FLOW OF A SUPERSONIC J E T INTO CHANNELS OF VARIOUS SHAPES B. A. Balanin Inzhenerno-Fizicheskii UDC

Zhurnal,

Vol.

15, No.

i, pp. 91-97,

1968

533.601.15:62-225

The author examines the process of flow from one or more supersonic nozzles into a chamber with a diffuser or a wide cylindrical tube. The characteristic regimes are established. An analysis of the experimental data shows that the chamber pressure is at a minimum when critical flow conditions exist at the diffuser outlet. In c e r t a i n e q u i p m e n t , such as wind tunnels, s u p e r sonic gas e j e c t o r s , etc., a s u p e r s o n i c j e t e j e c t s a i r from a certain closed space thereby creating reduced p r e s s u r e . At the outlet f r o m t h e s e s y s t e m s t h e r e is usually a supersonic diffuser or, m o r e commonly, a c y l i n d r i c a l e x h a u s t channel. When a i r is not supplied f r o m the outside, a p a r t f r o m the n o z z l e (i. e., when the e j e c t i o n c o e f f i c i e n t is z e r o ) , a t h e o r e t i c a l a n a l y s i s is difficult, s i n c e it is n e c e s s a r y to take into a c c o u n t the v i s c o s i t y on the in it ia l s e c t i o n of the j e t and c o n s i d e r the p r o c e s s of r e c o n s t r u c t i o n of the f r e e j e t into a channel flow. To e s t a b l i s h the c o r r e s p o n d i n g p h y s i c a l flow m o d e l , we c a r r i e d out a l a r g e n u m b e r of e x p e r i m e n t s [1] on m o d e l s of the type i l l u s t r a t e d in Fig. 1. The e x p e r i m e n t s w e r e p e r f o r m e d with cold a i r (k = 1.41). The i m m e d i a t e o b j e c t of the e x p e r i m e n t s was to d e t e r m i n e the e f f e c t of the g e o m e t r i c p a r a m e t e r s F e , l, L, and the M a n u m b e r on the d e p e n d e n c e of the c h a m b e r p r e s s u r e Pc on the p r e s s u r e u p s t r e a m f r o m the n o z z l e P0 and a s c e r t a i n the p r e s s u r e r e d u c t i o n mechanism. A t y p i cal r e s u l t is p r e s e n t e d in Fig. 1, w h e r e Pc is p l o t t ed a g a i n s t P0. T h e s e c u r v e s take d i f f e r e n t f o r m s d ep en d i n g on the length of the channel F. At I < / o p t ( c u r v e A on the graph), as P0 i n c r e a s e s the c h a m b e r p r e s s u r e f a l l s s l i g h t l y ( b r a n c h I of the c u r v e ) until P0 r e a c h e s a c e r t a i n value. Then, any s m a l l i n c r e a s e in P0 r e s u l t s in a n o n s t e a d y p r o c e s s of v a r i a tion of P c (i. e., Pc f a l l s c o n t i n u o u s l y at P0 = const), which c o n t i n u e s until ( b r a n c h II of the c u r v e ) Pc r e a c h e s a m i n i m u m v a l u e lying on the b r a n c h of the c u r v e c h a r a c t e r i z i n g the nli m r e g i m e c o r r e s p o n d i n g to the condition P c / P 0 = c o n s t ( b r a n c h III of the cu r v e) . Then the p r e s s u r e in the c h a m b e r b e c o m e s p r o p o r tional to the p r e s s u r e u p s t r e a m f r o m the n o z z l e . A t / o p t (Fig. 1, c u r v e B), the p r e s s u r e Pc f a l l s m o n t o n i c a l l y as P0 i n c r e a s e s , r e a c h i n g a m i n i m u m v a l u e ( P c ) m i n , and the nli m r e g i m e s e t s in. The quantity nli m does not depend on the length of the outl e t c h a n n e l / . The m a x i m u m v a c u u m a l s o c o r r e s p o n d s to l >-/opt; m o r e o v e r , when l >-/opt the m i n i m u m of Pc is r e a c h e d at the l e a s t (as c o m p a r e d with l < / o p t ) pressure upstream from the nozzle Pure.

