moto dei fluidi reali nei condotti

8/10/2019 Moto Dei Fluidi Reali Nei Condotti

1/15

Oomp*ttera Ma th. App lic.

Vol. 24 ,

N o .

1 1 , p p . 4 5 - 5 9 , 1 9 9 2 0 0 9 7 - 4 9 4 3 /9 2 5 . 0 0 + 0 . 0 0

P r i n t e d i n G r e a t B r i t a i n . A l l r i g h ts r e s e r v e d C o p y r i g h t ~ 1 9 92 P e r g a m o n P r e s s L t d

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

P . GL ISTER

D e p a r t m e n t o f M a t h c ~ n a t i c a U n i v e r s i t y o f R e ac l in g ~

P . O . B o x 2 2 0, W h i t e k n i g h t s , R e a d i n z , R G 6 2 A X , U . K .

Receieed Noeember 1991)

A b s t r a c t - - A n u m e r i c a l s c h e m e is p re s e n t e d f o r t h e s o l u t i o n o f t h e E u l e r e q t m t i o n s o f c c a np r es s ib l e

f lo w o f a r e a l g M i n a s i n g l e s p a t i a l c o o r d i n a t e . T h i s i n c l u d e s f l ow i n a d u c t o f v a r i a b l e c r o e s - s e c ti o n

a s w e l l a s f l o w w i t h c y l i n d r i c a l o r s p h e r i c a l s y m m e t r y , a n d c a n p r o v e u s e f u l w h e n t e s t i n g c o d e s f o r

t h e t w o - d i m e n s i o n a l e q u a t i o n s g o v e r n l n f f c o m p r e s s i b l e f lo w o f a r e a l g a s . T h e s c h e m e i s a p p l i e d w i t h

s u c c e s s t o a p r o b l e m i n v o l v i n g t h e i n t e r a c t i o n o f c o n v e r g i n g a n d d i v e r g i n g c y l i n d r i c a l s h o c k s f o r f o u r

e q u a t i o n s o f s t a t e a n d t o a p r o b l e m i n v o l v i n g t h e r e f l e c t i o n o f a c o n v e r~ i 'n ~ s h o c k .

1 INTRODUCTION

In 1988 , G la i s t e r [1 ] p rop osed an ap prox ima te Pd emann so lve r f o r t he one -d im ens iona l E u le r

equ a t ion s fo r compress ib l e fl ow o f a r ea l gas , i .e . , w i th s l ab sym me t ry . Th e r e su l t i ng sche me

was app l i ed w i th success t o a t e s t p rob le m invo lv ing the r e f l ec t ion o f a shock , and a s ign i f i can t

i m p r o v e m e n t i n t h e e f fi c ie n c y o f th i s s c h e m e h a s r e c e n t l y b e e n m a d e [ 2 ]. F o r a p p l ic a t i o n s w h e r e

the f low t akes p l ace in a duc t o f va r i ab l e c ross - sec t ion , i t i s aga in d es i r ab le t o have such a Pd ema nn

s o l v e r f o r s u c h f l o w s . I n p a r t i c u l a r , i t c o u l d b e u s e d t o c o m p a r e w i t h r e s u l t s o b t a i n e d f r o m a

two-d im ens iona l code fo r com press ib l e f l ows when p r esen ted wi th a cy l ind r i ca l o r spher i ca l ly

s y m m e t r i c p r o b l e m . I n t h i s p a p e r w e p r o p o s e s u c h a n u m e r i c a l s c h e m e , a n d t e s t t h e s c h e m e o n

a p rob lem invo lv ing the in t e r ac t ion o f converg ing and d ive rg ing cy l ind r ica l shocks .

In Sec t ion 2 we rev iew the d i f f e r en ti a l equa t ions govern ing th e f low und er co ns ide r a t ion and in

Sec t ion 3 we d i scuss how these e qua t ion s ca n be so lved us ing flux d i f f e rence sp l i t t i ng . I n Sec t ion 4

we g ive the de t a i l s o f t he r e su l t i ng num er i ca l scheme , w h i l e i n Sec t ion 5 we look a t t h e p rop er t i e s

o f t he sc heme . F ina l ly , in Sec t ion 6 , we desc r ibe two t e s t p rob lems , and in Sec t ion 7 we p r esen t

s o m e r e s u l t s f o r t h e p r o b l e m s o f S e c t io n 6 u s i n g t h e s c h e m e o f S e c ti o n 4 .

2 . E Q U A T I O N S O F F L O W

T h e o n e - d i m e n s i o n a l e q u a t i o n s g o v e r n i n g c o m p r e s s ib l e f lo w o f a r ea l g a s ca n b e w r i t t e n a s

1

p , + ~ r ) S r ) p u ) ,

= O , 2 . 1 a )

p . ) , + = -p .

2 . 1 b )

1

e , + - f ~C S C r )u Ce

+ p ) ) , = O , 2 . 1 c )

whe re the eq ua t ion s o f s t a t e r e l a t ing p r essu re , p , t o t he dens i ty , p , and the spec i f ic i n t e rna l ene rgy ,

i , i s g iven by

p = p p , i ) , (2 .1d)

and the to t a l ene rgy , e , i s g iven by

1 2

e p i J r ~ p u

, 2 . 1 e )

T y p e s e t b y . A . A ~ - T F X

45


2/15

4 6 P . G L A I S T E R

w h e r e u i s t h e v e l o ci ty . Eq u a t i o n s ( 2 .1 a )- ( 2 .1 c ) r e p r e s e n t a s y s t e m o f h y p e r b o l i c e q u a t i o n s f o r

