Some Problems on Fluid Motion, With Special Reference to the Flow of Compressible Fluids, and With Additional Papers

SOLE PnOBLTTE CHI FLUID '07101%

V.1TH SPECIAL EEFHEEMCE 70 TIE

YWE OF COHPfStSS IELI ’ FLUIDS,

AID V.H1I ADDITIONAL PAPERS.

IV

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7li o t h e s is i s m ainly concerned w ith problem s r e la t in g to th e tw o-d im en sional flo w o f a co m p ressib le

f lu id a t hixSi sp eed . Chapter I g iv e s a b r ie f resume o f c e r ta in standard r e s u lt s which are fr e q u e n tly used l a t e r , w h ile Chapter I I d e a ls w ith th e s o -c a lle d

hodograph method o f s o lv in g problem s on com p ressib le flo w . T his method h a s , in acme r e s p e c t s , been

t r e to d in f a i r d e t a il because an adequate account i s o n ly to be found in jo u r n a ls which a re compara­ t i v e l y in a c c e s s ib le . °h cse two ch a p ters co n ta in n o th in g o z ig in a l and e r e in clu d ed in order to make

th e work rea so n a b ly s e lf-c o n ta in e d . In Chapter H I a known r e s u lt , concerning th e convergence o f c e r ta in s e r ie s a s s o c ia te d w ith th e hodograph m ethod, i s

extended and, a t th e same tim e , appcpodnate s o lu tio n s o f th e hodograph eq u ation s are obtained* These

approxim ations a re th en a p p lie d in Chapter IV t o th e so u tio n o f a p a r tic u la r problem . Chapter V i s

on a body (moving throu^jh a co m p ressib le f lu id ) due t o th e p resen ce o f a shock wnve. T his

ch ap ter ends th e work on th e flo w o f co m p ressib le f l u i d s .

In Chapter VT a problem on th e siow m otion o f a v is c o u s f lu id i s d is c u s s e d .

The f i r s t o f th e a d d itio n a l p a p ers, "On th e FXuxgato P r in c ip le " , attem p ts to p la c e on a f a i r l y r ig o r o u s b a s is a c e r ta in r e s u lt in electron ngnetiom which i s now b ein g w id ely usod in th e d e sig n o f a ir c r a f t ccsapasses and a ccu ra te m agnetom eters and which lia s so fa r had v ery l i t t l e t h e o r e t ic a l

j u s t i f i c a t i o n . I was le d to co n sid e r t h i s problem d u rin g th e war y e a r s , v /!iile working in th e la v ig a tio n

S e c tio n o f th e lo y a l A ir c r a ft r sta b lish m e n t,

Farnborou^i, and com pleted th e work a f t e r retu rn in g to Glasgow.

Hie paper on lir a c 1 s eq u ation c o n ta in s an

enquiry in to th e c o m p a tib ility o f v a r io u s su g g ested method© o f exten d in g t h is eq u ation t o G eneral

r e l a t i v i t y .

of the Gitrte o f Constant Boar-irigf5 scps lapgsly

©Kereiises on B e s s e l Wmo^tfsm em& SSpfoe^icaX

Tvigfmm&i&yf 3?eepeetiv©ly*

F in a lly ^ I wiafe t o thatsfc X^ofGBSQS1 1U O ,Street o f th e 1Royal, Teo1ia:lea‘i C o lle g e s Gaas®3»9 flos*

SBp^wisaaiE Egr r esea rc h and f o r lielp isag mo i s

I . gtapq o y d r e s u lt.' o f th e t lo o t y o f f iQ flo w

a t a c o c ty e n sib le f lu i d i n one ana. two

ja ia ito M L

v;e b eg in by c o lle c t in g h e r e , f o r r e fe r e n c e , th e start iraportant form ulae fo r th e an e-d im en sion al flo w

o f a f T ic t io n lo s s eo a q p rtttib le f lu id v/hich are d eriv ed from B e r n o u lli9e Theorem and v/hich w i l l be req u ired la t e r in our v/ork#

a m s a m j a Theoroo fo r stea d y gttaB B dL tet n o tio n

t

C onsider a s n a il e o c tio n o f a stream tu b e , o f le n g th dg, and croG o-aoction al area S . Then, s in c e th e n o tio n i s ste a d y , i t fo llo w s from th e Second Low o f "ibtion tlia t

-i f-S

where i s th e v e lo c it y o f th e elem en t, p i s i t s d e n s ity and jo and dk a re th e rjressu res a c tin g a t th e en d s,

= 0

aitQfe?*atinc a lo n e th e stro a n tu b e , we have

w hich i s B o r n o u lli, s Theorem. Ih g e n e r a l, th e c o n sta n t o f in te g r a tio n v a r ie s from one strea m lin e t o a n o th er.

The v e lo c it y o f sound c a t any p o in t in th e f lu id i s g iv e n by

c -

*k

" ^

and, i f a d ia b a tic c o n d itio n s p r e v a il, i . e .

where y* i s th e r a t io o f th e s p e c if ic h e a ts o'* th e f lu id a t c o n sta n t p ressu re and c o n sta n t v o lu n e ,th en

cl = Yp/pco

I f zero su b s c r ip ts r e f e r t o th e s t a t e o f th e f lu id a t r e s t , i . e . a t th e sta g n a tio n p o in t vrtien we a re conturaplutiin; th e flo w o f th e f lu id paGt a s o lid body, th en ( 1 .1 ) i# v e s

t . i a U »M » (1.2)

£

C M 1

T tU • ['

Hr®)'-I t i c o fte n co n v en ien t to w r ite ( 1 .4 ) in th e form

= Ajl(c.v- c l ) ( (1.5)

where Ji = = 3 *5“ f o r a i r .

3

.

when c = o , o = o and {> =- o . 'Thus

W ~ 2 /^ ° ' ( 1 .6 )

The s o - c a lle d c r i t i c a l o ood o f tlie f lu id <fys , iftdefc i s a tta in e d when th e lo c a l speed o f th e f lu id and th e lo c a l speed o f sound ore e q u a l, i s o f g r e a t importance* i y (1 * 5 ) we s e e th a t

- ^ c 0 s . ( 1 .7 )

At t h i s p o in t, i t i s u s e fu l to in tro d u ce3* th e M cW iiiM M lanal p e n M to r r , g iv e n by

r * 1 l 7 * V >

For on in c o o p r e s s ib le f lu id T i s zero a t every p o in t o f th e f i e l d o f How* Formulae (1 * 2 ), ( 1 .3 ) and ( 1 .4 )

may now be w r itte n

f , (1 .8 0

p = J)O(<~ T)^ > (1. 0

c ^ c%( \ - z ) ' k . (1 .1 0 )

T} t i e v a lu e o f T c o r re sponding t o th e c r i t i c a l speed i s c iv c n fcy

C ¥ +»

Thus tliO su b son ic r e g io n o f th e f i e l d o f flo w i s i

c h a r a c te r ise d by 0 h t c m d th e su p erso n ic r e g io n yryy- ^ u £ I.

