thrust incrs by flow otation

8/8/2019 Thrust Incrs by Flow Otation

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F L U I D D Y N A M I C S 2 3

N O Z Z L E T H R U S T I N C R E A S E B Y F L O W R O T A T I O NI . N . N a u m o v a a nd Y u . D . S h m y g l e v s k i iI z v . A N S S S R . N [ e k h a n i k a Z h i d k o s t i i G a z a , V o l . 2 , N o . 1 , p p. 3 4 - 3 7 , 1 9 6 7Unti l recent ly, in cons truct ing opt imum supersonic axisymmetr ic noz-zles , gas motions without rotat ion about the axis have been consid-ered. This ques tion has been s tudied rather com plete ly for equi l ibr iu mgas flows in [1-6] . Avo idance of f low rotat ion is a l imita t ion whichmay red uce the noz zle thrust , other condit ions b eing equal . I t is easyto show that for zero no zzle leng th or for a length that perm its ob-taining uniform f low at the exis t that maximum thrus t is provided byunifo rm f low para l l e l t o the nozz le ax i s. I f t he l eng th l imi ta t ion doesnot mak e i t poss ible to do this, then the use of f reedom in the gasro ta t ion m ay increas e the n ozz le th rus t. Us ing a very s imple ex ampleit is shown that this possibility is rea lize d.

A x i s y m m e t r i c i s o e n e r g e t i c i s e n t r o p i c m o t i o n o f ap e r f e c t g a s i s d e s c r i b e d b y t h e s e t o f e q u a t i o n s

O r p u O r p v O v O uOx ~ Or O, Ox Or O,

p - ~ a 2 / ( u - - l ) ,

d , = r p ( ~ d r - v d x ) . (1 )H e r e x , r a r e C a r t e s i a n c o o r d i n a t e s i n t h e f l o w

m e r i d i o n a l p l a n e , r e f e r r e d t o s o m e l i n e a r d i m e n s i o nl ; u , v a r e t h e v e l o c i t y c o m p o n e n t s , r e f e r r e d t o t h ec r i t i c a l v e l o c i t y a . , a l o n g t h e x , r a x e s , r e s p e c t i v e l y ;p i s t h e g a s d e n s i t y , r e f e r r e d t o t h e c r i t i c a l d e n s i t yp . ; a i s t h e v e l o c i t y o f s o u n d , r e f e r r e d t o a . ; T i s t h ec i r c u l a t i o n , r e f e r r e d t o l a . ; ~ i s t h e s t r e a m f u n c t i o n ,r e f e r r e d t o / 2 p . a . ; ~ i s t h e g a s a d i a b a t i c e x p o n e n t .

T h e p r e s s u r e p , r e f e r r e d t o p . a . 2, a nd t h e c i r c u m -f e r e n t i a l v e l o c i t y c o m p o n e n t w , r e f e r r e d t o a . , a r ed e t e r m i n e d b y t h e e q u a t io n s

v ( , )p = - - a 2 w = - - ( 2 )~4 rT h e s e c o n d e q u a t i o n o f s y s t e m (1 ) m a y b e r e p l a c e d

b y th e e q u a t i o n0 0-~x r (p -~ pu S) -~- ~-r rPu v -~ - O . (3 )

T h e f o l lo w i n g r e l a t io n s a r e s a t i s f i e d a l o n g t h ec h a r a c t e r i s t i c s o f E q . ( 1 ):

U2 - a 2d r ~ ~ d x , b ~ ~ u 2 ~ v 2 - a 2 , (4 )u v ~ a b(a v +__ a b ) d u - }- ( - - u a -~- v b )

a 2 r2 - f - ~ v d r ) = O .x d r - 4 - a Z r Z _ y 2 r

( 5 )

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

O I g , ,~ .W i t h a v i e w t o w a r d i n c r e a s i n g t h e n o z z l e t h r u s t b y

f l o w r o t a t i o n , l e t u s c o n s i d e r a v e r y s i m p l e p r o b l e m .A s s u m e t h a t t h e g a s f l o w ( s e e t h e f i g u r e ) e n t e r s a n o z z l ew i t h t h e g e n e r a t o r a b t h r o u g h a c y l i n d r i c a l p i p e w i t ht h e g e n e r a t o r s a . F u r t h e r , a s s u m e t h a t t h e n o z z l el e n g t h X = x b - x a i s s p e c i f i e d . T h e g a s f l o w r a t e a n dt h e r a d i u s O a a r e a l s o g i v e n . A m o n g a l l p o s s i b l en o z z l e g e n e r a t o r s a b w e m u s t f i n d t h a t w h i c h p r o v i d e sm a x i m a l t h r u s t .

