o lu tio n .

B.

S o lu tio n by e lim in a tio n of ^ - i n t e g r a l .

C.

P ressu re c o e f f i c i e n t , l i f t and d ra g .

Busemann B iplane of F i n i t e Span.

7.

Source i n t e n s i t y over

8.

V e lo c ity p o t e n t i a l .

9.

w ing.

^

10.

5 ^ /and

11.

P re s su re c o e f f i c i e n t ,

12.

L i f t and d ra g .

^c

~

(o r ^

Appendix 1.

P r in c ip le of f i n i t e n a r t .

2.

A - d i s t r i b u t i o n f o r a r e c ta n g u la r wing o f symmetric s e c tio n a t in c id e n c e , (w ithout wave i n t e r a c t i o n )

3.

D eterm in atio n o f so u rce i n t e n s i t y over wing in th e in t e r a c t i o n re g io n .

U.

P o te n tia l c o n tr ib u tio n by n eig h b o u rin g s u rfa c e

3.

V e lo c ity p o t e n t i a l i n t i p zone o f symmetric wing a t zero in c id e n c e and o f f l a t p la te a t in c id e n c e .

B ib lio g rap h y

SuBEnary

A stu d y i s made of th e l in e a r i z e d th e o ry of

th re e -d im e n sio n a l su p e rso n ic flow about a m u ltip la n e wing in a uniform flow fie ld o

Based upon th e concept o f a su p e rso n ic

.source d i s t r i b u t i o n , a g e n e ra l method i s e s ta b lis h e d by which th e wave i n t e r a c t i o n can be ev alu ated *

The method fin d s

immediate a p p lic a tio n in th e d e te rm in a tio n o f t h e i n t e r a c t i o n between t i p Mach cones fo r a su p erso n ic m u ltip la n e of f i n i t e span*

A com putation i s a c c o rd in g ly made f o r th e wave d rag o f

a Busemann b ip la n e of f i n i t e s^an a t zero in c id e n c e ; t h i s i s e q u al to t h e wave drag in d u ced by t h e t i p Mach cone i n t e r a c t i o n , s in c e an i n f i n i t e span Bu.SGraa.nn b ip la n e i s known t o be w ith o u t wave drag*

The r e s u l t i s compared w ith th a t of th e co rresp o n d ­

in g monoplane*

I t i s found t h a t , w h ile th e spanw ise d i s t r i b u t i o n

o f wave d rag v a n ish e s everywhere ex cep t in s id e th e t i p re g io n i n case o f Busemann b ip la n e , i t rem ains c o n s ta n t, and indeed ta k e s on th e tw o-dim ensional v a lu e th ro u g h o u t th e span i n case of a monoplane*

M oreover, even in th e t i p re g io n where th e e n t i r e

wave drag o f Busemann b ip la n e i s c o n c e n tra te d , t h i s drag amounts t o on ly about h a l f t h a t o f th e same span i n t e r v a l of th e corresponding monoplane.

Thus i t i s ev id en t t h a t , as

re g a rd s wave drag a t zero l i f t , th e Busemann b ip la n e i s always s u p e rio r to a monoplane o f th e same t o t a l th ic k n e s s and planform *

Introduction In th e developm ent of lin e a r iz e d su p e rso n ic m u g th e o ry , we f i n d t h a t in 1936 o c n lic h tin g (R ef0 10) gave iiis s o lu tio n f o r th e f i n i t e - s p a n wing based on an e x te n s io n of P r a n d t l 's l i f t i n g l i n e theory®

In 19lUi L i g h t h i l l (Ref® 7)

in tro d u c e d th e id e a of a s u p e rso n ic source d i s t r i b u t i o n b; sed on an e x te n s io n of von Karman's (R e f0 12) and T s ie n 's (Refo 13) s o lu tio n s f o r s le n d e r bodies o f revolution®