I t i s u s e fu l a ls o t o o b ta in th e r e la tio n s h ip bctereen T ?8iy?eJBMiquJ^Q^9SSfi s cKtra*^ * lo u ^ .d ^ l^ C o ie

"I/S — _J - 1 /

4

.

onJ th e Tfech nunber ' , vM c’i i s d e fin e d b y

m « 1//c • IV ( 1 .5 ) and ( 1 .1 0 ) , t

i f ( $ - 0 - * f [ — - ' )

r i Rr ( l . i i )

5 7 -tT

e p asc on n a ; to co n sid er th e tw o -d in o n eio n a l ir r o t a t io n a l n o tio n o f a co u p reo o ib le f l u i d .

^ o -D l^ c n o io tia l I r r o ta tio n a l b t io n o f a conorcGsibik

i s m

L e ttc , v/ d en ote th e * f ^ components r e c e c t i v e l y o f th e v e lo c it y i • Then th e ab sen ce o f v e r t i c i t y im p lie s th a t

— —* ~ Q

Bit

Bo, ~

»

(1. 12)

w h ile th e c o n tin u ity c o n d itio n g iv e s

£(^) +

I t fo llo w s from (1 .1 2 ) th a t ^ * i/^ io th e com plete d if f e r e n t ia l o f so o e fu n c tio n (f o f * and 'M, 9

i . e . ^Anc + zr dcf

and D©

j t 1 1/5 ^ ; a *1 4 >

^ io c a lle d th e v e lo c it y p o t e n t ia l. S im ila r ly from (1 .1 3 )

' £ r <u

where ^ i s an oth er fu n c tio n o f x and ^ 9 c a lle d th e stream fu n c tio n , and

£o H v /* - £ i

^ (0 ^ 5 f> (1 .1 5 )

I^om (1 .1 4 ) and (1 .1 5 ) we conclude th a t

3qi o* ^ - °o ^

= ^ ^ ^ ~ " f B * ‘ (1 .1 6 )

h i t i e c a se o f an in co m p ressib le f lu id p

-everyw liere and eq u ation s (1 .1 6 ) a re sim p ly th e C andy 1*1 enarm 'Equation© w hich ex p ress th e f h e t th a t u/ h <p+Lf i s a h o lco o rp h ic fu n c tio n o f

^ ^ x ♦ t ^ . U n fo rtu n a tely , in th e g m m ro l c a s e ,

vs i s n o t a holorjorphic fu n c tio n o f y , and so th e pow erful methods o f t i e th eo ry o f th e comploc

v a r ia b le a re n o t a t our d is p o s a l f o r th e e x a c t s o lu tio n o f problem s on co m p ressib le flo w .

He now ro ceed t o d e r iv e th e b a s ic p a r t ia l

d if f e r e n t ia l eq u ation which i s s a t i s f i e d by (f> and ^ Maki ng u se o f th e v e c to r c a lc u lu s , we may w r ite th e eq u ation o f c o n tin u ity (1 .1 3 ) in th e foxn

p d\V +. <\j „ 0 ^ 0 y

o r , by (1 .1 4 )

H f + ^ Y * ^ f - Y * 1 f> ~ 0 • ( l . i ? ) ITom9 from B ernoulli*© Theorem ( 1 .1 ) ,

6

or , , z

= - 3? v ^ i a j 8 ,

Hence, by (1 .1 7 ) and (1 .1 0 )

a„ | -

jl

( £ (*)1} * * &

K

fcM

£>*j]

( \ - +- i l r i - ^ 2 j l _ Q

^ v - t1 TW ^ c - ; c 2- 3**3 - °. ( 1 * 1 9 )

3 in ila r ly f Ly e x p ressin g k 9 v In terras o f ^

In stea d o f <p v;e a r r iv e a t th e eq u a tio n

^ ^ / U1- \ * V f* ^ \ . _ /N

■>,A 'c~) >,» ^ c*- ^ ~ (1 ^ m (1 .1 3 ) and (1 .2 0 ) k , e and c o u s t be earpr—Bid In teru a o f th e d e r iv a tiv e s o f <f v|^, and I t

la

b ecau se

w.

v

^

«j>

I I . :^ > o a s -f s o lu tio n o f tl-e

d ilT o r a n tir l equation o f two d im en sion al catJ^ reeslb lo flo v /

The £\jndamental eq u ation (1*19) o r (1 .2 0 ) i s a p a r t ia l d if f a r e n t iu l eq u ation o f th e second order and o f mixed ty p e . I t i s o f e l l i p t i c a l ty p e when

cu 3 u > v c ( i . e . In tlio cu b sanic r e g io n )* i s

2- 2.

ry_ ^ b o l lc Vfefli 1/ > c ( i . o . in th e su p w sc n ic

r e g io n ) and i s p a r a b o lic when * £ * The d i f f i c u l t i e s e n ta ile d in th e in te g r a tio n are due p a r tly t o th e non­ lin e a r ch a ra cter o f th e eq u ation and p a r tly t o th e change o f ty p e a t some c r i t i c a l v a lu e o f th e v e lo c it y o f sound v/hich a t f i r s t i s s t i l l unknown and has to be determ ined in th e cou rse o f th e c a lc u la tio n i t s e l f .

Vs n<*. co n sid er twro o f th o more im portant methods w hich have been developed t o or ercomc th e s e d i f f i c u l t i e s .

For a reouno o f th o o th er methods ( e . g . th e Jonzen-P a y leiiJ i I te r a tio n hethodf G .I . Tsaylor’ s e le c t r ic a l method e t c . ) r e fe r e n c e should bo made t o a paper by

s o r .

Trio L inear crtu rb a tio n Theory

T his th eo ry p ro v id es an ai proxim ate nothod o f

d eterm in in g th o su b son ic flo w o f a f lu id p a s t a

1 o

sle n d e r p r o f ile and i s due to P randtl * and G lauert As a v ery f u l l accou n t can be found in a r e c e n t r e p o r t by G o ld stein and Young c 9 v/e s h a ll g iv e hero

o n ly a v er y b r ie f o u tlin e o f th e th eo x y and m ention tv/o form ulae which we s h a ll r e q u ir e l a t e r .

L et th e ?c - a x is be ta k e s a lo n g th e d ir e c tio n o f th e u n d istu rb ed flo w and l e t th e v e lo c it y a t an

I n f i n it e d is ta n c e fTara th e p r o f ile b e R ; U, nay be o f th e order o f th e l o c a l v e lo c it y o f sound. 8m

th en assume t lia t in th e neighbourhood o f th e p r o f ile (a ) the?*.- component o f th e v e l o c it y , u. f d if f e r s

o n ly s l i g h t l y from ^ ,

(b ) th e ^ -c o o p o M B t o f th e v e l o c it y , v , i s n e g lig ib le f

( c ) th e v e lo c it y o f sound c way be re p la ced by i t s v a lu e in th e undistrubod stream .

jq u ation (1 .2 0 ) f o r th e stream fu n c tio n th en red u ces

, bV ^ f'

^ V -

° ’