W e t a k e t h e r a d i u s O a e q u a l t o l ; t h e n r a = i . W es h a l l c o n s i d e r t h e q u a n t i t y T c o n s t a n t . I n t h i s c a s e t h eg a s f l o w i n t h e m e r i d i o n a l f l o w p l a n e w i l l t a k e p l a c eo u t s i d e s o m e l i n e t g f , b e l o w w h i c h a s t a g n a n t z o n ew i t h c o n s t a n t p r e s s u r e i s f o r m e d .

I f t h e f l o w i n t h e p i p e i s i n d e p e n d e n t o f x , t h e n t h es t r e a m i s d e f i n e d b y t h e e q u a l i t i e s

u = u0 = cons t , v = 0 . ( 6 )F o r s i m p l i c i t y w e s h a l l a s s u m e t h a t c h a r a c t e r is t i c s

( 4) a r e r e a l o v e r t h e e n t i r e f l o w r e g i o n c o n s i d e r e d .W e d e s i g n a t e t h e p r e s s u r e i n t h e s t a g n a n t r e g i o n

n e a r t h e x - a x i s a n d in t h e e x t e r n a l m e d i u m b y p~o.T h e g a s f l o w r a t e Q t h r o u g h t h e p i p e is d e t e r m i n e d b yi n t e g r a t i n g t h e e q u a t i o n f o r r f r o m (1) w i t h a c c o u n tf o r ( 6 ) ,

2r~1! ( ~ ' - ~ - ! ~ 2 \ i] ( x - - l)

uo - ; - i u ~ r d r ,( 7 )

w h e r e t h e v a l u e o f r 0 c o r r e s p o n d s t o p = P oe a n d i s d e -t e r m i n e d b y t h e f i r s t o f E q s . ( 2) . F o r g i v e n v a l u e s o fQ a n d p ~ E q s . ( 7) e n a b l e f i n d i n g u 0 a n d r 0 . T h e c h a r -a c t e r i s t i c a g o f t h e s e c o n d f a m i l y m a y b e c o n s t r u c t e du s i n g e q u a l i t i e s ( 4 ) a n d ( 6) w i t h r 0 _< r _ < 1 .

T h e t o ta l n o z z l e t h r u s t T i s d e t e r m i n e d b y t h e e q u a -t i o n

2n ( t + u02) ~ - - t ~2- 2 ~ - ~ p r d r +r e


2/3

2 4 M E K H A N I K A Z H I D K O S T I I G A Z A

Xv ~ O -~ = 0 ,3 ~ '= OA

Ib T / 2 ,~ r b T / 2 ~ r b T / 2 ~ X r b T / 2 n r b T / 2 n4 1 2 . t 6 0 . 7 7 65 ~ : 7 ~ 0 . 7 9 26 0 .8027 3.09 0.810

I2 . 3 4 0 . 7 7 8 [ 2 . 4 5 0 . 7 7 9 82 . 6 8 0 . 7 9 3 2 . 7 8 " 0 . 7 93 93 . 0 t 0 . 8 0 4 3 . t t 0 . 8 0 4 t 03.32 0.812 3.43 0.812 tt

I3.39 0.8t61 3.62 0 .8t83.67 0.822] 3.9 t 0.8233.94 0.826 4. t8 0.8274.21 0.830 4.44 0.830

Y ~ O Arb T / 2:~

3 . 7 2 0 . 8 ! 74 . 0 1 0 . 8 2 24.28 0.8264.54 0.830r b

~ - S p r d r - - ~ - ~ ( r b 2 - - r 02 ) .t ( s )T h e f i r s t t e r m o f t h e r i g h t s i d e o f t h i s e q u a l i t y i s

t h e r e a c t i v e f o r c e o f t h e s t r e a m i n t h e p ip e s a , t h es e c o n d t e r m y i e l d s t h e i n t e g r a l o f t h e p r e s s u r e f o r c e sa l o n g th e g e n e r a t o r a b , a n d t h e t h i r d t e r m i s d e t e r -m i n e d b y t h e b a c k p r e s s u r e p ,o .

