Limiting Gasdynamic Theory and Its Applications

Limiting Gasdynamic Theory and Its Applications

Wave Phenomena: Modern Theory and Applications C. Rogers and T.B. Moodie (eds.) (North-Holland), 1984 319 0 Elsevier Science Publishers B.V. LIMITI...

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Wave Phenomena: Modern Theory and Applications C. Rogers and T.B. Moodie (eds.) (North-Holland), 1984

319

0 Elsevier Science Publishers B.V.

LIMITING GASDYNAMIC THEORY AND ITS APPLICATIONS W . H . Hui a n d H.J. Van Roessel

Department of Applied Mathematics University of Waterloo Waterloo, Ontario N2L 3G1

An important area of hyperbolic waves i s t h a t of a n unsteady f l u i d flow with shock waves past a s o l i d body a s governed by the gasdynamic equations. This paper considers the l i m i t i n g case of such a flow when the shock wave i s extremely s t r o n g . I t i s shown t h a t in the l i m i t as t h e f r e e stream Mach number M_ approaches i n f i n i t y and independently a s the r a t i o of s p e c i f i c heats of the gas y approaches unity, the gasdynamic theory reduces t o unsteady Newton-Busemann flow theory in which the surface pressure c o n s i s t s of the Newtonian impact pressure plus a centrifugal c o r r e c t i o n . With the use of the Lagrangian method, the ll'mitlng gasdynamic equations a r e solved a n a l y t i c a l l y a n d applied to many practical problems a r i s i n g from the study of aerodynamic properties o f hypersonic vehicles. INTRODUCTION

According t o Newton's original impact theory 1 , f l u i d p a r t i c l e s do not i n t e r a c t with each o t h e r b u t , upon s t r i k i n g a body, l o s e t h e i r normal component of velocity r e l a t i v e to the body s u r f a c e , r e s u l t i n g i n an impact pressure force pimpact . After impact the p a r t i c l e s continue along t3e f r i c t i o n l e s s surface with zero tangential a c c e l e r a t i o n . However, Busemann pointed o u t t h a t the Newtonian impact pressure represents t h a t a t the t o p of t h e shock layer whereas the surface pressure must contain a centrifugal correction due t o t h e curved p a t h followed by the fluid particles. We shall f i r s t show t h a t in the most general case, unsteady Newton-Busemann flow m theory i s derivable from modern gasdynamic theory i n t h e double l i m i t as M, and y 1 , where M, i s the f r e e stream Mach number a n d y the r a t i o of s p e c i f i c heats of t h e gas. The governing equations of motion of a p e r f e c t gas are -t DD tP + p v . v = o (la) -+

-+

[

where,

-

-+

V denote the pressure, density and v e l o c i t y of the p a r t i c l e a n d denotes time d e r i v a t i v e following a p a r t i c l e . Now, i n the l i m i t as the p,

p,

-

Dt , which i s equivalent t o the shock density of the gas behind the shock p l a y e r becoming i n f i n i t e s i m a l l y t h i n (as required by conservation o f mass across the shock), the momentum equation ( l b ) may be s i m p l i f i e d . Thus when ( l b ) i s -+

320

W. H. Hui and H.J. Van Roessel

+ -+ resolved along t h e tangential d i r e c t i o n s y1 and y 2 and along the normal of the body surface ( F i g . 1 ) i t reduces to* , i n the l i m i t direction

+

-

,

(2a)

P

an

*

This i s s o , because the pressure gradient in the tangential d i r e c t i o n i s determined by the shock pressure j u m p a n d must remain f i n i t e , while in the normal direction the pressure gradient may approach i n f i n i t y as the shock l a y e r becomes infinitesimal l y t h i n . We thus see from above t h a t in the l i m i t

p + m the momentum equation of gas dynamics s t a t e s t h a t the f l u i d p a r t i c l e s move independently on the body surface with zero acceleration (Eqs. ( 2 a ) and ( 2 b ) ) in agreement with Newton's assumption, b u t the pressure change across the shock layer may be f i n i t e and i s due t o the centrifugal e f f e c t as seen from E q . ( Z c ) , as a n t i c i p a t e d by Busemann. Since the l i m i t i n g process as density p -+ i s the equivalent t o the l i m i t i n g process as Mm -+ m a n d , independently as y -+ 1 , we conclude t h a t unsteady Newton-Busemann flow model, a n d n o t the Newtonian impact model, i s derivable i n the most general case from modern gas dynamic theory i n t h a t double l i m i t . , The imp0 t n e o f the centrifugal correction i n unsteady flow has been we1 1 demonstrated.

