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:DI DEFT C O A .

Report A e r o No. 172

^iiiUüïïittïs

T H E C O L L E G E O F A E R O N A U T I C S

C R A N F I E L D

THE LINEARISED FLOW FIELD OF A RELAXING GAS THROUGH

A NON-UNIFORM CHANNEL AND IN A J E T AT SUPERSONIC SPEEDS

by

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CoA Report Aero No. 172

THE ^COLLEGE _0F_AERONAUTICS CRANFIELD

The linearised flow field of a relaxing gas through a non-uniform channel and in a jet at supersonic speeds

by

-G.M. Lilley, M . S c , D.I.C., A.M.I.Mech.E., F.R.Ae.S., and

M. Anne Stevenson, B . S c , D.C.Ae.

CORRIGENDA Equation (5-l) should read as

follows:-y z

c B r ~ r ^t' - t' - 3y + M ^t" - t" - Pz

+ ^•

2e tn=oj_

Page 1 5 . The lefthand side of equation ( 3 ' 2 ) should r e a d :

-2€

Page lli-. The left-hand side of the equation following equation (5«2) should

read:-cp B

-^w

^

2e

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CoA. Report Aero No. 172

T H E C O L L E G E OF A E R O N A U T I C S

C R A N F I E L D

The linearised flow field of a relaxing gas through a non-uniform channel and in a jet at supersonic speeds

b y

G. M. Lilley, M . S c , D . I . C . , A. M. I. M e c h . E . , F . R . A e . S . , and

M. Anne Stevenson, B . S c , D . C . A e .

Summary

A study is made of the linearised differential equation for supersonic flow of a gas relaxing in one mode, assuming a linear rate equation, in a two-dimensional non-uniform channel. An exact solution to this equation is found which includes the corner flow problem a s a special case. This solution clearly demonstrates the exponential decay of disturbances along the frozen characteristics associated with the relaxation p r o c e s s . The r e s u l t s obtained for the corner flow problem agree with the earlier r e s u l t s of J . F . C l a r k e and J . J . D e r , Approximate solutions a r e also obtained which a r e shown to be adequate for most practical values of the ratio of the equilibrium to the frozen speed of sound.

Similar exact and approximate solutions a r e also found for the linearised case of a two dimensional jet expanding into a uniform p r e s s u r e field.

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CONTENTS

Summary

List of Symbols Introduction

The differential equation, boundary conditions, and solution by the Laplace transform

Exact evaluation of the p r e s s u r e coefficient

Approximate evaluation of the p r e s s u r e coefficient J e t expanding into a uniform p r e s s u r e field

Discussion Conclusions

Acknowledgements References

ces

Isentropic flow of a non-relaxing gas in a two-dimensional channel

M o r r i s o n ' s method applied to the corner flow problem Evaluation of L (t , y )

Alternative method of solution of the exact differential equation One-dimensional analysis using a linear r a t e equation

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LIST OF SYMBOLS ^ e / B /

a , a . Equilibrium and frozen sound speeds bjj, Cjj Defined in equation (2,2)

c_ P r e s s u r e coefficient e(T) Internal energy h Halfwidth of channel

p Laplace operator; also p r e s s u r e u' Perturbation velocity in x-direction V* Perturbation velocity in y-dlrection

a - 1

X' - y '

Defined in section 2

Direction of axis of channel, x' normalised co-ordinate Normal to x, y' normalised co-ordinate

Defined in section 2 r t

*k-X y ^n' B Bf, K L , t" n z n ^ e I, . L .

VNT-I for the perfect gas • V ( M p i L ) , / ( M | - 1)

Modified Bessel functions of the first kind Relaxation length (proportional to rU^ ) Functions defined in section 2

M Mach number; Mf, Mg, freestream Mach numbers based on frozen and equilibrium soundspeeds

Pj^(t) Function defined in Appendix 3, equation (3) T Temperature

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List of Symbols

c A small positive quantity: angle of the corner « 1 + p

r Relaxation time

^ Perturbation velocity potential

(~) Denotes a transformed quantity, except c which

CnBf ^

denotes - j . •

Subscript

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1

-Introduction

Non-equilibrium effects in gas flows may a r i s e from chemical reactions between the various species comprising the gas or from a redistribution of energy among the internal energy modes of the molecules after the gas has been perturbed from an equilibrium condition. Kirkwood and Wood (Reference 1) have shown the basic similarity between the two types of relaxation. Both p r o c e s s e s introduce a source of dissipation into the flow ( i . e . the flow i s no longer isentropic), and if the processes take a time of the same order of ntiagnitude a s the time for a typical molecule to pass through the flow field considered, the relaxation effects become important.

In the present work, a simplified model of the gas. in which only one type of relaxation is present, is used in order to formulate the problem. In many c a s e s , this approximation to the r e a l gas behaviour is not unreasonable a s one type of relaxation is found to dominate all the others. For instance, in a dissociation relaxation region the change in energy associated with the internal energy modes is very small compared with the change in energy

associated with the dissociation; and similarly in the case of a gas in chemical equilibrium, in certain temperature ranges one internal mode is found to have a much longer relaxation time than the others which are treated a s active modes, i . e . reach equilibrium instantaneously.

Gunn (Reference 2) investigated the effect of heat capacity lag in one-dimensional nozzle flows by linearising the equations to find the loss in available energy due to the temperature lag and hence the loss in total energy and found it to be a small effect. Chu ( Reference 3) indicates how the problem of a relaxing gas may be solved by a step-by-step numerical calculation using the method of c h a r a c t e r i s t i c s . Bray (Reference 6) and others showed that in the case where the amount of energy in the lagging mode is small, a criterion could be established for the "freezing" position in the nozzle. This sudden "freezing-out" of the flow where the lagging mode, having folowed the equilibrium distribution closely at first, suddenly breaks away and rapidly approaches an apparently steady non-equilibrium value (frozen flow), is also borne out by the numerical calculations of Stollery and Smith (Reference 5) for vibrational temperature lag, F r e e m a n (Reference 7) and Hall and Russo (Reference 8) for atomic recombination together with the recent analjrtic formulation of the problem by Biythe (Reference 4).

The governing linearised differential equation satisfied by the two-dimensional perturbation velocity potential <t> (x,y) has been derived by Vincenti (Reference 9) and Clarke (Reference 10) in the form

K (Bf * 1^ - <t> ) + B^* # - # = o ^ ^ °f ^xx yy x e »'xx '^yy

where K is the relaxation length, B^ * = Mf« - 1, Be* = Me* - 1,

where Mf and Me are the f r e e - s t r e a m Mach numbers based on the frozen and equilibrium speeds of sound respectively. The importance of the two speeds of sound was shown by Chu (Reference 3) and Clarke (Reference 10) while Vincenti solved the equation for flow past wavy walls. The method of Laplace transforms has also been used by Clarke (reference 11) to solve the equation for flow past a corner, and by Der (Reference 12) for flow past an a r b i t r a r y boundary. Moore and

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2

Gibson ( R e f e r e n c e 13) a p p r o x i m a t e to t h e t h i r d o r d e r equation by the s e c o n d -o r d e r l i n e a r t e l e g r a p h equati-on f-or fl-ow p a s t a wedge and a wavy w a l l . C l a r k e and C l e a v e r ( R e f e r e n c e 14) find s o l u t i o n s t o the t h i r d - o r d e r equation for

# ( x , y ) b y u s e of a G r e e n ' s F u n c t i o n t e c h n i q u e for the flow p a s t thin a e r o f o i l s . C l a r k e h a s a l s o used the a x i s y m m e t r i c f o r m of the equation to i n v e s t i g a t e r e l a x a t i o n effects on s l e n d e r b o d i e s ( R e f e r e n c e 15).

In t h e s e c t i o n s t h a t follow, the m e t h o d of L a p l a c e t r a n s f o r m s i s u s e d t o solve the equation for r e l a x i n g flow t h r o u g h a t w o - d i m e n s i o n a l channel with s h a r p c o r n e r s a t x = o. T h e solution i n c l u d e s , a s a s p e c i a l c a s e , the flow round a n i s o l a t e d c o r n e r , and t h u s C l a r k e ' s solution for the l a t t e r p r o b l e m which will apply up to the f i r s t r e f l e c t e d c h a r a c t e r i s t i c f r o m the opposite c o r n e r i s obtained d i r e c t l y . T h e solution for a g e n e r a l point in the flow field in the c a s e of the i s o l a t e d c o r n e r i s c o m p a r e d with that obtained from t h e a n a l y s i s of M o r r i s o n

( R e f e r e n c e 16). A p p r o x i m a t e s o l u t i o n s a r e a l s o obtained for m o s t p r a c t i c a l v a l u e s of the r a t i o of the f r o z e n and e q u i l i b r i u m s p e e d s of sound.

At the s u g g e s t i o n of P r o f e s s o r N . H . J o h a n n e s e n of M a n c h e s t e r U n i v e r s i t y , the m e t h o d i s a l s o a p p l i e d to the c a s e of a t w o - d i m e n s i o n a l jet expanding into a

uniform p r e s s u r e field, and exact and a p p r o x i m a t e s o l u t i o n s a r e o b t a i n e d .