(p, p u , e ) - p , p u , e ) ( r , t ) , w h e r e p u i s t h e m o m e n t u m , a t a g e n e r a l p o s i t i o n r a l o n g t h e d u c t ,

( o f c r o s s -s e c t io n a l a r e a S ( r ) ) , a n d a t t i m e t. Th e p a r t i c u l a r e x a m p l e S ( r ) - r N , rep resen t8 s lab

s y m m e t r y , ( N = 0 ) , c y l i n d r ic a l s y m m e t r y , ( N = 1 ), o r s p h e ri c a l s y m m e t r y , ( N - - 2 ) . ( N .B . Th e

c o o r d i n a t e r is g i ve n b y r = z , y / ~ + y 2 a n d ~ z 2 + y 2 + z 2 , i n t h e c a s e o f c e n t r a l s y m m e t r y

wi th N = 0 , 1 and 2 , r e spec t ive ly , wh ere z , y , z rep resen t Car te s ian coord ina tes . ) A n idea l gas i s

rep re sen te d by equ a t ion (2 .1d ) wi th p = (7 - l )p i , w here 7 i s the ra t io o f spec if ic hea t capac i t i e s

o f the f lu id .

I n t h e n e x t s e c t i o n w e fi n d a p p r o x i m a t e s o l u t i o n s o f e q u a t i o n s ( 2 .1 a ) - ( 2 .1 e ) ; b u t f i r st o f a ll w e

r e w r i t e t h e m i n s t a n d a r d ' c o n s e r v a t i o n f o r m ' a s

w h e r e

_ Wt + F _ W ) ) , = g _ W ) ,

_ w = ,~ , _ F W _ ) - p + p u 2 ,

L

toge the r wi th (2 .1d ) , (2 .1e ) , and

~ p u

I r

g _ w )

S ~ r U e

+,

i s a s o u r c e t e r m a r i si n g f r o m t h e g e o m e t r y .

2 . 2a )

(2 .2b)

2 . 2 c )

3. F L U X D I F F E R E N C E S P L I T T I N G

I n t h e c a s e o f s l a b s y m m e t r y , f o r w h i c h S = 1 , e q u a t i o n s ( 2 .2 a ) - ( 2 .2 c ) r e d u c e t o t h e o n e -

d i m e n s i o n a l Eu l e r e q u a t i o n s f o r a r e a l g a s i n a s i n g le Ca r t e s i a n c o o r d i n a t e , w h i c h c a n b e s o l v e d

b y f l u x d i f f er e n c e s p l i t t i n g u s i n g t h e a p p r o x i m a t e l i n e a r is e d R e i m a n n s o l v e r d e v e l o p e d b y G l a i s-

te r [1 ] . We sha l l u se the s im i la r i ty o f equa t ion s (2 .2a ) - (2 .2c ) to th e C ar te s ia n case to deve lop a

c o r r e s p o n d i n g m e t h o d f o r d u c t f l o w s k e e p i n g a s f a r a s p o s s ib l e t h e v a l u a b le p r o p e r t i e s p r e v i o u s l y

f o u n d .

W e c o n s i d e r a f i x e d g r i d i n s p a c e a n d t i m e w i t h g r i d si ze s A t , A t , r e s p e c ti v e l y, a n d l a b e l t h e

p o i n ts s o t h a t r j = r j _ l + A t , t , = t n - z + A t a n d ~ d e n o t es t h e a p p r o x i m a t i o n t o _ W ( r / ,t n ).

Co n s i d e r a l s o t h e i n t e r v a l [ r j - z , r j ] a n d d e n o t e b y W / ; , _WR t h e a p p r o x i m a t i o n s t o W a t r j - z , r j ,

r e spec t ive ly . We now rewr i te equa t ions (2 .23 ) a s

O F

w _ , + ~ - _ w _ ) w _ , = g_ w _ ) 3 . 1 )

a n d s o l v e a p p r o x i m a t e l y t h e a s s o c i a t e d Re i m a n n p r o b l e m

w _, + 2 w _ L , w _ . ) w _ . = _ g _ w ) , 3 . 2 )

w ith da ta _WL, _WR ei th er s id e of the po int r j_ , l inear is ing b y tak ing A(_Wz;,_WR) to be s co nst an t

m a t r i x . W e s h a l l u s e t h e a p p r o x i m a t e f o r m o f e q u a t i o n ( 3 .2 ),

- - - ~ W j - 1

W ~ +z W ~ + .4(_WL, _WR) ~ = ~ ( ~ ) , (3 .3)

A t - 7 x ~ -

wh e r e

A(_WL,_WR) is th e m at r ix (3 .4) , _~ is an ap pro xim atio n to _g an d P m ay be L or R. T he

m a t r i x A (_W L,_W R) i s a n a p p r o x i m a t i o n t o t h e J a c o b i a n m a t r i x A - ~ r ( I Z ) a n d i s c o n s t r u c t e d

so tha t _FR - _FL - - A(_WL,_WR)(~/R- W ) fo r any f in i te change o f s ta t e and i s g iven by

[ o 1 1


3/15

Real gas

f l ow s

T h e a v e r a g e s o f t h e f lo w v a r ia b l e s a r e g i v e n b y

v ~ ; + v ~ 7

w h e r e H = ~ i s t h e t o t a l e n t h a l p y , t o g e t h e r w i t h

P

A = u , i , H ,

47

( 3 . 5 a ) - ( 3 . 5 c )

( 3 . 6 )

( 3 . 7 )

T h e a v e r a g e s o u n d s p e e d a i s g i v e n b y

PPi

( 3 . 8 )

2 = / 3 ~ ~ ,

w h e r e t h e a v e r a g es / 3 p ,/ 3 i o f t h e d e r i v a t i v e s o f t h e e q u a t i o n o f s t a t e P p , P i ) a r e g i v e n b y

Pi- = Pi ~, 3) J

i f - - z -- p _< 10 - m a n d - - = - -t - 10 - m ' ( 3 . 9a )

I ~ v l 6 p }

/3 , p o / ~, ) + [ A p [ [ A i [

A p

o t h e r w i s e , 3 . 9 b )

[ A i l 6 p

/3 ~ = P ' ( P ' I ) + I A v l + J A i l A i

6 v

= ~ p

- v p ~ , , ) a p - v , ~ , ) A i ,

w i t h

a n d w h e r e m i s m a c h i n e d e p e n d e n t ( s e e [1 ,3 ]).