(2.15

whsps M i s t b s Bfech. number o f th e u n d isturbed stream . T his eq u ation nay bo w r itte n

t ± +- t ± - = a

________________________________

Hence t e com p ressib le f i e l d o f flo w around th e p r a fU e can be ob tain ed f rom th e in co m iirescib le

f i e l d o f flo w around a p r o f ile whose th ic k n e s s r a t io d if f e r s x^oa th a t o f th e g iv e n p r o f ile by a fa c to r

( i - 'W' # The th eo ry b r is k s down n ear ray stagna­ t io n p o in t on th e p r o f ile f o r f a t such a p o in tf th e d iffe r e n c e betw een u. and U, i s eq u al t o 21 i t s e l f *

I f th e p ressu re a t some p o in t in th e f i e l d o f flo w p a st a g iv e n p r o f ile i s j=> and th e p ressu re in th e u n d istu rb ed stream i s 9 th en i t can be shown th a t

' ^ ^ “ y tl-r P o 1 ( 2 .2 )

where ■ i s th e Mach, number o f th e u n d isturbed Stream* F u rth er| th e th eo ry ;provides an approxim ate

method o f c a lc u la tin g ; th e c r i t i c a l Mach. number o f th e p r o f il e f i.e * t o Mach number ( o f th e undiaturbed flo w ) a t w hich th e v e lo c it y o f to und i s f i r s t a tta in e d a t some p o in t o f th e p r o file * The c r i t i c a l Tfech

number i s u s u a lly exp ressed a s a fu n c tio n o f Vhere i s th e laxinun v e lo c it y on th e p r o f ile

The Method

\;e s h a ll t r e a t t h i s method in co n sid e ra b le d e t a il b ecause a com plete account o f i t i s o n ly to be found in memoirs v/hich are com p aratively in a c c e ss­

i b l e . The p r in c ip le o f th e method was ex p la in ed in

%

1890 by Tioleribroeck * fo r a g a s f a r w hich y = -1 and th en extended and made a p p lic a b le t o any g a s by T ch ap ligu in e in 1 0 4 . Ikywever, l i t t l e heed was

3 .

p aid to th e s e papers u n t il Ucmtschonko drw?

a tte n tio n to them in 1 9 3 2 . The account g iv e n below 4 .

fo llo w s c lo s e ly th a t o f Cuius Jacob , who a ls o

a p p lie d th e method t o th e th eo ry o f g a s j e t s . Use 5 .

lias a ls o been made o f a paper by T"in|£Leb in v h ich some e x a c t s o lu tio n s o f th o fundam ental eq u a tio n s are d eriv ed by th e method.

The e s s e n t ia l fe a tu r e o f th e method i s th a t th e b a s ic p a r t ia l d if f e r e n t ia l aq u ation C l.2 0 ) d e fin in g

i

th e flo w o f a c o a p r e ssib le f lu id cor. be tranoform ed In to a lin e a r eq u ation by chancing th e independent v a r ia b le s f t " c m t o I , & where i s th e r e s u lta n t

U V'T (itiVffl-iOfft- » J r c b iv . d« 'la th .u . Pliys* Orursnert-dopiBQ a c ie n lif iq u e de l fTjhiv* de ^bocou 2

y . DGn^SceftlfiW ^C.R.1 9 4 (1 9 0 2 ),1 2 1 8 ; C .P..(1 9 3 2 ),1 7 2 0

Jacobs r s u ll.S c i. do l * ' ’c o le olytecftrdr-ue d e Titd-soaro

v e lo c it y o f th e f lu id a t any p o in t and 0 ^ ^ i s th e a n g le which t h is r e s u lta n t v e lo c it y make© w ith th e X, - a x i s . For th e subsequent a p p lic a tio n

o f th e theory* i t i s more co n v en ien t t o ta k e th e independent v a r ia b le s a s T & & ra th e r t'aan ^ and £ .

rrcra th e eciuations ( c f . (1 .1 4 ) and ( 1 .1 5 ) ) f ^ Uc(/K +- ^ ^ 1*~p ( “ +* u<(v^) ,

we have

= T l—L( u ^ - & vl(y) = A<j>

-ll +V I l/AW^ * J

^ ( v ^ + ^ K<L^ ' ) ~ ^ v ( S l" e ^ + !7ow, In an ob viou s n o ta tio n ,

^ K 4), f .

f ) ~

("''*1

f - ^ ) r ^

w hence, by ( 2 .4 )

-/>

( 2 .5 )

V ' ,v w r

19 * = i J T 'h + I f **>e V * ),

% - f c - f r n 9 Wq) '

The c o n d itio n s i<Tp - fcp < i y ^ = tb en y ie ld

Siv(9fT - ~ <fe - - £ ± tn e y z ;

^ 6 ^ + — ^ = +*>! >[ ( f ) t ' T F f ] f o

whence " - , ^ ( 2 .7 )

12

.

o0

and f t 1 ' 3 r y TT • ( 2 .8 )

Ly ( 1 .3 ) th e s e eq u a tio n s nay be w r itte n <p = 0 /3 -h)t - I

^ S t i ^ T T * * ( s a )

ami *fe = • (2 .1 3 )

Eciuations ( 2 .7 ) and (2*8) are th e g e n e r a l hGdograph eq u a tio n s eorresponding t o eq u a tio n s (1*16) f and are tr u e f o r a l l p ressu re - d e n s ity r e la t io n s h ip s ; th ey oGSune th e forno (2*9) and (2*10) when a d ia b a tic

c o n d itio n s a re p o stu la ted * a s w i l l be done in th e problem s vrith vrtiich we a re concerned here*

I t fo llo w s im m ediately from (2 *9) and (2 .1 > )

tlsa t v

a r , _ n

f e [ ; x ^ ‘ T)

^J

^

T h is eq u ation d e te r o ln e e when ce <jj> can be d e te m in o d b y ( 2 .0 ) and ( 2 .1 0 ) ,

Vs

no \i

.0300

o f

t and

&

alone, we have

_

u -t (i - t ')/**1

j_ i L f _/•.

W 1 ~ i-(2./3+i)t <*rj.

fu n c tio n s o f t and & a lo n e , we have Arguing in th e u su a l wtiy, we p u t each s id e o f t h is eq u a tio n oc;ual t o a co n sta n t •»'*? * and g e t

& ~ f a /v- ' ( * y &

where and are c o n sta n ts, and

13

.

Vz.

*e n e x t put T * a whence

a t , t t f +

d - t A t 1

d V _ z *k +- ^ t ^ _ I ^ + £ * ( £ " - 0 z

----

—

i

■+-

i —

i.*-Z u.