L e t u s in t e g r a t e E q . (3) o v e r t h e r e g i o n a g f b a a n dc o n v e r t u s i n g G r e e n ' s f o r m u l a t o a c o n t o u r i n t e g r a l .U s i n g e q u a l i t y ( 8) , t h e c o n d i t i o n d@ = 0 o n g f, a n d r e -l a t i o n ( 4) o n t h e c h a r a c t e r i s t i c o f t h e f i r s t f a m i l y , w eo b t a i n

% [ s i n l x - t- e o s O .]T_=2~ I q ( ~ t , r ) z ( ~ , r ) L s i n ( ~ - 4 - ~ ) r d r +A - ( r / 2 - - r b 2 ) , I~ = arc s in ] /u2 @__~,

[ l "a r c t g - - , a =u c o s 2 ~ A 'i ~+__A

h t - - c ~ ~ u - tz = 2 . .. .. ~ - - - - - J o ~ ) '

h _ - - . (9 ) r 2S i m i l a r l y , t h e f i r s t o f E q s . (1 ) w i t h a c c o u n t f o r

(7 ) a n d ( 4) o n t h e c h a r a c t e r i s t i c o f t h e f i r s t f a m i l yy i e l d s

r bQ _ ~ p q s i n2 u " s i n - ~ - ~ ) r d r . ( 10)

T h e n o z z l e l e n g t h X , w i t h t h e u s e o f E q s . (4 ) o nt h e c h a r a c t e r i s t i c s a g a n d f b , m a y b e e x p r e s s e d b yt h e e q u a t i o n

[ ' / . ( x - ~ t ) ( a o z - - t ) r 2 - 4 - ( x - - t )Y z - '~ ' l~dr-{ -X = I [ -/ - t - - ( ~ ' - - t ) t t 0 z ] r 2 - - 0 r t ) y 2 )r b

- b x l - - x e - - ~ c t g ( O + ~ ) d r . ( 1 1 )r e

T h e v a r i a t i o n a l p r o b l e m f o r a g i v e n v a l u e o f T m a yn o w b e f o r m u l a t e d a s fo l l o w s . F i n d t h o s e f u n c t i o n sN r ) a n d ~ ( r) o n t h e c h a r a c t e r i s t i c f b w h i c h r e a l i z e am a x i m u m o f f u n c t i o n a l ( 9 ) f o r t h e i s o p e r i m e t r i e c o n -d i t i o n s ( 1 0 ) , ( 1 1 ) , w i t h t h e d i f f e r e n t i a l r e l a t i o n ( 5) o nt h e c h a r a c t e r i s t i c o f t h e f i r s t f a m i l y , a n d w i t h t h e i n i -t i a l c h a r a c t e r i s t i c a g d e f i n e d by E q s . ( 6) a n d ( 7) .

T h e f u n c t i o n s u ( x , r ) , v ( x , r ) , p ( x , r ) i n t h e r e g i o na g f b a o b e y E q s . ( 1) , t h e k n o w n c o n d i t i o n s o n a g , t h e

conditions d~b= 0 and p = p~ on gf, and the conditionson the characteristic lb. The nozzle generator ab isthe streamline @= @a" This problem is analogous tothe problem of constructing the Optimum nozzle gen-erator for T = 0 and may be solved by the same method [7].We shall avoid the details and present the final results.

The generator sab has a corner at the point a. Theflow in the region acfgas determined by the knowncharacteristic ag, condition (5)for the characteristicof the first family at the point a for dr -- 0, and byEqs. (i). The calculations may be made using themethod of characteristics.

The functions ~ (r) and d (r) on the characteristiccb are determined by the equalities

s in 2 ~} cos ( ~ -- ~t)z q = ~l, q --~ ~ 9 ( 12)cos ~ cos ~xw h e r e k 1 a n d ?t 2 a r e c o n s t a n t s . T h e i r v a l u e s a r e c a l -c u l a t e d u s i n g E q s . ( 12 ) a t t h e p o i n t c s o t h a t th e f u n c -t i o n s # a n d ~ a r e c o n t i n u o u s a t t h i s p o i n t . T h e f u n c t i o nx ( r ) o n c b is c a l c u l a t e d b y i n t e g r a t i n g E q . (4 ) f o r t h ec h a r a c t e r i s t i c o f t h e f i r s t f a m i l y w i t h t h e b o u n d a r yc o n d i t i o n x ( r c ) = x c .

