1. In tro d u c tio n . T h e m e th o d fo r tre a tin g co m p ressib le flows, as d ev e lo p e d b y C h a p ly g in,1 v o n K a r m a n2 a n d T s ie n3 le a d s to a successful s o lu tio n of th e flow p a tte r n p a s t solid b o d ie s w hen th e flow h as no c irc u la tio n . W h en th e flow h a s a fin ite c irc u la tio n , as in th e case of airfoils, th e profile sh a p e s fu rn ish e d b y th is th e o ry are n o t closed.
I t is d o u b tle s s d e sira b le to d e v e lo p th e th e o ry so a s to rem ove th is d ifficulty.
R e c e n tly , B ers4 su cceed ed in o b ta in in g flows w ith c irc u la tio n a ro u n d closed p ro files. A s is u su al in th e case of a first success, th e new m e th o d h as a few d is a d v a n ta g e s . In th e first place, th e m a p p in g b e tw e e n th e a c tu a l c o m p ressib le flow a n d th e asso c ia te d in co m p ressib le flow is n o t re g u la r a t th e s ta g n a tio n p o in ts. T h u s , if th e profile in th e asso c ia te d in c o m p re ssib le flow is re g u la r e v e ry w h e re, a n g u la r p o in ts w ould a p p e a r in th e profile in th e c o m p ressib le case; a n d vice v e rsa . T h e a p p lic a tio n of th e m e th o d is f u r th e r c o m p lic a te d b y th e fa c t t h a t th e a n g le th u s g e n e ra te d d e p e n d s on th e fre e -stre a m M a c h n u m b e r. F o r th e e n g in eer, th e tr e a tm e n t h a s th e a d d itio n a l in c o n v e n ie n c e of in v o lv in g th e c o n c e p ts of R ie m a n n ia n g e o m e try (w hich a re a v o id e d in th e p re s e n t tr e a tm e n t) .
In th e p re s e n t a rtic le we shall d esc rib e a m e th o d w hich is free from th e d is a d v a n ta g e s m e n tio n e d a b o v e. T h e d e r iv a tio h 'is very sim ple, a n d no referen ce is m a d e to R ie m a n n ia n g e o m e try . Y e t th e re s u lt in clu d es all th e p re v io u s ones a s special cases.
In d e e d , th e tr e a tm e n t seem s to be now in th e m o st n a tu r a l a n d th e m o st general fo rm w h ich is o b ta in a b le from th e line of s tu d y of C h a p ly g in , von K â rm â n a n d T sie n . I t also h a s g re a t flexibility. G iv en one in co m p re ssib le flow, th e re is still a n a n a ly tic fu n c tio n a t o u r d isp o sal fo r c o n s tru c tin g co m p ressib le flows. T h is freed o m of choice e n a b le s us to a v o id m u c h u n n e c e ssa ry n u m e ric a l la b o r in c o n s tru c tin g flows of c e rta in g en eral ty p e s. A larg e n u m b e r of co m p ressib le flows can be d e riv e d from a given in c o m p ressib le flow b y th e p re s e n t m e th o d w ith o u t n u m erical in te g ra tio n .
A p a r t from g iv in g a useful m e th o d fo r c o n s tru c tin g c o m p ressib le flow p a tte r n s , th e p re s e n t d e v e lo p m e n t h a s th e follow ing significance. F irs t, th e freed o m of d isp o s
in g of o n e a n a ly tic fu n c tio n lea d s to th e so lu tio n of th e d ir e c t p ro b le m ,— n a m e ly to
* Received May 18, 1946.
1 S. A. Chaplygin, On gas je ts. Scientific Memoirs, Moscow Univ., Math. Phys. Sec. 21, 1-121 (1902).
(English translation published by Brown University, 1944. Also NACA TM No. 1063, 1944.) 3 Th. von KArmdn, Com pressibility effects in aerodynam ics. Jour. Aero. Sci. 8, 337-356 (1941).