I o.o

4~

o

20

Fig. 1. C h a m b e r p r e s s u r e P c, of the p r e s s u r e u p s t r e a m f r o m at M a = 2.45, F e = 16, L = 4.0 solid c i r c l e s - - T = 2): 1) n o z z l e , channel.

.~o

Pore

4'o

Po

at m abs., as a function the n o z z l e P0 arm abs., (open c i r c l e s - - l = 6, 2) c h a m b e r , 3) outlet

As a r e s u l t of an a n a l y s i s of the e x p e r i m e n t a l data we e s t a b l i s h e d the p h y s i c a l p i c t u r e of the p r o c e s s e s in the c h a m b e r when I
631

c r e a s e in P0 l e a d s to a d e c r e a s e in the a r e a of the a n n u l ar gap, but, obviously, the v e l o c i t y v c cannot i n c r e a s e any m o r e . Starting f r o m this instant, the flow r a t e in the r e v e r s e d i r e c t i o n can no l o n g e r c o m p e n s a t e the additional m a s s of the jet, and a n o n s t e a d y p r o c e s s of r e m o v a l of m a s s f r o m the c h a m b e r e n s u e s . The c h a m b e r p r e s s u r e f a l l s at c o n s t a n t P0. This c o r r e sponds to b r a n c h II of the c u r v e in Fig. 1. This q u e s tion was e x a m i n e d in d e t a i l in [2]. The t r a n s i e n t p r o c e s s ends when the o u t e r edge of the j e t b e g i n s to touch the w a l l s of the outlet channel. A f u r t h e r i n c r e a s e in P0 l e a d s to a r e o r g a n i z a t i o n of the flow field in the outlet channel until the s t r e a m line in the j et boundary l a y e r on w h i c h the v e l o c i t y is equal to the c r i t i c a l v a lu e to u c h e s the edge of the ch an nel inlet. T h i s m a r k s the beginning of the nli m r e g i m e . This a r g u m e n t , a p p l i c a b l e to the s i t u a t i o n when the o u t l e t channel is not s u f f i c i e n t l y long (l < /opt) is i l l u s t r a t e d in Fig. 1 by the v e l o c i t y f i e l d s at the e x i t s e c tion of the outlet channel. It is c l e a r f r o m Fig. 1 that On b r a n c h I of c u r v e A t h e r e a r e two flows in the e x i t section: the j e t p r o p e r and a s u r r o u n d i n g a n n u l a r c o u n t e r j e t with v e l o c i t y v c. At l >- /opt, the v e l o c i t y f i e l d s c o r r e s p o n d i n g to P0 > P0m a r e u n i f o r m . Sharp n o n u n i f o r m i t y d e v e l o p s only at P0 > P0m. If l -> /opt, the je t a l m o s t i m m e d i a t e l y t o u c h e s the edge of the channel w a l l s ; b r a n c h II of the c u r v e is v e r y s m a l l and is n o t d e t e c t e d e x p e r i m e n t a l l y , although it undoubtedly e x i s t s . The p r o c e s s e s a r e all s i m i l a r in c h a r a c t e r when the c h a m b e r and o u t l e t channel a r e r e p l a c e d s i m p l y by a wide tube or when s e v e r a l n o z z l e s a r e i n s t a l l e d in any o r d e r in the b o t t o m of a wide tube. Obviously, then, the c h a r a c t e r i s t i c q u a n t i t i e s a r e as follows: the m i n i m u m c h a m b e r p r e s s u r e (Pc)rain, the total p r e s s u r e u p s t r e a m f r o m the n o z z l e P0m at w h i c h (Pc)rain is r e a c h e d , and the r a t i o nli m. M o r e o v e r , t h e r e is a c e r t a i n o p t i m a l length of the outlet channel /opt which e n s u r e s m a x i m u m p r e s s u r e r e d u c tion at the l e a s t p r e s s u r e u p s t r e a m f r o m the n o z z l e . The quantity lop t d e p e n d s only on the M a n u m b e r at the n o z z l e e x i t and can be found f r o m the e m p i r i c a l formula