T h e e i g e n v a l u e s o f A a r e

~1 ,2 "-- fi -I- a ,

w i t h c o r r e s p o n d i n g e i g e n v e c t o r s

( 3 . 9 c )

A s ---- fi,

( 3 . 1 0 )

~ 1 , 2 = _ ~ a , ~ =

L ~

~a

~ ~

a s in t h e s t a n d a r d C a r t e s i a n c a s e w h e n S = 1 ).

( 3 . 11 )

CN4J~ 24t11 ..0

U s i n g t h e a b o v e p r o p e r t i e s o f A w e c a n w r i t e e q u a t i o n ( 3 .3 ) i n s u c h a w a y t h a t t h e l e f t - h a n d

s i d e h a s i t s n a t u r a l c o n s e r v a t i o n f o r m , i . e . ,

W~ p" '1 - W / ~ + F j - F j _ 1 __~ ~ ( ~ v n ) , ( 3 . 1 2 )

A t

A r -

w h e r e 0 (_ W ) is a s u i t a b le a p p r o x i m a t i o n t o t h e t e r m _ ~(W ) o n t h e r i g h t h a n d s i d e o f e q u a -

t i o n ( 2 .2 a ) . W e t h u s o b t a i n

__W ~p 1 - ~ = A t _ 0 ( W ) - ~ r ( _A t _ _ _ ~ -1 ) . ( 3 . 1 3 )

B e f o r e w e d e s c r i b e t h e m e c h a n i s m u s e d t o u p d a t e I ~ / t o W ~j + 1 w e l o o k a t t h e a p p r o x i m a t i o n

0(LL n) u se d for _g(W) .

G i v e n t h e f o r m f o r _g (W ) i n ( 2 .2 c ) , a n d n o t i n g t h a t e + p =

p H ,

o n e s u i t a b l e a p p r o x i m a t i o n i s

-0 (L ~V n) = - P 'S - ~ r ' ( 3 . 14 )

u s i n g t h e a v e r ag e s i n ( 3 .5 a ) , (3 . 5 c) a n d ( 3 .6 ) , w h e r e ~ r = r j - r j - i , A S =

S r j ) -

S ( r j - 1 ) , a n d

i s a s u i t a b l e a v e r a g e f o r

S r )

o v e r t h e c e l l [ r j - x , r j ] , e . g . , ,~ =

~ / S r j - 1 ) S r j ) .

T h e r e a r e , o f

c o u r s e , o t h e r c h o i c es t h a t c o u l d b e m a d e .


4/15

4 8 P . G L A i S T E R

4. U P W I N D I N G

I n th e h o m o g e n e o u s ( g - 03 case t he p roc edu re now i s t o p ro j ec t A_F = _Fj - _F j-1 on to t he

e i g e n v e c to r s o f 2 i. E a c h p r o j e c t i o n r e p r e s e n t s t h e c o n t r i b u t i o n o f o n e w a v e s y s t e m t o A F . H e r e

we fo l low a p r oce dur e g iven in [3 ] , and f i nd the p r o j ec t io ns bo th o f A_F = _Fj - _~ -1 and a l so

~ ( _ W ~ ) . F o r l i n e a r c o n s e r v a t i o n l a w s t h i s p r o c e d u r e l e a d s t o a c o r r e c t l y ' u p w i n d e d ' t r e a t m e n t o f

s o u r ce t e rm s . W e t h e n u p d a t e W ~ t o W ~ + 1 . S u p p o s e

3

A W = W j - _Wj -1 = E & ' ~ ' (4 .1 )

i-----1

s o t h a t

3

i = l

Since .4 has e igenva lues Xi w i th co r resp ond ing e igenvec to rs ~ , and

4 . 2 )

3

.~ _W .,~)= 1 E A ~ , 4 . 3 )

-

A r

i----1

f o r s u i t a b l e / 3 i , w e m a y w r i t e e q u a t i o n s ( 3 . 1 3 ) a s

3

At E ~ i ~ i ~ , (4 .4 )

~ p l = w ~ -

i = 1

w h e r e

5 , = a , + _-3~ ( 4 .5 )

Ai

a n d P m a y b e L o r R . T o u p d a t e W t o V + i , w e u s e f i r s t o r d e r u p w i n d d i ff e re n c in g , i. e. , f o r

e a c h c e l l [ r j - i , r j ] , w e

A t - W ~

a d d - ~ r A i S i ~ t o _ j , w h e n Ai > 0

o r

ad d - ~ r A i 7 i ~ to Vv 'f_ l, wh en Ai < 0 .

T h i s g i v es e x a c t s h o c k r e c o g n i ti o n f o r th e K i e m a n n p r o b l e m g i v en b y e q u a t i o n s ( 3 .2 ) w i t h i n

t h e r e s o l u t i o n o f t h e g r i d, p r o v i d e d t h a t w e u s e t h e p r e v i o u s l y d e f i n e d l o c a l a v e ra g e s g iv e n b y

e q u a t i o n s ( 3 . 5 ) - ( 3 . 9 ) .