1. T h e d i f f e r e n t i a l equation, b o u n d a r y c o n d i t i o n s , and solution by the L a p l a c e t r a n s f o r m m e t h o d

D i f f e r e n t i a l E q u a t i o n

It h a s b e e n shown by V i n c e n t i t h a t for the t w o - d i m e n s i o n a l flow of an i n v i s c i d , n o n - h e a t - c o n d u c t i n g , n o n - r a d i a t i n g g a s , r e l a x i n g in one m o d e , when p e r t u r b a t i o n s f r o m an u n d i s t u r b e d uniform s u p e r s o n i c flow and d e v i a t i o n s f r o m e q u i l i b r i u m a r e both s m a l l , a p e r t u r b a t i o n v e l o c i t y p o t e n t i a l ^(x,y) can be defined by

9x ' 9y

u' = ^— , V =

w h e r e

V = (U„, o) + (u>, V-)

which s a t i s f i e s the l i n e a r i s e d d i f f e r e n t i a l equation

K(Bf*^xx - ^yy)x + B g ' ^xx " ^'yy = o ( 1 . 1 )

K i s a p a r a m e t e r p r o p o r t i o n a l to the " r e l a x a t i o n l e n g t h " , r l l o . and B e , Bf a r e the e q u i l i b r i u m and f r o z e n P r a n d t l - G l a u e r t f a c t o r s r e s p e c t i v e l y , r , the r e l a x a t i o n t i m e , i s a s s u m e d to be constant and the r a t e equation i s

Dei<Ti)

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3

-w h e r e ei(Ti) i s t h e i n t e r n a l e n e r g y of t h e i n e r t ( r e l a x i n g ) m o d e specified b y t h e t e m p e r a t u r e T^, and ei(Ta) i s t h e i n t e r n a l e n e r g y of the a c t i v e m o d e s specified b y the t r a n s l a t i o n a l t e m p e r a t u r e T ^ .

In " e q u i l i b r i u m flow", when the r e l a x a t i o n p r o c e s s e s a r e infinitely fast and a l l m o d e s r e a c h equilibriumi i n s t a n t a n e o u s l y , T -o, K •• o, and equation ( 1 . 1 ) r e d u c e s to the P r a n d t l - G l a u e r t equation

B e fxx " ^yy - o

At t h e o t h e r e x t r e m e , when the r e l a x a t i o n p r o c e s s t a k e s a v e r y long t i m e

T -ta, K •• 00, and equation ( 1 . 1 ) b e c o m e s

(Bf ^xx - ^vv) y y ' x = o

i . e . Bf* *xx - *yy = Hy)

But the equation m u s t hold for a l l X , including the r e g i o n of u n d i s t u r b e d flow, h e n c e f(y) = o, and the equation r e d u c e s t o the P r a n d t l - G l a u e r t equation

2 J.

Bf *: X X ^yy

= °

T h i s o t h e r l i m i t of i s e n t r o p i c flow i s known a s "frozen flow"

B o u n d a r y Conditions

y»h / I / / / 1

"-•' f f I ) I I ) ~r-/ -/ ~ "y

LU^

I • U.*«i

It i s a s s u m e d that the flow i s uniform and in e q u i l i b r i u m u p s t r e a m of the station s t a t i o n x = o, and the b o u n d a r y conditions a r e t h e r e f o r e

^. ^ x ' ^x:x = o , X < o

a^

V' = -s*- = o , y = o , by s y m m e t r y , and

— = - ^ on y = ± (h + e x) , o r with sufficient a c c u r a c y U +u' dx

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4

-1 3^ . dy • * u T T J " A o n y = ± h . U. dy dx

where the walls of the channel a r e given by y = ± (h + e x) and e is a small p a r a m e t e r .

Solution by the Laplace Transform Method

The Laplace transform 0(y ,p) of 0(x,y) i s defined by

*(y.p) - I ^ P^*(x,y)dx

0

Equation (1.1) transforms to

dp -p^f \4^l) *=°

where a = "Be* z and is greater than unity. /Bf

This equation has the solution for <t> (y,p) ,

J

Kp+a Ikp+a

Kp+1 "f^ + B(p)e''AJKp+l"fy The boundary condition

9y " transforms to

d?

dy and this implies

A(p) o on y = = o on y = = B(p); o o

while the condition

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5 -t r a n s f o r m s -to d ^ TT « U -7^ = U — on y = h dy " p ' T h u s . P ' ^ . ^ . h . K E ± 5

I

e c o s h PJT^ Bfy

^' lp« Pïfsinh plËïaB.h

^ JKp+1 »IKP+^ ^

w h e r e L d e n o t e s t h e L a p l a c e t r a n s f o r m i n v e r s i o n o p e r a t o r , v i z : +i«o

L-^ j^ ^(y,p)eP^ ] = i i r ƒ «^''^"^y-p) ^P

-loo In the l i n e a r i s e d t h e o r y 2u' c = - 77- w h e r e u'

p u„ , ax

M

Hence t h e p r e s s u r e coefficient i s given by

.Px „^^u „ I^P+a

" ^ Ujit--pJiiBfh

( 1 . 2 )

T h e above equation ( 1 . 2 ) r e d u c e s to the p r e s s u r e coefficient t r a n s f o r m for " e q u i l i b r i u m flow" on putting K = o, and for " f r o z e n flow" on putting K = » . T h e i n v e r s i o n for t h e s e c a s e s i s p e r f o r m e d in Appendix 1 by a s t r a i g h t -f o r w a r d contour i n t e g r a t i o n giving

^pB X 2 " (-1) mrx nwy

' 2 7 =m^-^nh ~ " ' " B T "°" h

w h e r e B = B e o r Bf.

T h e f r o z e n and e q u i l i b r i u m p r e s s u r e coefficients on the w a l l and on the a x i s a r e p r e s e n t e d in F i g u r e s 1 and 2 for c o m p a r i s o n with the r e l a x i n g c a s e .

H o w e v e r , the s a m e m e t h o d cannot r e a d i l y be applied to the i n v e r s i o n of ( 1 . 2 ) . T h e i n t e g r a n d h a s a n o n - i s o l a t e d e s s e n t i a l s i n g u l a r i t y at p = - / j ^ , and although t h e c o r r e c t a n s w e r w a s obtained by i n t e g r a t i n g round a " d u m b - b e l l " contour a r o u n d P = - 1 / K and p = - a / j ^ , it could not be p r o v e d r i g o r o u s l y that the i n t e g r a n d

r e m a i n e d w e l l - b e h a v e d at a l l points of t h i s contour; nor could a n a l t e r n a t i v e s u i t a b l e contour be found.

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In t h e c a s e of one w a l l , the p r e s s u r e coefficient for t h e flow r o u n d a s h a r p c o r n e r i s p lK2+a, NKp+1" f--— = L { LL '^ with y now m e a s u r e d f r o m

'^ L |Kp+a J^, •^„

( 1 . 3 )

and t h i s h a s b e e n solved by C l a r k e for C p ^ , the p r e s s u r e coefficient on the wall (y = o ) . T h e e v a l u a t i o n for a g e n e r a l point in the flow field can b e obtained frora a n a n a l y s i s of M o r r i s o n ( a s m e n t i o n e d above) which i s outlined i n Appendix 2 .

In s e c t i o n 2 below, a n a n a l y s i s b y s e r i e s e x p a n s i o n i s given for t h e channel flow, which i n c l u d e s the c o r n e r flow a s a s p e c i a l c a s e ; and i n Section 3 , a n a p p r o x i m a t e m e t h o d i s p r e s e n t e d .

2 . E x a c t e v a l u a t i o n of t h e p r e s s u r e coefficient

T h e p r e s s u r e coefficient for the r e l a x i n g gas in the two d i m e n s i o n a l channel i s given by equation ( 1 . 2 ) ,

Kp+1""-- P ^ j i ^ ^ f '

CpBf -1 r ^ ^ ° ^ ^ P ^ J K ^ ƒ f y

' '' ' Ujl^sinh p.|iE±ï-B,h

and t h i s can b e s i m p l i f i e d on t r a n s f o r m i n g t o t h e n o r m a l i s e d c o - o r d i n a t e s , x ' K • y • K w h e r e p ' = Kp. Let 5 = 1 + p ' , t h e n , -X» r x ' uiy i\ S+a-1 , CpBf _ i r « ^ c o s h ( S - l ) ^ — ^ y « ->

'^ ""^ I (4-l)J"^sinh(5-l)J^h.J

( 2 . 1 ) Write r = a - 1 , and noting that

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-(2.1) becomes

Ï -2nK-l)j^h'

n=o ^ ^ = '^P,^ '^P. where

= P r ^ " ' ^ " ' [ ^ ' ^ ' e-^«-^>^l¥<»^•-y•^2nh.)"

' ^ n=o "^

'P - «""' L"' [ IT' ^f^ j ^ ^-(,-1) J^(h^y.-.2nh.)_

For simplicity of notation, let y^ = h' - y' + 2nh» Zfi = h' + y» + 2nh' Then K,x'

l4TFJ^e-<^-^)#^.

-X. , - i f e^^ - X ' , - i f egx' r i ~ ~ F T 7 and -pBf -•5 = Cn + 2e V '^Pa, £ b„ + 2 c„ say n=o n=o where b„ = -x» L n e -1 f e ^ _g_ _ g - H g + r _^g-V2T5T?Ty„ 5+r I S+r •^ (2.2) (2.3) but ZiT^fH = V(;^+^72)- - ^ 7 4 .