F=I 78M~. A g e n e r a l i z a t i o n of the e x p e r i m e n t a l data has shown that when M a = e o n s t the p r e s s u r e u p s t r e a m f r o m the n o z z l e at which the c h a m b e r p r e s s u r e is a m i n i m u m (pro v i d ed that l > / o p t ) is d i r e c t l y p r o p o r t i o n a l to the r e l a t i v e c l e a r c r o s s s e c t i o n of the outlet channel FeAn e x a m p l e of this g e n e r a l i z a t i o n is shown in Fig. 2, w h e r e , as m a y s e e n f r o m the g r a p h s , the p r o p o r t i o n a l ity f a c t o r depends on M a. Thus, we have e x p e r i m e n t a l l y d e o m o n s t r a t e d the e x i s t e n c e of the condition P0m Te -- const whenM~ = const.

(1)

The r a t i o lgma/~" e is shown as a function of M a in Fig. 3, f r o m which it follows that at a g iv e n M a t_he length of the c h a m b e r }-, has a l m o s t n o e f f e c t on P 0 m / / F e (at any r a t e within f a i r l y wide l i m i t s , ~, = 0 - 7 ) . 632

#

~e

/2

F i g . 2. P r e s s u r e Pore, atm abs., at l -> /opt as a function of the a r e a of the outlet channel F e at L = 4: 1) M a = = 1.0; 2) 2.02; 3) 2.45; 4) 2 . 8 5 ; 5 ) 3 . 3 7 .

/

& o-! A-ll x-

3

0

!

2

3

Ma

Fig. 3. The r a t i o P 0 m / l ~ e as a function of the Mach n u m b e r at the n o z z l e e x i t M a for different models ( c u r v e - - c a l c u l a t i o n s b a s e d on Eq. (4); s t a r - - g r o u p of four n o z z l e s ) : 1) L = 1.5; 2) 4.0;

3) 7.0. E s s e n t i a l l y , P0m is a quantity d i r e c t l y p r o p o r t i o n a l to the flow r a t e t h r o u g h the n o z z l e (in s u p e r c r i t i c a l r e g i m e s ) . Consequently, (1) can be r e p r e s e n t e d as follows: ago Fe

-- a p e v ~ = const,

(2)

s i n c e the flow r a t e t h r o u g h the n o z z l e G Ois equaI to the flow r a t e through the outlet channel G e. H e r e , a is a c o e f f i c i e n t depending on the g as constant, the a d i a b a t i c exponent, and the s t a g n a t i o n t e m p e r a t u r e u p s t r e a m f r o m the n o z z l e . It is e a s i l y d e t e r m i n e d f r o m known r e l a t i o n s . A v a r i e t y of conditions m a y e x i s t in the c h a m b e r and the outlet channel. At s m a l l He the flow f r o m the n o z z l e will be c h a r a c t e r i z e d by o v e r e x p a n s i o n and the flow m a y s e p a r a t e f r o m the n o z z l e w a l l s ; on the o t h e r hand, at h i g h e r v a l u e s of F e the u n d e r e x p a n d e d j e t is c h a r a c t e r i z e d by a c o m p l e x s y s t e m of shock w a v e s that c a u s e s an e x t r e m e l y n o n u n i f o r m v e l o c i t y f i el d at the inlet to the outlet channel and c o n s i d e r a b l e total p r e s s u r e l o s s e s . Br o ad v a r i a t i o n of the c h a m b e r l e n g t h L should a l s o l e a d to c o n s i d e r a b l e d i f f e r e n c e s in the c h a r a c t e r of the p r o c e s s . H o w e v e r , as follows f r o m Fig. 2 r e l a t i o n (2) a l w a y s holds w h a t e v e r the conditions.

Comparison of Calculated and Experimental Values of P0m and (Pc)min Pore, atm obs. Ma

(Pc)rain, atm, abs,

E calc. from I

experim.

APom, %

(4)

a(Pc )re,in, %

experim.