3 Hsue-Shen Tsien, Two-dim ensional subsonic flow of compressible flu ids. Jour. Aero. Sci. 6, 339-407 (1939).
4 L. Bers, On a method of constructing two-dim ensional subsonic compressible flow s around closed pro
files, NACA T N No. 969 (1945); On the circulatory subsonic flow of a compressible flu id past a circular cylinder, NACA T N No. 970 (1945). See also S. Bergman, On two-dim ensional flow s of compressible flu ids, NACA T N No. 972 (1945).
292 C. C. L IN [Vol. IV, No. 3 c a lc u la te th e co m p re ssib le flow p a s t a given profile. In d e e d , the solution of problems of compressible flow, either direct or inverse,— n a m e ly , th e c o n s tru c tio n o f flows a ro u n d profiles e ith e r given b e fo re h a n d o r n o t— is now an a parallel footing with the incompres
sible case. I n e ith e r case, th e d ire c t p ro b le m re q u ire s a m e th o d of su ccessiv e a p p ro x i
m a tio n s.5 S e co n d ly , th e a p p lic a tio n of th e p re ssu re coefficient fo rm u la of v o n K â r m â n a n d T sie n to flows w ith c irc u la tio n is ju stifie d on th e sam e b asis a s in th e c irc u la tio n - free case. E x p e rim e n ta lly , th e fo rm u la h a s b een fo u n d to b e su ccessfu l ev en w h en th e re is c irc u la tio n , a lth o u g h th e th e o ry h a s so fa r been in c o m p le te . T h e o rig in a l d e v e lo p m e n t of v o n K â rm â n a n d T sie n leav es th e b o d y n o t closed, w h ile th e profiles given b y th e m e th o d of B ers do n o t h a v e th e sam e a n a ly tic n a tu r e in th e in c o m p ressib le a n d th e c o m p ressib le cases. T h e p re s e n t m e th o d re m o v es th ese d ifficulties.
F u r th e r in v e s tig a tio n s of th e sig n ifican ce of th e p re s e n t m e th o d a re b e in g c a rrie d o u t. T h e p re s e n t a rtic le c o n ta in s o n ly th e e sse n tia l re s u lt a n d its p roof. I t is h o p e d t h a t a c o m p le te d iscu ssio n of f u r th e r d e v e lo p m e n ts m a y b e p u b lish e d v e ry soon.
2. M e th o d of c o n s tru c tin g tw o -d im e n s io n a l s u b so n ic flow s w ith c irc u la tio n a ro u n d pro files. L e t p, p, u, v be th e p re ssu re , d e n s ity a n d c o m p o n e n ts of v e lo c ity of a s te a d y tw o -d im e n sio n a l ir r o ta tio n a l flow in th e x, y p la n e . L e t p b e a fu n c tio n of th e d e n s ity p o n ly (given e ith e r b y th e is e n tro p ic re la tio n o r a n y o th e r a p p ro x im a te re la tio n ). T h e n th e re e x is t th e v e lo c ity p o te n tia l <j> a n d th e s tre a m fu n c tio n ip defined b y th e follow ing d iffe re n tia l re la tio n s:
d<j> = udx + vdy, (2.1)
dp = — pvdx -j- pudy. (2.2)
T h e v e lo c ity c o m p o n e n ts u, v a n d th e d e n s ity p a re f u r th e r c o n n e c te d b y B e rn o u lli’s e q u a tio n
—---- b
f
— == co n st., (q2 = u- + v2). (2 .3 )2 J p
I t is c o n v e n ie n t to re fe r th e d e n s ity of th e gas to t h a t a t th e s ta g n a tio n p o in t a n d to re fe r all th e v elo citie s to th e v e lo c ity of so u n d a t s ta g n a tio n . T h e c o o rd in a te s x, y m a y b e re g a rd e d a s re fe rred to th e size of th e b o d y , a n d th e p re ssu re a s re fe rre d to th e p r o d u c t of s ta g n a tio n d e n s ity a n d th e s q u a re of th e v e lo c ity of so u n d a t s ta g n a tio n . T h r o u g h o u t th is a rtic le , th is p ro ce ss sh all be im p lied , a n d all th e q u a n titie s u n d e r d iscu ssio n a re d im cn sio n less.