I f w e f o ll o w t h e a l g e b r a t h r o u g h , w e o b t a i n

A p

a2

4 . 6 a ) - 4 . 6 c )

a n d

1 _ _ A S /~ 3 = 0 . ( 4 . 7 a ) - ( 4 . 7 c )

~ . ~ = ~ p ~ ~

( N o t e t h a t t h e q u a n t i t y ~ i s i n d e p e n d e n t o f t i m e a n d t h e r e fo r e h a s to b e c a l c u l a t e d o n l y o n c e .

I f t h e a r e a c h a n g e s a r e d u e t o c e n t r a l s y m m e t r y , s o t h a t S ( r ) = b e ~ , ~ = 1 , 2 f o r c yl i nd r i c al o r

spher i ca l f l ow, t hen

r ~ - - r j - t

i f O - - 1 ,

A S ( r j ~ j _ ~ ) ~ l =

~ -- ~ r~ _r 2

S : i-~ if ~ = 2.

r j r j - x


5/15

P e a s a s f lo w s 4 9

Since these express ions a re s ingu la r when a g r id po in t i s p laced a t the o r ig in , we use g r ids tha t

s t r a d d l e t h e o r i g in . )

Th e e x p r e s s i o n s in e q u a t i o n s ( 4 .6 ) a n d ( 4 .7 ) h a v e b e e n w r i t t e n i n t e r m s o f p r i m i t i v e v a r i ab l e s

fo r s impl ic i ty . Th e express ions fo r & , , &=, &3 fo llow those g iven in [1 ] and s ign i fy wa ves t ren g ths

w h i c h w o u l d b e g e n e r a t e d b y t h e d a t a 1 ~ - 1 , _ ~ ] i n a p a r al le l d uc t . T h e e x p r e ss io n s f o r / ~ , ,

f1 2, f ls r e p r e s e n t m o d i f i c a ti o n s t o t h o s e w a v e s t r e n g t h s d u e t o a r e a c h a n g e s a n d o n l y o c c u r i n t h e

acous t ic waves .

To so lve equa t io ns (2 .2a ) - (2 .2c ) us ing the f in i te d i f fe rence appro x im at ion g iven by equa-

t i o n s ( 4 .4) a n d ( 4 .5 ) , w e u s e t h e m e t h o d o f u p w i n d d i f f e re n c i n g o n t h e t h r e e w a v e s w i t h w a v e s p e e d s

A:, A2, A3 and wav es t reng th s , , (wh ich wi l l d i f fer f rom the usua l wav es t reng ths in s lab

s y m m e t r y d u e t o t h e v a ri a t io n o f S ( r ) ) . Th i s g iv e s t h e f i r s t o r d e r a p p r o x i m a t i o n m e n t i o n e d e a r -

l ie r . A s i n [ 1 ], w e c a n m o d i f y t h i s s c h e m e t o b e e n t r o p y s a t i sf y i n g a n d d e f i n e t r a n s f e r s t h a t c a n

b e l i m i t e d u s i n g f l u x li m i t e r s t o o b t a i n a n o s c il l at i on f r e e s c h e m e . M o r e o v e r , th e u s e o f p a r t i c u l a r

f lux l imi te r s [4 ] wi l l sha rpen up ce r ta in fea tu res tha t wou ld be smeared by us ing the f i r s t o rde r

m e t h o d a lo n e.

5 . P R O P E R T I E S O F T H E S C H E M E

I n t h i s s e c t i o n w e d i s c us s b r i e fl y s o m e p r o p e r t i e s o f t h e s c h e m e p r o p o s e d h e r e f o r s o l vi n g

e q u a t i o n s ( 2 .2 a ) -( 2 .2 c ) i n r e la t i o n t o t h e f e a t u r e s t h a t w e e x p e c t t o o c c u r i n th i s t y p e o f p r o b l e m .

i ) I f w e c o n s i d e r t h e s p e c i a l c a s e o f c o n s t a n t d a t a p 7 =

p n i 7 -

i n ( so p7 - pn ) and

n = u n = 0 n o f l o w ) , fo r a l l j a t t im e t = n A t , equa t ions (4 .4 ) - (4 .5 ) reduce to

j

- w ? - -

o , 5 . 1 )

t

y i e l d i n g t h e c o r r e c t p h y s i c a l s o l u ti o n , n a m e l y u n+k = O, pn+k = pn , (pn+k = pn ) .

( ii ) Th e s c h e m e ' re c o g n i z e s' s t e a d y s t a te s . I f th e d a t a s a t i sf i es t h e c o m p a c t r e s i d u a l e q u a t i o n

- - - t j _ 1 = A r ( W ' ) j _ I / 2 , f o r .~ ]l j , ( 5 . 2 )

the n ~ = ~2 = ~s = 0 fo r a ll pa i r s o f cel ls and no upda t i ng takes p lace . Eq ua t io n (5 .2 )

c a n b e r e g a r d e d a s a t w o - p o i n t d i s c r et i s a ti o n , s e c o n d - o r d e r a c c u r a t e a t r j - t / 2 o f t h e o r -

d in a ry d i f fe ren t ia l equa t io ns govern ing s teady , compress ib le f low. Th is p rop er ty appe a rs

espec ia l ly va luab le whe n shock re f lec t ion shou ld leave beh in d a un i fo rm unch ang in g flow.