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8 -so that if and X = 5 + ^ / 2 . s = V V - r 2 / ^ (2. 3) becomes

- ^-i'^^ m^-"-^>]

. - " " ^ - ' [ ^ ^ J ^ [e<''->yn. j] e-^yn ,^È^y„j

+ e - a x • -1 r e IX-/S -Xy„ n S ^ ~j ( 2 . 4 )

where a = 1 + ^/2 , /9 = ^/2 •

Define ^n ~ ^' " yn • ^n = x' - z^ , and let the right hand side of Cp transform to s e m i - c h a r a c t e r i s t i c co-ordinates from the "top" corner and for each n, along the (n + 1)*^" characteristic parallel to the first; and similarly for

^P.= -h } I I I I I A ^ X

A° . '>l/t'

.e. ..„..„, ..-[^^JS|,J||,„

.]

La (tji . yn) = L -1

j^^thx|^^(X-s)yn_^j

So that if L , and L^ a r e known, bn is given by the convolution formula from equation (2.4), or

bn = e L,(tJ^ - yn^ + ^ yn)L,(t'i - r , y n ) d r ( 2 . 5 )

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- 9

The inversion for Lg(tn . yn) is given in Reference 17, or

^ 2 V (tfi + yn) - yn

where I, i s a modified Bessel Function of the first kind.

Therefore -ax bn = e

- e--' [L (.•„ . ,,) . g^ /"'L.^,.-., y„) i ^ ^ dr] ,2.6)

yn

Now when n= o, corresponding to the flow field between the corner and the first characteristic from the opposite corner, it is found from (2.6) that the solution in this case is similar to that found by Morrison (see Appendix 2). However, the solution on the wall y = h, corresponding to yo = o , found from (2.6) is

-ax», ,^, _, . - a x ' / e ° "^ | T ^ e"**" L , ( t i .o) = e^"*^ / ^-^ l ^ ^ ^ d X

c' f e ^ IX^Z

i x - « J x + /

since the second t e r m vanishes, and from Reference 17. x'

ffx'r -ax' / -a^ 1 L,(t^ . o) = e Ie lo ( ^ x ' ) + j e Io(^<i)d/iJ H(ti),

where IQ i s the modified Bessel function of the first kind. Hence Clarke's result,

. ^ f __ ^-ax' ^^^^^,j ^ r'e-''%o(^.)d;z, (2.7)

is recovered directly.

For duct flow it is seen from (2.6) that t e r m s corresponding to the double system of reflected characteristics for the frozen flow a r e added to the solution when n = o. In other words the duct flow solution is constructed from a s e r i e s of isolated corner flow solutions. It remains to evaluate L,(t^ . y^). This r e q u i r e s some manipulation to reduce it to a standard form. The inversions a r e obtained in Reference 17 and the evaluation is performed in Appendix 3, giving

.1 '» m

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10

-vhere Pjy,^*^ ^^ * function defined in Appendix 3 which satisfies the r e c u r r e n c e relation

P (t) = P „(t)+.re"* / e ^ P „(r)dT 2,9

m m - 2

and the infinite s e r i e s £ —r P ( t ' ) is absolutely and uniformly convergent

m'. m n J O

for all y and t^ since m-1

m=o

I 1 ~ 2 - " 1 (y/i)™

I P™(t') I * a for all m and t|^ and £ f= -^—— i s absolutely convergent

m=o

for all y^.

The exact expression for the p r e s s u r e coefficient i s therefore from (2.2), (2.6) and (2.8),

CnBf

£

n=o

e-"^'e ^n r ^ P (f)

^ ^ m'. ^m*^n' m=o

X '

- a x ' / - " - « ( x ' - r ) r yn" ^ , , U^r' - y^) yn + e m -ov» a t " *" ^n "" e *n £ ^ P (t") m=o "^- "^ " + e - a x ' /-^ m n ml m m=o ml m vtr -z I n

Because P (t) = 0 for t < o for all m, b_ = 0 for all n > k + 1 whenever t', ,, < 0 and m ' n K+1 c = 0 for all n > Ic + 1 whenever t'/ , < 0. Thus the first two t e r m s a r e summed from

n k+1

n=o to n=k only, and the last two from n = o to n=k' only, in the region where t'> 0, t" > 0 so that 2e k £ n=o m

e-«yn r ^ , P (t')

m=o ml m n + ^ y

nf

m I i ( ^ r > C - ^ ) a r " Vn .r -e l ^ P ( x ' r ) rt m». m Vr' -m=o d r 'n k ' + £ n=o m n

«Zn » "' f -ar r z"" I ( / 9 / ? ^ )

" £ 5 " P (t") + /9zn e ""^ £ ! l l , P ( x ' - r ) ^ . ^ a d m'. m n " ; ml m Vr -z„ m=o J m=o ^n 2.10

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11 -( -( I t I / I ) I f / ) f{ I

'^-'^'^TK'^/J

\ / U-ot \ . / « , V / _ _ \ 7-7—7-7-7 / \ — * ^ *

It should be noted thatk ' = k , k + 1 , o r k - 1 , so that in the region ABC in the diagram above the first sum is from o to k, the second from o to k - 1. In the region CDE the first sum is from n - o to k - 1, and the second from n<= o t o k . In the region BCEF, both sums a r e from n = o to n = k,

The p r e s s u r e coefficient on the wall and on the axis is shown in Figures 1 and 2 respectively, together with the frozen and equilibrium values. In computing these p r e s s u r e coefficients, the half-width of the channel was taken to be h' = 0 . 5 , a = 1.5, and it was found that ten t e r m s of the s e r i e s

~ y„m

£ , Pm(t) were sufficient to obtain a value accurate to two decimal m l cn

m=o m l

places for values of yn (or Zn) up to 3.

The p r e s s u r e coefficient at a general point in the flow round a corner i s , from (2.10) -£cBl ^ g-ayo ^ 2e m m=o ,. m Zo p

n.(t.o)+.yo r ; - £ ^ : P ^ ( x ' - . ) i < £ S ) d .

Vo

m=o

° (2.11) which is equivalent to

^ -. mV [ e-^-' +

. e - £ ' r e - ^ ^ ^ d.

Jyo K tiH ' V ( x ' +<i)« - ( / J + r ) 2 J

^ ^ - a x '

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12

B o t h e x p r e s s i o n s show t h a t c = o for y < x' s i n c e t h e d i s t u r b a n c e f r o m t h e c o r n e r i s confined t o the r e g i o n d o w n s t r e a m of t h e Mach w a v e a s s o c i a t e d with the f r o z e n M a c h n u m b e r Mf, a r e s u l t which i s w e l l known. F o r y = o, M o r r i s o n ' s m e t h o d g i v e s , + ^e

•^7 '"7

.-iS»i (<i+r)Vrr , I , ( / 9 / ( x ' + / i f - ( r + * i ) " ) L (2/rrM ) /, . >a ' , , a — d*J d r ^ V(x'+/i ) - (T + ur

and although t h i s i s e q u i v a l e n t t o ( 2 . 7 ) the e q u a l i t y can only b e e s t a b l i s h e d with s o m e difficulty. ( 2 . 1 1 ) above h a s the m e r i t that t h e s i m p l e f o r m for c p ^ found by C l a r k e , equation 2 . 7 , i s obtained d i r e c t l y in p l a c e of t h e r a t h e r i n t r a c t a b l e r e l a t i o n a b o v e .

3 . A p p r o x i m a t e e v a l u a t i o n of p r e s s u r e coefficient

T h e solution for s m a l l d i f f e r e n c e s b e t w e e n the f r o z e n and e q u i l i b r i u m sound s p e e d s .

It i s p o s s i b l e t o obtain a v e r y m u c h s i m p l e r e x p r e s s i o n than ( 2 . 1 0 ) for t h e p r e s s u r e coefficient if it i s a s s u m e d t h a t r = a - 1= ( B g / g * ) - 1 < < 1. E q u a t i o n ( 2 . 2 ) for t h e p r e s s u r e coefficient i s fe+r IS+r c B . . - ^ - L ' ^ 2e r &c' | — "

U - 1 ^J4+ 1

£ n=ó L + e Now s u p p o s e r < < 1, so that 1 S + r 4 —^ = (1 + ^ / S ) = 1 + 1 ^ + 0 ( r « ) .

T h e equation for c when t e r m s 0(r* ) a r e n e g l e c t e d then b e c o m e s , r e m e m b e r i n g t h a t ^ = ^ . c B , 2 6 = L n=o •Z (b + c ) a s b e f o r e , w h e r e now n=o n n b = e n -t» -/9v ••n '^•'n

^-'[-%^^-14]--1^^^!?^^]

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13

-Equation (3.1) retains all the important features of equation (2.1). It possesses an isolated essential singularity at 5 = 0 and simple poles at S = ^ , ^ = 1.