(s)

2,45

38_ 1

36

5.5

O. 197

O. 200

1.5

2.85

31.2

29

7

O. 171

0~ 160

5.5

M e a s u r e m e n t of the v e l o c i t y f i e l d s at the e x i t f r o m the outlet channel in v a r i o u s t y p ic a l r e g i m e s (Fig. 1) sho ws that in the r e g i o n w h e r e the p r e s s u r e Pc r e a c h e s a m i n i m u m (i. e., at P0 = P0m), the n a t u r e of the f i el d changes sharply. It f o l l o ws that the only point w h e r e the flow c o n dit i o n s r e m a i n unchanged is the exit s e c t i o n of the outlet channel. It is n a t u r a l to a s s u m e that this point is c h a r a c t e r i z e d by a flow with c r i t i c a l p a r a m e t e r s when P0 = P0m. We can now w r i t e the e q u a t i o n r e l a t i n g the flow r a t e s t h r o u g h the n o z z l e and the exit s e c t i o n of the outlet channel f o r the r e g i m e in which Pc r e a c h e s its m i n i m u m value:

~m

calc. from

PomF~ Fe .~,~q(Xa) % m P~7..q(~.e~)

(3)

The v a l u e of nli m can be found by the m e t h o d p r o p o s e d in [3], but with c e r t a i n r e f i n e m e n t s . In [3] it is a s s u m e d that the n l i m r e g i m e d e v e l o p s when the i de a l b o u n d a r y of the j e t t o u c h e s the edge of the i n l e t to the outlet channel. It a p p e a r s , h o w e v e r , that c l o s e r a g r e e m e n t with e x p e r i m e n t can be obtained by a s s u m i n g that the n l i m r e g i m e is r e a l i z e d when the s t r e a m l i n e in the b o u n d a r y l a y e r of the j e t on which the v e l o c i t y is equal to the c r i t i c a l v a l u e t o u c h e s the edge of the inlet. The v e l o c i t y d i s t r i b u t i o n in the b o u n d a r y l a y e r can be found f r o m the f o r m u l a

:t -

,

and the condition nil m c o r r e s p o n d s to

(6)

I" e = r t Jr- tj l ~ , = l .

E x p e r i m e n t a l i n v e s t i g a t i o n s of the p r e s s u r e d i s t r i b u t i o n along the w a l l s of the outlet channel show that in the exit s e c t i o n it is a l w a y s equal to the a m b i e n t p r e s s u r e Pn" M o r e o v e r , f o r the c r i t i c a l flow r e g i m e at the channel e x i t q(Xem ) = 1. Then, s e t t i n g To = Toe, fio~ __ Po.~

F~ __ %

Te

Fe

P~

~

1

(5)

Since the p o i n t P0m l i e s on the b r a n c h of the Pc = : f(P0) c u r v e a l m o s t c o r r e s p o n d i n g to the nli m r e g i m e , knowing n l i m , we can e a s i l y d e t e r m i n e the m i n i m u m c h a m b e r p r e s s u r e (Pc)rain o r the b a s e p r e s s u r e . Since

(4)

P~ _ P o ~ ( ~ )

z~(~.~m)q 0~)

H e r e , ~ a and ~e a r e the flow c o e f f i c i e n t s of t h e n o z z l e and the outlet channel. E x p r e s s i o n (4) c o r r e s p o n d s q u a l i t a t i v e l y to e x p r e s s i o n (1) obtained on the b a s i s of a g e n e r a l i z a t i o n of the e x p e r i m e n t a l data. C a l c u l a t i o n s b a s e d on (4) a r e in good a g r e e m e n t with the e x p e r i m e n t a l data at ~e/~a = 0.5 irresp_ective of the M a n u m b e r and the v a l u e s of F e and L. V a r y i n g the p a r a m e t e r s o v e r a wide r a n g e l e a d s to c o n s i d e r a b l e ch an g e s in the total p r e s s u r e l o s s e s due to i m p o r t a n t c h a n g e s in the s y s t e m of shock w a v e s in the jet. The only e l e m e n t to be r e t a i n e d in the e x p e r i m e n t s is the s h ap e of the e x i t s e c t i o n of the outlet channel and, obviously, the c o e f f i c i e n t (9e is c o m p l e t e l y d e t e r m i n e d by this e l e m e n t . This is r e i n f o r c e d by the f a c t that in r e l a t i o n to m o d e l s of o t h e r s h a p e s (flow into a wide tube f r o m a s i n g l e s u p e r s o n i c n o z z l e or a group of n o z z l e s ) the n a t u r e of the v a r i a t i o n of b a s e p r e s s u r e as a function of the p r e s s u r e P0 is the s a m e as in the b a s i c s i t u a t i o n c o n s i d e r e d above. In t h e s e c a s e s , too, the p r e s s u r e P0m is in good a g r e e merit with the c a l c u l a t e d v a l u e (Fig. 3) at ~e/~a = 0.5. It should be noted that in e x a m i n i n g flow into a c h a m b e r o r a wide tube f r o m a group of s e v e r a l n o z z l e s , the a r e a of the outlet channel F e m u s t be r e l a t e d to the total a r e a of the e x i t s e c t i o n s of all the n o z z l e s .