-

5,w

From a mathematical point of view the problem of c a l c u l a t i n g hypersonic flow past a given body i s greatly simplified in the Newtonian l i m i t . T h u s , i n t h e gas dynamic theory one needs t o solve the f i v e coupled non-linear p a r t i a l d i f f e r e n t i a l equations (1) subject to boundqry conditions o n the unknown shock position f o r the f i v e unknowns p , p a n d V , each of which i s a function of three space v a r i a b l e s and t h e time v a r i a b l e . By c o n t r a s t , i n the Newtonian l i m i t , the tangential momentum equations ( 2 a ) a n d ( 2 b ) are decoupled from the normal momentum equation (2c) and the continuity equation ( l a ) . One can t h u s c a l c u l a t e t h e flow by ( a ) f i r s t determining the p a r t i c l e t r a j e c t o r i e s o n the two-dimensional surface from ( 2 a ) a n d (2b) which i s only a problem of solving two ordinary d i f f e r e n t i a l equations, ( b ) then finding t h e density v a r i a t i o n across the shock l a y e r by solving the f i r s t order p a r t i a l d i f f e r e n t i a l equation ( l a ) which i s a l s o equivalent t o solving ordinary d i f f e r e n t i a l equations, and ( c ) f i n a l l y obtaining t h e pressure change across the shock l a y e r by i n t e g r a t i o n . We see then t h a t the problem of c a l c u l a t i n g a flow i n the Newtonian l i m i t i s reduced t o t h a t of solvi n g ordinary d i f f e r e n t i a l equations. The success of t h i s a p roach i s well demonstrat d in the e a r l i e r s t u d i e s f o r o s c i l l a t i n g a i r f o i l s ' , bodies of revolution and higher frequency e f f e c t s and indicia1 response6.

s

I t should be noted t h a t the g r e a t mathematical s i m p l i f i c a t i o n s mentioned above a r e achievable only when Lagrangian method of description of f l u i d m o t i o n i s used i n solving the tangential momentum equations to find t h e p a r t i c l e t r a j e c t o r i e s . If the Eulerian method of description were used we have aa =

~

at

+

...

+ v

an

n av"

, a=l,2

(3)

f o r the tangential a c c e l e r a t i o n . For steady flow, v n = 0 , a n d the number of independent s p a t i a l variables i s reduced by 1 . This i s , however, not the case f o r unsteady flow such a s t h a t a r i s i n g from t h e motion of a s o l i d body i n a uniform hypersonic flow as v n # 0 i n general. I n t h a t case no such s i m p l i f i c a t i o n s *

Superscripts 1 , 2 , components.

and

n

of velocity and acceleration denote t h e i r respective

Limiting Gasdynamic Theory and its Applications

321

can be made even i n the Newtonian l i m i t . We thus encounter here a r a r e example in f l u i d mechanics f o r which the Lagrangian method i s superior to Eulerianmethodof description.

x3

J

Fig. 1 Three-dimensi ona 1 shapes showing no t a t i on

S p e c i f i c a l l y we consider a uniform hypersonic flow p a s t a three-dimensional body o f a r b i t r a r y shape performing a combined pitching plunging motion. We b r i q f l y i n d i c a t e how the s t a b i l i t y d e r i v a t i v e s may be c a l c u l a t e d e x h i b i t i n g r e s u l t s for two p a r t i c u l a r examples.

322

W. H. Hui and H.J. Van Roessel

NEWTONIAN IMPACT PRESSURE Consider a body 62 o f a r b i t r a r y shape p e r f o r m i n g a s m a l l a m p l i t u d e p i t c h i n g m o t i o n about an a x i s t h r o u g h t h e c e n t e r o f g r a v i t y C i n a u n i f o r m hypersonic flow 3 , ( F i g . 1 ) . S i m u l t a n e o u s l y t h e p i v o t a x i s undergoes a p l u n g i n g m o t i o n . Let

OX1X2X3

be an i n e r t i a l r e f e r e n c e frame o f C a r t e s i a n c o o r d i n a t e s , w i t h

rl, t2,t3,

i n which

9- = U_eiEi

where

corresponding u n i t vectors

a x i s and o r t h o g o n a l t o t h e f r e e stream v e l o c i t y the plungino motion.

Then

OX3

i s p a r a l l e l t o the p i v o t

fl, , and

OX2

i1 = C O S ~,

i s parallel to = sinn

L'

t h e a n q l e o f a t t a c k a b e i n g t h e a n g l e t h a t fim makes w i t h OX1 l a t i n i n d i c e s denote a sum f r o m 1 t o 3 and Greek f r o m 1 t o 2.