A s is w ell-k n o w n , th e p ro b le m sim plifies g re a tly if th e p re s s u re -d e n s ity r e la tio n is a p p ro x im a te d b y
p — A — — ■ (2 .4 )
P
T h is is th e b a sis of th e m e th o d of C h a p ly g in , v o n K â r m â n a n d T sie n . A d iscu ssio n of its p h y sic a l in te r p r e ta tio n m a y b e fo u n d in th e p a p e rs of th e se a u th o rs . E q u a tio n (2.3) le a d s to
5 For the incompressible case, see T. Theodorsen, Theory of w ing sections of a rb itrary shapes, NACA Rep. No. 411 (1931).
T. Theodorsen and I. E. Garrick, General potential theory of arbitrary w ing sections, NACA Rep. No.
452 (1933).
c = 1 = V I - f q \ (2 .5 ) P
w h e re c is th e local v e lo c ity of so u n d . In d e e d , B m u s t be e q u a l to u n ity if th e referen ce s ta g n a tio n q u a n titie s a re c a lc u la te d from (2.3) b y th e use of (2.4)
U n d e r th e a p p ro x im a tio n (2.4), th e follow ing m e th o d m a y be used fo r c o n s tr u c t
ing tw o -d im e n sio n a l su b so n ic flows w ith c irc u la tio n a ro u n d closed profiles.
Given an incompressible flow past a profile Po i n the f-plane ( f = { + «7) described by the complex stream junction Fifl) and the complex velocity w o(f), choose a junction
¿(f)
> regular i n the region exterior to P« and including the point at infinity, having no root i n Rq, and such that R 0I itt'o(f) I < I ¿(f) I < W on Po, (2.6) and that
/ I r V(f)
H i ) = <2' 7>
where the integration is performed along any contour enclosing Po. Then
c 1 r ^u (f)
* +
i y= ¿ (f)# - - (2-8)
J 4 J kit)
1
¿(f)
2
qw«(f)
--- , (2 .9 )
1 + V I + ?2 ¿(f)
4> + t y = F (
f) (2.10)
gives the parametric representation oj a compressible flow past a profile P in the x, y plane with f as parameter, where P has the same general analytic nature (e.g. same number oj corners, etc.) as the original profile Po- I n these formulae, <j>, j ore the velocity potential and stream junction defined by (2.1) and q, 6 denote the magnitude and direction oj the velocity oj the compressible flow.
3. P ro o f. T h e p ro o f c o n sists of tw o p a rts . F ir s t, i t is n e ce ssa ry to show t h a t, a fte r th e a u x ilia ry v a ria b le f is e lim in a te d from (2.8) - (2.10), we o b ta in p ro p e r fu n c tio n a l re la tio n s b e tw e e n cf>, p, q, 6 a n d x, y. S eco n d ly , we m u s t show t h a t th e profile F is a clo se d c u rv e a n d is m a p p e d in to P 0 b y a re g u la r m a p p in g su ch t h a t th e re g io n s R, ex
te r io r to P , a n d Ro, e x te rio r to Po, a re m a p p e d in to each o th e r in a o n e -to -o n e m a n n e r.
(a) T h e first p a r t of th e p ro o f is sim ple. I t is well k n o w n*1 t h a t u n d e r th e a p p ro x i
m a tio n (2.4), th e re la tio n s
dF 1 _____
= ---w*dF, F — F(w*) (3 .1 )
w w ith z, F, w* defined b y
z = x + iy, F = <f> + i f , 2?
1 + V i + ?2
(3.2)
