comp fluids veld man

8/8/2019 Comp Fluids Veld Man

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Computers & Fluids, Vol. 1, pp. 251-271. Perga mon Press, 1973. Print ed in Grea t Britain,

THE NUMERICAL SOLUTION OF THEN A V I E R - S T O K E S E Q U A T I O N S F O R L A M I N A RINCOMPRESSIBLE FLOW PAST A PARABOLOID

OF REVOLUTIONA. E. P. VELDMAN

Dept. o f Mathem atics, University of Groningen, the Netherlands(Received 9 February 1973)

Abstract--A numerical method is presented for the solution of the Navier-Stokes equationsfor flow past a paraboloid of revolution. This method is based upon the ideas of van de Voorenand collaborators [1,2]. The flow field has been computed for a large range of Reynolds num -bers. Results are presented for the skinfriction and the pressure together with their respec-tive drag coefficients. The total drag has been checked by means of an application o f themomentum theorem.

1 . I N T R O D U C T I O NT h e n u m e r i c a l s o l u t i o n o f th e N a v i e r - S t o k e s e q u a t i o n s f o r l a m i n a r i n c o m p r e s s ib l e fl o wp a s t a s e m i - in f i n it e f i a t p l a t e h a s b e e n o b t a i n e d b y v a n d e V o o r e n a n d D i j k s t r a [1 ]. L a t e r ,t h e i r m e t h o d w a s i m p r o v e d a n d a p p l i ed t o t h e p r o b l e m o f fl o w p a s t a p a r a b o l i c c y l i n d e r b yB o t t a , D i j k s t r a a n d V e l d m a n [ 2]. I n t h e p r e s e n t p a p e r t h e a x i s y m m e t r i c v is c o u s f l o w p a s ta p a r a b o l o i d o f r e v o l u t i o n i s i n v e s t i g a te d .

T h e p r o b l e m d e p e n d s o n a R e y n o l d s n u m b e r R e , w h i c h i s b a s e d u p o n t h e s em i n o s er a d i u s o f c u r v a t u r e o f th e p a r a b o l o i d . T h e f o l lo w i n g t h r e e c a se s c a n b e d i s ti n g u is h e d :

( i) R e = 0 . T h i s c a s e c o r r e s p o n d s t o t h e s e m i - i n fi n i te n e e d l e w h i c h h a s n o i n f l u e n c e o n a n yo n c o m i n g f lo w .(ii) R e - - , o o . I n t h i s c a s e t h e f l ow is g o v e r n e d b y t h e b o u n d a r y l a y e r e q u a t i o n s . T h e s e

e q u a t i o n s h a v e b e e n s o l v ed n u m e r i c a l ly b y S m i t h a n d C l u t t e r [3 ]. A n a p p r o x i m a t es o l u t i o n h a s b e e n g i v e n b y D a v i s [ 4 ] w h o h a s u s e d a l o c a l s e r i e s t r u n c a t i o n m e t h o d .

( i i i) 0 < R e < o o. T h e g o v e r n i n g e q u a t i o n s t h e n a r e t h e f u l l N a v i e r - S t o k e s e q u a t i o n s . I t ist h e p u r p o s e o f t hi s p a p e r t o c o v e r t h is r a n g e o f R e y n o l d s n u m b e r s . A s o l u t io n v a l i d f a rd o w n s t r e a m h a s a l r e a d y b e e n g i v e n b y M a t h e r [ 5 ] , L e e [ 6 ] , C e b e c i , N a a n d M o s i n s k i s[ 7] a n d M i l l e r [ 8, 9 ]. T a m [1 0] h a s p r o v e d t h e e x i s t e n c e o f s u c h a s o l u t i o n f o r a h e a t e dp a r a b o l o i d .

T h e b a s i c i d e a o f t h e m e t h o d s u s e d i n [ I a n d [ 2] s h e s u b t r a c t i o n o f t h e b e h a v i o u r a t i n -f in it y. i t h t h e a i d o f a n a ly t i c a l a r g u m e n t s a n e x p r e s s i o n i s d e r i v e d f o r t h e s t r e a m f u n c t i o na n d t h e v o r t i c it y a l i d f o r l a r g e v a l u e s o f t h e c o o r d i n a t e s . T h e q u a n t i t i e s u s e d i n t h e a c t u a ln u m e r i c a l c a l c u l a t i o n s t h e n a r e t h e d e v i a t i o n s o f t h e f u l l s o l u t i o n f r o m t h i s a s y m p t o t i c b e -h a v i o u r a t in fi ni ty . n o t h e r i m p o r t a n t f e a t u re o f t h e m e t h o d i s t h e t r a n s f o r m a t i o n o f t h ei n fi ni te e g i o n o f i n t e re s t t o a f i ni t e e g i o n . T h i s t r a n s f o r m a t i o n i s c a r e f u l l y a d a p t e d t o t h eb e h a v i o u r o f t h e n u m e r i c a l l y c a l c u l a t e d q u an t it i es . s i n g t h i s m e t h o d o n e o b t a i ns r es u lt s o rt h e f u ll s o l u t i o n w h i c h a r e e x t r e m e l y a c c u r a t e f a r d o w n s t r e a m .

251C A F Vol. I No. 3---B


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252 A .E .P . VELDMANI n th i s p a p e r a n i t e r a t i o n s c h e m e i s p r e s e n t e d w h i c h d i ff e r s f r o m t h e o n e u s e d i n [ 2] .

T h e s c h e m e is b a s e d u p o n t h e a l m o s t p a r a b o l i c b e h a v i o u r o f t h e N a v i e r - S t o k e s e q u a t i o n sa n d i t l ea d s t o m u c h f a s t e r c on v e r g e n c e t h a n c o u l d b e o b t a i n e d w i t h t h e m e t h o d f r o m [ 2].E s p e c i a l l y f o r l a r g e a n d s m a l l v a l u e s o f R e v e r y f a s t c o n v e r g e n c e i s o b t a i n e d .

F o r t h e p a r a b o l i c c y l i n d e r a n a n a l y t i c ex p r e s s io n c o u l d b e f o u n d f o r t h e t o t a l d r a g a c t i n gu p o n t h e p a r a b o l a ( s e e B o t t a , D i j k s t r a a n d V e l d m a n [2 ]). F o r t h e p a r a b o l o i d o f r e v o l u t i o na l s o s u c h a n e x p r e s s i o n c a n b e d e r i v e d . T h e t o t a l d r a g t h e n i s g iv e n b y a n a s y m p t o t i c s e r ie sv a l i d f a r d o w n s t r e a m . T h e f ir s t t w o t e r m s o f t h i s s e ri e s a r e g e n e r a t e d b y t h e s u b t r a c t e d b e -h a v i o u r a t i n fi n it y , s o t h a t t h e m o s t i m p o r t a n t t e r m t o w h i c h t h e n u m e r i c a l l y c a l c u l a t e dq u a n t i t i e s g i v e a c o n t r i b u t i o n i s t h e t h i r d t e r m o f t h e s e ri es . T h e a n a l y t i c v a l u e o f t h is t h i r dt e r m h a s b e e n u s e d a s a c h e c k f o r t h e n u m e r i c a l r e s u l t s a n d g o o d a g r e e m e n t i s o b t a i n e d .

B y t h e t i m e t h i s p a p e r w a s f i n i s h e d a p a p e r b y D a v i s a n d W e r l e [ l 1 ] w a s p u b l i s h e d w h i c ha l s o t r e a t s f l o w p a s t a p a r a b o l o i d . D a v i s a n d W e r l e s o l v e t h e N a v i e r - S t o k e s e q u a t i o n s b ym e a n s o f a n i m p l i c it a l te r n a t i n g d i r e ct i o n m e t h o d . T h e y s o lv e p a ra b o l i c b o u n d a r y l a y e rt y p e e q u a t i o n s i n o n e i t e r a t i o n s t e p , a n d c o r r e c t f o r th e e l l ip t i c b e h a v i o u r o f t h e N a v i e r -S t o k e s e q u a t i o n s i n th e n e x t i t e r a t i o n s t e p . T h e u s e o f b o u n d a r y l a y e r t e c h n i q u e s i n t h ei t e r a t i o n p r o c e s s le a d s t o a c o n v e r g e n c e o f t h e n u m e r i c a l c a l c u l a t i o n s w h i c h i s c o m p a r a b l ew i t h o u r s . F o r l a r g e v a l u e s o f t h e c o o r d i n a t e s t h e i r s o l u t i o n i s b e l i ev e d t o b e l es s a c c u r a t e ,s i n c e D a v i s a n d W e r l e d o n o t s u b t r a c t t h e b e h a v i o u r a t i n fi n it y . M o r e o v e r t h e i r t r a n s f o r -m a t i o n o f th e i n f i n it e r e g i o n o f i n te r e s t is n o t o p t i m a l . B e c a u s e D a v i s a n d W e r l e p r e s e n td i a g r a m s b u t n o t a b l e s , o n l y a r o u g h g r a p h i c a l c o m p a r i s o n w i t h t h e i r r e s u l t s c a n b e m a d e .G o o d a g r e e m e n t i s i n d ic a t e d.

2. B A S I C E Q U A T I O N ST h e N a v i e r - S t o k e s e q u a t i o n s f o r a n i n c o m p r e s s i b l e v i s c o u s f l u i d c a n b e w r i t t e n a s

d iv q = 0t ( 2 . 1 )

g r a d q 2 _ q x r o t q = - - g r a d p - v r o t r o t qPw h e r e q d e n o t e s t h e v e l o c i t y , p t h e p r e s s u r e , p t h e d e n s i t y a n d v t h e k i n e m a t i c v i s c o s i ty .T h e p r e s s u r e c a n b e e l i m i n a t e d f r o m t h e s e c o n d e q u a t i o n o f (2 . 1) b y t a k i n g i t s r o t. W h e nto ffi r o t q w e ca n wr i t e

d iv q = 0 (2 .2)r o t ( q x to ) = v r o t r o t t oS i n c e w e w a n t t o s t u d y a x i s y m m e t r i c f lo w , w e i n t r o d u c e c y l i n d r i c a l p o l a r c o o r d i n a t e s

( x , r , 0) . L e t t h e p a r a b o l o i d b e g i v e n b y r 2 = 4 a ( x + a ) , w h e r e a i s h a l f t h e n o s e - r a d i u s o ft h e p a r a b o l o i d . T h e o n c o m i n g f l ow i s s u p p o s e d t o b e u n i f o r m w i t h v e l o c i ty U o a n d p a r a l le lt o t h e x - a x i s . N o w t h e b o u n d a r y c o n d i t i o n s t o e q u a t i o n ( 2 . 2 ) a r e

q = 0 a t t h e p a r a b o l o i d , (2.3)q " * U o ix f o r x ~ - o o .T o s a t is f y th e f i r s t e q u a t i o n o f ( 2 .2 ) w e i n t r o d u c e a s t r e a m f u n c t i o n ~k a c c o r d i n g t o

u = r - 1 ~ - r - t d ~" ~r an d v = ~x'x '


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The numerical solution of the Nav ier-Stokes equations 253wh ere u and v a r e the ve loc i ty com pon en t s in x - and r -d ir ec t ions . Ne x t we def ine nond i -mens iona l va r i ab les in the fo l lowing way

x + i r = v ( ~ / + i , , /r / ) 2 / U o , (2.4)~0 = 2vZ~F Uo .