T h e c o o r d i n a t e s x c a n d r c a n d t h e v a l u e o f r b a r ed e t e r m i n e d f r o m t h e c o n d i t i o n o f s a t i s f a c t i o n o f e q u a l -i t i e s ( 1 0 ) a n d ( 11 ) f o r t h e g i v e n Q a n d X a n d t h e e q u a t i o n

[ s i n ~ s i n ~ c o s ~}qz (, ~ c o s t ~ / , p ~

f o r x = x b , w h i c h i s a g e n e r a l i z a t i o n o f t h e B u s e m a n nc o n d i t i o n [1 ] t o th e c a s e o f r o t a t i n g f l o w s .

T h i s s o l u t i o n d o e s n o t e x i s t f o r a l l p o s s i b l e c o m -b i n a t i o n s o f t h e i n i t i a l v a l u e s o f > ~ , Q , X , a n d T . F o r ac o m p l e t e a n a l y s i s w e w o u l d h a v e to d e t e r m i n e t h er e g i o n o f t h e s o l u t i o n e x i s t e n c e a n d c o n s t r u c t o u t s i d et h i s r e g i o n s o l u t i o n s , p e r h a p s , o f t h e t y p e fo u n d i n[ 4] . T h i s a n a l y s i s i s n o t m a d e h e r e . I n t h e f o l l o w i n gw e p r e s e n t e x a m p l e s o f c a l c u l a t i o n s b a s e d o n t h i ss o l u t i o n . T h e c a l c u l a t i o n p r o c e d u r e c o r r e s p o n d s t ot h e c a l c u l a t i o n o f [ 7] f o r T = 0.

T h e m e t h o d o f [ 1 - 4 , 6 , 7 ] i s no t a p p l i c a b l e f o r t h e c o m p l e t ea n a l y s i s o f t h e e f f e c t o f t h e f u n c t i o n y ( r o n t h e n o z z l e t h r u s t . I norder to cons truct a no zzle with maxim um thrus t, the retat ion 7(rmust be determined on the bas is of the solut ion of the var iat ionalproblem in which 7(r is a f ree funct ion. In this case i t i s necessaryto us e the method o f [ 5 ] . T he der iva t ion o f the necess ary ex t r emumcondi t ions for f ixed y(~) reduces to repet i t io n of [5] . The var ia t ionof 7(r is also accom plished w ithout dif f icul ty. The correspondingex t r emal i ty condition for y is not presented here.

I f the c i rcul at ion y is taken cons tant through th e ent i re f low,bu t i ts magn i tude i s found f r om the ex t t emu m co nd i t ion , t hen theex t r em al i ty cond i t i on i s s impl i f ied , bu t r emains d i f f i cu l t to v i s ua l -ize, and for the ca lculat ion s us ing this condi t ion we must carry outadd i t i ona l t ed ious ca l cu la t ions o f t he L agrange mu l t ip l i e rs i n t her eg ion ag]c a . A t the sam e t im e, for ~ '(@) = cons t the extrem um with


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F L U I D D Y N A M I C S 2 5r e s p e ct t o y m a y b e f o u nd n u m e r i c a l l y b y a m u c h s i m p l e r m e t h o d .I f we m a ke c a lc u la t ions fo r va r ious va ine s o f y in a c c orda nc e w i ththe s o lu t ion ind ic a te d a bove , the n th i s e na b le s f ind ing the va lue s o fT fo r the s e le c te d X . T he r e l a t ion T ( ) ,) found in th i s way fo r fixe dX y ie lds the pos s ib i l i ty o f f ind ing num e r ic a l ly the m a x im um va lue o fT a nd the c o r r e s pond ing va lue o f ) / -