nli2

p~

P~

(7) '

we h a v e (Pc)rnin --

P~

nlim

~Pe~ (~'a)~'e n lirnq (}~a) "

(8)

V a l u e s of P0m and ( P c ) m i n d e t e r m i n e d f r o m the r e l a tions obtained above, and t h e i r e x p e r i m e n t a l v a l u e s , a r e p r e s e n t e d in the table. F r o m the t ab l e it is c l e a r that on a v e r a g e the d i s c r e p a n c y d o e s not e x c e e d 5%. The r e g i m e c o r r e s p o n d i n g to (Pc)rain is the m o s t i n t e r e s t i n g f r o m the standpoint of a p p l i c a t i o n s . Using the above r e l a t i o n s we can a l s o find the total p r e s s u r e l o s s e s in this r e g i m e : Poe - ~ l q(~o) := P0~-= % Fe

(9)

It should be noted that the c a l c u l a t i o n s and e x p e r i m e n t s show that in the P0m r e g i m e the total p r e s s u r e l o s s e s r e a c h a m a x i m u m and at n = nli m r e t a i n this value. C a l c u l a t i o n s b a s e d on Eq. (9) a g r e e with the e x p e r i m e n t a l d a t a to within 15%. Thus, it is p o s s i b l e to d e t e r m i n e the c h a r a c t e r i s t i c q u a n t i t i e s n l i m , P0m, (Pc)rain, and #.

633

NOTATION F e is the clear cross section of the outlet channel; F a is the area of the nozzle exit section; Fe = Fe/Fa is the relative flow section of the outlet channel; L is the length of the chamber; I is the length of the outlet channel; d a is the diameter of the nozzle exit section; d e is the diameter of the outlet channel flow section; L = L/d a is the relative length of the chamber; l= = I/d e is the relative length of the outlet chambers r e = de/2; Fop t is the relative length of the outlet chamber ensuring minimum chamber pressure; rt is the coordinate of the boundary of an ideal jet; b is the thickness of the jet boundary layer; y is the variable coordinate (the y-axis is directed from the boundary of the ideal jet toward the outer edge of the boundary layer); k is the adiabatic exponent; g = 9.81 m/sec 2 is the acceleration of gravity; R is the gas constant; acr is the critical velocity; M is the Mach number; = V/acr is the characteristic velocity; P is the pressure; T is the temperature; p is the density; v is the velocity; p is the total pressure loss. k

7:(k)=

1

k+l

mq ('Aa) Fa

Subscripts: 0--stagnation parameters upstream from nozzle; a--parameters at nozzle exit; c--parameters in chamber or wide tube (outside jet); m--parameters in regime corresponding to minimum Pc; n--parameters of ambient medium; e--parameters in exit section of outlet channel; t--parameters at boundary of ideal jet.

REFERENCES i. B. A. Balanin, "Study of the effect of the geometry of the outlet section on the flow regime in the chamber of a supersonic wind tunnel," Vestnik LGU, no. 7, 1965. 2. B. A. Balanin, IFZh [Journal of Engineering Physics], no. 5, 1965. 3. B. A. Balanin, "Propagation of a supersonic jet in a bounded space," Vestnik LGU, no. 7, 1965,

l

, q(X)=),

•

i

•

634

1

~+1

18 S e p t e m b e r 1967

L e n i n g r a d State U n i v e r s i t y