6

ROUND

M E K H A N I K A ZHIDKOSTI I GA ZA

TURBULENT

JET

IN A CROSSFLOW

N. I. A k a t n o v Izv. AN SSSR. M e k h a n i k a Z h i d k o s t i i G a z a , Vol. 4, No. 6, pp. 1 1 - 1 9 , We examine the discharge of a turbulent jet from a round opening into an infinite uniform erossflow and find the form of the jet centerline and the distribution of the maximum velocities in the jet ainngtheaxis. It is shown that the calculated jet axes and velocity distributions agree well with the experimental values for different ratios of the velocity at the source exit to the crossflow velocity and for different angles of entry of the jet into the crossflow. The study [1] formed the basis for the proposed semiempirical theory, 1. M o m e n t u m v a r i a t i o n i n a j e t in a e r o s s f l o w . We c o n s i d e r a f l u i d j e t i s s u i n g f r o m a r o u n d p o r t of d i a m -

1969

D r a w i n g t h e Z - p l a n e t h r o u g h d i f f e r e n t p o i n t s of t h e y - a x i s , we c o n s t r u c t t h e s u r f a c e 0 5 , w h i c h we c o n s i d e r to b e t h e b o u n d a r y s u r f a c e of t h e j e t . We u s e Oy to d e n o t e t h e p a r t o f t h e E - p l a n e w h i c h i s i n c l u d e d within 05 . We d r a w t h e p l a n e s 221 a n d 22~ at t h e d i s t a n c e dy f r o m one a n o t h e r a n d c o n s t r u c t t h e c o n t r o l s u r f a c e (1), c o n s i s t i n g of t h e p l a n e s Oy1 a n d Oy2 and t h e p a r t of t h e s u r f a c e 0 5 w h i c h is e n c l o s e d b e t w e e n t h e p l a n e s E 1 a n d Z 2, h e r e a f t e r d e n o t e d b y A o 6, a n d t h e n we w r i t e t h e m o m e n t u m c h a n g e of t h e f l u i d l o c a t e d i n s i d e t h e c o n t r o l s u r f a c e in t h e f o r m

(~v~

+ jAp2)~-

~Y2

! (p~tvi + ~Ap~)d~, = 6YE

=

f (Pn-- oV~V)dA~ A(16

(Ap = p - p~). In (1.1) we h a v e d r o p p e d t h e t e r m s

(1.1) ~ Pxyi 'd (Yyl gyi

"x X Fig.

1

e t e r d 0 into a u n i f o r m f l u i d s t r e a m f l o w i n g w i t h v e l o c ity V~ a n d h a v i n g t h e s a m e p h y s i c a l p r o p e r t i e s a s t h e j e t m e d i u m . We a s s u m e t h e f l u i d f l o w to b e i s o t h e r mal. We d e n o t e t h e i n i t i a l j e t m o m e n t u m b y I0 a n d a s s u m e t h a t 410/Trd 2 >> pV~, w h e r e p is t h e f l u i d d e n s i t y . We d e n o t e t h e a n g l e b e t w e e n I 0 a n d V~o b y ~ 0. We d e s c r i b e t h e f l u i d m o t i o n in t h e x y z r e c t a n g u l a r f r a m e w i t h o r i g i n a t t h e c e n t e r of t h e p o r t f r o m w h i c h the jet is discharged and we direct the x-axis para l l e l to V~ a n d t h e y - a x i s so t h a t i t l i e s in t h e p l a n e of t h e x - a x i s a n d t h e v e c t o r I0. We d e n o t e t h e u n i t v e c t o r of t h e y - a x i s b y j, a n d t h e p r o j e c t i o n s of t h e v e l o c i t y v e c t o r onto t h e x - , y - , a n d z - a x e s b y u, v, a n d w, r e s p e c t i v e l y . T h e g e o m e t r i c l o c u s of t h e p o i n t s a t w h i c h v = Vma x is c a l l e d t h e j e t a x i s . We m e a s u r e t h e s - c o o r d i n a t e a l o n g t h e j e t a x i s f r o m t h e c o o r d i n a t e o r i g i n O, a n d

the angle ~ is the angle between the tangent to the jet axis, defined by the unit vector ~, and the vector V~. We draw the plane y = const, which we denote by the letter E (Fig. i). We consider the edge of the jet in the Z-plane to be the geometric locus of the points of this plane at which V / t)max ~

(sV0 / V~ ~ t,

6 ~

const

V0 =- (410 / p~d0z) V~).