( ii i) Th e schem e a lso ' r ecogn izes ' shock-waves . At a shock A_F = S A W fo r som e sca la r shock

speed S and by equa t ion s (4 .1 ) and (4 .2) , S i s an e igenva lue o f A. Th e p ro jec t ion o f A W

onto the loca l e igenvec to rs o f A wi ll be so le ly on to th e e igenvec to r wh ich co r respond s to S .

In th i s spec ia l case , the so lu t ion o f the l inea r i sed Riem ann p rob lem g iven by equa t ion s (3 .3 )

i s exac t .

Th e r e a s o n t h a t t h e f o r m u l a t i o n ( 3.3) r e c o g n iz e s s h o c k s i s t h a t t h e r i g h t - h a n d s i d e t e r m g ( W )

d o e s n o t c o n t r i b u t e t o t h e s h o c k w a v e, e s se n t ia l l y b e c a u s e i t d o e s n o t c o n t a i n a n y d e r i v a t i v es in

W , a n d t h e r e f o r e d o e s n o t j u m p w h e n i n t e g r a t e d w i t h r e s p e c t t o r . To d e m o n s t r a t e t h i s p o i n t ,

suppose th a t the va r iab le q jum ps f ro m qL to qR a t r = ro > 0 , the n

lira / d r = l i m qL _~ S( r) d r + qR S ( r ) d r = 0 ,

- O J o _ S r ) - . o o

s ince S ( r ) / S ( r ) i s con t inuous . Thus ,

f r o+~

lim / _g(_W)

r

= O, (5.3)

(by ch oos ing q =

p u , q = p u 2

and q =

u ( e + p ) ,

in tu rn ) , i . e . , the shock speed i s g iven by [~ ,

w i t h t h e r i g h t - h a n d s i d e m a k i n g n o c o n t r i b u t i o n .


6/15

50 P. QLAISTER

M o r e o v e r , i n t e r m s o f t h e t h r e e s c a l a r p r o b l e m s o b t a i n e d b y d i a4 F m a l is in g t h e s y s t e m g i v e n b y

e q u a t i o n s ( 2 . 2 a ) - ( 2 .2 c ) w i t h .4 = ~ _ a c o n s t a n t m a t r i x , w e h a v e

OVi

~ i O Vi h~ (W ) , i = 1 ,2 ,3 , ( 5 .4 )

a t + 0 , =

w h e r e

Y - - X - I~ w ~ b = X - I g , 5 . 5 )

a n d X i s t h e m o d a l m a t r i x c o n s i s t i n g o f t h e e i g e n v e c t o r s o f A w i t h e i g e n v a l u e s h i . S o l u t i o n s to

e q u a t i o n s ( 5 .4 ) a n d ( 5 .5 ) c a n b e r e p r e s e n t e d i n t e r m s o f a ' s h o c k s o l u t i o n , ' th i s b e i n g a s o l u t i o n o f

t h e h o m o g e n e o u s e q u a t i o n , i .e . , e q u a t i o n s ( 5 .4 ) w i t h _h (W ) - O , a n d a 's o u r c e s o l u t i o n , ' b e i n g t h e

p a r t i c u l a r s o l u t i o n o f t h e i n h o m o g e n e o u s e q u a t i o n s ( 5. 4) a n d ( 5 .5 ) . I n e f fe c t t h e s c h e m e w e h a v e

d e v e l o p e d so l v es e q u a t i o n s ( 5 . 4 )- ( 5 .5 ) a p p r o x i m a t e l y , a n d t h u s t h e i m p o r t a n t ' s h o c k so l u t i o n ' is

m o d e l l e d a s a c o n s e q u e n c e o f t h e c o n s t r u c t i o n o f A .

6 . T E S T P R O B L E M S

I n t h i s s e c t io n , w e l o ok a t t w o t e s t p r o b l e m s u s e d t o v a l i d a t e t h e p r e v i o u s l y d e s c ri b e d a l g o r i t h m

o f S e c t i o n 4 f o r s o l v i n g e q u a t i o n s ( 2 . 2 a ) - ( 2 . 2 c ) .

T h e f i r s t t e s t p r o b l e m i s c o n c e r n e d w i t h a c o n v e r g in g c y l i n d r ic a l s h o c k a n d w e c o n s i d e r a r e g i o n

0 < r < 2 00 f o r t h e c y l i n d r i c a l ly s y m m e t r i c c a se g i v e n b y e q u a t i o n s ( 2 . 2 a )- ( 2 . 2 c) w i t h S ( r ) - r .

I n i ti a l l y , a c y l i n d r i c a l d i a p h r a g m o f r a d i u s r : 1 00 s e p a r a t e s t w o u n i f o r m r e g i o ns o f g a s a t r e s t .

T h e i n i t i a l c o n d i t i o n s a r e p -- 4 , p - - 4 i n t h e o u t e r r e g i o n , a n d p - - 1, p - 1 i n t h e i n n e r r e g i o n .

W h e n t h e d i a p h r a g m i s r e m o v e d a t t : 0 , a c o n v e r g i n g s h o c k w a v e f o l lo w e d b y a c o n v e r g in g

c o n t a c t d i s c o n t i n u i t y m o v e s t o w a r d s t h e a x i s, r - 0, a n d a d i v e r g in g r a r e f a c t i o n w a v e m o v e s

o u t w a r d s . T h e s h o c k a c c e l e r a t e s a s i t a p p r o a c h e s t h e a x i s o f s y m m e t r y a n d i s r e f le c t e d f r o m t h e

a x i s , i n t e r a c t s w i t h t h e c o n t a c t d i s c o n t i n u i t y ( s t i l l c o n v e r g i n g ) , w h i c h r e s u l t s i n a t r a n s m i t t e d

s h o c k , a c o n v e r g i n g c o n t a c t d i s c o n t i n u i t y a n d a w e a k c o n v e r g i n g r e f le c t e d sh o c k .