The following r e s u l t s a r e obtained from Reference 17, with the aid of the convolution formula y / p - I I / •=••=• * * Y - T dt "' L' P , o f px y/p -1 [^ 1 i .px J / P ^ - l f"" i i x-t p _ 1 - . J -->-. ~^)e" dt L' ^ o and •' , . P-1 J X dp = e L' P

-Therefore, using the above r e s u l t s ,

b = e"'^ e " ^^" r r<^yn>^'' ' I,(2V?igr ) e ' ^ ' V + e**^

^ ^o

- ^ ƒ y z V ^ l T i D e ^ ^ ' ^ d J 'o

'- ^ ' L ƒ Hi7 I o ( ^ 2 r y n ' ' ) e " ' ' dM + 1 - ^ ƒ I ^ t ^ ^ ) e ' % ^ J

.-/9yn, , , , , - , -tK ,, ^, -^yn ""

e" ^'^ y / 2 Ï ^ « j ) e ' ' ^ + (1 -/9)e ' ^"^ ƒ y V l i ^ ^ ) e ' " d»i o

after integration by p a r t s .

Therefore, substituting in equation 3 . 1 , k

- \ - ^ = £ fe" ^ " Io(/2'?Srt{;) e'^^ + (1 - ^ ) e ^" ƒ Io( ^ ^ 2 ? ^ ) e"" d*ij

n=o *- o -^ t "

k' ^ _ ^ 2 -t" -/5z /"^ u "I

£ e '"lo(V2rZj^t;;)e " + ( 1 - / ? ) " / y ^2rz^*i) e" d,i J 3.1 +

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14

-w h e r e t h e s u m m a t i o n i s o v e r a finite n u m b e r of t e r m s for t h e s a m e r e a s o n s a s in s e c t i o n 2 .

T h i s a p p r o x i m a t i o n to t h e p r e s s u r e coefficient g i v e s e x c e l l e n t a g r e e m e n t with t h e e x a c t r e s u l t , and r e m a i n s a good a p p r o x i m a t i o n even for v a l u e s of r a s high a s

0 . 5 a s shown on F i g u r e s 1 and 2 . It i s p a r t i c u l a r l y good up t o the f i r s t r e f l e c t e d c h a r a c t e r i s t i c and t h e r e f o r e for the c o r n e r flow p r o b l e m ( 3 . 2 ) r e d u c e s to

to

C B . py -t« - ^ y r „

- - ^ = e ° e °y/277;rj,) + (l - ^ ) e ° ] lj.^2f^) e' 6u 3.2a

o and when y = o, (on t h e wall y = h)

3 . 2 b CpwBf 2e w h e r e a s Cn. B , Pw f = (1 -f r o m (2

= 1 n

/9) . 7 ) . + « + ^ e - ' with r < - ox'. p 1 < 1 .

but s i n c e a = 1 + /9, it i s e a s i l y v e r i f i e d that t h e s e a r e t h e s a m e to 0(r*), provided x' < < o ( 1 / / 9 ) .

T h e a p p r o x i m a t i o n ( 3 . 2 b ) for the p r e s s u r e coefficient on t h e w a l l i s c o m p a r e d with the e x a c t e x p r e s s i o n ( 2 . 7) for v a r i o u s v a l u e s of r in F i g u r e 3 .

F i g u r e 4 shows the v a r i a t i o n of the p r e s s u r e coefficient with t for different v a l u e s of y (at different d i s t a n c e s f r o m the c o r n e r ) for a = 1 . 1 , c a l c u l a t e d f r o m equation ( 3 . 2a), Since for l a r g e v a l u e s of t ,

m - 1 P ^ ( « ) = (1 + r ) " 2 -and I , ( / g / r ' - y ^ ) - a r _ - ^ r y ^ ^ - a y o

i V r ' - y o - '

^ y o / ' ./-a „ 'a e dr = e yo

it follows that the e x a c t e x p r e s s i o n ( 2 . 1 1 ) g i v e s ,

. f E ^ f . £ 1 ^ ! ^ g " ^ ^ ^o ^ _ 1 _ ^ - ( 1 - / 2 - VTT-r)y 1 - ( l + r / 2 - ^ ^ l ^ ) y o - ( l + r / 2 - T^l1^)yo, (e + l - e ) a s t . , o o , '1+r showing t h a t , for a l l y, c B . B . 2e B ' e

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15

-Thus the flow reaches equilibrium far downstream of the initial frozen Mach wave, corresponding to large values of t for a fixed y, and this is demonstrated in Figure 4 where the p r e s s u r e coefficient approaches (1 - fi), for all y. (1 - ^ ) is

1 Bf the first order approximation to r— = D ~ '

It is ^ o w n below that for small values of t, the disturbance decays exponen-tially a s e" yo away from the corner. The approximation to Cp given by (3. 2) is equivalent to replacing (1.1) by

in the normalised co-ordinates. But if (1.1) (in the normalised co-ordinates) i s differentiated with respect to x', it becomes

ax«* ^ <^^^' 3x1» Ï7W' 3 V 3 7 « = ° r* 3*0

and adding a t e r m — -3-77 to both sides and rearranging gives

, 3 « 3*0 -, 3 . 3*0 r* 3*0 „ .

so that (3. 3) is a form of the exact linearised equation when r < < 1.

It cannot be assumed that inclusion of the additional t e r m in (3.3) modifies the rate equation. In fact no justification on physical grounds can be found for the addition of this t e r m . It therefore appears to be purely fortuitous that the

addition of a t e r m of smaller order than the existing t e r m s in the equation simplifies the resulting modified equation. Because the additional t e r m is of small order, solutions to this simplified equation give reasonable agreement with these obtained from the original equation.

2. Solution for small values of t

Certain useful r e s u l t s can be obtained from an investigation of the flow in the vicinity of the leading c h a r a c t e r i s t i c s , i . e . for small values of t' or t".

F r o m section 2, since P (0) = 1 for all m, - «t' yn

L«(t' , y ) e n - e a s t' - 0,

n n n so, for snaall values of t' ,

n

JK

' ""' U K. y j = e - ''y- [ e - «^{^ i^( ^ t O + ƒ e- "^ y ^,1 )d.3 n "n

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16

--^ y -« y •^n •'n

+ e - e 3 . 5

a s t ' * 0. n

Hence from (2.6) together with (3.5),

'I

-av« - a y n f b^ = e "^ IJfiVj + e -^^ J o o n -au

e Io(^*i )d<i + e " - e -/9yn - » y n

3 -ax' /" M^^?[Zyn) TT / . / , .X «(x'-r) [ ^ \ ^ i V r ' - y T } M / 3 ( x ' - r ) ) + e j l ^ i Vn '^ o r ) a ( x ' - r ) " ] , - e 1 d r . (^M )d/i ^n a(x' -+ e e But M/gA-*-yi;) ^ 1 d , ( i S / F T ^ ) Therefore /9r d r

-/5yn -«^yn r - tfi b = e + e n

[e"*^I„(^t{,)+ ƒ

*Ji e " ""^ y /9/i ) d/i

-]

t' + y e •'n - «v., r "^ ^-«^T •'n / e

ƒ f77„ h [ • o < ^ < - ^ . * ' - „ • ] [

e '^ y/9(t|^-r))+ tli-^ + / e Io(/9/i )d/i + e - 1 d r Therefore a s t ' •• 0, n '!, -ar O t' •^yn -'^yn r - a r d f, = e + e / « "d? U ^ V 2 T F ) d r

^^n)]

/9y = e (1 + o(t)). Similarly c^^ * e - ^ z ,

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17

-showing that along leading c h a r a c t e r i s t i c s , the disturbance decays exponentially with distance from the appropriate corner, a s found previously by Clarke and others. In addition the reflected disturbances decay exponentially with distance from their points of reflection. As already noted above, it is éeen that the nozzle flow solution is obtained from a s e r i e s of isolated corner solutions. The r e s u l t s of numerical solution by Der (Reference 19) justify this approximation.

4 . Jet expanding into a uniform p r e s s u r e field

Following a suggestion by Professor N. H. Johannesen of the University of Manchester, the case of a jet expanding into a uniform p r e s s u r e field is also considered

The free boundary, for a linearised problem, is assumed to be at y «= i h, and the boundary condition is

P = Pa where pa is the external p r e s s u r e .

If po is the stagnation p r e s s u r e , p » is the static p r e s s u r e at the nozzle exit, and Cn is defined a s

P - P.. Po • P - ' Pa - P«. Po - P - '

then the boundary condition on the free boundary is

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18

-The case considered i s when p,„ > p . Thus the first Mach wave is one of expansion, followed by one of compression and so on, but the analysis could equally apply to the case when pbo * P„. when the first wave i s a compression, provided that the p r e s s u r e difference p» - P^ i s small enough to prevent the formation of shocks beyond the mouth of the nozzle. This condition must apply anyway to keep the problem within the scope of a linearised analysis.

Applying the method of the Laplace transform a s in section 1, the equation for 0(y, p) i s iKp+a ^p^|Kp+l B,y 0(y,p) = A ( p ) e ' ' ^ " ' " ^ ^ +B(p) e 30

IE

-p^jK Kp+1 ^f^

The boundary condition v' = - g ^ = 0 on y = 0 still applies and gives A(p) = B(p)

o r

0(y,p) = 2A(p) cosh p J | E i | B ^ .