N o w t h e p a r a b o l o i d is gi ve n b yr~ = R e = W o a / v (2.5)

w h e r e R e is th e R e y n o l d s n u m b e r b a s e d u p o n t h e c h a r a c te r i st ic l e n g t h a .In non-v i scou s p l ane f low, the vor t i c i ty co r emains co ns tan t a long s t r eaml ines . The no n-d imen s iona l qu an t i ty .w hich has th i s p roper ty in ax i symm et r i c f low i s g iven by

2v2 c0- - - . ( 2 . 6 )G = U o 3 r

W i t h t h e s e n e w v a r ia b l e s t h e N a v i e r - S t o k e s e q u a t i o n s ( 2.2 ) b e c o m e_-':'~. _--r_ _2 G d G d 2 G 2 d G = d G d ' f d G d ~ ( 2 . 7 a )dr/ de d r/ dr/d

d 2 ~ d 2 ~~ + r / ~ = - ~r /(~ + r /)G. (2 .7b)W h e n R e i s ve ry smaU, G has an a lm os t s ingu la r beh av iou r near the nose o f the parab o lo id .

T h i s c a n b e s e e n a s f o ll o w s :F r o m s y m m e t r y w e h a v e ~ ( 0 , r / ) - - V ( ~ , R e ) = 0 and the no- s l ip cond i t ion y ie lds

dW/dr/(~, R e ) = 0 . There fo re a Tay lo r se ri es fo r t he s t r eam func t ion near th e nose o f the pa ra -b o l o i d m u s t b e g i n w i t h V ~ . 4 ~ ( r / - R e ) 2 , whe re .4 i s some co ns tan t wh ich i s un l ike ly to b ez e ro . F r o m e q u a t i o n ( 2 .7 b ) w e f in d t h a t t h e c o r r e s p o n d i n g t e r m f o r G is g i v en b y G ~

- 2 .4 ( ~ + R e ) - 1 , and th i s is a s ingu la r t e rm when R e - - 0. N o te the an a logy w i th the vor t i c i tyin plan e f low (see [1] an d [2]) .F u r t h e r m o r e , w h e n ~ = 0 i n t h e r ig h t h a n d s i d e o f ( 2.6 ) b o t h n u m e r a t o r a n d d e n o m i n a t o r

a r e ze ro , w h ich l eads to an undef ined v a lue fo r G . W e there fo re in t roduce a new var i ab leL = -~ (~ + r /)G. (2 .8)

The f a c to r (~ + r/) i s u sed to r em ove the s ingu la r i ty and the f ac to r ~ makes L van i sh on ~ - - 0.E q u a t i o n ( 2.7 ) c a n n o w b e e x p r e s s e d a s

~ + r / d r / 2 ~ + r / ~ - N = d ~ & / 0r/ d~ + ~ + r / [ 0 ~ - 2 + ~ ~ (2.9a)d2~F d2~F - ~ + r / ~ = e L ( 2 .9 b )

w i t h b o u n d a r y c o n d i t i o n s~ = 0 : ~F = L = 0 ( 2.1 0a )= R e : ~ F a ~ F= d - '~ " = 0 ( 2 . ] 0 b )


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254 A .E .P . V~LD~AN

an d/o r r / - - . oo : ~u ~ ? . f (r /) + ~-~--~fo07), L , ,- ~f"(r / ) + ~ " ~ f o (~/ ) . (2 .10c)Th e c o n d i t i o n s (2 .1 0 c ) wi ll b e d i scu ssed in t h e n ex t sec t i o n .

T h e q u a n t i ti e s ~P a n d L a r e u n b o u n d e d a t i n f in i ty . W e t h e r e f o r e i n t r o d u c e t h e d e p a r t u r e so f t h e s o l u t i o n f r o m t h e b e h a v i o u r a t i n f in i t y

~ = ~ - ~ f ( r / ) - ~ ( ~ + ~ ? ) - l f o ( r / ) (2 .11a)L1 = L - ~f~07) - ~(~ + r/)- lfo '( r/) + 2~(~ + ~7)-2fo'(r/). (2.1 lb )

Th e l a s t t e rm in (2.1 l b ) i s an ex t ra t e rm , i n se r t ed t o k ee p eq u a t i o n (2 .9 b ) in i t s s imp les t fo rmaf t e r su b s t i t u t i o n . Th i s su b s t i t u t i o n ch an g es eq u a t i o n s (2 .9 ) i n to

[ 0 2 L 1 + C 9 + , I + - + c 3 - - - -\ ~42 \ ~n2 + ,7 \ o an= ~ '~ + \ "~ '~ + \ d r/ + \ d e + (2 .1 2 )

+ + I + " /+ ~ + , i l o t ~ I ~o ,1O 2 ~ t 0 2 ~ t

+ . 7 p - - - - . L . -T h e b o u n d a r y c o n d i t io n s a r e n o w c o m p l et e ly h o m o g e n e o u s

~ - ~ 0 : W I - - L 1 - - 0 ,17 = R e : ~ F 1 = = 0 ,

drl (2.13)4 ~ : q J l - ' 0 , L 1 ~ 0r /~ .o o : W1 ~ 0 , L t ~ 0 (ex p o n en t i a l l y )

T o d e r iv e t h es e c o n d i t i o n s u s e h a s b e e n m a d e o f r e s u lt s f r o m t h e n e x t s e c t io n .T h a t t h e v o r t i c it y d ec a y s e x p o n e n t i a l ly a s r / ~ oo h a s b e e n p r o v e d f o r f l o w p a s t f i n it e

b o d i e s b y C l a r k [ 1 2] . F o r f l o w p a s t i n f in i te b o d i es n o f u ll m a t h e m a t i c a l p r o o f is a v a i l a b leu n t i l n o w .Th e q u an t i t i e s C~ , i - 1 , . . . , 7 ap p e a r in g i n eq u a t i o n s (2.12 ) can b e ea s i l y ex p ressed int h e s u b t r a c t e d f u n c t i o n s f a n d f o .

3 . T H E A S Y M P T O T I C B E H A V I O U R F A R F R O M T H E N O S EMi l l e r [9] h as g iv er i a v e ry d e t a i led i n v es t i g a t i o n o n t h e b eh av io u r o f t h e so lu t i o n fo r l a rg ev a lu es o f t h e co o r d in a t e s . In t h i s sec t i o n we wi ll p re sen t h i s m o s t im p o r t an t re su l ts .

T h e m e t h o d u s e d b y M i ll e r t o f i n d t h e b e h a v i o u r a t in f i n it y is b a s e d u p o n t h e m a t c h i n go f two a sy m p to t i c ex p an s io n s . On e i s v a l i d fa r f ro m th e p a rab o lo Jd su r face (~ , 17 ~ co )wh e re we h av e p o t en t i a l f l o w. Th e o th e r o n e i s v a li d n ea r t h e p a ra b o lo id su r face (~ --* o o,r / f i n i t e ) wh e re t h e fu l l Nav i e r -S to k es eq u a t i o n s mu s t b e u sed . Th i s l a s t r eg io n we ca l lb o u n d a r y l ay e r.


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The num erical solution of the Navier-Stokes equations " 255F i r s t c o n s i d e r th e p o t e n t i a l r e g i o n . I n t h e m e r i d i a n p l a n e w e d e f in e n o n d i m e n s i o n a l p o l a r

c o o r d i n a t e s ( p , ~ ) b yU o x / v = p c o s d p, U o r / v = p s i n ~ .

I n t h e p o t e n t i a l r e g i o n L is e x p o n e n t i a L l y s m a l l, h e n c e e q u a t i o n ( 2 . 9 b ) r e d u c e s t od2~p d2~p

W r i t te n i n p o l a r c o o r d i n a t e s w e h a v ep 2 ~ 2 ~ ~ 2~ j~ p 2 + ( 1 - / z 2 ) ~ --- 0 ( 3 .1 )

w h e r e /z = c o s ~ . T h e r e l a t i o n b e t w e e n ( p , / ~ ) a n d ( , r/) is g i v e n b y p - - + f f a n d /~ =( - , 7 ) / ( + , 7 ) .T o s t a r t w i t h , w e l e t th e a s y m p t o t i c s e ri e s o f ~u f o r l a r g e v a l u e s o f p c o n s i s t o f i n t e g e r

p o w e r s o f p . T o g e t h e r w i t h t h e c o n d i t i o n = 0 a s ~ = 0 (~b = n ) e q u a t i o n ( 3 . 1 ) h a s e x a c t l yo n e s o l u t i o n f o r e a c h p o w e r o f p . T h e b e g i n n i n g o f t h e s e ri e s t h u s b e c o m e s

---- A 2 p 2 ( l - - /~ 2 ) + A l p (1 + / t ) + A (I + / a ) + A - I \ T / + " " (3 .2 )T h e c o e f fi c ie n t s A ~ m u s t b e d e t e r m i n e d b y m a t c h i n g . T h e f i rs t t e r m m u s t m a t c h t h e o n c o m i n gf l o w --- ~ /, w h i c h r e s u l t s i n A 2 - - . E q u a t i o n ( 3 .2 ) r e - e x p a n d e d f o r l a r g e a n d f i n i t e r /g i v e s

1- - ~F/ 2 A 1 ~ + 2 A o + ( - 2 A o ~ / ) ~ + " ' " ( 3 . 3 )I n t h e b o u n d a r y l a y e r r e g i o n ( r/ f in i t e ) w e a s s u m e t h e a s y m p t o t i c e x p a n s i o n t o b e

I= O f (r /) + f o ( r / ) + ~ f ~ ( r /) + . . . ( 3 .4 )S u b s t i t u t i n g t h i s in e q u a t i o n ( 2 .9 ) a n d e v a l u a t i n g t e r m s w i t h e q u a l p o w e r s o f ~ w e d e r i v ee q u a ti o n s f o r t h e u n k n o w n f u n c t i o n s f a n d f i , i = 0 , 1 . . . .