T h e c a l c u l a t i o n s w e r e m a d e w i t h ~ = 1 . 4 f o r t h eb a c k p r e s s u r e p~o = 0 . 0 0 0 2 2 5 9 . T h e q u a n t i t y Q / 2~ r w a st a k e n a s 0 . 3 6 5 4 , w h i c h c o r r e s p o n d s t o u 0 -- 1 . 5 f o r7 = 0. T h e n o z z l e l e n g t h X a n d t h e v a l u e s o f 7 w e r ev a r i e d . T h e c o m p u t a t i o n a l r e s u l t s a r e s u m m a r i z e di n t h e t a b l e , f r o m w h i c h w e s e e t h a t t h e n o z z l e w i t hr o t a t i n g f l o w h a s a t h r u s t e x c e e d i n g t h a t o f t h e n o z -z l e s w i t h o u t r o t a t i o n o f t h e g a s .

In o rde r to i l lu s t r a te the d i f f e r e nc e in the fo rc e d i s t r ibu t ion innoz z te s w i th ro ta t ing a nd nonro ta t ing flows, l e t u s in t rodu c e them a g n i tude I o f the th rus t c r e a te d by the f low in the s e c t ion Oa , a ndthe in te g ra l i o f the p re s su re fo rc e s a long the noz z le w a l l

r b

1 = i ( p -- p ~ -4 - p u 2 ) d r , I ~ I (p - -p ~ ) r d r ,r o i

where T = 2~r(I- t- J ) in acco rdan ce with Eq. (8) .For y = 0 we have I = 0,678, J = 0.098, for y = 0.4 w e hav eI = 0 .610 , J = 0 .169 , i f X = 4 . T he s e num be r s ind ic a te tha t the

c on t r ibu t ion o f the qu a n t i t i e s I a nd J to the qua n t i ty T i s m a rke d lyre d i s t r ibu te d a s a func t ion o f y .R E F E R E N C E S

1 . G . G u d e r l e y a n d E . H a n t s c h , " B e s t e F o r m e nf t i r a c h s e n s y m m e t r i s c h e U b e r s c h a l l s c h u b d i i s e n , " Z .F l u g w i s s , 3 , n o . 9, S e p t e m b e r 1 9 5 5 .

2 . Y u . D . S h m y g l e v s k i i , " S o m e v a r i a t i o n a l p r o b -l e m s o f g a s d y n a m i c s o f a x i s y m m e t r i c s u p e r s o n i cf l o w s , " P M M , v o l . 2 1 , n o . 2 , 1 9 5 7.

3 . G . V . R . R a o , " E x h a u s t n o z z l e c o n t o u r f o ro p t i m u m t h r u s t , " J e t P r o p u l s i o n , 2 8 , n o . 6 , 1 9 5 8 .

4 . Y u . D . S h m y g l e v s k i i , " V a r i a t i o n a l p r o b l e m s f o rs u p e r s o n i c b o d i e s o f r e v o l u t i o n a n d n o z z l e s , " P M M ,v o l . 2 6 , n o . 1 , 1 9 6 2 .

5 . K . G . G u d e r l e y , a n d J . V . A r m i t a g e , " A g e n -e r a l m e t h o d f o r t h e d e t e r m i n a t i o n o f b e s t s u p e r s o n i cr o c k e t n o z z l e s , " P a p e r p r e s e n t e d a t t h e S y m p o s i u mo n _ E x t r e m a l P r o b l e m s i n A e r o d y n a m i c s , B o e i n gS c i e n t i f i c R e s e a r c h L a b o r a t o r i e s , F l i g h t S c i e n c e sL a b o r a t o r y , S e a t t l e , W a s h i n g t o n , D e c e m b e r 3 - 4 ,1 9 6 2 .

6 . A . N . K r a i k o , I . N . N a u m o v a , a n d Y u . D .S h m y g l e v s k i i , " O n t h e c o n s t r u c t i o n o f b o d i e s o fo p t i m u m f o r m i n a s u p e r s o n i c s t r e a m , " P M M , v o l .2 8 , n o . 1 , 1 9 6 4 .

7 . Y u . D . S h m y g l e v s k i i , " S o m e v a r i a t i o n a l p r o b -l e m s o f g a s d y n a m i c s , " T r . V T s A N S S S R , 1 9 6 3 .

2 0 M a y 1 9 6 6 M o s c o w

thrust incrs by flow otation

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