The boundary condition c = Cp on y = x h gives on y '^x 2 which transforms to . p 0(y.p) = giving

?(y.p) =

^Pa

-_ Ü

2

u„

on y ± h

l^^^ B V

cosh p N K P + 1 f^ Therefore

0(x.y)

U c_, , «» Pa -1 Kp+a

K?Tï ^fy

e^ cosh p^J Kp+1^ f^ j P '^««^ P ^ ) K ^ V in the notation used above, or

c _E_ 'P= iKp+a - l [ eP^^coshpNjKp+l fy (

P ' = ° " ^ P J K ^ ^f^

4 . 1

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19

T h e f r o z e n and e q u i l i b r i u m s o l u t i o n s a r e obtained b y putting K = «> , and K = 0 r e s p e c t i v e l y , giving

! E _ = 1,-1 r eP^coshp By | (g = B , or B )

Cp L p cosh p Bh J ' ^ f e which i s e v a l u a t e d in a s i m i l a r way a s in Appendix 1, giving

c 1 2 " ( - 1 ) " (n+i)7rx . . _ (n+i)try c„ n- " n+i Bh Pa n=o On t h e a x i s of the j e t , m a k i n g u s e of t h e w e l l - k n o w n r e s u l t ( R e f e r e n c e 18), c o s 6 - T c o s 36 + r- c o s 56 . . . . = ^ , - j < e < ^ , c p . '^Pa = 0 = 2 e t c . - B h < X < B h B h < X < 3 B h E q u a t i o n 4 . 1 i s e v a l u a t e d in e x a c t l y t h e s a m e way a s 1 . 2 . On t r a n s f o r m i n g to n o r m a l i s e d c o - o r d i n a t e s x' = x / K , y' = ^^ , and putting 4 = 1 + p , K % ^ - I f e ^ ' ' e " ' ' ' c o s h ( 4 - l ) N S y^ "Pa i r e ^ e cosh( ^ - D J v ^ y*]

I (S-l)cosh(<;-l)J-^ h' J

Since coshx X e n=o " , , ,n -2nx £ (-1) e c _E_ = Cp^ PI P« - x ' c = e c^ + c^ , w h e r e now

5x' „ _ _/x_,uIiir

- I f e ^ * - , , , n - ( 5 - l ) ^ ^ ^ , L l ^ ^ y £ (-1) e ^ y^ *-* n=o J

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20

-t^~rï 2 (-1) e '>J 5 nj

- x ' , -1 c„ = e L Pz n=o E (-1) b ^ + £ ( - 1 ) " c^ s a y n=o n=o w h e r e y , i^+r _x- -II o ^ ^ JYÏ V 4. ^\^r "ST' yr b = e n

^' L-^r ^ e^^?^«^>yn e " ^ ^ "j

Let

+1 -x [ x ^ T ^.1 ^ T - l f e " ^ ^ ^ ' ' ' ^ i w h e r e X = <S + r / 2 , s = A * - ^*/4 T h e n , • n . ' t - , . ' M n n ' n but now ^^l

-1

e m=o - a t ' „ - ' "•• " " ' = e £ m=o • «*k " yn = e " £ ^ P.AV) m'. m + l n m=o

w h e r e t h e P (t) a r e t h e functions defined in Appendix 3 . H e n c e t h e e x a c t m

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21

-c k r - a y

•f- - £ (-ine ^"

*^Po n=o 2 ^ P , ( t ' ) i l n - i j - l n m = o m'. m + l n

• . / •

m + Py ƒ e £ - ^ P , ( x ' - r ) - 2 — > - j g - — d r •'" ' m'. m + l Vr" y ' m = o n yn k' + 2 ( - 1 ) " n=o -"^^n - Zn"" e £ - ^ P ^ , ( t " ) '' m l m + l n m=o

-'\j

X' - a r .0 z H i m i m + l m=o I , ( ^ V r * - z * )

( x ' - r ) —J-p——a d

4 . 2

A s in t h e c h a n n e l flow, a m u c h s i m p l e r e x p r e s s i o n than ( 4 . 2 ) i s obtained if it i s a s s u m e d t h a t t h e r e l a x a t i o n p a r a m e t e r r « 1, s o t h a t a n d b e c o m e s ^ c P "Pa c P ^Pa = 1 + ^ / g + 0(r*) • ' .

•1

L £ n= n=o ^ n=o (-1)" (b f c ) D d t a s b e f o r e , but now b = e n St' fiy t f i - ^ y n f / S - 1 d S +

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22 --^yn r r" e = e after integration by p a r t s , Hence

I I dJT I ^o^ V2ryjj<i)J e ' " d>i + 1 J (Reference 17) o

" l ^ ( / 2 r 3 ^ ) e ' * n + e ^" ƒ y ^2r yn<i)e'*' d/i

r -t» -t'

Ly^2ry^t{^)e " + ƒ " y ^2?!^) «•'^d/ij

• ^ - z (-1) e Pa n=o k' - ^ z „ r -t" *-^ + L (-l)*" e n=o

^'^fy V27v";;)e " + / " y / 2 T ^ ) e - ' ' d , i ^

4 . 3

for sufficiently small values of r .

The exact solution (4.2) and the approximate one above (4.3) a r e presented in Figure 5 together with the frozen and equilibrium values of the p r e s s u r e

coefficient along the axis of the jet for a » 1. 5, h' = . 5 .

The summation over n in both (4. 2) and (4. 3) is over a finite number of t e r m s a s in the channel flow, and it can be verified that

c

- E - = 1 on y' = ± h ' , "Pa

all t e r m s but the first cancelling out.

5. Discussion

An exact linearised solution has been found for the flow of a relaxing gas in a two dimensional channel, and in a jet, assuming a linearised rate equation where the relaxation time, r , is assunned to be constant along the channel. Of course it might be expected that r will vary with temperature and p r e s s u r e and will increase with increasing distance downstream in a diverging channel.

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23

-In the one-dimensional analysis which is performed in Appendix 5, and the resulting p r e s s u r e coefficient on the axis is shown in Figure 2. This shows no evidence of 'freezing' found by Biythe who used a rate equation of the form

de.

d T = " < " ' T ) ( i i ( T ) - e . )

(where e. is the equilibrium value and u = p 0 ( T ) where n ( T ) is assumed to be n a T^) for a one dimensional analysis (see Reference 4).

Figures 1 and 2 show that the p r e s s u r e approaches the equilibrium value between each reflected disturbance, but does not remain between the frozen and equilibrium isentropic solutions a s may at first be expected. This is because of the exponential decay of the disturbance along characteristics found in section 3. In the isentropic case, disturbances a r e reflected with the same strength at each intersection with the channel wall, but by the relaxation process each reflection is weaker than the preceding one by an amount corresponding to the exponential decrease e ' where 2h' i s the distance between reflections. This effect can also be observed for the flow in a jet in Figure 5,

The approximation for small values of the relaxation parameter of section 3.1 follows the same trend, and the reason why it remains a good approximation even for values of r a s large a s 0. 5 is because it is equivalent to adding a t e r m of smaller order than the existing t e r m s to the original differential equation, a s explained in section 3.

The method of solution of the isentropic channel flow by contour integration is immediately applicable to other wall shapes. It is just a matter of calculating residues at the poles of the integrand which a r e the origin, ± iüï., and any others introduced by the transform of the wall shape; see Appendix 1. Bh However, tiie method of solution of the relaxing flow given in section 2, is not easily extended to other wall shapes, a s additional t e r m s in the integrand will change the form of L , (t,y) radically. However, when the wall boundary condition produces an

additional t e r m such as — (for walls i (h + e x*) ) or — - (for walls - (h + e ( l - c o s x))), P P+1 the solution is immediately obtained from the exact solution above for the walls ± (h + ex), and the transform of the additional t e r m with the aid of the convolution formula.

In addition an alternative method of solution of the equation in the form (3.4) i s outlined in Appendix 4 for the corner flow problem which gives r e s u l t s correct to 0(^' ) and it is shown that this method can also be extended to channels with different wall shapes with the aid of the convolution formula.

The linearised duct theory used in this report as well as being r e s t r i c t e d to values of x not too far fown this nozzle from the corners will only be valid for

eMe'

values of „ ^ jss^r < < 1, since for larger values of this p a r a m e t e r , non-linear e

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24

-inertial effects will introduce a streamwlse displacement of the equilibrium characteristics of the order of the spread between the equilibrium and frozen characteristics in the linear problem. Thus the present analysis is r e s t r i c t e d to not too large x', small values of e, and values of Me not close to unity.

6. Conclusions

An exact linearised solution for supersonic flow in a two-dimensional diverging channel of a gas relaxing in one mode, assuming a linear r a t e equation, has been obtained which contains a s a special case the solution for flow round a sharp c o r n e r .

The p r e s s u r e coefficient for the relaxing gas, compared with the two limits of isentropic flow in the same channel, demonstrates the effects of damping introduced into the flow by the relaxation p r o c e s s .

An approximate solution, assuming small values of the relaxation p a r a m e t e r , which involves much simpler algebraic expressions and which remains in good agreement with the exact for values of r up to 0. 5, i s also obtained.

The solution for other wall shapes is indicated. Similar solutions a r e found for the case of a two-dimensional jet which also demonstrate the danaping effect.

7. Acknowledgment

The authors a r e grateful to Professor G.N. Ward, Dr. J . F . Clarke, and Mr. J . W . Cleaver, for helpful criticism and discussion.

8. References

1. Kirkwood, J . G . and Wood, W.W. 1957 Dynamics of a dissociating gas. J n l . F l . Mech. 2, 1.

2. Gunn, J . C, 1952 Relaxation Time Effects in Gas Dynamics,

ARC, R & M 2338.

3. Chu, B . T . 1957 Wave propagation and the method of characteristics in reacting gas mixtures with applications to hypersonic flow.

WADC. TN 57-213.