6 Cf., for example, Eqs. (23), (26) of Tsien’s paper quoted in Footnote 3.
294 C. C. LIN [Vol. IV, No. 3
w ould give a so lu tio n of th e differential e q u a tio n s of co m p re ssib le flow. In s te a d of tr y in g to e sta b lish a re la tio n b e tw e e n F a n d w* d ire c tly , w e in tro d u c e an auxiliary variable
f
su c h t h a t F i f)an dw*(f)
a re a n a ly tic fu n c to in s. I t is w ell kn o w n t h a t g re a t sim p lific a tio n is o b ta in e d b y ta k in g F ( f) to b e th e co m p lex p o te n tia l of a n in c o m p ressib le flow s im ila r to th e c o m p ressib le flow w e d esire. H o w ev er, th e e x te n t of a rb ita rin e s s in th e choice of w * (f) h a s n o t b e e n c a re fu lly e x a m in e d . I t is c le a r t h a t a n y choice o f w * (f) will be su fficien t so fa r a s sa tisfy in g th e differential e q u a tio n s is c o n c ern e d .In th e p re s e n t case w e d isp o se of th e a r b itr a r y fu n c tio n b y w ritin g
w * (f) = w o ( f ) / f e ( f ) . ( 3 . 3 ) T h e o n ly re q u ire m e n ts on ¿ ( f ) a re th e gen eral c o n d itio n s of re g u la rity a n d th e re la tio n s (2.6) a n d (2.7), w h ich will be d iscu ssed im m e d ia te ly .
(b) T h e second p a r t of th e p roof is also v e ry sim ple. In th e first p lace, th e profile P is closed b y v irtu e of (2 .7).6 T h e re g u la r ity of th e m a p p in g is e s ta b lis h e d if th e J a c o b ia n of th e tra n s fo rm a tio n m a in ta in s th e sa m e sig n a n d d o es n o t v a n ish o r b e com e in fin ite in th e region Ro, in c lu d in g th e b o u n d a ry Po a n d th e p o in t a t in fin ity . I t c a n be e a sily verified t h a t th e J a c o b ia n is
wo
Ti
( 3 . 4 )F ro m th is ex p ressio n , i t is c le a r t h a t th e re q u ire m e n t is satisfied w h en k sa tisfie s th e r e s tric tio n s specified in th e la s t sectio n .*
4. D is c u s s io n , (a) The function ¿ ( f ) . T o m a k e use of th e freed o m in c h o o sin g th e fu n c tio n ¿ ( f ) is th e esse n tia l im p ro v e m e n t m a d e in th e p re s e n t p a p e r. V on K â rm â n a n d T sie n gav e a n in te r p r e ta tio n of w *(f) b y id e n tify in g i t w ith th e c o m p le x v e lo c ity w0(f) in th e a sso c ia te d in c o m p re ssib le flow. T h is m e a n s t h a t th e y p u t
¿ ( f ) - a 1 . ( 4 . 1 )
T h e y w ere th e re fo re u n a b le to m e e t th e re q u ir e m e n t (2.7) fo r closing th e profile, for flows w ith c irc u la tio n . B ers o v e rc a m e th is d ifficu lty b y v irtu a lly ta k in g
¿ ( f ) = c o n s t . { w o ( f ) j ‘ ( 4 . 2 )
w h ere is th e fre e -stre am M a c h n u m b e r. H o w ev er, a t th e s ta g n a tio n p o in ts of th e in c o m p re ssib le flow
| ¿ ( f ) ¡ = 0 , ( 4 . 3 )
a n d th e m a p p in g b e tw e e n th e profiles P a n d P0 is n o t re g u la r th e re . In a p p ly in g his m e th o d to th e c a lc u la tio n of co m p re ssib le flow p a s t a circle, B ers h a d to s t a r t w ith th e r a th e r c o m p lic a te d p ro b le m of finding th e in c o m p re ssib le flow p a s t tw o in te r s e c t
in g c irc u la r a rc s w ith th e s ta g n a tio n p o in ts a t th e p o in ts of in te rs e c tio n a n d w ith th e
* Note added in pro o f: Professor K. O. Fried ricks pointed out to the author that, in the compressi
ble as well as the incompressible case, a mapping with non-vanishing Jacobian does not always lead to a useful result: the region obtained may be simply-connected but multiply covered. This difficulty has not been experienced so far in some numerical examples which have been worked out.
an g le s p ro p e rly a d ju s te d in re la tio n to th e fre e -stre am M ach n u m b e r. In d e e d , it seem s t h a t a f te r re a c h in g th e re la tio n
(IF 1
dz —---w*dF,
w* 4
th e m o st n a tu r a l d e v e lo p m e n t of th e K a rm a n -T s ie n m e th o d is to leav e i£>*(D q u ite free, a s w e h a v e d o n e here, in ste a d of c o n n e c tin g it d efin ite ly w ith wo, as w as d o n e b y p re v io u s a u th o rs . T h e p re s e n t m e th o d of le a v in g k ( f) free seem s to be th e m o st g e n era l schem e.