- i / V ' ' / J

d x ’ A x

and (2,113) bocomue, on s in p lif ic a t io n

°* ( s .1 3 ) Comparing t h is eq u ation w ith th e -^pergeojaetrtc

Equation

one o f whom fundam ental solu t io n s I s ^ ^ F yf *c)^ we s e e th a t a s o lu tio n o f (* .1 3 ) I s

^ ( t ) * F f a * , C * , T )

\t':xa.'Q a„ + ^ = ^ - p ■>

A.^ r - d -K(-h + l) (3 t

^4v - X + I '

Thus, a form al s o lu tio n o f ( 2 .U ) i s

^ G , T ) « A+ Z B uI tJ 1 ^ ) ( s . 1 4 )

where A-, 8 , 0*, e r e constant© , t ( i o t' e 'c o n sta n t) v a lu e o f T f t r th e main stream (we a r e co o ftesp la tin g th e flo w o f th o f lu id ix is t s o lid b o d ie s) and th e

summation i c tak en over v a lu e s o f n n o t y e t s p e c if ie d . The t v m M> a re ob tain ed by ta k in g ^ and T b oth z e r o , and th e r e i s no lo s s o f g e n e r a lity in ta k in g f\

eq u al t o z e r o . The reaeon fo r in tro d u cin g th o

14

How, toy (2 .1 0 )

2><f I t y-p^ [ X V I %( *) 1 - t O

* j7 ~ j* ~ ^ •' L ^ r > ' ’

H I

a-c y

B

^

J

' o ^ ^ t r * ^ r*J L * %(t.) ^fT ,)J • '

whene# <f

care, by ( 2 .9 ) F (x ) i s Bucfo txiat ^ ( a / U Q x - l

»T

and io d eriv ed from ( 2 .1 4 ) . Hsus we fin d t i n t P M = t ( t J 1M T - '

" 4 t« .* >

-X tmfa&l-f**) f (2 .1 5 ) * * r* M x ) - I ‘

Lot us n-j*./ d e te x iiln e ?fcat (2 .1 4 ) and (2 .1 5 ) reduce to when th e f lu id i s in c o m p re ssib le . i o n a / i n f i n i t e ,

T and Z( a re b oth se r o although T/ r f I s rep la ced by (Vfy)

9 K ( 0 ) = I > ( 0 5 r - i ard

i r . T r f H - o i * ' A x ^ f U u ~

'ianoe, Jj? f\- 0 *

f ( 9 , r ) ~Bfr + Z (2 .1 6 )

<?(.$,-) ~ - S l o ^ l ^ - Z ^ ( ^ )

i . e . , i f th e p hases o f th e c ir c u la r fu n c tio n s are taken a s s e r o ,

(2 .1 7 )

15

Thus* f o l l o d a g r in gleb * we s e c th a t eq u a tio n s (2 .1 6 ) a re com p letely e q u iv a le n t t o th e exi^ansior* of* th e

1 .; e

co n p lex p o t e n t ia l fu n c tio n w in powers o f */ *

± JL* U'

i . e . in powers o f ^ Z t s in c e

j ^

— - x ( < ? + i f ) = H - i v = fye.'*- . ( 2 ,IP ) lienee* i f th e complex p o te n tia l fu n c tio n i s fcnom fa r a f i e l d o f In eo tsp ressib l e flo w and 5© expended in th e form (2*17)* th e c o e f f ic ie n t s B, B**. oo determ ined can h e s u b s titu te d in (2 .1 4 ) (2 .1 5 ) to determ ine th e

v e lo c it y p o t e n t ia l and stream fu n c tio n f o r a correspo­ nding f i e l d o f com p ressib le flow *

The q u e stio n o f th e cosr/ergence o f th e s e r ie s

(2 .1 4 ) and (2 .1 5 ) w i l l be d isc u sse d in th e n ex t s e c t io n . The s o lu tio n o f eq u a tio n s (2 .1 4 ) and (2 .1 5 ) g iv e s <f a n t ^ iB t m o f P and r . On th e strea m lin e vp — c o n s t. 9 v/e th e r e fo r e have

f ( e , x ) = (2 .1 0 )

In ord er to determ ine th e flow In th e p lan e we th en have reco u rse to th e r e la tio n s h ip

w hich fo llo w s e a s i l y from eq u a tio n s (1 .1 4 ) and (1 .1 5 ) d e fin in g th e v e lo c it y components in term s o f th e

d e r iv a tiv e s o f cf end ^ . Hence* e lim in a tin g £ and x betw een (2 .1 9 ) and th e two eq u a tio n s ob tain ed fVom

I l l The Convergence uJ th o i*oj.» «y a~ d W

ir: v .e Ix d o tja fc : vthoa. ar*! tViG D etern ittu tio n

Q.-: .• .jro & y X ? .r!* '~ p V '> ^ £.& JgjL >.

I t h as been proved by Jacob* on th e b a s is o f a l i c c a t i equation* which i s s a t i s f i e d b y *♦*(*}§ th a t th e s e r ie s (2 .1 4 ) and ( 2 . 1 5 ) g iv in g f and <f fo r th e com p ressib le flo w converge u n iform ly and a b s o lu te ly in th e range 0 x % r, provided th e same i s tr u e o f th e s e r ie s (2 .1 6 ) fo r th e corresp on d in g incm cpre-s cpre-s i b l e flo w and th a t th e cpre-s e r ie cpre-s ore in a cpre-scen d in g l^ov/srs o f r/T. . T’e s h a ll now show how Jacob*© argument can be extended t o prove th a t t h is r e s u lt

i s v a lid in th e w ider range 0 h z t . P h y s ic a lly , t h is io much more s a tis fa c to r y * fo r th e c r i t i c a l

v e lo c it y g iv en by (2-ji + l ) ' 1 i s o f much g r e a te r s ig n if ic a n t s ttvan th e i n in stream v e lo c it y g iv en by

T “ T, •

T o b eg in by pityving th r e e lerria s due to Jaccto and tak e ^ 0 th rou gh out.

In th e in te r v a l 0 ^ x ^ , 0 ^ a id o c **(T ) t I.

Since

s a t i s f i e s th e eq u atio n (2 .1 3 )

T ( i - r ) ^ -h i - z ( H - f l i /> f t <^K - c>;

17

.

( 3 .1 ) w h e r e U x ) H r ^ ( x ) .

L et t - a }>q th e f i r s t zero o f in th e In te r v a l con sid ered # Then | k( t ) v a n ish e s f o r r - o and t r a. and s o 9 hr/ n o lle *s Theorettj th<are e x i s t s a v a lu e ^ where o ^ 4 ^ a f o r \$iidh ^ ( r ) ~ o • The

fu n c tio n z ( «“ X y / < t ) i s th e r e fo r e zero f o r r “ o and r v £ r and s o 9 lay P o lle ’s Theorem i t s d e r iv a tiv e v a n ish e s f o r sorae vdL ue t - c f where

\

o u c 4 > . Hence* by ( 3 * l ) f ^H( r ) v a n ish e s when T * c (^ a )j aixi t h i s c o n tr a d ic ts th e i n i t i a l

assu iu ytion . Hence ^v fx> and y * (r ) hove no ze ro s in th e in te r v a l o fe t § # n in ce « | i t f ilo w s th a t % CO >D and ^ ( i ) = o in th e range consSdared*

F urtherf (r ) cannot have a se r o in th e in t e r v a l9 o th erw ise by ( 3 .1 ) ^ ^ CO would have a zero in th e in t e r v a l. Hot/

18

.

l a th e I n te r v a l O h ~ m in e e (z!f

tlirou^hout th e in t e r v a l9 i t f o lio v a th a t th e M&e i s tr u e o f z COt_{-K *}