T h e s e c o n d t e s t p r o b l e m i s c o n c e r n e d w i t h a c o n v e r g i n g a n d a d i v e r g i n g c y l i n d r i c a l s h o c k i n

t h e s a m e r e g i o n . I n i t i a l l y , t w o c y l i n d r i c a l d i a p h r a g m s o f r a d i i r -- 5 0 a n d r - 1 50 s e p a r a t e t h r e e

u n i f o r m r e g i o n s o f g a s a t re s t . T h e i n i t i a l c o n d i t i o n s a re p - 4 , p : 4 i n t h e i n n e r a n d o u t e r

r e g i o n s , a n d p - - 1, p - 1 i n t h e m i d d l e re g i o n . W h e n t h e d i a p h r a g m s a r e r e m o v e d a t t -- 0 , a

c o n v e r g in g s h o c k w a v e , f o ll o w e d b y a c o n v e r g i n g c o n t a c t d i s c o n t i n u i t y m o v e s t o w a r d s t h e a x is ,

r - 0 , a n d a d i v e r g i n g s h o c k w a v e , f o l lo w e d b y a d i v e r g i n g c o n t a c t d i s c o n t i n u i t y m o v e s a w a y

f r o m t h e a x i s. T h e s h o c k s s u b s e q u e n t l y i n t e r a c t , r e s u l t i n g in a d i v e r g i n g s h o c k w a v e w e a k e n i n g

i n s t r e n g t h , t o g e t h e r w i t h a c o n v e r g i n g s h o c k w a v e i n c r e a s in g i n s tr e n g t h . E a c h o f t h e s e s h o c k s

t h e n i n t e r a c t w i t h t h e c o r r e s p o n d i n g c o n t a c t d i s c o n t i n u i t y , a s i n P r o b l e m 1 , a n d t h i s re s u l t s, f o r

e a c h i n t e r a c t i o n , i n a t r a n s m i t t e d s h o c k , a w e a k re f l e ct e d s h o c k a n d a c o n t a c t d i s c o n t i n u i ty .

E a c h p r o b l e m i s t e s t e d f o r f o u r e q u a t i o n s o f s t a t e :

( a ) i d e a l g a s

p 3 ` - 1 ) p i,

w h e r e w e t a k e t h e r a t i o o f s p e c i fi c h e a t c a p a c i t i e s 3` t o b e 1 .4 c o r r e s p o n d i n g t o a d i a t o m i c

gas ;

( b ) s t i f f e n e d g a s ( s o m e t i m e s u s e d f o r a l iq u i d )

p = B ( ~ - - 1 ) + (3 ` - l ) , i ,

wh ere w e tak e B = 1 , Po = 1 , 3 ' = 1 .4 ;

( c) c o v o l u m e ( u s e d in c o n n e c t i o n w i t h c o m b u s t i o n e n v i r o n m e n t s )

3 ` - 1 ) p i

P - - 1 - p b '

w h ere w e t ak e b - - 0 .2 , 3` = 1 .4 ; an d

( d ) t w o m o l e c u l a r v i b r a t i n g g a s

3 ` - 1 ) p p = 3 ` - 1 ) p i ,

p t e x p ( p p / p ) - 1

w h e r e w e t a k e 7 = 1 .4 a n d ~ s - - 0 . 5 90 2 w h i c h c o r r e s p o n d s m o l e c u l a r o x y g e n a t 2 0 0 0 K .


7/15

Rea l gas flows 51

3.99

2 . ~

-1.33

L ~ T J

I I I I I

4 0 8 0 1 2 0 I J O

I r

I I I I I

~

1 2 0 1 5 9

I

- 1 . 3 2

- 2 .~

U

o w

0 . ( ~ ,

0 . 9 ,

. I I eJ k I

J

-0.ll8 l

-0o~

0.99 ,

At time t = 50.0

U

o . ~ 2

o . a 8

0.14

r

-o,14

-0.a8

-0.42.

/

I

in ~ ltlO

r

S . 2 5

~ ,

-I ,?5

4,$9

At t ime t ---- 80 .0

8 0

1 2 0 1 6 0

I r

U

0.I6

0 .~

0 . 1 2

- 0 . 1 2

-0.~t

- 0 .N

:

1 2 0 1 4 0 ~ r

At t ime t = 110.0

Figure 1 . Solut ion of Problem 1 with equat ion (a) .


8/15

5 2

P

: t ~ o

t . m t .

- . 2 . 5 6 ,

___/

I I I I

4 0 8 0 I I O 1 4 0

P . G L A I S T E R

U

0.01

0 . S t

0 . 2 7 .

I

3OO r

- 0 , 2 ? .

-0.01

. I I I / ;

4 a O 0 1 8 / I

I

3OO r

3 . 9 9

2 6 6 j .

- I 3 3

~ t ~

4 1L~

I

40

- I . ~

A t t i m e t = 4 0 . 0

I I I I I

W 1 20 I N I P

U

0 . ~

0 . 3 4

0 . 1 2

80 I m 140 300 r "

- 0 , 1 2

- 0 . ; ~

- 0 . 3 6

A t t i m e t = 7 0 1 0

U

0 . 3

0 . 2

o .

..0.1

- 0 . 2

- 0 . 5

/

/ .

I I

m 13o

\

I ~ ... r

,a

i

J

I I I

1 8 I J O m r

I

A t t i m e

t = 1 0 0 0

F i g u r e 2 . S o l u t i o n o f P r o b l e m 1 w i t h e q u a t i o n ( b ) .