.

,

i3=

0

,

Here, repeated

0 1 x 1 x 2 x 3 be a b o d y - f i x e d system o f C a r t e s i a n c o o r d i n a t e s w i t h c o r r e s p o n d i n g + + + u n i t v e c t o r s el, e2, e3, i n which O ' x 3 i s k e p t p a r a l l e l t o OX3 and t h e Let

c e n t e r o f g r a v i t y C l i e s on O ' x ' a d i s t a n c e o f h u n i t s f r o m 0 ' . A l l l e n g t h s a r e s c a l e d by t h e body l e n g t h a. , v e l o c i t i e s by U, , d e n s i t y by p, , p r e s s u r e by pmU2 and t i m e b y i/U, The p i t c h i n g m o t i o n o f t h e body may be

.

r e p r e s e n t e d by t h e d i s p l a c e m e n t a n g l e 8 ( t ) which i s t h e a n g l e between O'xl and 1 2 3 OX1 . L e t (Xc,Xc,Xc) be t h e c o o r d i n a t e s o f C Then t h e p l u n g i n g m o t i o n may

.

be r e p r e s e n t e d by that

O(e,6,yEPc.{)

1 Xc = Xc = c o n s t e t c . are a l l

2

, Xc3

Xc = Y c ( t ) <<

1

=

.

0

I t w i l l be assumed

so t h a t t h e i r q u a d r a t i c terms and h i g h e r

may be n e g l e c t e d . The v e l o c i t y and a c c e l e r a t i o n o f a f l u i d o a r t i c l e r e l a t i v e r e f e r e n c e frame a r e g i v e n by

r e s p e c t i v e l y , where

W:

= +1

,

U'

2

= -1

,

and

0

i j

= 0

L e t t h e e q u a t i o n s d e f i n i n g t h e body s u r f a c e be g i v e n by where

c1c2

to

the i n e r t i a l

otherwise. x

i

=

'

1

2

H3 ( 5 ,c ) , i

i s some c u r v i l i n e a r c o o r d i n a t e system o f t h e body s u r f a c e .

corresponding vectors tangent t o t h e surface then aret) + + -f '1 X T 2 i+ w h i l e t h e normal i s g i v e n by n = ? = n e . . + 1 '21

: = Hi a ,a

'

= 1,2,3,

The

a = 1,2,

We now c a l c u l a t e t h e Newtonian impact p r e s s u r e . By s u b s t i t u t i n g t h e e q u a t i o n f o r + t h e s u r f a c e i n t o ( 4 ) and ( 5 ) and r e s o l v i n g fim , v and i n t o components + normal and t a n g e n t i a l t o t h e s u r f a c e ( i . e . a = aa? t an; e t c . ) we g e t

where

baB

surface, ra

=

i i n H

,“B

i s t h e i n v e r s e o f t h e m e t r i c t e n s o r (gaB), and

(4“’)

= gaaHi

Hi

BY ,BY i s given by ’impact

a r e t h e components o f t h e second f u n d a m e n t a l f o r m o f t h e

,o

a r e t h e C h r i s t o f f e l symbols.

=

cu; -

=

ninjeiI?j

The N e w t o n i a n i m p a c t p r e s s u r e

vnI2

-

i j j i k 2n n o I? I? 8 k

-

i i

2n R (njw$lk

-

n2 h ) 6

-

2n i n2 Q i y

.

(3)

The p a r t i c l e m o t i o n i s d e t e r m i n e d f r o m ( 2 a ) a n d ( 2 b ) , i . e .

For t h e s p e c i a l case o f steady f l o w

(6

: y :0 ) , t h i s r e d u c e s t o

i m p l y i n g t h a t t h e p a r t i c l e t r a j e c t o r i e s a r e t h e geodesics o f t h e surface. Since o u r u l t i m a t e o b j e c t i v e i s the determination o f t h e s t a b i l i t y d e r i v a t i v e s f o r a t h r e e - d i m e n s i o n a l body u n d e r g o i n g a p i t c h i n g - p l u n g i n g m o t i o n , we o u t l i n e a method b e l o w f o r t h e e v a l u a t i o n o f t h e s e s t a b i l i t y d e r i v a t i v e s w h i c h does n o t e x p l i c i t l y r e q u i r e d e t a i l e d c a l c u l a t i o n o f t h e unsteady pressure.