U- ^ ( x ) > 0 ; O t T % (2-/S+I)'' Al&o, (2 .1 3 ) nay be w r itte n

£ f t ^ O - c T ' S m ' ^ ) ] + + - t ) “£ ' ^ /'r ) = <?. AX

I^r d n i l c r rea so n in g to th e ab ove, i t fo llo w s th a t

(x ) cannot b e se r o in th e in te r v a l 0 r r ^ .

low ^ (0) = ^ 0 f whence ^ H( r ^ 9

e in c e ^ ( 0) = ( t X ( T) = 1 • Thus y j z ) ~ I , 0 ^ T r (&jU0~'*

I t fo llo w s to o th a t \ ( t j j ^ | m d oof by th e above

0 ^ M T> * I » 0 = T * CTf+t)~\

^

• locuta llnfe^tion fo r

Fl\aa ti e d e f in it io n o f ^ ( T j v/e im e d ia te ly H iv o th a t

[ ^ v,(t) - | ] (3#2)

..banco 3j ' ^ ,'fT) 4- ^ ( T)J r - n ^ f r ) [ * J t ) - i J +■ -h

S u b e titu tin g fo r \>H( t ) in towns o f £ (T) and •yK( x ) (C f. ( 3 . 1 3 ) ) , and then f o r (O In t a m e o f

fr a .: ( 3 . 2 ) and d iv id in g throughout by ( ~ ) , we

Lem a I I L et t (x ) tie a con tin u ou s fu n c tio n o f x w ith con tin u ou s f i r s t d e r iv a tiv e in th e in te r v a l

0 H n h m ft(x) ^ o and ft(< 0 ~ 1 • I f in th e in te r v a l c> k T % ( zP4l )~ff

th en ft (x ) S. ^ ( x ) t in th e same in te r v a l; s im ila r ly i f ^CftO :)] ^ 0 , then k( z ) 4 % ( x ) f o £• x £

L et f j j i ( x Y\ § 0 f then

? T ( \ - T ) [ k ,( T ) - * „ Y x ) J

( 3 . 4 ) Uo\. suppose th a t ft(*0 ^ *K(f ) a t some p o in t in th e

in te r v a l Then "R(x) - ^ ( r ) must have a n e g a tiv e low er bound a t san e p o in t ^ c ^ in th e in te r v a l* I f T = o~ io an in t e r io r p o in t o f th e

in t e r v a l, i t must correspond t o a tu rn in g v a lu e o f ft ( r ) - ^ K( r ) and k '(a) - - O # Hence, s in c e th e c u r ly b rack et in ( 3 . 4 ) i s p o s it iv e by Lenina

I , v/e must have

ft(V ) - 2l o

v/Mch con trad ict© th e f a c t th a t th e low er bound i s n e g a tiv e .

I f x - ex. i s an end p o in t o f th e in t e r v a l, then s in c e f t ( o ) - r: 0 * A lso ,

f t ( 0 - ~ o , Whence

f t 6 0 - * + , ( * ) > o ?

Hence K (x ) ~ ^ ( r ) throughout th e in te r v a l

0 | | x # (z|J+i)J andf we can prove in a t a i l o r manner t h a t , i f f th en ft ( s ) £ n j r ) tfeNM0fc» ou t th e sa*e in te r v a l*

bcarxa I I can be used to ehcw th a t

i ! - * - ( 3 . 5 )

L et fc ( x ) H ( i - y 1 , th en ft (x ) a b is f ie s th e c o n d itio n s o f th e lemma and

f f t M l M " ) t z

F g f c M j •> - ( , . T)^

Ilonce ft(x ) ^ \ ( x ) and th e r e s u lt follow ® •

Leona I I I r) i s a d ecrea sin g fu n c tio n o f rt in th e in t e r v a l 0 ^ r - t1^ ' ) f.

Lot j? > o , th en

f \ . [ w ^ (Td * - F U |,[**♦).w ]

= - . t f

\fj ( 3 . 3 ) . Ileico v ( 3 . 5 ) , i 0 , end by L a r a I I ,

= * n M .

The whole o f th e fo r e g o ir g v a s provod by Jacob v/ho, a f t e r p ro vin g two o th er le c u a o , d e r i/e d th e r e s u lt s ta te d a t th e b egin n in g o f t h i s se c tio n *

d if f e r e n t way* v/e can e s ta b lis h th e raore g en era l r e s u lt t o which we r e fe r r e d .

• e co n tin u e t o regard >v a s p o £ ib iv e and r e a lis e th a t to e S le e a t l eq u ation ( 3 . 3 ) can h e so lv ed

e x a c tly in two s p e c ia l caf*eof v i z . o f 'K “> o6# ‘ hen ^ o f we m ve

r ( t - - t ) %J(x) +- fix 'Xo(x) - 0

whence

oc^fx) - K

(i~

x)P

>

K a consta

■Xofx) - (3.G)

1t e \ * ( x ) and *** (x ) ren a ln f i n i t e ( c f . Lemraa I ) and x ( r ) -*> _{f H} x (x) f v/here by ( 3 . 3 )_*

i . e . x J T) = I — (Zf 4 , ) r 1'/z ( 3 . 7 )

th e p o s it iv e s ig n b ein g taken to en su re th a t xo6( ° ) ~ I Thus, by Leona I I I

- ^ § 0 - t ) ^

"i - rvj J + O r l ^ ^ )4. 2r yj cx)

. i - x J r ' ^ '

I t fo llo w s f^om t h is in e q u a lity th a t i f r > t.

t

o i *

<

The in te g r a ls t o t h is in e q u a lity a re b oth n e g a tiv e

♦ *

22.

% T 7

—

k

and Thua

■ % (t )

U tt)

tyn

x \£ ~ V i.fi)

£ _{(3 .1 0 )}

% (T<)

StVv

I

( *

0+^0

% J

d^Y{)

1*«)

J ^

t t m c e if . fo llo w s f r u i th e ©esiaral p r in c ip le o f

convergence th a t tfc« s(H « b ( 2 . 1 4 ) i s a b s o lu te ly anti

u n ifcrafty con vergen t in tu e in te r v a l 0 § t ( % x ^ ^ p rovided th e sane i s tr u e o f th e cocrroanonding s e r ie s

< 2 .1 6 ) .

I f

x

T,

9

th e in eq u a lity

i c reversed

end becomes

Ct (,_t ) A_i . •y„(T) - > u r h , f i 1 ,

‘11 = h v c > = * J , M -

J -

^ 3 .