(i = 1, 2), in v i e w of t h e f a c t t h a t Pxyi = - P ( u % ' ) i << << -PUiV i, w h e r e p~y i i s t h e t u r b u l e n t f r i c t i o n s t r e s s on t h e e l e m e n t ~yi. Here p and p~ are, respectively, the pressures at a g i v e n p o i n t a n d at i n f i n i t y ; Pn is t h e s t r e s s a c t i n g on t h e a r e a dAo 6 w i t h t h e n o r m a l n. We c o n s i d e r t h e q u a n t i t y P n f = - P V n V in t h e i n t e g r a n d of t h e r i g h t h a n d s i d e of (1.1) t o b e a f i c t i t i o u s s t r e s s " a p p l i e d " to t h e a r e a dAo 5. We c a l l t h e q u a n t i t y AR --~ S (P~ + P ~ ) d A ~

(1.2)

t h e p r i n c i p a l v e c t o r of t h e p r e s s u r e f o r c e s , i n e r t i a f o r c e s (Pnf), a n d t u r b u l e n t v i s c o s i t y f o r c e s a p p l i e d t o t h e s u r f a c e AoS. The bounding s u r f a c e o6 is d r a w n so t h a t on t h i s s u r f a c e we c a n a s s u m e to w i t h i n s t h a t t h e v e c t o r V is p a r a l l e l to t h e E - p l a n e . T h e r e f o r e , t h e p r o j e c t i o n of P n f o n t o t h e y - a x i s , j u s t a s t h e p r o j e c t i o n of t h e t u r b u l e n t s t r e s s e s a c t i n g on t h e s u r f a c e A~5, c a n b e t a k e n e q u a l to z e r o . C o n s i d e r i n g t h i s , we w r i t e t h e p r o j e c t i o n of (1.1) o n t o t h e y - a x i s in t h e f o r m

S pU22d~y2-- S PvJZdo~i = ~y2

= S (Yyt

(Yyi

t"Ap2 %2-S Apoo ( y)dA . 6y2

(1.3)

&(~

We n o t e t h a t

S pv2~d%2_ X pviZd~y~.~S OAp dg d(~yo %z %~ %r ay

(\ ~

__~yl ~ + ~y2.) .

(1.4)

FLUID

DYNAMICS

7

Let u s e x a m i n e the jet axis s e g m e n t on which the angle a is not small. It is a s s u m e d that V0 >> V~ and that t h e r e f o r e the s e g m e n t is r a t h e r long. Let us u s e the hypothesis analogous to the hypothesis of plane sections, which is applicable for the c a l c u l a tion of wings of l a r g e aspect r a t i o [2] and a m o u n t s t o the fact that the flow, in the plane E ~= Z - ay which is e x t e r n a l in r e l a t i o n to Oy, can be c o n s i d e r e d p l a n a r and independent of the flow in the neighboring p l a n e s Z ' . In other words, while c o n s i d e r i n g the v e locity and p r e s s u r e in the s t r e a m which is e x t e r n a l to Oy to be dependent on y, at the s a m e t i m e we n e glect the d e r i v a t i v e s of these quantities with r e s p e c t to y. Since Ap, which a p p e a r s in the integrand of the r i g h t - h a n d side of (1.4), is c r e a t e d by the outer flow, after u s i n g the hypothesis of p l a n a r sections, we set @Ap/3y = 0. T h e r e f o r e (1.3) yields

S (pu'v')dEz'-- S
}2~'

in a c c o r d a n c e with the hypothesis f o r m u l a t e d above. Moreover, S ~)U2U2 d ~ 2 z - Eet

vf 'OL~IUt d ~ ' l ' ~t'

~1t

=

E_J

+

S pV=v2dY2"--f oV=vi d~V ..~ E2t

Zl'

in a c c o r d a n c e with the hypothesis of plane sections. Considering (1.8) and (1.9), we obtain f r o m (1.7) ARx = 9Voo~6oody @ pV~0 dg,

qyi

fly2

B

The p r o j e c t i o n of (1.1) onto the x - a x i s has the f o r m

~ ptt2v2d(yy2-- ~ pulvt d%t = fly2

~ ( P ~ -- pV=a)dAa~ = hRx.