9/15

I :

lLd4.

2 , ~ 8 1 . .

I . I I

- 1 . 2 2

4O 8O 11o ItO

R e a l g a s f l o w s

U

0~ .

0 . ~

r

- 0 , ~ o .

- 0 . 6 6

- 0 .66

5 3

II.TII

2 I I ~ .

1 . 2 6

3 , 7 1 1 ,

41~.05

2 . 7 0

| ' I I I ,

- I o 1 5

- 4 . 1 1 l

I I 1 : :

I 0 I i O ll ll 2 o o

. _ _ . . , - - - ,

I I I I I

4 O 8 O l ~ O 1 4 0 ~ m O

A t t i m e t ~ 5 0 . 0

U

0 .S

0.1

r

- 0 .

- 0 . 2

i - 0 . 3 .

A t t i m e = 8 0 . 0

U

0 . 2 1

0 14

O.OT

r

-0.07

- 0 . 1 4

-.0 11

; I

. I I f i ; ~

r

A t t i m e t - - 1 0 0 . 0

F i g u r e 3 . S o l u t i o n o f P r o b l e m 1 w i t h e q u a t i o n ( c ) .


10/15

54 P. GLAISTER

3 . W

2 , ~ 1 6

- l A B .

-,%66

. . . ~ . . ~

I I I I

w W I m 1 4 0 1 0 0

P

3 , 1 4 .

I ~ 7

- I . ~ 7 ,

4 . 1 4 .

-4.71

I I I I I

4 0 8 0 I B 1 4 0 ~ r

U

0 . 1 M ,

O. t

. I t , ' , I i J I

, . ~ m M O r

i i

At time ~

=

50.0

U

0 . ~ ,

0.110,

0 . 1 5 .

- 0 . I S

- 0 . W

t ?

/ .

. I

4 O

t r - I

1 0 Im y l i l 0 I~

- i

1 . 8 4 ,

- I . 1 ~ ,

4 . 4 0 .

I I I I

4 0 8 0 1 11 0 1 4 0

At time t = 80.0

U

0.119

0.~t

0 , 1 3

I

P

4 . 1 3

- 0 , W

f

|

4O gO

I I I

I M 1 4 0 ~

t

t

r

At time t =

I I 0 0

Figure 4. Solution of Pro bl~ n 1 with equation d) .


11/15

3 .9 9

1 . n

. -2~6

- 3 . 9 9

3 .9 9 .

2 t ~

: I I I

40 IO 120 Im

Real gas flows

U

o . P J .

0. 0.

0 . 2 5 ,

2OO r

. . o . 2 5

o . $ D

o.75 ~

: ; -

4o go

. I

- I . 1

~, 99 ,

"At time

i I I t I

40 8O 120 140 20O r

- - -

30.0

U

0 . 6 1

- -o54

i

t

4O W

l aO 14o r

i

5 5

2 aQ

1.21

- I . 2 1

At time t = 40.0

,

L

_ f

I

t r

U

0 . 6 4 .

0 . ~ 6

o .m

. . o m

- O * S ~ ,

- 0 , 8 6

I I I I :

o ~ I m t r

At ti me t --- 60.0

Figure 5. Solution of Problem 2 with equation (a) .


12/15


13/15

R e a ST

- p

S . 4 2 .

1 . 1 4

- I , 1 4

. ~ . 2 1 i

..3.42

I

t o 80 I W 140 aO 0 r

g M f l o w 8

U

O . W ,

o . 1 6

o . 1 3

- 0 . 1 3 ,

- 0 . ; ~

- 0 . m

I I t

t o g o

I I I

IW 140 " ago r

:

j

1 . 1 5

A t t i m e t = 3 0 . 0

U

_

4 o

g o

- I . I S

. 2 . 1 0

4 . 4 S

I I

laO 14o

0 1 1 6

0 . a 4 .

0 . 1 2 ,

I

II00 r

- 0 . 1 2

- 0 . ; 16 .

- 0 . 1 6 .

I

4O

I I I I

3.43, rt

3.42, /

1 .2 1 - . , : - -

- I . a t

- i t=

-.3.43,

A t t i m e t

=

4 0 . 0

I I I I I

40 gO 120 160 300

f "

U

0 . 5

0 . 2 .

0 . 1

0 . I

- 0 . 2

- 0 . 3

.'t

I I I I

r

At time t - -- 60 . 0

F i g u r e 7 . S o l u t i o n o f P r o b l e m 2 w i t h e q u a t i o n ( c ) .


14/15

58

I I I I I

W

l a O 1 6 0

P. C..-.LA STER

U

0,?8

0 , 9 2

0,~

r

. . 0 , w ,

. . 0 . 5 2

. . 0 . N

At time t = 30.0

3 . ~ 9 ,

- I . l m

I I I I I

0 , I t .

0 . ~

O . m

.-0 ~

- 0 . ~

-0,114

\

8

I 0 r

y

3 . 6

2 . 4

- I . 2

--2,4

4 ,6

At ti me t --- 40.0

p

I

/

r

U

0 . 9 .

0 . 6 ,

0 3 ,

- O i l ,

-0 .6

-0.9

/

. ~ \

t

I

I

4O

o

8 0 I m I ~ r

S

At time t -- 60.0

Figure 8. Solution of Proble m 2 with equation (d) .


15/15

Real gas flows 59

7 . N U M E R I C A L R E S U L T S

I n t h i s s e c t i o n w e e x h i b i t t h e n u m e r i c a l r e s u l t s o b t a i n e d f o r t h e t e s t p r o b l e m s d e s c r i b e d i n

S e c t i o n 6 u s i n g t h e s c h e m e d e s c r i b e d i n S e c t i o n 4 .