,

Hence

w f * , £ y . > i

n T( t - t r r ^ j w - M t / l J •

Doth I n te g r a ls arc again n e g a tiv e o r

zero

" . ^ k(t) <- I ( 3 . 1 2 )

V 1" ■ "M m ) " ■

IIowf i f th e a e r ie s (2*16) i s a b s o lu te ly convergent in th e in te r v a l 0 § r - *€, ( i . e * in o ^ V ^( # I J th en th e s e r ie s 2! | f**o|

ia c o n v e rg e n t* Hence by ( 3 .1 2 ) i t f o U o i S

a b s o lu te ly con vergen t In th e in te r v a l 0 ^ t ± Tf ~

Thus, our r e s u lt h o ld s Whether r t ^ f or r ? r, i . e * th e a e r ie s (2*14) i s a b s o lu te ly and u n iform ly con vergent in th e in te r v a l 0 ^ t § fiA* i) "provided

th e oai jo i s tr u e o f th e corresp on d ng s e r ie s ( 2 , 1 6 ) • Hr (d*P)f | \ ( r ) | ^ I so th a t th e oane r e s u lt h o ld s f o r ( 2 . 1 5 ) f the s e r ie s expansion o f <p ( £,T)*

. , r o y l : c • o - i o r > ( t) and

ih o in o c iu a lity ( 3 , 8 ) and a s in ili- r one which w i l l he givers below onabl u s to o b ta in f a i r l y c lo s e

approxiiaatio^s t o ^ ( r ) M ft \ ( t)* T his method o f approxim ation w i l l be a t 1; a s t eo ood f i f n ot b e t t e r ,

X .

than tn&t due to St o aaa and ?sf<arw

G iving I th e v a lu e V i 9 i t f o l l o w from ( 9 # l) th a t

o f i , . £ r . 1 L

-" T ^ T T '*• - - 1 T L r c “2.

1 r ... - - 2- r " * > l « c *

2^/ r _ _ \ c *wx> H / . r + !T r > ^ V. A * r ■ " / =• - A L A r

:ien ce, i f T > t (

'’V - r - - ft£ _*■

\ / /

/ v ^

)

* (. i r T - ) s ^ W r ) = l l 1 *■ Z *" / , th e co n sta n t o f in te g r n tic a i b ein g taken a s zero to

en su re th a t ^ ( o ) = I ; i f Tt* T( , th e in e q u a lity I s r e v e r se d .

S in ce D | t '/(, t we have apprw drevtely ( in 1 .

y q p lw ^ kQT ^ TTTT^T— ?r

•

Thus* although

b o tii c a s e s )

/ 5^ fS" 2- \

^ 4) d T) ~ X ( T r ~

L / ,

* « • • ,. , . • r l ! S' ( 3 . 3 3 )

lb e stim a tin g th e error in cu rred by th e

approxim ation* l e t us ra&*iJL>er th a t a f t e r we have found ^ { t j v/© determ ine th e stream fu n c tio n fo r th e oanprecsib l? flo w by r e p la cin g th e to n s

i

%r

S Z r s t o f « q i a l l m (2 .1 6 ) by V%) Hu (z,) • H snce, i f we t th e e g r e s s io n (3 .1 3 ) fo r % ( r ) . we

1 . -rrj, J. £'£ f ~

our approxim ation laiy lo a d to an e r r o r or 12" (When

~CT 1ipj in th e iiid ox o i each exp on en tial* th e r e s u lta n t o n w in %:?<*• the com p ressib le flo w la a t most o n ly

r ' n,

— ^ ZH- I . e . S \ The ay'mndbsi t io n can t h G r e f o r s

be regarded

s a ti sfactory .

To © sable u s t o c a lc u l a te (f ( £, r ) we now seek an approxim ation to %H ( t ) . The I n e q u a lity (3 .^ ) la n o t o u f f ic ia n t ly strong* ao we t r y to fin d a

fu n c tio n rela ted , to th e fu n c tio n s in th e s e r ie s (2 .1 5 ) fo r cf (e„T) in th e same way aa (x ) i s r e la te d to

th e fu n c tio n s appearing in th e s e r ie s (2 .1 4 ) fo r ^ (<9,r) alid ti.en r e p e a t th e above proced u re.

2 5

.

W O 1 + -K 1 (< -t ) ' / 4 -K^Ct)

S im p lifyin g* we e a s ily fin d th a t t h i s red u ces t c

Wt , = '-

-w -1 Ii—t) l i . l t )

I n v i t i n g ard. d iffe r e n tia tin g * we fin d th a t i - f j M t K V t) _ I

^ *“ | - x

whence* s u b s titu tin g fo r \ ( l ) and x * !(r) in (3*3)* we nee th a t ^n (^) s a t is f ie d th e ~ ic c a t i eq u ation

, V x 6 . ?h<tvM

K ft) + 7 — ^ 1' ^

(l-t)|l-f^4l)x} ^ g ,-^4!)t „ n (3.14)

2- T(l-T)

As before* th is

can be solved

o sm s 1x * 0 and ^ « v/q have

i , M =

airi _ _ g ± _ A8+»

^olx) *~x 1 ^P4')'

i« e# (~

W O ~

(,-xyi*i ’

the c«iat& n t o f in te g r a tio n b ein g else sen t o make

>0 Co> - I

.

The equation (3.14) has the sane properties as

(3.3)* the argument of Lqubs II still being valid*

iM jan d ic. Hence

2 6

.

2 - [ ( i - r ) ' ^ - K „ ( r ) ^ 5 -____JOT v

r

4-C T.

4. 14 — - - - _ t r 1 2

^ x

^ I T ~ * r “ *• ~ 4* {(1-x)^ /m^K) As our approsdiaation v/e ta k e

2 t 1 V r _ Z£ t z *

~ y-y

~

t i e erro r I n tr o d u c e here i s about V ie sorae a s in th e p rev io u s c u s e . ihue

L r J * (3*15)

iancaf fcjy (3 .1 3 ) (3 .1 b ) wo have th e r u le *

to obxain th e s e r ie s (2 .1 5 ) fTcra th e corresponcH ng

s e r ie s (3.1G> r e p la c e f%t in (2 .1 6 ) y j/ *w-|o ^ x ~ ~ )

1-. * + H s - n < )

IV, Pome a o i-H ca tio n a o f th e '. Wxurmvh -othod

In Vie

ce l u t i o r o f any problem by th e fcodograph

n eth o d ,

th e

s te p i s to GxponO

th e coopleoc

ITcv t h i s lr.ttea* fm o tio n i s extrem ely u s e i\il (as

an

s&sdLlieay

calutlcn

o f a f r e e jot, & l i e & is cor:: tan t a lo n g r.ry boundary « Tt. i s n a tu ra l th e r e fo r e to tr y to

apply the hodogrepb method to such p ro h la a s ao

t h e s e , ^rofclans

an

jo ts

o th e r s but^ac f a r a s I an aw are, th e r e i s no

m ention In

th e

any

a -loos? otyoai-i ovor a r c c t i l l n o a r

hoi

1 .

V’e co n sid er a f lu id f l a v i n ; A to D which a re i n f i n i t e l y d ista n t. .’rota and on e it h e r s id e o f t

o

#

v p e r s id e o f t he bouxdory,

VJe b egin by assuraing th a t th e f l u i i i s in co m p ressib le .

L et 30 Toe ofiLengtfcXand l e t ABC ~ .

L et th e v e lo c it y a t i n f i n i t y be U. and l e t ua taico th e v e lo c it y p o te n tia l (jpae '« 6 a t A a s too a t Df s in c e th e d ir e c tio n o f flo w i s th e d ir e c tio n o f in c r e a sin g v e lo c it y p o te n tia l*

. vertical.,** d id ..: o f £faw in t:

o

£

The

Scfcraers and

3.