(1.6)

We draw the control surface 2 in the f o r m of a p a r a l l e l e p i p e d f o r m e d by the planes Et and Z z, the f o r w a r d and aft planes Ef and E a, r e s p e c t i v e l y , p a r allel to the O y z - p l a n e , and the two l a t e r a l planes E61 and Y'62. We denote the part of the ~1,2 - p l a n e which is occupied by the t u r b u l e n t wake f o r m e d b e hind the j e t by Okt,2. Edge AD of the p a r a l l e l e p i p e d must be long enough so that in the Z b - p l a n e we can a s s u m e that Ap = 0, and that the length of edge AB must be g r e a t e r than the t h i c k n e s s of the t u r b u l e n t wake, m e a s u r e d in the d i r e c t i o n of the z - a x i s . We find the m o m e n t u m change of the fluid e n c l o s e d b e tween the c o n t r o l s u r f a c e s 1 and 2, we project this onto the x - a x i s , and after some t r a n s f o r m a t i o n s we obtain B A

(p~'v'))dE,').

(1.7)

El"

Here B

f

A

(1.1o) Et"

Aft8

nu

dz,

Qey=Q dy-l-

(ryi

-- S (,ou,v,

1--

~

D

I

A

is the difference p e r unit length between the amount of fluid e n t e r i n g the c o n t r o l s u r f a c e 2 p e r unit t i m e through the faces Ef, Eot, and ~o 2 and the amount of fluid leaving through the s u r f a c e Z a. We note that

Eo"

Here 5"~* is the wake m o m e n t u m t h i c k n e s s , Qdy is the rate of fluid flow into the jet through its side surface Ao 6. Thus, under the a s s u m p t i o n made on the s m a l l n e s s of the derivative with r e s p e c t to y, the p r o j e c t i o n s of (1.1) on the x - and y - a x e s have the f o r m

Iy ~

.f pvZ d% ~ Io~ , ay

l~:~ - l ~ l = S o~-'z d z ~ - - ~ p~v~ dz~, = qy2

ffyi

=- pY~Z6:*'clg -{- pV~Q dg.

(1.11)

Thus the m o m e n t u m change in the jet takes place u n d e r the influence of the profile drag, r e p r e s e n t e d by the t e r m uv~oo~, and the "source" drag which a r i s e s as a r e s u l t of the fact that the effective bounda r y of the jet is p e r m e a b l e . The last t e r m defines the difference between the jet drag and the drag of an i m p e r m e a b l e solid body. 2. Equation of j e t axis and d i s t r i b u t i o n of m a x i m u m v e l o c i t i e s along the jet axis. To obtain f r o m (1.11) the d i f f e r e n t i a l equation of the j e t axis, we must make s e v e r a l a s s u m p t i o n s about the connection between 5"*oo, Q, and Ixy and the angle ~ and the other given flow c h a r a c t e r i s t i c s , such as I0, ~0, and V~. We a s s u m e that 6~** ~ cb

(b -~- "~a~, (y~ "~ C~ysin a).

(2.1)

Here b is the c h a r a c t e r i s t i c d i m e n s i o n of the j e t section a r e a o s d r a w n p e r p e n d i c u l a r to the jet axis tangent r (Fig. 2); c is an e m p i r i c a l constant. E x p e r i m e n t s show [3] that the j e t expands m a r k e d l y initially over a length equal to a few d0, and then the rate of b r o a d e n i n g d e c r e a s e s and the dependence of the j e t

8

MEKHANIKA ZHIDKOSTI I GAZA

width on the distance f r o m the s o u r c e b e c o m e s l i n e a r [3, 4]. We t e r m the s e g m e n t of l i n e a r expansion of g

vz-(

l

except in the aft region, w h e r e t h e r e is the " c o r r i d o r " mn through which the fluid of the o u t e r s t r e a m flows into the j et flow region. We take the width of c o r r i d o r mn equal to kb, w h er e k is an e m p i r i c a l constant. We take the velocity Vr on s e g m e n t m n to be equal to Vr = -NVoo sin a , w h e r e N is the n o r m a l to the s e g ment m n and the p r i n c i p a l n o r m a l to the j et axis. Thus, the second t e r m on the r i g h t - h a n d side of (2.3) can finally be w r i t t e n in the f o r m

i (n.Vo)dAcy6=kaV~sina(so+s)ds.