T h e f ir s t e q u a t i o n i st l f " + 2 f '~ + f ' f " + f f ~ ' = 0

w h i c h c a n b e i n t e g r a t e d o n c e , o b t a i n i n g~ /f~ ' + f " + i f " = 0 . ( 3 .5 a )

T h e i n t e g r a t i o n c o n s t a n t v a n i s h e s b e c a u s e f o r l a r g e ~ / a l l s e c o n d a n d h i g h e r d e r i v a t i v e s o f fm u s t b e e x p o n e n t i a l ly s m a l l b e c a u s e o f th e e x p o n e n t i a l d e c a y o f v o r t i c it y . B o u n d a r y c o n -d i t i o n s a r e g i v e n b y

f ( R e ) = f ' ( R e ) = 0 a n d f ' ( ~ ) = 1 . ( 3 .5 b )T h e fi r st t w o f o l lo w f r o m ( 2 .1 0 b ) . T h e t h i r d o n e c o m e s f r o m t h e m a t c h i n g o f th e f i rs t t e r m so f (3 . 3) a n d ( 3. 4) . E q u a t i o n ( 3 .5 ) is u s e d t o c a l c u l a t e f T h i s h a s b e e n d o n e b y m e a n s o f as i m p l e s h o o t i n g m e t h o d . S o m e n u m e r i c a l v a l u e s c o n c e r n i n g f a r e p r e s e n t e d b y V e l d m a n[ 13 ]. F o r l a r g e r / t h e b e h a v i Q u r o f f i s


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256 A .E .P . VELDMANf ~ ~ / - f l + ex p . sma l l t e rms . (3.6 )

H e r e ~ i s a c o n s t a n t w h i c h f o l lo w s f r o m t h e n u m e r i c a l c o m p u t a t i o n s . T h e s e c o n d t e r m o f(3.3) can n o w b e ma tc h ed w h ich re su l ts i n A1 = -f t .

T h e e q u a t i o n f o r f o c a n b e w r i t te n a sr / f o " + ( f + 2 ) f o " + 2 f 7 o " + f i f o ' = r lf 'f " - i f " . (3.7a)

T h e b o u n d a r y c o n d i t io n s a r ef o ( R e ) = f o ' ( R e ) = 0 an d fo ' (Oo) = 0 . (3 .7b)

A g a i n t h e f ir s t t w o f o l l o w f r o m ( 2 .1 0 b) a n d t h e t h i r d o n e f r o m m a t c h i n g w i t h (3 .3 ). A l s ot h e s e c o n d a n d h i g h e r d e r i va t iv e s o f f o m u s t d e c a y e x p o n e n t i a ll y . S in c e f o ' ( O o ) = 0 , f o a p -proaches a constan t when r / - - , oo . This constan t de termines Ao in (3 .3) , v iz . Ao = fo(OO).To ca l cu l a t e fo n u m er i ca l l y eq u a t i o n (3 .7 ) can b e i n t eg r a t ed twice re su l t i n g in

~lfo" + f ro '= (~ " _ f 2 + g~ f , + C l rl + C2). (3.8)C 1 a n d C 2 a re i n t e g r a t i o n c o n s t a n t s w h i c h c a n b e e v a l u a te d b y n o t i n g t h a t t h e l e ft h a n d s id eo f eq u a t i o n (3 .8 ) t en d s t o ze ro a s ~ ~ o o , a n d s o m u s t t h e r i g h t h a n d s i de . W h e n w e u s e t h eb eh a v io u r o f f g iv en b y (3 .6) we o b t a in b y p u t t i n g t h e r i g h t h a n d s i d e o f (3 .8 ) eq u a l t o ze ro ,C 1 = - ~ a n d C 2 = # ' . E q u a t i o n ( 3.8 ) c a n n o w b e i n t e g r a t e d d i r e c tl y s i nc e t w o b o u n d a r yc o n d i ti o n s o n t h e i n n er b o u n d a r y a r e k n o w n . S o m e i m p o r t a n t v a lu e s c o n c e r n i n g f o a r e

f o ' (R e ) = {#ZRe -1 - ~ + Re f " (Re )} a n d f o ~ ' ( R e ) = - ~ 2 R e - 2 . (3.9)T h e v a l u e s o f f o ( o o ) can a l so b e fo u n d i n Ve ld man [1 3 ] .

A p a r t f r o m t h e t e r m s w i t h i n t e g r a l p o w e r s o f p e q u a t i o n ( 3 . 1 ) a l s o h a s s o l u t i o n s w i t hn o n - i n t e g e r p o w e r s o f p . M i ll e r h a s p o i n t e d o u t t h a t t h e l e a d i n g t e r m o f s u c h a , s o l u t io n o fo rd e r p -k , wh en ex p an d ed fo r l a rg e ~ , h as t h e fo rm A ~ { -~ . Fo r a co r re sp o n d in g t e rm ~- kf~(r/)i n (3 .4 ) t h i s mea n s t h a t an o u t e r b o u n d a ry c o n d i t i o n fk ' (o o ) = 0 ex i s t s. Th e o th e r co n d i t i o n sa r e a s a l w a y s f ~ ( R e ) = f k ' ( R e ) = 0 a n d f k ' ( o o ) = 0 e x p . T h e f u n c t i o n f ~ (g ) s a ti sf ie s t h eh o m o g e n e o u s e q u a t io n

r / fk " + ( f + 2 ) fk ~' + ( k + 2 ) f ~ " + f ' f k ' - k f f' fk = 0. (3.10)I t a p p e a r s t h a t f o r s o m e v a l u e s o f k th i s h o m o g e n e o u s e q u a t i o n t o g e t h e r w i t h t h e h o m o -g en eo u s b o u n d a r y co n d i t i o n s h as a n o n - t r i v i a l so lu t i o n . Th e se v a lu es o f k a re ca l l ed e ig en -v a lu es . Ca l cu l a t i o n o f t h e e ig en v a lu es sh o w s t h a t t h e sma l l e s t v a lu e k l l ie s b e tween 0 an d 1fo r a l l R e . Th er e fo re t h i s c rea t e s i n (3.4 ) a t e rm o f o rd e r ~ -k , w h ich co m es d i rec t l y a f t e r

t h e t e rm fo (g ) . F o r d e t a i l s o n t h e ca l cu l a t i o n o f t h e fi r s t fo u r e i g en v a lu es see Ve ld ma n [13 ].In Tab l e 1 we p re sen t k x fo r sev e ral v a lu es o f R e . In t h e ap p en d ix we p ro v e t h a t a l l e i g en -v a lu es t en d t o i n t eg e rs a s R e a p p r o a c h e s z e r o .

Table 1. The sm allest eigenvalueklRe kL Re k x

10-5 0.08 8 I0 0.68610 3 0"1 42 102 0"9 50I0-~ 0"295 l0 s 0.995


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The numerical solution of the Nav ier-Stokes equations 257W e a r e n o w a b l e t o d e r i v e b o u n d a r y c o n d i t io n s (2 .1 0 c) . W h e n w e w r i t e ( 3 . 2 ) in t e r m s o f

and ~ we f ind a f t e r us ing the match ing r esu l t s f o r A2 , A1 and Ao~ ( n - ~ ) + ~ - ~ / o ( O O ) . ( 3 .1 1 )

Combin ing th i s wi th (3 .4 ) we can fo rm an expans ion va l id in bo th po ten t i a l r eg ion andb o u n d a r y l a y e r w h e n w e w r i t e

~ ~f(n) + ~--~--~fo(n). (3.12)F r o m t h is e x p a n s i o n t o g e t h e r w i t h ( 2 . 9b ) b o u n d a r y c o n d i t io n s ( 2 .1 0 c ) a r e f o u n d .

W e h a v e a l s o s e e n th a t t h e m o s t i m p o r t a n t t e r m a f t e r t h e t w o t e r m s i n (3 .1 2 ) i s o f t h eorder p -h . Th i s then i s t he o rder o f t he var i ab les VI an d L I defined in (2.11). S incethe smal l es t e igenva lue i s pos i tive we conc lude tha t f o r l a rge va lues o f the coo rd ina tes th ef u n c t i o n s V 1 a n d L ~ t e n d t o z e r o , t h u s y i e l di n g h o m o g e n e o u s b o u n d a r y c o n d i t i o n s a s w a ss t a t ed in sec t ion 2 .

4. B E H A V I O U R O F T H E S O L U T I O N F O R R e - - . o v A N D Re-.*OI f R e ~ o r , i . e . v - -, 0 t r ansfo rm at ion (2.4) loses sense and i t shou ld be r ep laced byx + i r f f i a ( ~ / A + i ~ # ) 2, ~ = 2 ~ U o ~ b a n d G - . ~ R e - 2 G b . (4.1)

T h e p a r a b o l o i d c a n n o w b e e x p r e ss e d b y / ~ = 1 . I n t h e s e n e w v a r i a b le s th e N a v i e r - S t o k e se q u a t i o n s a r e g i v e n b y

2 ~2Gb (~Gb (~2Gb aGb = /3 G b ~Gb.. ~t~ + 2 - ~ + ~ - ~ - ~ + 2 - ~ - , ~ ~ ~ . a ~ !( 4 . 2 )

+ ~ ~ = - ~#(~ + ~)G~.T o d e s c r ib e t h e f l o w p a t te r n p r o p e r l y f o r l a r g e R e w e m u s t u s e a s t r e tc h i n g t r a n s f o r m a t i o n

! ~ - 1 = R e - l / 2 1 z b , g 'b = R e - l / 2 u / b . (4.3)Sub s t i tu t ion o f the t r ansfo rma t ion (4.3) in to equ a t ions (4 .2 ) y i e lds in the l imi t R e ~ ~ afour th o rde r d i f f e ren t i a l equa t ion fo r the s t r eam func t ion ~Fb which can b e in t eg ra t ed on cewi th r espec t t o /~b" The r esu l t t hen becomes

I a ~ ' ~ ~' ~% a~l ){~ -~ (4.4)~ . ( ~ . + 1 ) [ ~ - ~ - ~ + #b . - - - - - ~ ~#~ ~ s ! + ( ~ ' + \ ~# ~ 1 =whe re the f igh t ha nd s ide has been d e te rmined by match ing ~Fb fo r #b ~ oo wi th the ou te rpo ten t i a l f low g iven by ).p} . The b ou nd ary con d i t ions a r e

~ I ' d 0 , ~ b ) = 0 ,

~bO-, O) = ~ (~., O) = O, (4.5)C~, ()., /~}) --* ;~ for /~b --, 00.~/~b


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258 A .E .P . VELDMANFo r l a rg e v a lu es o f 2 t h e so lu t i o n o f eq u a t i o n (4 .4 ) can b e w r i t ten a s ~Pb = 2 F(/~b ) wh ereF sat isfies

F " + F F " = 0 , F(0) = F ' (0 ) = 0 , f ' (o o ) = 1 . (4 .6)T h u s w e s e e t h a t F i s t h e w e l l -k n o w n B l a si u s f u n c t i o n .