In

19U6 P u c k e tt (Ref® 9) developed th e id e a o f th e so u rce d i s t r i ­ b u tio n from p o te n tia l analogy®

R ecen tly E w a rd (Refo 5)

showed th e p o s s i b i l i t y of so lv in g th e su p e rso n ic flow problem about a r b i t r a r y wings a t sm all in c id e n ce by an extended a p p li­ c a tio n of th e source d i s t r i b u t i o n only

The p ro o fs of th e

source d i s t r i b u t i o n th e o ry g iv en by L i g h t h i l l and P u c k e tt arc n ot e n t i r e l y co n v in cin g , so l a t e r H e asle t and Lomax (Ref® I;) developed t h i s th e o ry in d e p en d e n tly from G reen 's tra n s fo rm a ­ t i o n , by introducing* th e ”co-norm al” and u sin g Hadamard's p r in c ip le of th e " f i n i t e p a r to ”

Put l i k e L i g h t h i l l , th e y to o

lim ite d th e a p p lic a tio n of th e source d i s t r i b u t i o n to sym m etric w ings, because th e d ir e c t r e s u l t of th i s tra n s fo rm a tio n shows t h a t a s h e e t wing a t in c id e n c e corresponds e x a c tly to a p lan e d i s t r i b u t i o n o f doublets®

I t was E w a rd who f i r s t in tro d u c e d

a f i c t i t i o u s diaphragm ex ten d in g beyond wing t i p , by w hich he com pleted th e c lo se d boundary s u rfa c e f o r a Cauchy problem - 2 -

w ith "unduly in c lin e d ” datum p la n e , and th u s s im p lif ie d th e c o n s id e ra tio n o f i n te r a c t io n betw een th e to p and bottom s u rfa c e s in s id e th e t i p Mach cone®

This d ev ice makes i t p o s s ib le

t o re p la c e a wing by a p la n e source d i s t r i b u t i o n o n ly , w ith o u t in tro d u c in g d o u b le ts , which fe a tu r e i s e s p e c ia lly d e s ir a b le in c ase th e r e i s no re th a n one wing to be considered®

As

E w a rd based h is work on t h a t o f P u c k e tt, h is th e o ry i s open to th e same c r i t i c i s m as P u c k e tt’s , a lth o u g h i t c e r t a i n l y has g r e a t u tility ®

This stu d y th e r e f o r e endeavours to p re s e n t

th e p ro o f in a more convincing form , fo llo w in g Hadanard and H e a s le t’s work, th e re b y j u s t i f y i n g th e in tr o d u c tio n of our " e q u iv a le n t d is tu rb a n c e s h e e t” w ith E w a r d 's diaphragm as a p o rtio n o f it® In 1936 Pusemann (Ref® 11) a r r iv e d a t an im p o rtan t con­ c lu s io n re g a rd in g th e s u p e rso n ic b ip la n e , t h a t th e r e i s a c e r ta in arrangem ent f o r each Mach number t h a t makes th e wave drag of t h e system vanish®

This i s ach iev ed th ro u g h f u l l y

u t i l i z i n g th e i n te r a c t io n o f th e two le a d in g edge Mach wedges® I t is to be n o ted t h a t th e c h a r a c te r of a b ip la n e depends much on t h i s wave in te ra c tio n ®

At h ig h e r Mach num bers, th e i n t e r ­

a c tio n d e c re a s e s , u n t i l f i n a l l y th e system re d u c es to two s e p a ra te monoplanes®

At low er Mach numbers, th e in t e r a c t i o n I s

- 3 -

g r e a te r and r e s u l t s in in c re a s e d wave drago

The e f f e c t on

l i f t and drag of v a r ia tio n of Mach number from th e Busemann v alu e was s tu d ie d by L I g h th ill (Bed* S) in Id Uh• T his th e o r y , how ever, a p p lie s only f o r i n f i n itc - s p a ti (tw o-dim ensional) wingso