(2.6)

Aq 8

Fig. 2 of the j e t the main s e g m e n t and a s s u m e for this s e g ment b --~ a (s + s0),

(2.2)

w h e r e the coefficient a and the pole d i s t a n c e s o a r e e m p i r i c a l constants. We t e r m the s e g m e n t of nonline a r growth of b the initial s e g m e n t and denote its length by l. We r e s o l v e the v e l o c i t y of the fluid p a r t i c l e s l o cated in the Ors-Plane into the tangential component V~_~"and the component Va which lies in the Os-plane. Then the r a t e of fluid flow through the s u r f a c e Ao 6 may be w r i t t e n in the f o r m

eQdy=P

f(n.V)dAa~ ha8

The m o m e n t u m v e c t o r of the fluid flowing through the s u r f a c e Oy is d e t e r m i n e d by the magnitude and d i r e c t i o n of the m a x i m u m v e l o c i t i e s on this s u r f a c e , which have a d i r e c t i o n c l o s e to the d i r e c t i o n ~. T h e r e fore, in cal cu l at i n g the m o m e n t u m of the fluid o v e r the s u r f a c e Oy we a s s u m e that e v e r y w h e r e on this s u r face V -- V.ff. Then

I~u-~- Soavd%~ctga Ipv2d%~Ioyctga. •y

Denoting ctg a = d x / d y = t, we have

(t~2

(lyi

Substituting (2.5), (2.6), and (2.8) into the second r e l a t i o n of (1.11), we obtain the d i f f e r e n t i a l equation of the j e t axis i n t h e f o r m

Iou~s -~- apV~(So -4-s)[(c -4- k)sin a -4- 2a[ cos ~I]"

a~

We a s s u m e that V7 = Vco cos ~ on the side s u r f ace of the j e t , aft er which the f i r s t integral of the r i g h t hand side of (2.3) may be w r i t t e n in the f o r m

.f ( "

dA 0 =

A(~5

(2.9)

The flow r a t e Qdy, p a r t s of which ar e e x p r e s s e d by (2.5) and (2.6), is taken in absolute magnitude in

tg aa6

(2.7)

qy

:o

<.

/

I

/

, ?.o,~

dAo = A(~5

--

V~ cos a ( ~ -- o~).

(2.4) tO i"

Taking (2.2) into account, we obtain .f (n 9 z)V~dhas~

--2a~Vo~cosa(so+s)ds.

- - - -

--

(2.5)

/y M e a s u r e m e n t s of the v e l o c i t y field, and also v i s u al o b s e r v a t i o n s of j e t s issuing into a c r o s s f l o w , show that under the influence of the approaching s t r e a m the jet c r o s s s ect i o n takes on a h o r s e s h o e f o r m [1], with the fluid at the ed g e s of the " h o r s e s h o e " p e r f o r m i n g a v o r t i c a l motion in the o s - p l a n e (regions 1 and 2 in Fig. 2) as a r e s u l t of flow through t h e s e edges. The q u a l i t a t i v e p a t t e r n of the s t r e a m l i n e s of the velocity component Vcr in the Os-plane is shown in Fig. 2. We a s s u m e that the boundary of the j e t section o s is a s t r e a m l i n e f o r the v e l o c i t y component Va e v e r y w h e r e

20

30

Fig. 3 (2.9), si n ce the s o u r c e drag m u s t be p o si t i v e. introduce the d i m e n s i o n l e s s quantities x

Y

r-d0,

S

s

=-go'

so

So= do'

We (2.10)

and the s p e c i f i c m o m e n t a i~

~-~ 9]7002,

i0 = pV02

(2.11)

and noting that s i n s = (1 + t2)-1/2, we substitute (2.10) and (2.11) into (2.9), which a f t e r s i m p l e t r a n s f o r m -

F L U I D DYNAMICS

9

"-- qu

]

I

L

,.Io

9 gO=l?ff

/

///

/

f I cca~ 1 ~ o

/ X -20

-10

0

tg

Zg

Fig. 4 ations becomes

dt

4a

dS

a sin a0

\ (~ + c) + 2alt[ ioo) (S + So). (2.12)

an accuracy which is quite adequate in practice. t(S) h a s b e e n f o u n d , we f i n d 8

S

!* t d S

E q u a t i o n (2.12) m u s t b e i n t e g r a t e d w i t h k n o w n v a l u e s of t h e c o n s t a n t s k, e, a, and S O w i t h t h e b o u n d a r y c o n d i t i o n s p e c i f i e d at t h e e n d of t h e i n i t i a l s e g m e n t of t h e j e t , s i n c e (2.2) i s v a l i d o n l y in t h e i n i t i a l s e g m e n t . H o w e v e r , c a l c u l a t i o n s s h o w t h a t in v i e w of t h e r e l a t i v e l y s h o r t l e n g t h of t h e i n i t i a l s e g m e n t t h e b o u n d a r y c o n d i t i o n m a y b e s h i f t e d to t h e c o o r d i n a t e o r i s i n a n d t h e e r r o r i n t r o d u c e d in d o i n g t h i s m a y b e c o r r e c t e d b y s e l e c t i o n of a s u i t a b l e v a l u e of t h e p o l e d i s t a a c e So. T h u s , we w r i t e t h e c o n d i t i o n n e c e s s a r y f o r ! ' ~ t e g r a t i o n of (2.12) in t h e f o r m t-=t0----ctgao

for, S = 0 .

a---0.3,

S0=6.6.