F i g u r e s 1 - 4 r e f er t o t h e f i r st p r o b l e m o f t h e p r e v i o u s s e c t io n w i t h e q u a t i o n s ( a ) - ( d ) , r e s p e c-

t iv e l y , u s i n g a s e c o n d o r d e r s c h e m e w i t h t h e s u p e r b e e l i m i t e r [4 ] a n d 2 0 0 m e s h p o i n t s . T h r e e

o u t p u t t i m e s h a v e b e e n f e a t u r e d . F o r e a c h e q u a t i o n o f s t a te , w e o b se r v e th e d e v e l o p m e n t o f t h e

s h o c k a n d c o n t a c t d i s c o n ti n u it y , t h e r e f le c t io n o f t h e s h o c k , a n d t h e i n t e r a c t i o n o f th e s h o c k w i t h

t h e c o n t a c t d i s c o n ti n u it y . T h e r e s u l ts f o r e q u a t i o n ( a ) c o m p a r e f a v o r a b l y w i t h r e s u l ts p r o d u c e d

u s i n g a m o r e c o m p l e x a l g o r i t h m [ 5 ] .

T h e c . p . u , ti m e r e q u i r e d t o c o m p u t e t h e r e s u l t s on a n A m d a h l V / 7 w a s f o u n d t o b e a s f ol lo w s:

w i t h 2 0 0 m e s h p o i n t s i t t o o k 0 . 0 4 c .p . u , s e c o n d s t o c o m p u t e o n e t i m e s t e p , a n d a t o t a l o f 8 . 64

c . p . u , s e c o n d s t o r e a c h a r e a l t i m e o f 5 4 .0 u s i n g 2 1 6 t i m e s t e p s .

F i g u r e s 5 - 8 r e f e r t o t h e s e c o n d p r o b l e m o f t h e p r e v i o u s s e c ti o n w i t h e q u a t i o n s ( a ) - -( d ) , re s p e c -

t iv e l y , a g a i n u s i n g 2 0 0 m e s h p o i n t s a n d a s e c o n d o r d e r s c h e m e w i t h s u p e r b e e h m i t e r . T h r e e

o u t p u t t i m e s h a v e a g a i n b e e n f e a t u r e d . F o r e a c h e q u a t i o n o f s t a t e , w e o b s e r v e th e d e v e l o p m e n t

o f t h e s h o c k s a n d c o n t a c t d i s c o n t i n u i t i e s , t h e i n t e r a c t i o n o f t h e s h o c k s , a n d t h e s u b s e q u e n t i n t e r -

a c t i o n o f e a c h s h o c k w i t h a c o n t a c t d i s c o n t i n u i t y , r e s u l t i n g in t r a n s m i t t e d a n d r e f l e c t e d sh o c k s

a n d c o n t a c t d i s c o n t i n u i t i e s .

A m i n o r m o d i f i c a t i o n t o o u r a l g o r i t h m a l l o w s f o r a v a r i a b l e ( a d a p t i v e ) t i m e s t e p a n d g i v e s t h e

a b i li t y to d e c r e a s e t h e t o t a l a m o u n t o f c o m p u t i n g t i m e u s e d .

8 . C O N C L U S I O N S

W e h a v e e x t e n d e d t h e o n e - d i m e n s i o n a l v e rs i on o f G l a is t e r s s c h e m e [2] t o o b t a i n a n e w s c h e m e

t h a t i n c lu d e s c y l i n d r ic a l l y a n d s p h e r i c a l ly s y m m e t r i c p r o b l e m s . W e h a v e s h o w n h o w t h e s e p r o b -

l e m s b r e a k a w a y f r o m s t a n d a r d c o n s e r v a t i o n f o r m a n d t h u s g iv e r is e t o s o u r c e t e r m s , a n d t h a t

w i t h t h e a p p r o a c h o u t l i n e d i n S e c t i o n 4 w e c a n a c h ie v e s a t i s f a c t o r y r e s u l t s o n t w o s t a n d a r d t e s t

p r o b l e m s . T h e m a i n f e a t u r e s o f t h is s c h e m e a r e

( i) i t c a n b e u s e d f o r c y l i n d r i c a l l y o r s p h e r i c a l l y s y m m e t r i c p r o b l e m s w i t h c o n f i d e n c e , a n d

( ii ) i t c a n b e u s e d a s a c o m p a r i s o n w i t h r e s u l ts f r o m t w o - d i m e n s i o n a l s c h e m e s b y c h o o s i n g a

l a rg e n u m b e r o f m e s h p o i n t s f o r a c c u r a c y a n d n o t b e e x p e n s i v e o n c o m p u t i n g .

T h e s c h e m e c a n a l s o b e u s e d f o r g e n e r a l d u c t f lo w s .

REFERENCES

1. P. Glaister, An ap pro xim ate linearised Riem ann solver for real gases, J . Com p. Phys . 74,382-40 8 (1988) .

2. P. Glaister, An efficient Ftiemann solver for uns tead y flows with non-ideal gases, Com p~tera M~th . Appl ic .

24 (3), 7 7 9 3 (1992).

3. P. Glaister, Sec ond order difference schemes for hyperbolic conservation law s with source term s, Nume rical

Analysis Report 6-87, University of Reading, (1987).

4. P.K. Sweb y, High resolution schemes using flux limiters for hyperbolic conservation laws , S I A M J . N u m e r .

A~a/. 21,995-1011 (1984).

5. M. B en-Artzi and J . Falcovitz, A second order Go duno v-type scheme for compressible fluid dynamics,

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