A s o l u t i o n o f e q u a t i o n (10)

i s sought o f t h e form Sa = Ea(t,s,CQ,SQ) 1 2

324

where

W. H. Hui and H.J. Van Roessel

1;

s = -f

(i.e.

vr

=

Q

+ -d t , vr

*a+

5

T~),

layer a t

(5

1

2

since

and

y

9

Q’ 5 Q ) .

and

t

Since

Q

being t h e f l u i d ve loc ity r e l a t i v e to t h e body

the time when t h e f l u i d p a r t i c l e s e n t e r s t h e shock

t

appears i n ( 1 0 ) only through

e and

y

, and

a r e s mal l , the s o l u t i o n may be expressed a s

The scaled r e l a t i v e speed i s expressed as

together with i n i t i a l conditions obtained by demanding c o n t i n u i t y of ta nge ntia l v e l o c i t y a t t h e shock s u r f ace. I n s e r t i n g (13) a n d ( 1 4 ) i n t o ( 1 0 ) r e s u l t s i n the following system of o . d . e . ' s t o be solved successively

ds d2E:

ds2

(16a )

+

dE1 dEA

(E*ra ) -- = 0 0 By ds ds

w i t h s i m i l a r expressions f o r

F yj

and

En yj

,

325

Limiting Gasdynamic Theory and its Applications

where

we

j

and

r:

j

f

a r e known f u n c t i o n s , 7

=

1

,(-I

0 and E$I'&

= igy(EA,Ei).

Equations (16a) and (16c) a r e e a s i l y s o l v e d t o g i v e

1:

1 2 I i s d e f i n e d as I [ f ] = f ( u , t Q , c Q ) d u and where 1 2 1 " a r e determined f r o m t h e i n i t i a l c o n d i t i o n s . ( E ~ , $ ) = F, (0,~Q,~4)

where t h e o p e r a t o r

F,

j

j

The

s t r e a m l i n e s a r e determined b y s o l v i n g ( 1 6 b ) f o r t h e geodesics and (16d) f o r s u c c e s s f u l l y ?or a s i m i l a r way

.

.

j = 0,1,2,...

:E

j The c o n t i n u i t y e q u a t i o n may a l s o be s o l v e d i n

To c a l c u l a t e t h e s t a b i l i t y d e r i v a t i v e s t h e p i t c h i n g moment f o r a f l u i d element i s i n t e g r a t e d a l o n g a s t r e a m l i n e t o determine t h e c o n t r i b u t i o n t o t h e t o t a l p i t c h i n g moment o f t h a t s t r e a m l i n e , and t h e n one i n t e g r a t e s o v e r t h e s u r f a c e t o t a k e i n t o a c c o u n t t h e e f f e c t s o f a l l t h e s t r e a m l i n e s . The p i t c h i n g moment coefficient

1 Cm = M/2

i s d e f i n e d as

P

U S P. m-b

where

and

Sb

c h a r a c t e r i s t i c area and l e n g t h i n t h e body r e s p e c t i v e l y and p i t c h i n g moment a b o u t C W r i t i n g t h e p i t c h i n g moment as

.

we g i v e examples below o f

k

a r e some

M i s the t o t a l

m

-Cmi

f o r various bodies.

I n f i g . 2 we p l o t t e d d a m p i n g - i n - p i t c h d e r i v a t i v e

-Cmi

versus a n g l e o f a t t a c k

a ( r a n g i n g form -120' t o 80") o f a r o c k e t whose c r o s s s e c t i o n i s an e q u i l a t e r a l t r i a n g l e . A t h i g h a n g l e o f a t t a c k o n l y t h e l o w e r s u r f a c e has a c o n t r i b u t i o n and a c c o r d i n g l y -C i s proportional t o s i n u A t l a r g e n e g a t i v e a , on t h e o t h e r

.

mi hand, o n l y t h e two upper s u r f a c e s have a c o n t r i b u t i o n . For moderate a n g l e o f i s thus continuous b u t a t t a c k a l l t h r e e s u r f a c e s c o n t r i b u t e . The c u r v e o f -Cmi n o t smooth as shown i n t h e f i g u r e . I n f i g . 3 i s p l o t t e d the damping-in-pitch d e r i v a t i v e

-C

"i

versus a n g l e o f a t t a c k

a o f a 20" s h a r p cone t o g e t h e r w i t h experiments' u s i n g a i r ( y = 1.4, h e r e y i s t h e r a t i o o f s p e c i f i c h e a t s o f t h e gas) a t M_ = 8 Good agreement i s seen between t h e o r e t i c a l p r e d i c t i o n and e x p e r i m e n t s when due allowance i s g i v e n t8 t h e f a c t t h a t t h e experiments correspond t o y = 1.4 I t has been shown b y Hui

.