~;quation ( 2 . 1 2 ) i s e a s i l y s o l v e d b y m e a n s of s u c c e s s i v e a p p r o x i m a t i o n s if it i s i n t e g r a t e d o v e r S a n d

~ l

I

I~.z

',

!

y~

]/1 + t 2'

!

dS

(2.15)

f l + t2

Figure 3 shows curves of the jet axes calculated using (2.14) and (2.15) for c% = 90 ~ and different values of i~o/io = B, which are indi-

[ ' x,, -,.~

0

e

/5

2~

s/zo

~o

Fig. 6

(2.13)

C o m p a r i s o n of t h e c a l c u l a t e d j e t a x e s w i t h t h e e x perimental axes shows that the empirical constants in (~:.12) must have the f o l l o w i n g values: c-t-lr

X -~

cared on the curves; the open and filled circles correspond to the experimental data of [5, 63. Figure 4 shows a comparison of the jet axes calculated by means of these same formulas with the experimental data of [5J for the values ~ = 0.0025 and 0.01 (open and filled circles, respectively). The injection angles a0 are indicated on the curves. We see from the figures that the agreement of the calculated and experimental jet axes is satisfactory. T h e i n c r e m e n t of t h e f l u i d m a s s f l o w r a t e in t h e j e t o n a n e l e m e n t d s of t h e j e t a x i s xs

dM ~- pQdg --~ (pkbVo~ sin a q+ p V . cos ad(~ / ds) ds.

8

/5

74

s/d o

Fig. 5

(2.16)

U s i n g (2.2), w e i n t e g r a t e (2.16) w i t h r e s p e c t to s and, r e f e r r i n g t h e r e s u l t i n g e x p r e s s i o n t o t h e f l u i d flow r a t e M 0 = 14zTd~ p V 0 t h r o u g h t h e s o u r c e o p e n i n g we obtain

w r i t t e n in t h e f o r m

4a

After

S

(t~)j

t=t~

i+O.6ltl ~

(S+&)dS. (2.14)

m*+4ak~ zt

~sina(S+So)dS+ L

j/t -~ t z

Ln t h e z e r o a p p r o x i m a t i o n we s e t

f(t) - -

M m ~Io

t+O.6ltl ]/l+t z

-- t.

T h e n we c a l c u l a t e t (~ f i n d f ( t ) , a n d s o on. T h e caiculations show that the second approximation yields

+ 8 a ~ f co~~(s + So)dS L (L = l/do, m" = M"/Mo).

(2.17)

H e r e m * i s t h e d i m e n s i o n l e s s flow r a t e at t h e e n d of t h e i n i t i a l s e g m e n t . T h e j e t flow in t h e i n i t i a l s e g -

10

M E K H A N I K A ZHIDKOSTI GAZA

m e n t is d e t e r m i n e d t o a c o n s i d e r a b l e d e g r e e b y t h e c o n d i t i o n s of the j e t d i s c h a r g e f r o m t h e s o u r c e . T h e r e f o r e we c a n a s s u m e t h a t t h e s a m e r e l a t i o n s h i p s

] + 8aZl] [,~ j cos a (So + S) dSj -0

-

-~

,71

[cos -cos '(

,~ / j .

(2.22)

I

6'

i

For reasons of computational convenience the integrals in the bracket in the right-hand side of (2,22) are taken from 0. On the segment 0 -< S _< L we take AVOn= 1. To determine L, in (2.22) we set S = L and AV~ = 1 and solve the resulting equation for L. Comparison with experiment shows that we should take K = 0.15 and )r = 0.67. Figure 5 shows a comparison of the calculated (by means of (2.22)) and measured [5] distributions of the quantity