Fo r sma l l v a lu es o f 2 t h e so lu t i o n can b e wr i t t en a s ~Pb = 2 G(# b) , wi th G sa t i s fy in gG " + G G " - ( G ' z - 1) = 0, G(0) = G'(0 ) = 0, G'(o o) = 1. (4.7)

T h i s i s a s p e c ia l c as e o f t h e F a l k n e r - S k a n e q u a t i o n .Fo r a l l v a lu es o f 2 eq u a t i o n (4 .4) h as b een so lv ed b y Smi th an d C lu t t e r [3 ] an d b yDavis [4] .F r o m t h e s e s o l u ti o n s w e i n fe r t h a t t h e v a r i a b le s 2 ,/~ b a n d ~ b a r e t h e p r o p e r o n e s t o w o r k

w i t h w h e n w e d e a l w i t h la r g e v a l u e s o f R e . T h e r e l a t io n b e t w e e n t h e s e v a r i ab l e s a n d t h e o n e sin t ro d u ce d i n sec t i o n 2 can b e fo u n d f ro m (2.4 ), (4 .1 ) an d (4 .3 ) re su l t i n g i n

= R e 2 , q - R e = R e l / 2 1 t b , u t t = R e a / 2 u / b , L = R e l / 2 L b , (4.8)w he re Lb is define d by Lb = 02qJb/tg#b2 , w h i c h is t h e b o u n d a r y l a y e r f o r m o f ( 2. 9b ) .

F o r s m a l l v a l u es o f R e t h e r e i s a n e i g h b o u r h o o d o f t h e p a r a b o l o i d w h e r e w e h a v e S t o k e sf l ow. Th e v e lo c i t y i n th i s reg io n i s so sma l l t h a t w e m ay n eg l ec t th e n o n - l i n ea r t e rms i n t h eN a v i e r - S t o k e s e q u a t i o n s . I t is a p p r o p r i a t e t o i n t r o d u c e S t o k e s v a ri a b le s a c c o r d i n g t o

x + ir = a (x /2 + ia/# ) 2, ~k = 2a2U0 ~ks,H e n c e t h e N a v i e r - S t o k e s e q u a t i o n s b e c o m e

02G, 0G~ 6 ~ 2 G s dG ~2 " ~ ' + 2 " ~ + # - ~ " 2 + 2 " ~ - ~ = 0

B o u n d a r y c o n d i t i o n s a re

G = R e - 2 G s . (4.9)

O t k , , , ( 0 , ~ ) = , / , , ( L 1 ) = ~ ( L l ) = O .

(4.10)

Th e so lu t i o n o f eq u a t i o n (4 .1 0 ) is g iv en b y~ , = C 2 ( # l o g # - / a + 1 ) , (4.11)

w h e r e C h a s t o b e d e t e r m i n e d f r o m a n o u t e r b o u n d a r y c o n d i t i o n . T h is c o n d i t i o n c a n b ef o u n d b y m a t c h i n g ( 4.1 1) w i t h a n o u t e r s o l u t i o n , f o r i n s ta n c e t h e O s e e n s o l u t i o n . T h e l e a d i n gt e rm o f su ch an o u t e r s o lu t i o n i s a lway s ~ / . W h en we wr i t e (4 .1 l ) i n o u t e r v a r i ab l e s ~ , r/a n d ~ t h e m o s t i m p o r t a n t t e r m f o r s m a l l R e i s g iven by qJ ~ - C et / l og R e . T h i s m u s t m a t c hth e o u t e r f l o w, re su l t i n g i n C = - ( l o g R e ) - t . (4.12)

W e n o w o b s e r v e th e e x i s te n c e o f th r e e r e g i o n s :( i) Th e S to k es reg io n w h ere t h e S to k es ap p ro x im a t io n is v a li d . In t h i s reg io n /~ = O(1 )

w h i c h m e a n s q = O ( R e ) .( ii ) A t r a n s i t i o n r e g i o n w h e r e o n e m i g h t u se t h e O s e e n a p p r o x i m a t i o n . H e r e r / = O ( l ) .

( iii) F a r a w a y f r o m t h e s u r f a c e o f t h e p a r a b o l o i d w h e r e t h e v o r t i c it y h a s b e c o m e z e r o w eh a v e t h e p o t e n t i a l r e g i o n .


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T h e n u m e r i c a l s o l u t io n o f t h e N a v i e r - S t o k e s e q u a t i on s 2 59

l f '( i ) ~ t I/ ,l i O s e e n . ~ .~I

i S to k e s ;0. ~ f

i

0 .20 ~ " t g T-S - t - 3 -2 -1 0 1

F ig . 1 . Do wn s t r eam v e lo c i ty p ro f iles in S to k es ap p ro x im at io n , Oseen ap p ro x im at io n an dex ac t fo r R e - - - I O - s .T h e s e r e g i o n s a r e v i s i b l e i n F i g . 1 w h e r e t h e d o w n s t r e a m v e l o c i t y p r o f il e f ' ( r / ) i s d r a w n

f o r a R e y n o l d s n u m b e r o f 1 0 - s . W e s e e t h a t i n t h e S t o k e s r e g i o n w h e r e ~ / = O ( 1 0 - 5 ) t h e r ei s a n i m p o r t a n t v a r i a t i o n i n f ' ( P l ) . T h e r e f o r e w e m u s t t a k e t h is S t o k e s re g i o n i n t o a c c o u n ti f w e w a n t t o s o l v e t h e N a v i e r - S t o k e s e q u a t i o n s n u m e r i c al l y .A l s o s h o w n i n F ig . 1 a r e t h e S t o k e s a p p r o x i m a t i o n a n d t h e O s e e n a p p r o x i m a t i o n f o r th ev e l o c i t y p r o f il e . T h e S t o k e s v a l u e c a n b e o b t a i n e d b y c o m b i n a t i o n o f (3 . 4) a n d ( 4. I 1 ) w h i c hr e s u l t s i n

f s ' ( r / ) = 1 - log ~ / /log R e .T h e O s e e n v a l u e i s f o u n d m o s t s i m p l y b y l i n ea r i za t io n o f e q t t a ti o n ( 3 .5 ) a r o u n d t h e o n -c o m i n g f l o w . W e t h u s w r i t e

r t f " + ( n + 1 ) f " = O , f ( R e ) = f ' ( R e ) ffi 0 , f ' ( o ~ ) ffi 1 . ( 4 . 1 3 )F r o m t h e s o l u t i o n o f t h i s e q u a t i o n w e d er iv e

I ;o , ' ( ~ ) = 1 - E l ( t l ) / E 1 ( R e ) , w h e r e E l ( x ) - - e - i t - 1 d t .T h e t h i c k n e s s o f t h e b o u n d a r y l a y e r is i n F i g . 1 s e e n t o b e o f t h e o r d e r ~ -- O ( 1 0 ) . H e n c e

t h e b o u n d a r y l a y e r o f a t h i n p a r a b o i o i d i s m u c h t h i c k e r t h a n t h e l ~ r a b o l o i d i ts e lf . B u t t h e r ei s o n l y a l i m i t e d r e g i o n [ 7 - - o ( 1 ) ] i n t h e b o u n d a r y l a y e r w h e r e t h e f l o w d i f f er s S i g n if i c a n tl yf r o m t h e o n c o m i n g f l o w . I n th a t r e g io n ( 4 .1 3 ) m a y b e a p p r o x i m a t e d b y

t l f " ' + f " ----0 ( 4 .1 4 )


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2 6 0 A . E . P . V EL DM AN

i n which the non- l inear t e rm i s neg lec ted . There i s a lmos t no d i f f e r ence in tha t r eg ionb e t w e e n

( i) t he so lu t ion o f (4 .14), wh ich i s t he S tokes so lu t ion ,( ii ) t he so lu t ion o f (4 .13), wh ich i s t he O seen so lu t ion and

( ii i) the so lut ion of (3 .5) , whic h is the ex act solut ion .5. R E F O R M U L A T I O N O F T H E P R O B L E MIn the p reced ing sec t ion we have seen tha t t he var i ab les in t roduced in sec t ion 2 , i . e . ~ , q ,

~P and L , a r e n o t t he bes t one s to use fo r a l l va lues o f R e . F o r l a r g e R e t he var i ab les 2 , / zb ,kvb a n d L b a r e t h e p r o p e r o n e s . F o r s m a l l R e w e h a v e a S t o k e s r e g i o n w h e r e S t o k e s v a r i a b l e ss h o u l d b e u s e d a n d a t r a n s i t i o n r e g i o n w h e r e t h e v a r i a b l e s f r o m s e c t i o n 2 s e e m t o b e t h eb e s t o n e s in o r d e r t o k e e p a l l u s e d q u a n t i t ie s o f o r d e r u n i t y .

T o c o m b i n e t h e s e c a s e s w e i n t r o d u c e t h e f o l lo w i n g t r a n s f o r m a t i o t l= A 2 ~ , q - R e = A l l (5.1)

wi thA = 1 + R e 1 /2 .

F o r l a r g e R e w e se e t h a t ~ a n d f / t e n d t o t h e b o u n d a r y l a y e r v a ri a b l e s 2 a n d / l b . F o r s m a l lR e w e h a v e c h o s e n ~ to b e a p p r o x i m a t e l y e q u a l t o ~ . A l t h o u g h ~ n o w i s n o S t o k e s v a r ia b l ei t a p p e a r e d f r o m t h e n u m e r ic a l c a l c u l a ti o n s t h a t ~ w a s t h e b e s t v a r ia b l e t o u s e . I n t h eq-d i r ec t ion we mus t accep t a smal l d i f f i cu l ty , namely tha t f / i s O ( R e ) i n the S tokes r eg ionand O(1) in the t r ans i t ion r eg ion .