4. Biythe, P . A . 1962 Non-equilibrium flow through a nozzle.

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25

-5. Stollery, J . L . and Smith, J . E .

6. Bray, K . N . C .

7. F r e e m a n , N . C .

8. Hall, J . G . , and Russo, A . L .

9. Vincenti, W.G.

10. Clarke, J . F .

1 1 . Clarke. J . F .

12. Der, J . J .

13. Moore, F . K . , and Gibson, W . E .

14. Clarke, J . F . , and Cleaver, J . W .

15. Clarke, J . F .

16. Morrison, J . A .

1962 A note on the variation of vibrational temperature along a nozzle.

Imp. Coll. of Sc. & Tech. , Aero. Dept., Report No. 111. 1958 Atomic Recombination in a

hypersonic wind tunnel nozzle. J n l . F l . Mech. 6, 1.

1959 Non-equilibrium theory of an ideal dissociating gas through a conical nozzle.

A . R . C . C P . 4 3 8 .

1959 Studies of chemical non-equilibrium in hypersonic nozzle flows.

Cornell Aero. Lab. Rep. AD-1118-A-6 AFOSR. TN. 59-1090.

1959 Non-equilibrium flow over a wavy wall.

Dept. of Aero. E n g . , Stanford Univ. SUDAER No. 85.

1958 The flow of chemically reacting gas mixtures.

College of Aeronautics Rep. No. 117. 1960 The linearised flow of a dissociating

gas.

Jnl. F l . Mech. 7, 577

1961 Linearised supersonic non-equilibrium flow past an a r b i t r a r y boundary.

N . A . S . A . TR. R-119.

1960 Propagation of weak disturbances in a gas subject to relaxation effects. J n l . Aero/Space Sci. 27^, 117. 1963 Green's functions and the

non-equilibrium equation with applications to non-equilibrium free s t r e a m s . College of Aeronautics Rep. Aero No. 163.

1961 Relaxation effects on the flow over slender bodies

College of Aeronautics Note No. 115. 1956 Wave propagation in rods of viogt

material and visco-elastlc m a t e r i a l s with three p a r a m e t e r s models.

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- 26 17. Erdelyl, A . , Magnus, W. Oberhettinger, F . , and Tricomi, F . G . 1954 Tables of Integral T r a n s f o r m s , Vol. 1,

McGraw-Hill Book Co. Inc. , New York.

18. JoUey, L . B . W . 1925 Summation of S e r i e s .

Chapman and Hall L t d . , London. 19. Der, J . J . 1963 Theoretical studies of supersonic

two-dimensional and axisymmetric non-equilibrium flow including calculations of flow through a nozzle. N . A . S . A . Tech. Report No. R.164.

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27

A P P E N D I X 1

I s e n t r o p i c flow in a t w o - d i m e n s i o n a l channel

T h e p r e s s u r e coefficient for t h e flow of a p e r f e c t g a s in the t w o - d i m e n s i o n a l c h a n n e l i s given b y

p _ I - I j e c o s h B p y

' 2e "" L p sinhBph J'

w h e r e B = Bg for t q u l U b r i u m flow, = B - for f r o z e n flow .

T h e s i n g u l a r i t i e s of t h e i n t e g r a n d a r e a m u l t i p o l e at t h e o r i g i n , and s i m p l e p o l e s a t p = t i ü ï l , n = l , 2

Bh

T h e path of i n t e g r a t i o n can be c l o s e d by a s e m i c i r c l e in t h e half p l a n e in which R e p i s n e g a t i v e , the c o n t r i b u t i o n f r o m t h i s p a r t t e n d i n g t o z e r o a s t h e r a d i u s t e n d s t o infinity, p r o v i d e d ( x , y ) l i e s d o w n s t r e a m of the c h a r a c t e r i s t i c s t h r o u g h ( 0 , - h) w h e r e R " ^ and R * » t h r o u g h i n t e g r a l v a l u e s of n, so t h a t t h e contour d o e s Bh not p a s s t h r o u g h a p o l e . • - R < T h e n , by C a n c h y ' s t h e o r e m , the i n t e g r a l = 2iri x ( s u m of r e s i d u e s at s i n g u l a r i t i e s e n c l o s e d by C ) . T h e r e s i d u e at t h e o r i g i n i s obtained by expanding t h e i n t e g r a n d a s a L a u r e n t s e r i e s about p = o and i s the coefficient of 1/p in t h i s e x p a n s i o n . It i s

Bh T h e r e s i d u e at p = inir Bh inw Bh i s ing-By cosh Bh in»»" Bh ^ ( s i n h B p h ) !

dp J P

in<r Bh inwx e c o s h (-1) inir cosh inw imr

inirx Bh

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28 -i n ^ x o- . . , xu . ^ X in*" . ( - 1 ) " ' B h n f f y S i m i l a r l y t h e r e s i d u e a t p = - -^r- i s - -r-—— e c o s —r^ , s o t h a t t h e s u m of t h e r e s i d u e s at p = + -^r for a l l n i s an 2 - ( - 1 ) " , rarx niry - £ ^-~- s i n - g r - c o s -rf-IT , n Bh h n=l T h e r e f o r e p X 2 r (-1) . ntrx ntry - - ^ = =r- + - E ^—— sin -r^r cos —r^ . 2« Bh TT , n B h h n=l H e n c e , on t h e a x i s y = o, u s i n g t h e w e l l - k n o w n r e s u l t , " (-1) . nwx «"x »rx /T> * i o\ ^ - J T ^ ^ " - B h = - 2 B h ' - ' < B h < ^ ' ( R e f e r e n c e 18) n=l c B 2 ' Bh ' B h . g ^ , ^ , ^ ^ ^ T h e r e s u l t a n t of t h e two t e r m s i s a s t e p function. T h e p e r t u r b a t i o n p r e s s u r e coefficient i s z e r o up t o the f i r s t M a c h l i n e s f r o m t h e c o r n e r s , and c h a n g e s

d i s c o n t i n u o u s l y b y an a m o u n t - z£., w h e r e t h e Mach l i n e s

X = t B ( y + (2n+l)h)

(the c h a r a c t e r i s t i c s of t h e d i f f e r e n t i a l equation)cut the a x i s .

S i m i l a r l y t h e p r e s s u r e coefficient on t h e w a l l y = h i s given by 2* Bh Bh ' o < x < 2 B h a g a i n u s i n g a known r e s u l t " 1 , nirx »i(Bh-x) ^ n «^" -Bh = - 2 B h - • n=l o < ^ < 2«", f r o m R e f e r e n c e 1 8 . B h On t h e w a l l , the f i r s t p r e s s u r e d r o p at t h e c o r n e r i s half t h e m a g n i t u d e of t h e f i r s t p r e s s u r e d r o p on the a x i s , but the s u b s e q u e n t s t e p s a r e of m a g n i t u d e '.

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29

-2c

i . e . there is a p r e s s u r e drop of magnitude — for each characteristic the flow p a s s e s through.

The discontinuity in the p r e s s u r e coefficient is the result of linearisation, which effectively approximates to the expansion fan by one of zero thickness parallel to the Mach line.

For a different wall shape, the contour remains the same but there will be additional poles introduced by the transform of the wall shape, e . g . for the walls y = t ( h + ex*) - ! E _ = / eP^'coshBpy 2« J p*sinhBph P X* 1 „ ^ 4Bh " (-l)'^"^^ nfl-x nwy =T: - - Bh + -—z- £ ^—r cos -—- cos —-f- . Bh 3 W , n Bh h n = l

On the axis, y = o, and the infinite s e r i e s reduces to the well-known F o u r i e r s e r i e s for

.2 . B « h ' , „ ^ rrx

(- X + r—), -«• < r^r < IT (Reference 18). 4 B ^ v - x + — 3 - ; . -.. . —

The two t e r m s cancel out for x < Bh, and in general, when x = 2n Bh + x ' , - Bh < X' < Bh,

c B

- - ^ = 4n Bh + 4nx'. 2e

There i s a continuous linear p r e s s u r e drop down the axis between x = (2n-l)Bh and X = (2n+l)Bh with discontinuities in slope at odd multiples of Bh.