If we d e lib e ra te ly w a n t to in tro d u c e som e s in g u la r p o in ts in P b y s t a r t i n g from a profile P0 w ith o u t a s in g u la r p o in t, | | should be allow ed to ta k e th e lim itin g v a l
ues specified in (2.6) ; e.g., k(£) =0 w h ere w >o(f)=0.
A lth o u g h &(D c a n n o t be ta k e n to be u n ity w hen th e flow h as a c irc u la tio n , it sh o u ld n o t d e p a r t from u n ity v e ry m u ch if th e profiles P a n d P0 are n o t to differ m u ch from each o th e r. T h is is easily seen from a co m p ariso n of te rm s in (cf. (2.8))
1 ~ ~
1 »0 ,
dz = k d t---di. (4 .4 )
4 k T h e y h a v e th e r a tio (cf. (2.9) a n d (2.5))
M 2
- ( ■\ 1 + \ / l + — = )q V
= , (4 .5 )
+ V I + q*/ (1 + V I M T
-w h ere M is th e local M ach n u m b e r q/c. T h is v alu e is m u c h sm a lle r th a n u n ity , e x c e p t fo r v a lu e s of M close to u n ity.7 H e n c e , (4.4) is a p p ro x im a te ly th e id e n tity tra n s f o rm a tio n if k is v e ry close to u n ity . T h is a p p ro x im a te ly p re se rv e s th e sh a p e of th e profile d u r in g th e tra n s fo rm a tio n .
(b) Conformal m apping of compressible flows. If we m ak e a con fo rm a l tra n s fo rm a tio n of th e f-p la n e in to th e f - p la n e b y th e a n a ly tic re la tio n
r = M r ) , (4-6)
w e a rc m e re ly m a k in g a c h an g e of th e a u x ilia ry v a ria b le in (2.8)—(2.10)- In d e e d , we h a v e
w here
r ___ 1 r w5('f) ~
« + i y = kd'ç - - df,
J 4 J M r)
(4 .7 )
2q Wo(r)
i + V i + M r)
(4 .8 )
4> + i f = P ( r ) ; (4 .9 )
H f ) = m , (4 .1 0 )
wn(~n = F Ê ) = n r ) %
« r
(4 .1 1 )
Cf., Fig. 3 of Tsien’s paper quoted in Footnote 3.
296 C. C. LIN [Vol. IV, No. 3
a n d
K f ) = ¿ ( f ) i • ( 4 .1 2 )
" f
W e n o te t h a t th e e q u a tio n s (4 .7 )-(4 .9 ) a re o f th e sam e n a tu r e a s (2 .8 )-(2 .1 0 ). H o w e v e r, th e profile Po, in to w h ich th e profiles P a n d Po a re m a p p e d , m a y b e a r no re se m b la n c e a t all to th e o rig in al profiles. In d e e d , th e r e is no loss in g e n e r a lity in ta k in g Po to be a circle. The relations (4 .7 )-(4 .9 ) thus serve to transform the incompressible flow past a circle into a compressible flow past a profile of a quite arbitrary shape. R e
fe rrin g to (4.6) a n d (4.12) a n d to th e f a c t t h a t ¿ ( f ) sh o u ld b e ta k e n n o t fa r from u n ity , w e see t h a t ¿ ( f ) sh o u ld be ch o sen so t h a t i t is n o t v e ry m u c h d iffe re n t from th e d e riv a tiv e of th e fu n c tio n m a p p in g P in to a circle.