T h e c o r r e s p o n d i n g n e w v a r i a b le s f o r s t r e a m f u n c t i o n a n d v o r t ic i t y a r e g iv e n b yW = A a ~ a nd L = A L . (5 .2a)

Ins t ea d o f W1 and L1 def ined in (2 .11) we use~ 1 = A - a W l a nd t = A - X L I (5 .2b)

An oth er d i f fi cu l ty a r i ses f rom the in f in i t e ex ten t o f t he r eg ion o f in t e res t w hen we so lvee q u a t i o n s ( 2 .1 2 ) n u m e r i c a ll y . T o o v e r c o m e t h is d i ff ic u l ty w e f o l l o w t h e m e t h o d u s e d b yvan de Vooren and Di jks t r a [1 ] by t r ansfo rming the in f in i t e r eg ion in the (~ , 0 ) -p lane to af in i t e r ec t ang le in the ( a , Q-p lane . As f in i t e r eg ion we t ake the square

0__


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The numerical solution of the Navier-Stokes equations 261I n / / - d i r e c t io n t h e t r a n s f o r m a t i o n is ta k e n s u c h t h a t t h e b o u n d a r y l a y e r re g i o n i s t r a n s-

fo rm ed to 0 < ~ < and the po ten t i a l r eg ion to < z < 1 . Fo r R e > 1 the s imple express ion0 = D (5.5)1 - - '

g ives exce l len t r esu lt s . The con s tan t D i s eva lua ted such tha t t he edge o f the bo un dar y l ayeri s t r ansfo rm ed to ~ - - . Th i s edge i s de f ined as the va lue o f 0 w here L1 becom es O(10 -7 )com pare d to O(10 -1 ) a t t he parabo io id su r f ace . Al so a t t h i s va lue o f 0 we h ave 1 - f ' ( t / ) - -O ( 1 0 - a ) .

F o r v a l u e s o f R e _ 1 t r a n s f o rm a t i o n ( 5.5 ) c a n n o t b e u s ed . T h e a p p e a r a n c e o f t h e S t o k e sreg ion beco me s imp or tan t an d (5 .5 ) then g ives too f ew po in t s in th i s r eg ion . N ow ano thert r a n s f o r m a t i o n i s u s e d w h i c h g iv e s m o r e p o i n t s n e a r t h e p a r a b o l o i d s u r f a c e,.z2f / = Ct + C2 ~ , z 2 > 1 (5 .6)The t e rm C2 z is needed becau se d f/ /dz mu s t be unequ a l t o ze ro w hen z = 0 . The ex pon en tz2 and the con s tan t s C1 and 6"2 a r e chosen su ch tha t we have a r easona b le sp read ing o f theS tokes r eg ion and the t r ans i t ion r eg ion o ver 0 < x < . A goo d cho ice fo r m os t o f t heRe yno lds num ber s i s Ct = 200 and 6"2 = 5 R e . The va lue o f z2 vari es f rom 3 to 7 fo r R eb e t w e e n I a n d 1 0 - s .

A f t e r th e t r a n s f o r m a t i o n t o ( a , r ) a n d t h e s u b s t it u t io n o f ( 5. 2 b ) t h e N a v i e r - S t o k e sequa t ions (2 .12) a r e wr i t t en as02"~1~ O ,2 t 0 2 L I 0 L I [ # ( ~ 0 ~ 1 ) }b~2 7 7 : " ' ' 2 + W ~ " ' + A~+' + ~ ~' + Cs

+ : '

a ' l A 2 " t ' { ~ ' ~ l ' + ~ ' - ( ~ ' 7t 3 r ~ 2A 2 + ~ ) }OLI [ ~ o " 2 ~ + 0 ~ 1 : AC , _ C +

2~ _ + 2A ~ + a ~ - 4 t ~ s- C ' ~ - C ~'1 + '4 2 3 T ' ~ + , tC , { , + C , - ( 2 A '+ ~ ) C 6 } ]+T7-

0 2 9 ~ ~o '2 + 0 2 ~ , o c t , ~ o q ' l( 5 . 7 a )

(5 .7b)In the po ten t i a l r eg ion we may t ake /~1 numer ica l ly equa l t o ze ro and f rom equa t ions (2 .12)t h e r e o n l y r e m a i n s

~ o ~ ~ nT ~ + ~ '7~ -'- O o


12/21

2 6 2 A . E . P . V EL DM A NT h e b o u n d a r y c o n d i t io n s a r e g ive n b y

o ' = 0 : ~ P ~ = / ~ t = 0 ,~" = 0 : t'I'/1 = C~k~/1/C~'t"= 0 ,u = l : ~ t = L l = 0 ,z = l : " ~ / 1 = 0 ,

"~ = + A ~ : L ~ = O .

( 5 . 8 )

T h e b o u n d a r y c o n d i t i o n f o r L t a t i n f in i t y ( ~ ~ o o ) c a n b e t a k e n a t t h e e d g e o f t h e b o u n d a r ylayer . I t has be en take n a t z = + Az where Az is the m eshs ize in z -d i rec t ion .6 . T H E N U M E R I C A L P R O C E D U R E

Equ at ions (5.7) have been rep lace d by a sys tem of d i f fe rence equa t io ns based on a ne to f wh ic h t h e n e tp o in t s h a v e c o o r d in a t e sa = p h ( p = O , 1 . . . . N ; N h = I ) , (6.1)" r = q k ( q = 0 , 1 . . . , 2 M ; 2 M k = I ) .

Der iva t ives have been re p laced by ce n t ra l d i f fe rence express ions. The n eq ua t ions (5.7) canb e wr i t t e n i n t h e f o l l o win g f o r mat l /~ la , m - t + b t tL tn , m + C l l L l n , m + l + a1 2 ~ ln , , . - z + b12 ~ ln , . , + c12 ~ l . , .+ t

= d l tL ln - t .~ , + e l lL tn + l ,m + d l 2 ~ i n - l , r a + e l2~t tln+l ,m " + f t (6 .2a)a 2 2 ~ t l n , . - 1 + b 2 2 ~ l n , m + c 2 2 ~ l ln , m + l + b 2 1 L l n , . = d 2 2 ~ l n - t , m e 2 2 + ~ ' t l n + l , m (6.2b)

T h i s f o r m i s v a l i d i n th e b o u n d a r y l a y e r m = 1 . . . . M . In the po ten tia l r egion i t s implif ies toa t P i ., , ~ - 1 + b ~ P l n, + e ~ P 1 . , m + 1 = d ~ P 1 . _ 1 , m + e ~ P i .+ t , , . ( 6 . 2c )

w i t hm = M + l . . . . 2 M - I.

I n t h e s e e x p r e s s io n s / ~ i. , . - t d e n o t e s L 1 ( n h , ( m - l ) k ) a n d a n a l o g o u s l y f o r t h e o t h e r t e r m s.T h e c oe ff i ci en ts H , b H , . .. a r e f o u n d b y w r i t i n g o u t ( 5. 7) . T h e t e r m b 1 2 ~ P t . ,~ w i l l b e

i n t r o d u c e d i n t h e s eq u e l. H o w t h e n o n - l i n e a r t e r m s i n ( 5. 7 a ) r e t r e a t ed is s u g g e s t e d b y t h ef o r m i n w h i c h t h i s e q u a t i o n is w r i tt e n . F o r i n s t a n c e i n th e t e r m

+ ~ ' P ~ a ' C 5 ~ IO 1 { " * " + A 2 " ' ( 2 A ' ( + r l ) - t + / twe r e g a r d t h e e x p r e ss io n b e twe e n p a r e n th e s e s i n c lu d in g O ~ l /d a a s a c o e f f ic i e n t a n d c a l c u la t ei t f r o m p r e v io u s ly f o u n d v a lue s o f ~ t . I n ( 6 .2a ) it t h e n g ive s a c o n t r i b u t io n t o a l t a n d e ~l .T h e b o u n d a r y c o n d i t i o n s t o e q u a t io n s ( 6.2 ) a r e f o u n d d i r e c t ly f r o m ( 5.8) e x c e p t f o r t h econd i t ion 0~J~9(~ , 0 ) = 0 w hich m us t be wr i t ten in a d i f fe ren t fo rm . To do so we comb inei t wi th equa t ion (5 .7b) w hich a t the para bo la su r face reduces to z '2 (d2~P1/dz2 = L t . W eth e n c a n d e r iv e t h a t

Ll,,, o = 8~ /t"' 12k- ~/x~, 2 x"2This fo rm has a l so been used in [2 ] .


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The numerical solution of the Navier-Stokes equations 263A f t e r s u b s t i t u t io n o f t h e b o u n d a r y c o n d i t i o n s , e q u a t i o n s ( 6.2 ) h a v e b e e n s o l ve d u s i n g a

l i n e -i t e ra t i o n m e t h o d a l o n g l in e s a = c o n s t a n t . T h e c o m p u t a t i o n s a r e s t a r te d a l o n g t h e l i n ea = h a n d p r o c e e d i n t h e d i r e c t i o n o f i n c r e a s i n g g . T h e u n k n o w n s L I ( ~ , ~ ) a n d ~ 1 ( ~ , ~ )a lo n g a l i n e t r = co n s t an t , ~ = q k , q = 1 , 2 . . . . . 2 M - 1 , a r e s o l ve d s i m u l t a n e o u s l y f r o mequ at io ns (6 .2) . F or th e va lues o f ~P1 an d r~1 ap p e a r in g i n t h e r i g h t h an d s i d e o f (6.2 ) a s we l la s i n t h e co e f f i c ien t s i n t h e l e f t h an d s i d e we t ak e t h e l a s t ca l cu l a t ed v a lu es .