APPENDIX 2

M o r r i s o n ' s method applied to the corner flow problem

Morrison (Reference 16) solved an equation analagous to (1.1) for the

problem of wave propagation in rods of voigt material and visco-elastic m a t e r i a l s with three rparameter models. M o r r i s o n ' s method for solving equation (1.3) is a s follows: if the Laplace transform of f(x,y) is written L [ f(x,y)], then (1.3) is written

[«-']=iJiFe-^l^'

(1)

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30

-w h e r e ( x , y ) no-w denote t h e n o r m a l i s e d c o - o r d i n a t e s of s e c t i o n 2 . T h e follo-wing g e n e r a l r e s u l t s for L a p l a c e t r a n s f o r m s a r e r e q u i r e d ; If G( X ) = e = L w( X - ^ ) and ^-yh(X) ^ ^

j^g(x,w)J

[0(x,y)J

w h e r e h(X) t h e n A * - ^* - X + /9. 2 G(y,h(X)) = e ^ ^^^^J

( ƒ 0(x,

= LI i 0(x,y')g(y'.y)dy'j

o A c c o r d i n g t o equation (1),

f(x,y) = L

•^{£1 |SI ^-p-l^y]

L p ^Jp+a J and on a p p l y i n g t h e t r a n s f o r m a t i o n X = p + a, equation (5) b e c o m e s f(x,y) = e L

if^:jè^'-^^-"^j

a x Xt-/9 /• - I f Xx - w ( X - a ) ^ X ^ ,

j L j^e e j

y dw (2) (3) (4) (5) (6) But M o r r i s o n showed t h a t X+/9

-.(x-.)J|a

y - ^ -w X-w( a-2^)-wrh(X ) - -j^)

2^ = e so t h a t e q u a t i o n (6) can b e w r i t t e n ax / - wt a - 2 ^ ) [ Xx wX L f(x,y) = e «o / e - - < / e " - " " G ( w , h ( X ) ) d X d w - a x ƒ " - w( a-2^ ) y w h e r e H(x) i s t h e unit H e a v i s i d e function. = e ƒ e y H(x-w) / 0 ( x - w , y ' ) g ( y ' , w ) d y ' d w o

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31

-Morrison gives the following r e s u l t s ,

Ux.y) = L-^[e^e-y'^<^)j

H(x)I, Ux^(x+2y)*J

= e"^y [6(x)+ ^ y xi(x+2y)^ (7)

where 6(x) i s the Dirac delta function,

and 2/9 g(x,w) = L e e

l [ , > - e - » [ ^ - T

]

1 I, [2V2/3w(x-w)] = 8(x-w) + w*H (x-w) V ^ Vx-w (8) where 2fi = r .

Thus from the general r e s u l t s (2), (3) and (4), with (7) and (8),

ƒ 0(x,y')g(y',w)dy' = L - ^ [ e ^ e H ^ ^ ^ ^ ' hfsöj J o = 6(x)e +/9 w e V x(x+2w) f - 5 v ' v'VT^ I 1/9 V x(x+2y')J r ^ J -•]

i ^ "^ ^ -fe^r—^i,[2«S^)Jdy'

+^ e w and X+f. , - I f Xx -w( X-a)NX:?"~i R, L e e = ö(x- w)e + /9wH(x-w)e /x** - w*

««/ V - w r " - ^ y ' V?w(y'+w) , ,„j« . J , ^LAx+y')' -(y'+w)']^... m x - w ) e j e . ; ^ " ^ i < 2 ^ ^ ^ ' V ( x + y ' y - (y'+w)" "^^

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32 -+ ^ e - « ^

I -/^e -^r ^ ^ ^

M2/^.)^r;;::jr^'^y'<^-APPENDIX 3 Evaluation of L (f . y ) n n t' X , f X + ^ y .

L,(t:,. y j = L j ^ — ^ J ^ e J

n ' n

Replace X = S + ^ / 2 , and for simplicity write t' = t, y = y.

Then ^ , . . . ^ e « L " ' ^ ^ J ^ e j ^ ' ' j (.)

1 S+r m v ^ T " " v™ / ' 5 + r \ T

Expand e-' * = £ « ^ V-T"/ for all values of y and 4 . so that (1) m'. \ fc / m=o becomes m-1

T /. ^ ''t - yü. T-lf e'* ( i ± £ ' ) ~ l

L , ( t , y ) = e j L . L ^ - ^ 3 - ^ \ ^ ) J m=o

(The order of summation and integration can be inverted, since the integral is shown to converge below).

(This expansion procedure can only be used to evaluate (1); it cannot conveniently be applied for the evaluation of (2. 3) say, because of the extra (^ - 1) t e r m within the exponential). If <i = 4 - 1, then

m-1

(2) m=o

Let P^(t) = L - { e ' ' * ( ^ ) ^ l ]

-if ^(^+A^l1 If- ^ t / ^ M + a ^ 2 1 ^ i__-]

(3)

so P (t) satisfies the r e c u r r e n c e relation, m

.t

P J t ) = P ^ „(t) + r e " * / e ^ ' p ( r ) d r (4) m m-2 J m-Z

o P (t) and P^ (t) a r e given in Reference 17,

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33

Po(t) = H ( t ) | e ' " ' y ^ ) + J e"'^y^M)d/i]

o

{e-«^^(^) + j

P, (t) = H(t)

where H(t) is the Heaviside unit step function, and from the r e c u r r e n c e relation (4), all the P (t) can be evaluated.

Thus L , ( t . y ) = e«* £ 2 - P (t) m=o (5) P (t) to P (t) a r e given below, P^ (t) = H(t) | e ^ y / 9 t ) + (1+ r ) J e y/9<i)d^ o P, (t) = H(t) a - r e / tm P^(t) = H(t) e " " * y f f t ) + a « | e"'*y^;i)dM - r « e " * | e'^^I t •\ - A l l {eu)dti P (t) 0 P. (t) H(t) = H(t)

-t t -t. »x -t 1

e + a ( l - e ) - r t e

l e " " * y ^ t ) + a ' f e'°"'y^M)d/i + ar* e ' M ar" e ; e I (/9*i)d(i -Pu,

t r + r ' e • * / / . o ' o e yff/i)d/idr P, (t) = H(t) P, (t) = H(t) - t , 2/1 - t , , , - t - t 1 Ï 2 - t a " i 2 / 1 - X . « . -X / , a . . -X 1 Ï . 2 -X a , - x . e + a (1-e ) - r t e + r ( l - a )te - f r t e + a r ( l - e ) -at / _OM.

e Io(/9t) + a / e I (/9M)d/i + r e V + a r - a ' ) / e I (fin)du o

+ r*e'^ j I e ' ^ ' ' y ^ M ) d / i d r - r * e ' M / j e'^^lJifii')dHdTd

""o ^ o o o o

r'

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34

-APPENDIX 4

Solution of the Exact Differential Equation in the form (3.4).

As shown in section 3, the exact differential equation can be rewritten in the form,

* " ^ dx' 3x ^^ ^ 3x' 3y* 4 ^ ^^' where (x,y) denote the normalised co-ordinates of section 2. This form of

the equation can be solved a s follows; writing the Laplace transform 0(y,p) of 0 ( x , y ) a s 00 0(y.p) = ƒ e ' P ' ' 0 ( x , y ) d x , o equation (1) transforms to ( a+ p)p 1+p where ' 0 - ^ = - A(y,p) (2) 3 y o

The boundary condition on the wall for the corner flow problem has the transform

^f ^f> e ^ ..V

T ; - -Ö- = — o n y = o ( 4 )

Uoo *y P ^

and since the disturbance must be bounded at y = oo ,

0 ( « . p ) = 0 (5)

The solution of (2) is an integral equation for 0 and has the form

0 ( y , p ) = i 0 ( o . p ) [e^y+ e - ^ y ] + i - 4 ^ £ - (e'ïy - e-^^)

qv . -Qv

£_1 / ^-qy' A/,., „N^,.. . e _ ! i / „qy'

' o

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35

In order to satisfy the boundary condition (5),

*(o,p)

'i£_

q B

f/ XJco

k[

e'^^y'A(y'.p)dy' (7)

and hence

j/y p) = . _ ! / £ _ e'^^y + / ' P . ! "°^hqy I

**y'P' qBf^^ " ^ i r n : ^ q J

e'"'^' '(y'.p)dy'

r*p* e

* 4(l+p)' q / cosh q y' ji(y', p)dy' (8)

where / \ (a+ p)p (9) Therefore (x,y) = -rr-^— L " ^ ^f/ U„ eP^e'qy 1

I qp .

p x 2

^ï-lil4ff^«"^'^'<^'-P)^^'j

r* ,-11 f eP%*e-qy ^ , I , . ,H .1

+ :r- L I I ,,, r^g„ coshqy' 0(y',p)dy'j

(l+p)"q

or since c 2u' 2 30 U. U.0 3 x '

c B .

_E_i

2e

.-'[^]

f e P y coshqy ^-qy', ! E j : ) d y i I U (i+p)"q ^ * 26 ' ° y J c B,

r« -if f eP^p*e-qy ^ , , V f ,^ , (

^ 4 ^ i J ( 1 + p ) ^ coshqy' ( - - ^ ) d y ' J

(10)

Now the last two t e r m s in (10) a r e in the nature of correction t e r m s and

CnBf

so a reasonable approximation to — ^ j ~ in them will be obtained if the first approximation (correct to 0(^)) is inserted in these t e r m s ,

(43)

36 -c B , -qy so (10) becomes c B, _E_L - T -1 2e ^

px^-qy-) ^a _if pPX,» fp-^iy

e'^'e 1 . /9* . -If eP^p' Te -qy - q y T

. ^ 2 ^ L ( ï T ^ [ q ^y^^'^JJ <12)

Replace p by 5 - 1, and write t = x-y to get

2e ^ us-i)(s+/s)5-i3 ^ 2 ^ Ue-ixs+zsyis-^:

+ ^

%-^y . - l ! e * V ^ / S

e ^ V '^ I

L-(W?-J <13>

But L-^[e*^(e^^/2S.i)J= Ë_i^(V55F^t)

^""^ ^ r (5-l)(5+^)J = ~ïTr "" ÏT^ from Reference 17.

Therefore - I f e*^ "^ e* e-^t ^ a .