(c) Formulation of the direct problem. If w e d isre g a rd th e in te rm e d ia te s te p of th e f -p la n e a n d d ro p th e tild e , we h a v e a m a p p in g of th e n a tu r e d esc rib e d in S e c tio n 2, b u t w ith ¿ ( f ) so ch o sen t h a t a circle will be m a p p e d in to som e profile P . T h e fu n c tio n
¿ ( f ) m u s t s a tis fy th e re q u ire m e n ts e s ta b lis h e d th e re , b u t i t sh o u ld n o t be v e ry m u c h d iffe re n t from th e fu n c tio n ¿ o ( f ) w h ich m a p s th e c ircle in to P0 b y th e re la tio n
s = f
¿ o (f )d f.T h u s fo r e ach profile P , th e d e te r m in a tio n of th e c o m p re ssib le flow p a s t i t is e q u iv a le n t to th e d e te r m in a tio n o f a p ro p e r ¿ ( f ) m a p p in g i t in to a circle b y (2.8), w h e re W o(f) is th e flow p a s t th e circle. T h e re is no q u e stio n a b o u t th e e x iste n c e of su ch a m a p p in g fu n c tio n . I t is c le a r t h a t to e a c h p u re ly s u b so n ic flow, w h e re th e a p p ro x im a tio n (2.4) is a c c u ra te e n o u g h a n d th e re fo re F = F ( w * ) , w e m a y find a c e rta in
¿ i ( f ) m a p p in g th e co m p re ssib le flow in to some in co m p re ssib le flow. B y co n sid e rin g su ccessiv e c o n fo rm a i tr a n s fo r m a tio n s , we ca n th e re fo re always m a p th e flow in to a circle. T h e a c tu a l m e th o d of fin d in g ¿ ( f ) is all t h a t re m a in s to be d o n e in th e d ire c t p ro b lem .
T h e th e o ry of su b so n ic co m p re ssib le flows (so lo n g a s th e a p p ro x im a tio n (2.4) is v a lid ) is now p u t on a n e q u a l fo o tin g w ith th e in c o m p re ssib le flows. T h e in v erse p ro b le m is c o m p le te , th e d ir e c t p ro b le m of fin d in g a m a p p in g fu n c tio n fo r a given profile c a n o n ly b e solved (p ra c tic a lly ) b y su ccessiv e a p p ro x im a tio n s , even in the i n compressible case.s T o d e v e lo p a m e th o d of su ccessiv e a p p ro x im a tio n s fo r th e d ir e c t p ro b lem in th e c o m p re ssib le case seem s to be a n a tu r a l n e x t step .*
5. A p p licatio n of v o n M is e s ’ m e th o d o f g e n e ra tin g a irfo ils. In th e in c o m p re ssib le case, von M ises tr a n s f o r m s a c ircle in th e f -p la n e in to a n airfoil of a q u ite general sh a p e b y th e tra n s fo rm a tio n
H - t X - t ) 0-7)
8 Cf., e.g. the papers quoted in Footnote 5.
* Note added in proof: The essential difference of the two cases lies in the existence problem. While the existence of the incompressible flow follows from well-known results concerning the Laplace equation, very little seems to be known about the existence problem for compressible flows.
w h e re Xi, • • • , X, a re p o in ts inside th e circle, a n d X0 is a p o in t on th e b o u n d a ry (w hich tra n s fo rm s in to th e tra ilin g edge of th e airfo il). A sim ila r m e th o d can be used h ere.
In (2 .8 )-(2 .1 0 ), we p u t
‘“»-(‘- t X'- t )' (‘ “ t ) <5'2)
w ith th e sa m e g en eral r e s tric tio n s on th e p o in ts X. T h e c o n d itio n t h a t th e p o in ts X a re in sid e th e circle is e x a c tly th e c o n d itio n re q u ire d of k ( £ ) a s s ta te d in S ectio n 2.
T h e c o n d itio n (2.7) fo r th e clo su re of th e b o d y h as also its e q u iv ale n ce in th e c a se co n sid ered b y von M ises. T h e re il isa
y k m =
o.H ere, it d iffers b y th e p resen ce of a n o th e r te rm . T h e c o n d itio n t h a t I H t ) | > I hwott) I
on th e circle is th e o n ly a d d itio n a l re stric tio n in th e p re s e n t case. As it is a m ere in e q u a lity , th e re is no g re a t d ifficu lty in e n su rin g it to be satisfied.
T h e in te g ra tio n re q u ire d in e sta b lish in g th e re la tio n b e tw e e n z a n d f ca n be re a d ily p e rfo rm e d , as it in v o lv es o n ly ra tio n a l fu n c tio n s. T h is ease of c a lc u la tio n , to g e th e r w ith th e flex ib ility of th e choice of th e p o in ts X in c o n tro llin g th e sh a p e of th e airfo il, a re th e a d v a n ta g e s of th e m e th o d of von M ises w h ich a re still p re se rv e d in th e p re s e n t a p p lic a tio n .