I n s p it e o f t h e u s e o f u n d e r r e l a x a t i o n t h e it e r a t i o n p r o c e ss d e s c ri b e d a b o v e a p p e a r e d t o b eu n s t a b l e f o r v a l u e s o f R e > 1 . T h i s i n s t a b i l it y w a s c a u s e d b y. t h e f o l l o w i n g p h e n o m e n o n :R e g a r d a v e r y s i m p l if i ed f o r m o f e q u a t i o n ( 5 .7 a )

1 t 3 2 L O 2 L - 2 ~ L _ 3 LR e c3~2 + ~ t3a 2 ~'z = 0 (6.3)Di sc re t i ze t h i s eq u a t i o n i n t h e (~ , z ) -p l an e i n t h e u su a l way w i th mesh s i zes h i n b o th d i rec t i o n s .T h e n w e g e t

( R e - 1 - h ) L n+ l ,m + ( R e - t + h ) L . - l , m + (1 - h ) L . , . , + l+ (1 + h)L. , ,~_~ - 2(1 + R e - I ) L . , . , = 0

W h e n t h i s e q u a t i o n i s s o l v e d b y p o i n t i t e r a t i o n - - J a c o b i o r G a u s s - S e i d e l - - t h e i t e r a t i o np r o c e s s m a y b e d i v e r g e n t i f t h e m a t r i x a s s o c i at e d i s n o t d i a g o n a l l y d o m i n a n t . I n f a c td i a g o n a l d o m i n a n c e i s a s u f f ic i en t c o n d i t i o n f o r c o n v e r g e n c e o f l in e a r s y s t em s . I n o u re x a m p l e d ia g o n a l d o m i n a n c e m e a n s

2 I R e -~ + 1 1 ~_ I I +h l+ l l - h i + I R e -1 + h i + I R e - ~ - h iWh en h < - 1 an d h


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264 A .E .P . VELDMANW e c a n r e w r i t e t h i s a s

c~L L a, - t) ,-(k) _ i:k ) _ L( k- ~)n+ l ,m - - / ' ~ - - l , m - - Pt , ~ t n , m0~ 2h + h (6 .6)I n t h i s f o r m w e s e e t h a t t h e u s u a l c e n t r a l d i f f e r e n c e e x p r e s s i o n f o r O L / O o i s u s e d w i t h ac o r r e c t i o n t e r m w h i c h h a s t h e f o r m o f a t i m e d e r i v a ti v e ( t h e i t e r a t i o n s t e p s c a n b e r e g a r d e das t im e s t ep s ). Wi th ex p ress io n s o f t h e f o rm (6 .6 ) fo r 0 /~1 /0 o a n d cge~i/Oa s u b s t i t u t e d i n( 5 . 7 ) t h e r e s u l t i n g e q u a t i o n s ( 6 . 2 ) w e r e m o d i f i e d s u c h t h a t t h e i t e r a t i o n p r o c e s s b e c a m es t a bl e . A r e l a x a t i o n f a c t o r o f 1 g a v e t h e f a s t e s t c o n v er g e n c e .

F o r s m a ll R e a p r o b l e m a p p e a r e d c o n c e r n i n g t h e s u b t r a c t e d t e r m s i n ( 2 .1 1 ). T h e p r e s e n c eo f a f a c t o r ( + r t ) - J i n t h e s e s u b t r a c t e d t e r m s m e a n s t h a t a t q = R e t h e s e t e r m s a r e c o n -s i d e r a b l y l a r g e r t h a n ~ a n d L . S i n c e f i n a l ly W a n d L m u s t b e c a l c u la t e d f r o m ( 2 .1 1 ) a f t e r W~a n d L t h a v e b e e n o b t a i n e d , w e a r e f a c e d w i t h l o ss o f s i g n if i c a n t f i gu r e s. T h i s c a n b e a v o i d e db y rep l ac in g (~ + g ) -~ b y (~ + q + 1 ) - t wh ich h as t h e sam e b eh av io u r fo r ~ an d rt --* o o.

Th e i n i t i a l so lu t i o n fo r t h e i t e ra t i v e p ro cess h as b een t ak e n i d en t i ca l l y ze ro , ex cep t fo rl a rg e R e w h e r e i t a p p e a r e d u s e f u l t o s t a r t t h e i t e r a t i o n s w i t h t h e s o l u t i o n o f e q u a t i o n ( 5 .7 )w i t h o u t t h e s e c o n d o r d e r ~ - d e r i v a t i v e s . T o c a l c u l a t e t h i s i n i t i a l s o l u t i o n t h e m e t h o d o fB l o t t n e r a n d F l i i g g e - L o t z [ 1 6 ] f o r t h e s o l u t io n o f t h e b o u n d a r y l a y e r e q u a t i o n s h a s b e e nu s e d .

7 . R E S U L T ST h e f l o w f ie l d h a s b e e n c o m p u t e d f o r t h e f o l l o w i n g v a l u e s o f t h e R e y n o l d s n u m b e r

R e = U o a / v :

R e = 1 0 ", n = i n t eg ra l v a lu es b e twe en - 5 an d 5T h e s e v a l u e s h a v e b e e n c h o s e n s u c h t h a t t h e y a r e r e p r e s e n t a t iv e o f th e e n t i r e r a n g e 0 < R e< o o. F o r e a c h o f t h e s e v a lu e s , c a lc u l a t i o n s h a v e b e e n p e r f o r m e d w i t h n e ts o f t h e f o r mN x 2 M (see (6 .1)). Th e use d nets izes were 8 x 32 , 16 x 64 an d 32 x 128 . Th e Pre sen te dr e s u l ts a r e t h e o n e s o b t a i n e d w i t h t h e f i n e st g r i d .

T h e i t e r a t i o n p r o c e ss w a s s t o p p e d w h e n i n t h e w h o l e f ie l d t h e c h a n g e i n t h e v a r i a b le sf f/ t a n d L x d u e t o 1 i t e r a t i o n s te p w a s n o w h e r e m o r e t h a n a g i v e n t o le r a n c e . C o n v e r g e n c ea p p e a r e d t o b e t h e b e s t f o r l a r g e a n d s m a l l R e . T h e s l o w e s t c o n v e r g e n c e o c c u r r e d a t R e =1 0. T o g i v e a n i d e a o f t h e r a t e o f c o n v e r g e n c e w e h a v e f o u n d t h a t , a t R e = 10, af ter 140i t e r a t io n s t h e v a l u e s o f ~ t i n t h e w h o l e f ie l d d i f f e re d n o w h e r e m o r e t h a n a b o u t 0 . 1 % f r o mt h e f i n a l l y - o b t a i n e d v a l u e s ( a f t e r 2 8 7 i t e r a ti o n s ) , w h e n t h e f i n e s t g ri d w a s u s e d . O n ei t e ra t i o n s t ep wi th t h e 3 2 x 1 2 8 g r i d l a s t ed 1 .3 sec o n a CDC Cy b er 7 4 -1 6 co mp u te r .7.1. P r e s s u r e

A t t h e p a r a b o l o i d s u r f a c e w e h a v e q = 0 w h e n t h e N a v i e r - S t o k e s e q u a t i o n s ( 2 . 1 ) c a n b ew r i t t e n a s1- g r a d p = - v r o t r o t qP

W h e n w e c o m b i n e t h i s w i t h ( 2. 6) w e c a n d e r i v e t h e f o l l o w i n g r e l a ti o n s v a l i d a t t h e p a r a -b o l o i d s u r f a c e

a -~ = - ~-~ ( ~ a ) , ~ ~ ( c ) ( 7 . 1 )


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The numerical solution of the Navier-Stokes equations 265where P i s the no ndim ens ional p res sure P = ( p - p ) /p U o 2 . Using (2.8) and (2.11) weobta in for the d i r ec t iona l d er iva t ive of the pres sure a long the sur face of the parab olo id theexpres s ion

= + + ~ " ~ f o " ~ f o a t r /= R e . (7.2)The p res sure P can b e foun d by in tegra t ion of equat ion (7.2) , s t a rt ing a t ~ = oo wi th a va lueof P equal to zero and in tegra t ing bac k a long the sur face .The pres sure P cons i s t s o f two pa r t s ; one par t i s genera ted b y the t e rms w i th f and foan d t h e o t h e r p a r t is t h e co n t r i b u t i o n o f th e t e r m i n v o lv i n g L , . T h es e p a r t s w e d en o t e b yP s and PN respec tive ly . Thu s w e have

R e - ~ . . . . .R e ( i f ( R e ) - f o ~ ' ( R e ) ) + ( $ ~ ~ e ) , ] o t ~ e ) ( 7.3 a)s(~, Re) = ~ + a--"'~an d

P N ( ~ , R e ) = - f ~ [ L I ( ~ ' R e ) R e a L 1 }[(~ + R e ) 2 + ~(~ + R-----~) ~1 (~ ' R e ) d~. (7.3b)F o r l a rg e R e the p res sure P appro ache s the p res sure P~ which fo l lows f rom inv i sc id theory .This inviscid pressure is given by

1 R eP ~ = 2 ~ + R e " (7.4)Fo r a d er iva t ion o f th i s resu l t s ee Veldm an [13] . He h as a l so shown th a t

l im P~ = Pi. (7.5)R es u l t s f o r P a l o n g t h e p a r ab o l o i d s u r face a r e p r e s en ted i n T ab l e s 2 a an d 2 b an d g r ap h ic -al ly in Fig. 2 . A n o r ma l i za t i o n f ac t o r R e ( 1 + R e l / 2 ) - 2 has been used to keep the r esu l t s o f

order u n i ty for a l l R e . In Table 2a w e have a l so t abu la ted th e va lues of the inv i scid pres sureP l a t R e = o o . These va lues can b e found f rom (7.4) by l e t t ing R e go to infinity. Using (5.1)we f ind fo r this pre ssure P t(~) = (4 + 1) -1.

~50.4

,,,,,;~ 0 . 3L ~ 0 . 2

T 0.10

R

R e = 1 0 ~ I I I I J I I

1 0 3 -

_, 101(1"10 j

I I I I I I I- 6 - ~ - 2 0 2 4 6

Fig. 2. Pressure at the para boloid surface.


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26 6 A. E. P. VELDMANTab le 2 a . P ressu re a t th e p arab o lo id

Re(l + ReII2)-2P(~, Re)R e = 10 R e = i0 " R e = 10 a R e = 10 ~ R e = 105 Re = oo

0 0'33 2 0"422 0"472 0.491 0"498 0"50"1531 0'275 0"363 0.408 0"426 0"431 0'4 340"3889 0"219 0.299 0"338 0'35 3 0'35 8 0"3600'78 00 0'165 0"233 0"264 0"275 0"279 0"2811"5000 O -ll 4 0.164 0.187 0.196 0'199 0"2003-0556 0"069 0"101 0.115 0"121 0.123 0'1237.5000 0-032 0.048 0"055 0"058 0"058 0'05931"5000 0.008 0"012 0"014 0"015 0"015 0"015

T a b l e 2 b . P r e s su r e a t t h e p a r a b o l o i dRe(1 + Re: /~)-2P(~, Re)

~/Re Re = 1 Re = lO - t R e = 10 -2 Re--- 10 -a R e = 10 -4 R e = 10 -50 0.264 0"227 0"192 0.152 0"118 0.0950" 01 0.260 0"225 0.190 0"152 0-117 0.0940"1 0"235 0"202 0'17 3 0.139 0"107 0"0861 0"147 0"113 0"088 0-075 0"059 0"04810 0'03 8 0"024 0"020 0"011 0'0 10 0.009I O0 0.004 0.003 0.002 0.002 0.002 0"002

T h e p r e s s u r e d r a g , i. e. t h e f o r c e in x - d i r e c t i o n o n t h e p a r a b o l o i d c a u s e d b y t h e p r e s s u r ei s g i v e n b y