.

W. H. Hui and H.J. Van Roessel

326

6-

2-

01 -120 -95 -70 -45

-20 Q

F i g . 2.

5

30

55

I

80

(degrees)

V a r i a t i o n o f damping d e r i v a t i v e o f a body w i t h t r i a n g u l a r c r o s s - s e c t i o n versus a n g l e of a t t a c k .

0’

4

8

1

12

,

16

I

20

(I(degrees)

F i g . 3.

V a r i a t i o n o f damping d e r i v a t i v e o f a 20' cone versus a n g l e o f attack.

327

Limiting Gasdynamic Theory and its Applications

t h a t d e c r e a s i n g y tends t o i n c r e a s e dynamic s t a b i l i t y . Thus Newton-Busemann flow t h e o r y t e n d s t o o v e r e s t i m a t e -Cmi , t y p i c a l l y by a b o u t 1 0 % . The remaining d i f f e r e n c e may be a t t r i b u t e d t o flow s e p a r a t i o n f o r l a r g e a n g l e o f a t t a c h which r e q u i r e s s p e c i a l t r e a t m e n t and has n o t been accounted f o r i n t h e p r e s e n t calculation. As shown by H u i and Tobak", the s t a b i l i t y d e r i v a t i v e s given i n thispaper f o r small a m p l i t u d e o s c i l l a t i o n c o n t a i n a l l t h e i n f o r m a t i o n f o r l a r g e a m p l i t u d e slow o s c i l l a t i o n o f the body. On the o t h e r hand, the t h e o r y depends c r i t i c a l l y on t h e shock l a y e r being v e r y t h i n . For flows f o r which t h e shock l a y e r i s n o t t h i n , t h e t h e o r y i s n o t e x p e c t e d t o g i v e a good a p p r o x i m a t i o n

Acknowledgement T h i s work was s u p p o r t e d by the Natural S c i e n c e s a n d o f Canada.

Engineer ng Research Council

Re f e r e nces Newton, I . , Mathematical P r i n c i p l e s o f Natural Philosophy, T r a n s l . by A . Motte ( 1 9 2 9 ) , r e v i s e d by A . C a j o r i , U n i v . o f C a l i f o r n i a Press, BerkBley, 1936, r e p r i n t e d 1946. Busemann, A . , " F l U s s i g k e i t s and Gasbewegung", HandwGrterbuch d e r N a t u r w i s s e n s c h a f t e n , Vol. IV, 2nd e d . , Gustav Fisher, J e n a , 1933, p p 244279. Hui, W . H . and Tobak, M . , "Unsteady Newton-Busemann Flow Theory, P a r t I : A i r f o i l s " , AIAA J o u r n a l , Vol. 1 9 , p p 311-318, 1 9 8 1 , H u i , W . H . and Tobak, M . , "Unsteady Newton-Busemann Flow Theory, P a r t 11: B o d i e s o f R e v o l u t i o n " , AIAA J o u r n a l , Vol. 1 9 , p p 1272-73, 1981, a l s o NASA TM-80459.

Mahood, 6 . E . and H u i , W . H . , "Remarks on Unsteady Newtonian Flow Theory", A e r o n a u t i c a l Q u a r t e r l y , Vol. 29, 1 9 7 6 , pp 66-74. Hui, W . H . , "Unsteady Newton-Busemann Flow Theory, P a r t 111: Frequency Dependence and I n d i c i a 1 Response", A e r o n a u t i c a l Q u a r t e r l y , Vol 33, November 1 9 8 2 .

.

H u i , W . H . and van R o e s s e l , H . J . , Three-Dimensional Unsteady NewtonBusemann Flow Theory", AIAA ( t o a p p e a r ) .

AEDC Report TDR-62-11

.

H u i , W . H., "An A n a l y t i c Theory o f S u p e r s o n i c / H y p e r s o n i c P i t c h i n g S t a b i l i t y " WTA AdARD Symposiun on Dynamic S t a b i l i t y P a r a m e t e r s " , CP-235, p a p e r #22, 1 9 j 8

[lo]

Hui, W . H . and Tobak, M . , " S i f u r c a t i o n A n a l y s i s o f A i r c r a f t P i t c h i n g Motions a b c u t Large Mean Angles o f A t t a c k " , J o u r n a l o f Guidance, C o n t r o l , and Dynamics, Vol. 7 , p p . 113-122, 1984.