U - i ) ( 4 + / 9 ) ^ s - n = ; r - - ^ ( i + « t - ^ t * ) (14)

and t «

'[(^J = '^'''

<15)

so that c B , -t - ^ y r t 2e

- = ^ ^ p.^e-^+j' [e^-%/.e-^<t-)]l-y,,3^,d.J

e-/^t

+ — 2 p j^e - d + a t - ^ — )

+ ƒ [ e * ' ^ - ( l + « ( t - r ) - - ^ (t-r))e-^<^-^> f^ y ^ 4 ^ )d;

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37

-ff«e-*e-^y

^ y [ t e - ^ * +ƒ (t-r)e- ^^""^ f^ y V47^)drj

(16)

After integration by p a r t s and rearrangement, (16) can be written c B^

2

where

J . l a , | . , U t , y ; . ) + [ | - - f ^ ( l + a t - ^ ) + ^ J j ( t , y ; . )

hTf^d -afft)-^ I M(t,y;/9) + ;^N(t,y;/9)

L(t,y;^) = e'^yfe'* ^ Vi^lFt) + ƒ e " ' ' y V 4 ^ ) d r j

- ^ y ^-t

J(t,y;/9) = e "^ e •• IJ^^iSf^t) - e

[•"'

-/9t

ƒ /'•yvï3^)drj

M(t

t -^

,y:/9) = fft J(t,y;/9) - ^e'^^^e'* ƒ e ' ' e ^ J( r ,y;ff )dr J

o

t

N(t,y;/9) = /9t M(t,y;^) - /9e'^*e'* j e^ e^^ M( '•,y;^)drj

For ^ < < 1, (17) reduces to (3.2a) and

c B

--f^ = ^L(t,y;/9) + f J(t,y;/9)

and when y = o, this reduces to

(17) (17a) Cn B , ^ ,. . a'

- ^ ' 4 ^ ^ ^ [ l - - i ? " - '

^^.]

- X -/Sx e e

which could have been obtained directly from (16), since

L(x,o;^) = 1.

T/ o\ "X - / 9 x

J(x,o;/9) = e e

(45)

38 -M(x,o;/9) = O N(x,o;/9) = O e q u a t i o n (18) g i v e s

W°)B^

2e and w h e r e t h e e x a c t r e s u l t i s , (given b y P (oo)) ''Pw<">B^ ^ ^ 3^2 5 , 1 0 5 ^ ^ Q , ^ )

s o t h a t the a p p r o x i m a t e t h e o r y i s c o r r e c t to OO* ). T o apply t h i s m e t h o d t o t h e c h a n n e l flow, t h e c o - o r d i n a t e s ( x , y ) in the above a n a l y s i s m u s t b e r e p l a c e d b y t h e n o r m a l i s e d c o - o r d i n a t e s ( x ' , y ) and ( x ' , z ) of Section 2 and t r e p l a c e d b y t{^, when t h e r e s u l t (16) i s now bj^ and c^^ and

c B k k'

- —^— = £ b + £ c„ a s b e f o r e . 2 e n n

n=o n=o

F o r a different w a l l s h a p e , e . g . h = t (h +ex*), t h e b o u n d a r y condition (4) b e c o m e s

^ 30 2e IL 3 y • ? "

w h i c h will modify t h e f i r s t t e r m on the r i g h t hand s i d e of equation (8) s o t h a t equation (12) b e c o m e s

^ - 4 ^ j - - - l ^ F ? - ^ - l J

% B f . i r , p x , - q y , . , - i f f i P ^ n Te-'^y

2e and (13) b e c o m e s

(46)

39

-S

^ / 5

^ " " y ^ U-1)(5+/9)«J

and these inversions can still be performed with the aid of the convolution formula in a similar way a s above.

APPENDIX 5

One-dimensional analysis using a linear rate equation

The p r e s s u r e coefficient on the axis can be obtained from a one-dimensional analysis as follows:

F i r s t , for a perfect gas, the basic equations a r e , TT du 1 dp „

Momentum: U -r- + T j = 0 dx P dx

Continuity: j - ( p U A ) = 0

Normalise the co-ordinates to x' = /K, A' = —rr- , and linearise, iv. dA' du' do ' p U B , ^ + p A B , ^ + U B A ^ = 0 eo » f d x ' ~ " f d x ' oo f B. d x ' du' 1 dp' ~ dx' P dx' 00

and eliminate - r ^ by writing -r-r = —;— "TT i on the assumption that the dx' •' " dx' a.* dx'

flow is isentropic, to get ••

A ; , B , ( M * -Dpj - U B , ^ = 0 " f f dx' « f dx' where

i . e .

2u' Take Aj;, = 1 , A' = 1 + 2cBji:', and solving for c = - ^7— the frozen solution is

^ f du' dx' U.0 ^ f o o ' ^ U^ dA AiBj^ dx'

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40

-T h e e q u i l i b r i u m solution i s s i m i l a r l y obtained f r o m t h e equation

so o r o r A i **"• ^ dx' C B ' - P e 2e c B . . P f 2c =

Ik. dA'

Bg* dx' = 2B X» 2 - ' B ; 2

^

" B T

= a^

e = 1.333 X» for a = 1.5. T h e s e a r e shown in F i g u r e 2 . F o r t h e r e l a x i n g g a s , t h e b a s i c e q u a t i o n s a r e , M o m e n t u m : U -:— + T— ^*—- = 0 (1) " dx P dx OB Continuity: ^ (pUA) = 0 p U ^ + P A^ ^ + U A ^ = 0 (2) - » dx • ^ " dx ~ " d x E n e r g y : u J i l + ^ = 0 (3) ^•^ " dx dx E q u a t i o n of State: h = h ( p , P ,q) w h e r e q i s t h e e n e r g y in the r e l a x i n g m o d e , s o t h a t

dhi

. hp JE:- + h ^ + h ^

(4)

dx Poo dx p dx q dx • • ^«o R a t e E q u a t i o n : U„ ^ = 2 ! _ 1 ^ (5) w h e r e q' = q ' ( p , p) = e q u i l i b r i u m e n e r g y in the r e l a x i n g m o d e , o r dq' - d p ' , - dp' , „ .

dï -

''^^ dT

^ %

dT

*''

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41

-Substitute (5) into (4) and t h e n (4) into (3) and e l i m i n a t e p r e s s u r e and d e n s i t y g r a d i e n t s using (2) and (1) to get

1 - "h

( A — ^ )

uj

-1]

^' - ?= ^ - ^

i^^ - 0

\ h p / « dx A„ dx h p \ o / Peo fl» o r B * ËH!. - ^ dA _^^^^ i^Hr^] - 0 (7) f dx Aoo dx h p ' '^ ' Poo OO w h e r e ( ^ ) B« = ^ - 1 . a,* = :j ^ " - r — by definition, I a - 2 f 1 • - n_ ^x. « /p P_ 00 «0

( 7 ) i s differentiated with r e s p e c t to x, and (4) and (6) used t o e l i m i n a t e g r a d i e n t s of q ' , q' and then (1), (2) and (3) used to e l i m i n a t e g r a d i e n t s of h ' , p ' , and p ' i n favour of v e l o c i t y and a r e a g r a d i e n t s . T h u s

„2 d^l v^ d'A , / V -^^^ %\ r u^'(V - %- \« - V ) ,

f dx» A „ d x \ hp .U„ r ; t hp + h qp 00 ^ 00 ^00 ^

- ^ ïï^ ] = 0

A „ dx J du' dx o r

. a d*u U.O d*A^ „ a du' I k dA

^<Bf d?-

" A :

dPV

+ B^ 5 ^ - ^

55^

- 0

(8)

w h e r e K - oo "oo o hp + \ % «0 ^ 0 0 oo and a Uo« a ^ P "^^qo-^^oo B* = ^ - 1, a* = '» - ^ ^ " e 1, - q h - h ' " p«. q » P -00 OO OO oo b y definition. T r a n s f o r m to n o r m a l i s e d c o - o r d i n a t e s x' = / K , A* = —^- t o get

(49)

42

-d»u' U d'A' du'

dP^ ' B * A i dx'* " ^ ^ d x '

where A' = A^ + 2eB x' H(x') and so A' = 1

oo

The equation for u' is therefore

d*u' „ du' li, _ , _ • , , - 0 + a -r-r - 3 " . 2 eB, = 0. d x ' * dx' BF t The solution for u' i s

2glL r , , , -ax'1 ^ 2eUoo ,, ^-ax

"' =

i [ ^ L^^

^ ' -^ ^ J -^ B ^ <' " ^

so

%^i 2 r , , ,

-ax'1

, 2 .,

-ax',

- 2 i ~ = I ^ [ax' - 1 + e J + - (1 - e )

This is also shown in figure 2.

Uo,

B*A' f ~

dA'

(50)

Exoci- ,fRO;EN

- C p B f 2 <

EOUtLIBRIUM

ï f

_Dlb)ANwE PROU T'.!"SAT

F I G . I . PRESSURE COEFFICIENT ON WALL OF DIVERGENT NOZZLE.

«.DISTANCE FROM TMBOAT ,

F I G 2 . PRESSURE COEFFICIENT ON AXIS OF DIVERGENT N O Z Z L E .

2 c

« ' c DISTANCE FROM THROAT K

E X A C T . APPOXIMATE

(51)

-CpBf 2 <

10 2 0 x ' i P I S T A W C E FBOM THROAT

FIG.4. VARIATION OF PRESSURE COEFFICIENT WITH DISTANCE

FROM WALL-EQUATION (3. 2o)

X = Ê 1 S

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