$

D p = 2 ~ f o rP * s i n y d s ( 7 . 6 )

w h e r e y is t h e s l o p e o f t h e p a r a b o l o i d s u r f a c e a n d s is t h e c u r v i l i n e a r d i s t a n c e f r o m t h e t o pa l o n g t h e s u rf a c e . T h e d i m e n s i o n a l p r e s s u r e P * is d e f in e d b y P * = p - p . A l o n g t h e p a r a -b o l o i d w e c a n w r i t e , u s i n g ( 2 . 4 ), s i n y d s = d r = v ( R e / ) I /2 d / U o a n d t h e n o n - d i m e n s i o n a lp r e s s u r e d r a g c o e ff ic i e n t b e c o m e s

D , = 4 n R e ~ P d = 4 n R e j o ( P u + P s ) d CDP = ~ ~o, , ~ + R e ~ d ~ ]= 4 t t R e[ ( R e ( f " ( R e ) - f o " ( R e ) ) - f o ( R e ) } l n ( ~ ) + 2 f o " ( R e ) ~ + ~ e + f o e N

J

( 7 . 7 )T h e p r e s s u r e d r a g c o e f f ic i e n t t e n d s t o i n f i n i t y a s g r o w s w i t h o u t l i m i t . T h e r e f o r e w e i n t r o -d u c e a m o d i f i e d d r a g c o e f f i c i e n t ~ ' o , b y s u b t r a c t i n g t h e l e a d i n g t e r m w h i c h i s g i v e n b y

4 n R e { R e ( f " ( R e ) - f o " ( R e ) ) - fo " ( R e) } ln ~ . ( 7 . 8 )F o r C o , t h e r e r e m a i n s a t ~ = o o

P u d ~ ( 7 . 9 )4 7tR e ( R e ( f " ( R e) - f o " ( R e ) ) - f o " ( R e ) } l n R e - 1 + 2 f o " ( R e ) + f ow h i c h i s a f i n i t e v a l u e .


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The numerical solution of the Navier-Stokes equations 2677 .2 . Sk in fr ic t ion

When z denotes the shear stress, the local coefficient of skin friction is given byz 2v a 2R e 1/2Cs = ~ = --~o2-~nv = ( + Re)~l/2 L(~, Re) (7.10)

In this expression (cg/~n)vdenotes the normal derivative of the dimensional velocity com-ponent re. Using (2.11) and (3.7b) we obtain

2 ( , R e ) t/ z f 1 , , 1 }C y = ~ + R e f ~ ( R e ) + ~ --'+ "R ef ( R e ) + ~ L I ( ~ , R e ) . (7.11)In Tables 3a and 3b and in Fig. 3 we present values of(~ + Re ) ( ~ Re ) - X / z C along the para-boloid surface. A normalization factor Re (l + Re1/2) -~ has been used. In Fig. 4 results for

Table 3a. Skin-friction at the parabol oid. A = 1 + Re ~/2A-~(Re/E)'/2(~ + Re)Cr

Re = 10 Re---- 102 Re~- 103 Re= 10+ Re~ l0 s0 0"598 0"775 0"871 0"989 0"9210"1531 0"586 0'747 0"833 0"866 0"8770'3889 0"571 0 '714 0"787 0"816 0"8250.7800 0'552 0"673 0"734 0"757 0.7641.5000 0"528 0-626 0.673 0.690 0"6963"0556 0.499 0"574 0.607 0"619 0"6237.5000 0"466 0"519 0'541 0.548 0"55131'5000 0'435 0"472 0-487 0"491 0"493oo 0.421 0.452 0.464 0.468 0.469

Table 3b. Skin-friction at the parab oloid. A = 1 + Re ~/2,4-~(Re/O~/2(~+ Re)Ca

Re= 1 Re = 10-* Re = 10 -2 R e = 10 -3 Re---- 10 -4 Re---- 10 - s0 0.413 0.284 0.199 0.145 0.112 0.09010 -1 0.413 0.284 0.199 0.145 0.112 0.0901 0.410 0-282 0.198 0.145 0.111 0.08910 0.390 0.277 0.197 0.144 0.111 0-089102 0.365 0.272 0,196 0.144 0.111 0-0890.357 0.271 0.195 0.144 0-111 0.089

the skin friction in the nose are presented together with the values found by Davis and Werle[11]. For large R e these values should tend to the value assumed in first order boundarylayer theory. This value is given by G"(0), the function G being defined in (4.7). From thenumerical solution of equation (4.7) we have G"(0) = 0.927680. For small R e the Stokesapproximation is valid. Therefore we present in Fig. 4 also the Stokes values of the presentedquant ity, which are found with the aid of the results of section 4.The friction drag Dy can be written as

$D f = 2n | rT cos ~ ds (7.12)ao

C A F Vol. I No. 3--C


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268 A .E .P . VELDMAN

J

7

O~

0'2

.... Re.=_ 0~" iO~

_ 10

10"

i I I I 1

104I f I I " . I- 2 O 2 4 6 8

= " t o gFig. 3. Skin friction at the paraboloid s u r f ac e .

0 " 6

0 " 6

r~ 0.4

0

t i i i I I i I I i i1 s t o r d e r b o u n d a r y t a ~ e r- - p r e s e n t r e s u t t s /

o D a v i s * W e r t e /

~ ' ~ . ' t o g R eI = l i = . . . . I f i = I r- 5 -~ -3 -2 - I 0 1 2 3 ~ 5

F i g . 4 . S k i n f r i c t i o n a t t h e n o s e O f t h e p a r a b o l o i d .F o l l o w i n g th e s a m e r e a s o n i n g a s a b o v e f o r t h e p r e s s u re d r a g t h e n o n - d i m e n s i o n a l f r ic t io ndrag coef f i c ien t i s wr i t t en in the fo rm

_ D /C D , - " ~ = 2 ~ f o ( ~ R e ) l / 2 C f d ~ .W i t h t h e a i d o f (7 .1 1 ) t h is b e c o m e s

d + ]


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T h e n u m e r i c a l s o l u t i o n o f t h e N a v i e r - S t o k e s e q u a t i o n s 2 6 9W e see th a t the f r i c tion d rag coef f ic i en t a l so t ends to in f in i ty as ~ - -, oo . Ag ain we su b t r ac tthe l ead ing t e rm s , v iz .

47~ReLf"(Re)~ - {Ref "(R e) - fo"(Re) )ln ~] (7.14)an d w e thus d ef ine a mo dif ied f r ict ion d rag coe ff icient Cvs which a t ~ ffi ov has a f ini te va lue

CD~(~) = 4~Re {Ref ' (Re ) - fo "(Re)}ln Re - fo '( Re ) + ~ + Re7.3. The momentum theorem

Fo r the f low pas t a parabo l i c cy l inder Bot t a , D i jks t r a and V eldm an [2] have der ived anana ly t i c expres s ion fo r the d rag coef fi c ien t s wi th w hich these num er ica l ly ca lcu la ted coef -f ic i en ts can be co mp ared . Th i s i s a l so pos s ib le fo r f low pas t a pa rabo lo id . W e can d er ive thef o l lo w i n g e q u a t i o n

12 "~ { C v , ( ~ ) C v , ( ~ ) } 2 ~ R e f ' ( R e ) + t 2 I n ~ + M ( R e ) + 0 ( ~ t ' ) ( 7 . 1 6 )

w h e r eM(Re) = _ 2Re2f.(Re) + 2fo(OV _ _~2 _ Re 2

I 23~R e - ~ - ~ In Re - [4f~- , + {(~ _ p)2 - f f } /~ ] d ~@A d er iva t ion o f th is r esu l t is g iven by Ve ldma n [13]. When we use (3.9) we s ee tha t the l ead ingte rms in (7.16) a r e p ro du ced by the sum of (7 .8 ) and (7.14), t ha t i s they a re genera ted by th es u b t r a c t e d t er m s . T h e f ir st t e r m t o w h i c h t h e n u m e r i c a ll y c o m p u t e d q u a n t i t i e s ~ a n d L Ig ive a con t r ibu t ion i s the t e rm M(Re) . This t e rm mus t s a t i s fy the fo l lowing equa t ion

1 { C v , ( o o ) C v , ( o v ) ) M ( R e ) ( 7 . 1 7 )2 nT a b l e 4 . T h e d r a g - c o e f f ~ e n t s a n d t h e t o t a l d r a g c o m p a r e d w i t h t h e e x a c tr e su l t f r om (7 . 17 ) . B ff i - - 2~ r (1 + R e1 /= ) ,

{~'ot(*~) ~ ' ~ , ( o o ) ) / Be x t r a -R e ~ v t ( o v ) l B C ~ ,( oo )I B 3 2 1 2 8 p o l a t e d 2~rM(Re)IB

10 - s 0"108 - -0"00 2 0"106 0"115 0"122] 0 - " 0 " 1 52 - - 0 " 0 0 2 0 " 1 5 0 0" 1 6 0 0 "1 6110 -3 0 . 221 - -0"00 3 0"218 0 . 225 0"22910 -2 0 . 369 - -0"015 0"354 0"366 0"37210 = 1 0"653 - -0"056 0"597 0"608 ~ ' 6101 0"974 0"111 1"085 1"094 1-09710 0"892 1 "383 2"275 2-276 2 .277102 0"518 3 .790 4"308 4"312 4"312103 0"2 34 6"465 6"699 6"701 6"70110" 0"094 9"012 9"106 9"108 9"10810 s 0"03 5 11 "431 11 "467 11.46 9 11"469

i

I n T a b l e 4 t h e c a l c u l at e d a l u e s o f C v ~ ( o o ) , C v , ( o o ) a n d b o t h s i d e s o f e q u a t i o n ( 7 .1 7 )a r e g i v e n . A l l v a l u e s h a v e b e e n n o r m a l i z e d b y ( l + R e l / 2 ) . F o r s m a l l R e t h e r e a p p e a r e d


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2 7 0 A . E . P . V EL DM ANt o b e a l a r g e d i s c r e t i z a t i o n e r r o r i n t h e n u m e r i c a l c a l c u l a t e d q u a n t i t i e s , w h i c h c a u s e s t h er a t h er l a r ge d i s c r ep a n c y b e t w e e n t h e v a l u e s o f b o t h s id e s o f ( 7 . 1 7) . T o s h o w t h a t t h is d i s -c r e p a n c y d e c r e a s e s w h e n w e u s e a f i n e r g r i d w e a l s o p r e s e n t i n T a b l e 4 t h e v a l u e s o f t h e l e f th a n d s i d e o f ( 7 .1 7 ) o b t a i n e d w i t h a R i c h a r d s o n e x t r a p o l a t i o n f r o m t h e t w o f i n e st g ri d sb a s e d o n a d i s c r e t i z a t i o n e r r o r o f O ( h 2 ) . T h e e x t r e m e l y g o o d a g r e e m e n t b e t w e e n b o t h s i d e so f e q u a t i o n ( 7 .1 7 ) f o r l a rg e R e y n o l d s n u m b e r s i s c a u s e d b y t h e f a c t th a t n o t o n l y t h e p r es -s u r e i s a p p r o a c h e d b y t h e c o n t r i b u t i o n o f t h e s u b t r a c t e d t e r m s a s R e - - , o o ( s e e e q u a t i o n( 7 . 5 ) ) b u t a l s o t h e m o d i f i e d p r e s s u r e d r a g C o p . F o r d e t a i l s s e e V e l d m a n [ 1 3 ] .A c k n o w l e d g e m e n t s - - T h e a u t h o r w i s h e s to e x p re s s h i s g r a t i t u d e t o P ro f . D r . I r . A . I . v a n d e V o o r e n f o rh i s s ti m u l a t i n g h e l p a n d e n c o u r a g e m e n t . H e a l s o w i s h e s t o t h a n k D r s . D . D i j k s t r a a n d D r s . E . F . F . B o t t af o r v a l u a b l e a d v i c e . T h i s w o r k w a s s u p p o r t e d b y t h e N e t h e r l a n d s o r g a n i z a t i o n f o r t h e a d v a n c e m e n t o fp u r e r e s e a r c h ( Z . W . O . ) .

