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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