T/ith a f i n i t e span b ip la n e , th e

t i p e f f e c t in e v ita b ly comes i n , so t h a t in a d d itio n bo th e s e r e l a t i v e l y sim ple c o n s id e ra tio n s , th e r e rem ains a n o th e r i n t e r a c t i o n problem to be c o n sid e re d , i©e©, th e in t e r a c t i o n between th e t i p Mach. cones©

I t i s th e n im m ediately seen th a t

Busemann*s i d e a l i z a t i o n w ill not be realized ©

However, s in c e

th e t i p e f f e c t i s only l o c a l , th e aivanta-^e of th e Busemann arrangem ent should be e v id e n t a t high a sp e c t ratio s©

The

problem s t i l l rem aining to be so lv ed i s a q u a n tita tiv e one, i*e© , to determ ine th e degree o f in flu e n c e of t h i s t i p Mach wave i n t e r a c t i o n on th e c h a ra c te r of a f i n i t e —snan biplane©

A

t h e o r e t i c a l stu d y of t h i s wave i n te r a c t io n forms th e main o b je c t of th e p re s e n t study© The b a s ic id e a of our s o lu tio n c o n s is ts in r e a l i s i n g t h a t fo r a l in e a r iz e d su p erso n ic flow f i e l d , th e d is tu rb a n c e v e lo c ity p o t e n t i a l o ’f

which s a t i s f i e s th e wave e q u a tio n =

/ s ' ?r f

-

jV

=

O

and th e boundary-" c o n d itio n on tu e s u r f a c e , can always be

-

li -

e x p re ssed as a w eighted i n t e g r a l , over c e r t a i n p o rtio n of th e s u r f a c e , of th e fundam ental s o lu tio n

~pr

i n th e

fo llo w in g form

The w eight fu n c tio n

-*-s re c o g n iz e d as th e su p e rso n ic

so u rce i n t e n s i t y . For th e d e te rm in a tio n o f , th e wing i s re p la c e d by an e q u iv a le n t d is tu rb a n c e s h e e t, which i n t u r n i s re p re s e n te d ,as a p la n e su p erso n ic source d i s t r i b u t i o n . The knowledge o f v e lo c i ty p o t e n t i a l

» which i s r e l a t e d to th e normal n ( )y>-~n,

J

n3)

by

( 2.U)

Kj - °.f

Then, i t is seen t h a t G reen’s theorem on surface-volum e in te g r a ­ tio n g iv e s

J L ( n sfChoosing f o r

;

^

12.

(2 .5 )

th e fundam ental s o lu tio n o f (2 ,1 .), i , e * s /7l A - o

/ / ' J f x - z f - - p > v^-Z)vJ f x - r j - c r - Y ) Vz-2 'jv

th e n th e re g io n R w i l l be bounded by s u rfa c e

( 2*^)

S , c o n s is tin g

of th e forw ard c h a r a c t e r i s t i c cone (Mach cone)

P

from p (x, y , 2 ,^)

a t which p o in t the v a lu e of $ i c to be deter'm ined, th e le a d in g Mach wedge- A > th e d is tu rb a n c e plane T , and th e sm all a re a Z in s id e P by v.hich th e s in g u la r p o in t p is excludedo be Itave, from ( o ) ,

JJS ( * H

-

Zp SLS

*s = 0

(2 -7 )

and, u sin g Hedu nard’s n o ta tio n fo r th e f i n i t e p a r t , (2 ofl)

r»+r 4-+ 1r Hi

where th e prime in d ic a te s th e s u rfa c d v alu es on o p p o site s id e -from P .

of t

'P ie in te g r a tio n over sm all £ can be e v a lu a te d as fo llo w s : % ta k in g Z

a t d is ta n c e

6

p u ttin g our o rig in a t P

upstream from p , and f o r convenience , th e co-norm al o f 2

and th e elem en tary a re a i s

where

9

is along