A P P E N D I XI n S e c t i o n 4 w e h a v e s e e n t h a t f o r s m a l l R e t h e O s c e n a p p r o x i m a t i o n g i v e s a g o o d d e s c r i p t io n o f t h e f l o wp a t t e r n . T h e r e f o r e i t m a y b e e x p e c t e d th a t t h e O s c c n a p p r o x i m a t i o n a l s o g i v e s g o o d r e s e t s f o r t h e e i g e n -v a l u e s m e n t i o n e d i n S e c t i o n 3 .T h e e i g e n v a l u c e q u a t i o n ( 3.1 0) c a n b e i n t e ~ a t e d o n c e

~ TA " + ( f + 1 ) f , " + ( k + 1 ) f ' f , ' - - k f ' f k = 0, (A.1)w h e r e t h e i n t e g r a t i o n c o n s t a n t v a n i s h e s b y r e a s o n o f t h e b o u n d a r y c o n d i t i o n s f o r 77 ~ o o. F o r c o n v e n i e n c ew e d o n o t u s e t h e o r i g i n a l O ~ e n a p p r o x i m a t i o n , i .e . a li n e a r i z a t i o n a r o u n d t h e o n c o m i n g f lo w , b u t w e l in - a r i z e a r o u n d t h e p o t e n t i a l f l o w p a s t t h e p a r a b o l o i d g i v e n b y t F = ~ 07 - R e ) . T h u s w e s u b s ti t u te i n e q u a -t io n (A . I ) f = r / - R e a n d w e o b t a i n

r l f k ' + ( ~ -- R e + 1)) ~* + ( k + l ) f d = = 0 .I n t h is e q u a t i o n w e s u b s t it u t e r / = - x a n d f d 0 7 ) = g ( x ), w h i c h re s u l ts in

x g " + (1 - R e - x )g" - ( k + 1)g = 0 , (A.2 )w i t h b o u n d a r y c o n d i t i o n s g ( - R e ) = 0 , g ( - o o) = g ' ( - o e ) = 0 e xp .E q u a t i o n ( A . 2 ) i s K u m m e r ' s e q u a t i o n a n d t h e fu l l s o l u t i o n i n t h e c o m p l e x z - p l a n e c a n b e w r i tt e n a s

g - - - - C t t F t ( k + l ; 1 - - R e ; z ) + C z z ~ ' t F t ( k + R e + l ; l + R e ; z )w h e r e C a a n d C 2 a r e c o n s t a n t s , w h i c h c a n b e e v a l u a t e d b y i m p o s i n g t h e b o u n d a r y c o n d i t i o n s . F i r s t w ee x a m i n e t h e b e h a v i o u r o f g as z ~ oo, arg z - -- r r. This b eh av iou r i s g iv en by (see [17] , eq ua t ion 13.5.1)

Ie ' ' ( ' ' z -~ ' + " e ~ + " " 1g ~ c , r ( 1 - R e ) ~ ~ - - ' B A -~ p - ~ - ~ T i B )I e l Z ( k + l + I c e ) z - ( a + t + l ( e ) e Z z , }+ C 2 z a ' l " ( 1 - - R e ) t F ( - - k ) A + F ( k + l + X e ) B _ ( A .3 )

w h e r eA - - - - ~ l ( k + l ) " ( l + k + R e ) " ( z ) n =N )n - o n ! - - + O ( [ z l

a n dB = ~ 1 ( - - k ) . ( - - k - - R e)nn . o n ! z - n + O ( I z l - u ) .

T h e e x p o n e n t i a l d e c a y o f g n o w i m p l i e s th e c a n c e l l i n g o f t h e f i rs t t e r m s b e t w e e n t h e p a r e ia t h es e s i n ( A . 3) .T h i s y i e l ds F ( 1 - R e ) F ( 1 + R e )+ C2 e lxa" - - = 0 . ( A . 4 )C t F ( - - k - R e ) F ( - - k )

N e x t w e d e m a n d t h a t g ( - - R e ) = O. S i n c e R e i s s m a l l w e c a n d e v e l o p g ( - - R e ) i n a pow er s e r i e s i n R eg ( - - R e ) ~ C x 1 - - 1 _ R - - - - ' ~ R e + O ( R e :) + C 2 ( - - R e ) ~" 1 l + R e R e + O ( R e 2 ) " "


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The numerical solution of the Navier-Stokcs equations 271Further we have ( - - R e ) a " = e~'a'(1 + R e In R e + O ( R e 2 In2 R e) ) . When we substitute this in the expansionfor g ( - R e ) and s et g ( - R e ) = 0 we obtain

C1 + C, e~a'(1 + R e In R e ) = 0. (A.5)The two equations for C1 and C2, (A.4) and (A.5), must have a nontrivial solution hence we must set itsdeterminant equal to zero

l~(--k)Y(l - ReXI + R e In R e ) = r ( - - R e - - k)I~(l + R e ) . (A.6)Since R e is small we can develop Y(1 - R e ) , 1"(1 + R e ) and Y ( - R e - - k ) in Taylor series whence equation(A.6) becomes r ' ( - -k)F(-k) = - In R e + 2F'(l). (A.7)For the left hand side of (A.7)a series expansion is available ([17], equation 6.3.16) and r' (1) = - 7 ' where Yis Euler's constant, so we can transform (A.7) to

~2 - 1 - k - - - I n R e - - ~ . ( A . 8 ). - o ( n + l ) ( n - - k )

The error in (A.8) is of the order R e In2 R e.As Re approaches zero, In Re grows without limit thus we conclude that -- k lies near a pole of F ( z ) .Suppose k ~ m, m integer, then the main contr ibution to the sum in (A.8) is given by the term with n -- m.Setting this term equal to the right hand side of (A.8) we finally obtaink - - m - 1/ln R e , m = 0 , 1 , 2 . . . . ( A . 9 )

with an error of the order (In R e ) - 2 . T h u s we see that the eigenvalues tend to integers as R e tends to zero.This result is completely in agreement with the numerically calculated eigenvalues by Veldman [13].R E F E R E N C E S

1. van de Voorcn, A. I., and Dijkstra, D., The Navier-Stokes solution for laminar flow past a semi-infiniteflat plate, ,7 . Eng. M ath . 4, 9-27 (1970).2. Botta, E. F. F. , Dijkstra, D. , and Veldman, A. E. P., The numerical solution of the Navier-Stokes equa-tions for laminar , incompressible flow past a parabolic cylinder, J . E ng . Math . 6, 63--81 (1972).3. Smith, A. M. O., and Clutter, D. W., Solution of the incompressible laminar boundary layer equations,Douglas Aircraft Corp. Engng. Papers 1525 (1963).4. Davis, R. T., Boundary layers on parabolas and paraboloids by methods of local truncation, n t . J . N o n -L i n e a r M e c h . 5 , 625-632 (1970).5. Mather, D. J., The motion of viscous liquid past a paraboloid, Quar t . J . Mech . A pp l . M ath . 14, 423--429( 1961 ) .6. Lee, L. L., Boundary layer over a thin needle, P hys . F lu ids 10, 820--822 (1967).7. C~beei,T., Na, T. Y., and Mosinskis, G., Laminar boundary layers on slender paraboioids, A I A A J . 7,1372-1374 (1969).8. Miller, D. R., The boundary layer on a paraboloid of revolution, Proc. Cambridge phi los . Soc . 65,285--299 (1969).9. Miller, D. R., The downstream solution for steady viscous flow past a paraboloid, Proc. CambridgePhilos . Soc . 70, 123-133 (1971).I0. Tam, K. K., On the asymptotic solution of viscous incompressible flow past a heated paraboloid ofrevolution, S l a m J . A p p l . M a t h . 3 0 , 714-721 (1971).11. Davis, R. T., and Werle, M. J., Numerical solutions for laminar incompressible flow past a paraboloidof revolution, A I A A J . 10, 1224-1230 (1972).12. Clark, D. C., The vorticity at infinityfor solutions of the stationary Navier-Stokes equations in exteriordomains, M a t h e m a t i c s J . 2 0 , 633-654 (1971).13. Vldman, A. E. P., The numerical solution of the Navier-Stokes equat ions for laminar incompressibleflow past a paraboloid of revolution, Report TW-122, Dept. of Math,, University of Groningcn (1972).14. Grccnspan, D., Lec tur e s on the Numer ica l So lu t ion o f L inear , S ingu lar and Non l inear D i f f e ren t ia l E qua-tions, Chapter 10, Prentice-Hall, Englewood Cliffs, New Jersey (1968).15. Dennis , S. C. R., and Chang, G. Z., Numerical integration of the Navier-Stokes equations for steadytwo..din~nsional flow, Phys . F luids Suppl . H 12, 88-93 (1969).16. Blottner, F. G., and Flfigge-Lotz, I., Finite difference computation of the boundary layer with displace-ment thickness interaction, J . Mdcan ique 2, 397-423 (1963).17. Abramowitz, M., and Stegun, I. A., ed., Handbook of Mathematical Functions, U.S. Department ofCommerce, Washington D.C. (1964).

comp fluids veld man

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