T h e a u th o r is g re a tly in d e b te d to D r. J . B. D iaz fo r v e ry helpful d iscu ssio n s in th e c o u rse of th e in v e s tig a tio n a n d to P ro fesso rs W . P ra g e r a n d K . O. F rie d ric h s for th e ir in te r e s t a n d discussions.*
s Cf. W. F. Durand, A erodynam ic theory, vol. 2, J. Springer, Berlin, 1933, p. 78, Eq. (20.4).
* N ote added in proof: After the paper was completed and presented in a colloquium at Brown University, Professor E. Reissner informed the author that Professor A. Gelbart had recently presented a somewhat similar approach in a lecture at M.I.T.
298
N O T E S
-O N T H E E L A ST IC D I S T -O R T I -O N -O F A C Y L IN D R IC A L H -O L E B Y A L O C A L IS E D H Y D R O S T A T IC P R E S S U R E *
By C. J. T R A N T E R (M ilita ry College of Science, Shrivenham, England)
W h e n a h y d r o s ta tic p re ssu re is a p p lie d o v e r o n ly a sm a ll p a r t of th e le n g th of a c y lin d ric a l h o le e x te n d in g th r o u g h a n in fin ite e la s tic solid, th e stre s s e s a n d d is p la c e m e n ts d iffe r c o n s id e ra b ly from th o s e c a u se d b y th e a p p lic a tio n of th is p re s s u re o v e r th e e n tir e le n g th of th e hole. T h is p ro b le m h a s b e e n d isc u sse d b y H . M . W e s te r g a a r d1 u sin g a n a p p ro x im a te m e th o d b u t i t is n o t e a sy to assess th e a c c u ra c y of h is n u m e ric a l re s u lts . I t is th e p u rp o s e of th e p re s e n t n o te to give a n e x a c t so lu tio n a n d to c o m p a re n u m e ric a l re s u lts w ith th o se g iv en b y W e s te rg a a rd .
T h e a n a ly s is u sed h e re is a sim p le a d a p ta tio n of t h a t g iv en b y A. W . R a n k in2 fo r th e s im ila r p ro b le m of a b a n d of u n ifo rm p re s s u re a p p lie d to a lo n g c y lin d ric a l s h a f t. T h e n u m e ric a l c a lc u la tio n s a re n o t so fo rm id a b le a s w o u ld a p p e a r a t first s ig h t a n d a m e th o d g iv en b y L. N . G. F ilo n3 fo r e v a lu a tin g tr ig o n o m e tr ic in te g ra ls h a s p ro v e d v e ry v a lu a b le in th is c o n n e c tio n . T h e re s u lts fo r th e m a x im u m ra d ia l d is p la c e m e n t show t h a t th e a p p r o x im a tio n used b y W e s te rg a a r d is r a th e r c ru d e .
1. T h e a n a ly tic a l so lu tio n . W e use c y lin d ric a l c o o rd in a te s a n d c o n s id e r th e p re s su re lo a d in g as b e in g g iv en b y a r — — p, \ z\ <c, <rr = 0, |z | > c on th e s u rfa c e of th e c y lin d ric a l hole r = a. W ith th e u s u a l n o ta tio n4 w e th e re fo re re q u ire to find a s tre s s fu n c tio n cj) s a tis fy in g
VV> = 0,
r > a, — x < z <co, (1)
w h e re V2 d e n o te s d2/ d r2 + ( l / r ) ( 3 / d r ) - l - d2/ d s2 a n d th e b o u n d a r y c o n d itio n s
3 ( 32 ) i i
" T F M < e '
= 0, I s I > c,
r = a, (2)
d ( a2 )
, (1 — v ) V - > <j> = 0, — x < 2 < c o, r = a , (3 )
dr t. dz2)
v b e in g P o isso n 's r a tio fo r th e m a te ria l of th e c la s tic solid.
v b e in g P o isso n 's r a tio fo r th e m a te ria l of th e c la s tic solid.