Ijet a, *>, % . . . . be *v p o in ts on th e r e a l a x is

in th e £ p la n e oueli th a t ^ c X^- ^ 4 . , . . and l e t A y ,* * * * be th e in t e r io r a n g le s o f a sim p le c lo se d polygon o f

Tv v e r t ic e s so th a t ot+fi+y+ * • . . S ( ^ X ) H . Thor, th e tran sform ation from U ie p la n e t o th e p la n e ,

d e fin ed by , sL_( i- - i £-1

£ =

« -

' Y < - « " « - o

£ L _______________________ ( 4 . 1 )

* 8 . ’i l n o - r ilVISOrT: T h e o r e tic a l :$ d ro d y n a n ies,

29

•

th e p e a l aads in tlie

p la n e in to th e

fecundassir o f a atseeci polygon t o th e

p lan e to

emiki

a way t h a t th e v e r tic e s o f th e polygon

correspond to th e p o in ts ar ^ , S ***** and th e

i n t e r i o r smgloa o f th e polygon ssro

^,7-,..** ***

Moreover $ when th e polygon :1o s:lr^le§ th e totezfcr

i s

mapped I

?y th e tx^nM-dMiattom on th e iippar

h a l f o f th e ^ p le s c B

9

xhe co n sta n t & may he caapiox: ancl. a l l

polygons eoxreapoM isg to glvon m lu a s o f ft, X, c

ar© siiB ilar^

I f th r e e o f th e

niiribem

o re chosen a rM trin * ily to

earaeepcod to th ro e o f th e tre r tle e s c f a

giwm

palygo% 'the reriatocloi* rsust tlion to a ai^aBged to

make th e polygon th e proper elmpe^

I t can he

ehoim* too

9

that

a vertex o f the polygon

oorwaspoMs to a p o in t a t i n f i n i t y on th e

£~axi

&9

th e ooiroopondisig fa c to r in (4*1) alm*M ho

o m itted $ th e aiigie o f th e polygon a t th e v o rte x

eogieearcied does s o t then appear^.

B w e rtto g

nmt

t o th e problcsn wvlop e o ^ id « ra tio n $

l e t &■% €|i I) oeiTOsponc! t o ^ ~ ~‘oOf of a ? <*)

re a p c o t iv ejy *

Binoe we ean choose on ly threes o f

9

3 0

.

th e corresp on d in g p o in t £ ~ & | ^ depends on th e v a lu e o r i . The Schucr»*-Chr ic T o f f e l ffeMMM then

t iv e a ^ ^ K i >

where K io a co n sta n t f

i .e * ^ * K ^ — ) . (4.2')

Ve have hare regarded th e f lu id a s en clo sed in a polygon in th e \ -* p lan e v /ith two v o r t ic e s a t i n f i n i t y , and we © ball d e te r n to e th e flo e / t o th e ^ p la n e w hich corresponds t o unifocm r e c t ilin e a r flfltf to th e %**. n la n e . P u ttin g i t r a th e r cn u d ely, we are c o n sid erin g a uniform r e c t ilin e a r flo w a lo n g th e r e a l a x is o f th e p lan e and th en deform ing th e boundary t o th e sln p e ABCD t o th e p la n e . 71>e v e l o c i t i e s a t p o in ts i n f i n i t e l y d is t a n t from 30 are u n a ltered by th e tra n sfo rm a tio n .

In th e zj- p la n e9 th e complex p o te n tia l ftm ctio n iv i s g iv en by

u/ s ~ H i (4 .3 )

w hence, by (4.2)

. « ( > - * ) * .

d

\l

h an i n f i n i t e d i sta n ce from K f iv « 06 ^nd Viw = do t lia t S s 1 . TJais

,

i \

(.

y%

3 1

.

ITow, hy ( 4 .2 ) and ( 4 .3 ) ,

du, s ( Uj

- l i f t - jf)~ ^

(Li

from w hich i t fo llo w s th a t Ux i s i n f i n i t e when i . e . th e f lu id has in f i n i t e v e lo c it y a t th e p o in t corresponding t o % * cl , i . e . a t C. When v/e pass on to th e ca se o f a com pressible flu id * th e flo w in th e neighbourhood o f C w i l l be v ery in te r e s tin g * s in c e th e r e i s an upper lim it to th e v e lo c it y which can be a tta in e d by a com p ressib le f l u i d .

I f we tak e th e boundary ABCD a s th e strea m lin e ^ - o , th en w ^ (p on ABCD and ( 4 . 3 ) shows th a t

w -

y

=r

c

w ^ = O a t r .

We are no-./ in a p o s itio n t.o determ ine th e r e la tio n s h ip betw een C and a .

On DC,

by ( 4 .4 ) , w hence, ta k in g B a s th e o r ig in in th e

y — p lan e and in te g r a tin g frora B to C, v/e have

o r i ~ cx{' Aoc

J 0

where - U<Knt • Hence, u sin g th e w ell-know n

3 2 .

X = <x B ( V % , I +

t t

)

* * r o - * ) r ( n 4 )

* r O - * ) r ( * ) ,

(\o^ __ ^

- ^ "** StvxOC 9 Ul* 0^ 5kncjL (4.*3y

Who flo w 1g co m p letely doborcalnacl by ( 4 .4 ) and ( 4 .3 ) .

Ho. * l e t cenowfi tIjj i kd a u ltM l f l u id v e lo c it y

?X nry p o in t in t i e f io ld and l e t £ bo th e a n g le

vliich it makes with the r e a l a x le . '♦’hen - /■„ - ‘0

^ - 1 * .

a ll

' ' - ( 4 - 6>

• o i'j’o oi o c iiilly In te r e s te d iii th e p o rt o f th e ■ioia o f flo w fo r which *U , (n ear C ). For t h is

r e g io n - / V * ^

f +Lvt' ~ aK Z. L u — )

%

33

.

Ye now pas© on to th e ca se o f a com p ressib le f lu id and u se th e holograph method t o determ ine f

w hich th e ”ach nurnber IT o f th e u n d istu rb ed stream i s 0 8 and oC = 45uf and s h a ll determ ine th e so n ic l i n e o f th e flo w , i . e . th e lo c u s o f p o in ts at'Vvhich L! i s u n ity .

•;e n o te by ( 1 .U ) th a t Tf • 0 • 1134 and t va t T - % on th e s c n ic l i n e .

Bow, by (3 i3 ) and ( l ' O f th e stream fu n c tio n and v e lo c it y p o te n t ia l fo r th e com p ressib le flow a re ob tain ed from ( 4 .7 ) and ( 4 .8 ) w hich ap p ly to and vh . to M U m m J tor th e 8 met .1 M M in

th e incom r e s s i b l e flo w by MfOtMftog u by

Hence, on th e so n ic li n e

v/here C i s a c o n sta n t, v/hose v a lu e we s h a ll n o t r e q u ir e to d eterm in e.

so th a t* on th e so n ic li n e

le- '

~ i l e

+ h ;

t o i .