,...,.-J o/

,,,,,d

~ r

ag

ff

/6

J2 s/do

Fig. 7

/B

between the characteristic flow quantities which aff e c t the i n i t i a l s e g m e n t of t h e c o n v e n t i o n a l s u b m e r g e d j e t a r e a l s o v a l i d f o r t h e j e t in a c r o s s f l o w . T h u s , w e a s s u m e t h a t t h e fluid m a s s e j e c t e d by t h e j e t in t h e i n i t i a l s e g m e n t e q u a l s [7] AM* - - M0(t + •

(2.18)

H e r e ~ is an e m p i r i c a l c o n s t a n t . The t o t a l f l u i d f l o w r a t e t h r o u g h an a r b i t r a r y s e c t i o n of the j e t i s M =

= ~s

I

o,

V cos ,

(2.19)

V~ -- V~

AVe'-- ~ (1 - cos n)

Vo--V~

i-~

along the jet axis for a0 = 90*~ points I, 2, and 3 correspond to the

values B = 0.2, 0.1, and 0.05. F~gure 6 compares the calculated and measured [5] distributions of the same quantity for 13 = 0.0025 but for a0 = 30 and 150" (poims 1 and 2, respectively). We see from Figs. 5 and 6 that the agreement between the theoretical distributions of the maximum velocity and the experimental values is satisfactory. Figure 7 shows curves for the total dimensionless flow rate m along the axis for a0 = 90*; points 1, 2, and 3 correspond to the values B = = 0.2, 0.1, and 0.05. The dashed line shows the flow rate variation along the axis of the conventional submerged jet.

REFERENCES

at

w h e r e AVe- is t h e e x c e s s t a n g e n t i a l v e l o c i t y c o m p o n e n t in r e l a t i o n to t h e c r o s s f l o w t a n g e n t i a l c o m p o n e n t V~ c o s a . The f i r s t t e r m on the r i g h t - h a n d s i d e o f (2.19) c a n b e w r i t t e n in t h e f o r m

f3pAV~d(h = 9AV,r

J \ AV,J ~

a s = oAV~,A~. (2.20)

qts

We t a k e t h e d i m e n s i o n l e s s f l o w r a t e A in (2.20) to be t h e s a m e a s in t h e c o n v e n t i o n a l j e t in a p a r a l l e l s t r e a m [1], i . e . , A = 0.26. The t o t a l f l u i d f l o w r a t e t h r o u g h the s e c t i o n ~s at t h e e n d of t h e i n i t i a l s e g m e n t , in a c c o r d a n c e with (2.18) and (2.19), is

M* - - OAVma~*A + pV~a~* cos a* ~-~ = M0(i + •

+ pVr

cos a*.

(2.21)

R e f e r r i n g (2.19) to M0, w e e q u a t e the r e s u l t i n g d i m e n s i o n l e s s f l o w r a t e to (2.17) and, t a k i n g (2.21) into account, we finally obtain the e x p r e s s i o n f o r d e t e r m i n i n g t h e m a x i m u m e x c e s s v e l o c i t y in t h e i n i t i a l s e g m e n t of t h e j e t i n t h e f o r m

A Vm AV,.'------

zt

Vo

4Aaz(So + S) 2

x

I. G. N. Abramovich, Theory of Turbulent Jets [in Russian], Fizmatgiz, Moscow, pp. 581 and 215, 1966. 2. L. G. Loitsyanskii, Mechanics of Liquids and Gases [in Russian], Gostekhizdat, Moscow, p. 421, 1957. 3. I. B. Palatnik and D. ~h. Temirbaev, "Propagation of an axisymmetric air jet in a uniform crossflow, " collection" Problems of Thermoenergetics and Applied Thermophysics, No. 4 [in Russian], Nauka, Alma-Ata, 1967. 4. J. Keffer and W. D. Baines, "The round turbulent jet in a cross-wind, " J. Fluid Mech., vol. 15, pt. 4, 1963. 5. Yu. V. Ivanov, Effective Combustion of Overbed C o m b u s t i b l e G a s e s in F u r n a c e s [in R u s s i a n ] , E s t g o s i z d a t , T a l l i n , 1959. 6. G. S. S h a n d o r o v , " D i s c h a r g e into a c r o s s f l o w f r o m p o r t s in a c h a n n e l w a l l and p r o p a g a t i o n of j e t s in a c r o s s f l o w , " T r . T s I A M , no. 263, 1955. 7. L. A. V u l i s and V. P. K a s h k a r o v , T h e o r y of V i s c o u s F l u i d J e t s [in R u s s i a n ] , Nauka, M o s c o w , p. 276, 1965.

N

•

Ssin o

(So+SldS+

27 M a r c h 1969

Leningrad