Thusf by (4 *9) and (4*10)

^ ^ si*%(kh0 * 4*l/K( £ ) ,2^)

ae c ’

/•Iso | by ( M0 ) ,

H _C r 0 _{\ I- L /} * [ ~ = ° and* by (4*5)

x

ft. s — - — > (C

Whence ^

£v _ t V ^ 5_f( 07^) - ( £ ) (l irC)su.P<v»<iHW

xe- ^ t 1/

+ 1 Itotegratin . , v/e o b ta in

alt V „ f/, „N'^5 C«(4»»l>© ^ Cv>f t w- 0£}.„ V^JcnfaMrtP

i * T 2 l H 7 ‘" ^ r - }

—L (i-o7*j)

aiid, on eq u atin g r e a l and li-iaginary p arts* liave th e fVeedoci eq u ation s o f th e so n ic li n e :

cO

£v>fan+\)& 6*3(<+n-i)&l

1 U-r\ - 1 ^

^ S Iv^y ♦ ( ) 6

[. 4l\+! - 1

♦ |£ V V

f 4 . r } ) £4n(bK* ()6 Xo >6 ll

1 ^h4l U^-i S\

^ >

1

(4.1?.) ) $U*pfH4 l)0 *K^ (Ury-l)@ \

The so n ic l i n e can now be drawn by c a lc u la tin g

* and Mr v a r io u s v a lu e s o f 9 in th e range 0 ~ ^ us °

\:j .on & = M tv e v S g t a 01*

c o -o r d in a te s l i e s soraewhere an CD. A rou£^i sk etch i s shown in th e diagram .

&L S v/ide otrisata p a st a_ th in o b s ta c le *ro;jectira; ,.~-r.-(3iiaiculjy t o a ctrair.'ht bank.

3>

I

A 3 C

36

.

A to C which are i n f i n i t e l y d is t a n t firon and on e ith e r s id e o f th e o b s ta c le HD. L et th e v e lo c it y a t in f in i t y be U. -e co n sid er th e f lu id a s

§

occupying th e in t e r io r o f th e p o ly g o n . • • • .ATPHC.. . , /

on which Bf B are zero d ista n c e a p a r tf and ? ta k in g B a s th e o r ig in

A

£ B'

in th e X -p la n e t sap th e polygon on t o th e £ -p la n e w ith th e fo llo w in g c rreo , ondence

A ^ s - oO, I' {j m - I f m y T ^ { • / f C ~C m 06.

iy Scbwars-Chri s t o f f e l theorem* th e tr a n s fo m c tio n

i& d e fin e d t y »/ k/

$ . «*.*)■*<«. 4fc

-whence ^ = K -t- c o n sta n t.

At Bf ^ s 0 and <£ s * so th a t th e c o n sta n t o f in te g r a tio n v a n ish e s and

1 *

At D| j » i l and £ s 0 f hence 1 * 1 and

1 = j t t K r ) ( 4 .i 3 )

>

#

from a uniform r e c t ilin e a r flo w In th e f, -p la n e 9 so th a t

^ ~ 11 % (4 .1 4 )

E|y (4.1C) and (4.14)

i(m _ <t<v | _ 71 v/( ^ —X ) (4 .1 5 )

or i i - — — —- r r r • (4 .1 6 )

Xu, n- kV )

. once a =• — J(uv- h rl l).

o IL

I t fo llo w s to o from (4 .1 5 ) th a t th e f lu id v e lo c it y is I n f in it e when £ • 0 , I . e . a t D.

Further* hy (4.1G )

TcX r ,

“ ’ Htfyf *

fli

Ve are s p e c ia lly in te r e s to d in th e r e g io n fo r v/hich ^ > 1 1 ^ in t h i s r e g io n

u *

- M

U

^ y + K h - - y + U b ' 7 * - \

~ - t i l l | ( u ) * ' + i ( J ) + f ( J - )

+ HI j ( £ ) ' 5H.© + z (u -)~ 3s - * e +

so th a t

k-f » /*?, \-3

*

I U | ( J )

i(k)

s

J

T k m compressible f i e M of flow ia then

detom lBea, ae b e fo re by rei& aetog

t o <4 *X8)

and (4*19) by th e a^i^roprtote E m otions o f r

aM x # *

fhe sonic Xtoe can be re a d ily

a body govlrx- tlg y g g h a eaB W B O lblo f lu id

fla g ftteg tfflH

'.Then th e speed o f flow o f a eot& prescible f lu id p e a t a body i s such th a t th e r e io a r e g io n o f su p

er-co n ic flo w around aaoe p a rt o f th e body ( i . e . vfc on

/

th e tech nunber exceed s i t s c r i t i c a l v a lu e ) th e n , in g e n e r a l, a shock wave eaan atee trosi th e p o in t on th e body xL ere th e tr a n s it io n fr o o su p erso n ic to

su b so n ic flo w i c ta k in g p lu c e . "hen th e f lu id

p a s se s through th e shock w ave, i t i s co o p ra ssed , i t s t a ^ e r a tu r e r i s e s and, th e p ro cess b e in g n o n -a d ia b a tie , th e r e i c an in c r e a se o f en trop y. H ence, o n ly p a r t o f th e m echanical work transform ed in to h e a t energy in th e coniine s c io n i s capable o f b ein g reco n v erted in to ineohanical energy and th e energy ?rl o s t fl in t h is way m a n ife sts i t s e l f ac p a rt o f th e drag cm th e body.

Soon a f t e r a shock wave h as form ed, boundary la y e r se p a ra tio n ta k e s p la c e and le a d s to a fu r th e r

■#

in c r e a se o f d ra g . These two cum ulative e f f e c t s cau se th e drag c o e f f ic ie n t to r i s e r a p id ly vrith

i

fo r th e c a s e s o f flo w p a s t a Icankiiie o v a l and an e l l i p t i c c y lin d e r , th e in c r e a se in drag c o e f f ic ie n t due to th e p resen ce o f a shock w ave. I t w i l l be shown th a t t h is in c r e a se i s much sm a ller tl*an th e t o t a l in c r e a se vh icth would be observed in p r a c t ic e , whence we con clu d e th a t i t I s th e boundary la y e r

s e p a r a tio n , rathei* th en th e chock w ave, w hich i s tl:e r e e p o n slb l e f o r th e n o s t o f th e d ra g .

1 . Slow th r o u h a njr.. . 1 :hock ave

. e t u s c o n sid e r a n o m u l shock wave and c a lc u la te th e clionge in entropy o f th e f lu id in p a ssin g through i t #

L et q u a n titie s measured on th e upstream s id e o f th e sh ock , th e dosm etreao s id e o f th e chock and a t th e sta g n a tio n p o in t be denoted by s u f f ix e s 1 , f , 0 r e s p e c t iv e ly . Thus l e t , a , ct , r e p r e se n t v elo city " , p r e ssu r e , d e n s ity , v e lo c it y o f sound and ?foch number u xctreu n o f th e chock. S in ce we are c rotee^dLatiug a Dliool: v/ave errjanatirp; tr a i l a body such a s a c y lin d e r on an a e r o f o il, ^ » f, > f , i c, ? (m easured j u s t u p streon o f th e shock) w i l l b e d if f e r e n t

firoci th e corresp on d in g v a lu e s v , f > , | 3 , c , ^Vl in th e main stream #