The Effect of Density on Jet Flow at Subsonic Speeds

Kluyv. 3 DELFT

THE COLLEGE OF AERONAUTICS

CRANFIELD

THE EFFECT OF DENSITY ON JET FLOW

AT SUBSONIC SPEEDS

by

1

J u l y . 1959

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

C R A N F I E L D

The Effect of Density on J e t Flow a t Subsonic Speeds

b y

-A, H. Craven, M.So,, P h . D . , D.C.Ae,

SUiaiARY

On t h e aasunptlon t h a t the v e l o c i t y and d e n s i t y d i s t r i b u t i o n s across a j e t of one gas i s s u i n g i n t o a stream of a second gas a r e cxf esponential foim, the momentum i n t e g r a l approach has been employed t o f i n d t h e v a r i a t i o n of c e n t r e - l i n e v e l o c i t y dovmstream of the p o t e n t i a l c o r e . The r e s u l t s f o r t h e particiolar case of a j e t i s s u i n g i n t o a gas a t r e s t a r e eqmvalent t o those found t h e o r e t i c a l l y by B l o t t n e r (Ref. 2) and experimentally by Keagy and t e l l e r (Ref. 1 ) , The r e s u l t s for t h e j e t i s s u i n g i n t o a moving stream agree v/ith t h e l i m i t e d experiments of t h e present author.

These c a l c u l a t i o n s shov/ the importance of j e t t o free-stream d e n s i t y r a t i o with respect t o t h e r a t e of decay of the j e t dovmstream of the mixing region,

CQOTEtCTS P a g e

L i s t of Symbols 3 1 , Introdviotion 4

2, The Axl-symmetric J e t 5

2 . 1 . The flow downstrea'n of t h e j e t 5 2 . 2 . The mass flow i n t h e j e t . 10 2 . 3 . The inflow v e l o c i t y 11 3# The Two Dimensional J e t 12

3 . 1 . The flow downstream of the j e t 12

3 . 2 . The inflow v e l o c i t y 15 4» R e s t r i c t i o n s of the Analysis l 6

5# Ccnclvisions 16 6, References l 6

3

-LIST OF SYI-SOI^

a half width of two-dimensional jet at exit c concentration parameter

o (x) centre-line concentration m^ ''

k constant defining the j e t velocity profile m mass flmr

m_, stream mass flow 1

m„ jet mass flow at exit r radial distance

Tp jet radius at exit from nozzle

u stream-vra.se velocity u stream velocity u jet velocity at exit

u (x) centre-line velocity in excess of free stream velocity

u m

v(x,r) inflow velocity X stream-wise distance

y distance normal to stream direction B r / in axi-symmetric case

•yx in tiYo-dimensional case

jLi constant defining jet concentration profile p density P. stream density P jet density „ u (x) / U m^ '/n. ~ 1 w (x) sirjk strength ^ stream function

1, Tntrnduotion

The v e l o c i t y exid d e n s i t y d i s t r i b u t i o n s i n an axi-symmetric j e t of one gas i s s t d n g djtto a second gas a t r e s t have been shown experimentally by Keagy and Y/el.l:;.r ( l ) t o foiloTir an e::ponential law downstream of the p o t e n t i a l c o r e , B l o t t n e r (2) lias used t h e s e exponential d i s t r i b u t i o n s i n a moraent'jm integr:..! method t o f i n d the v a r i a t i o n of c e n t r e - l i n e v e l o c i t y and d e n s i t y with distemce downstream of tlie p o t e n t i a l core. His r e s u l t s agree reasonably with t h e e:>-;.;v:'.bnents of Keagy ard Weller.

I n t h e present paper, en the assumption t h a t the same exponential forms of v e l o c i t y and d e n s i t y p r o f i l e s can be applied t o both axi-synmetric and two-dimensional j e t s of one ga3 exhausting i n t o a stream of a second gas, t h e maaentum i n t e g r a l approach i s used t o deduce equations s a t i s f i e d by t h e c e n t r e - l i n e v e l o c i t y a::.", - i ' n s i t y downstream of t h e p o t e n t i a l c o r e , Assuming t h a t t h e c o n s t a n t s givin i n (2) can be used f o r a j e t isstiing i n t o

a moving stream as well as for onoe i s s u i n g i n t o a s t i l l medium, the r e s u l t i n g equations Irnve been solved numerically and t h e r e s u l t s presented g r a p h i c a l l y ,

Sqixire and Trouncer (3) have used a s i m i l a r method with t h e v e l o c i t y p r o f i l e

u = u ^ . ^ O ^ c o s f : )

to solve the restricted problem of a round air jet issuing into a stream of air. This profile does not however give such good agreement with experiment as the exponential profile found by Keagy and Vfeller.

The author is indebted to Mr. P. J, Berry (A.R.D.E., Fort Halstead)

5

-2, The j^xl-symmetric J e t

Consider a subsonic j e t of d e n s i t y P - , speedUp and e x i t radius r^ i s s u i n g i n t o a surrounding stream of another gas of d e n s i t y P and speed u (Fig. 1) . The mass flavf m a t any c r o s s - s e c t i o n i s

CO

m = 2 w / p u r d r ( l )

where p, u are respectively the density and velocity at any point (x, r) downstream of the jet exit. The density P can be expressed in terms of the jet and stream densities and a concentration parameter c defined by

P = P^ + (^2 - ^ ) ° (2)

2 , 1 , The flovf downstream of the p o t e n t i a l core

I t has been shown by Keagy and Yfeller ( r e f , l ) t h a t , beyond t h e p o t e n t i a l c o r e , the v e l o c i t y and concentration p r o f i l e s for a j e t of foreign gas issidng i n t o s t i l l a i r are s i m i l a r and expressible i n terms of escponential functions i n t h e form

u = u j x ) e

c = c (x) e , m^ '

where a = r/x and H and k depend upon the relative densities of the jet and the sxorrounding medivun.

Some velocity distributions in a jet of air issuing into a moving stream of air have been measvired by the author and are shosm in Pig. 2 to be of the f onn

It is reasonable therefore to assume that, for a jet cf foreign gas issuing irto a moving stream of air, the velocity and concentration profiles can be written in the form

-k ^

^ = ^1 + "m^^^ ^^ * ^^^

The determination of /i and k experimentally i s tantamount t o solving t h e energy equation. The values of ju and k p l o t t e d a g a i n s t Pp/ i n F i g . 3

^ 1

a r e taken from ref, 2,

Eliminating P and u between equations 1 - 4 we have

n(x) = 2 vrx^ / ^ (p^ + [p^ - P^j c j x ) e - ' ^ ^ ^ ) ( u ^ 4- u j x ) e - ^ ^ ^ 2 d «

ƒ> 00 2

But / a e " ^ ^ ^"^ = fe (^) •'o

and defining m t h e mass flow i n t h e imdisturbed stream upstream of t h e j e t e x i t as

ĥ eo

m^ - 2ïr / p^ u^ r dr (?)

and m the jet maas flow before mixing as

mp = 2^ 1 1 j Pg Ug r dr = w Pg Ug r^ (8) TiTé may r e w r i t e (5) a s

"^^^^ = "^1 + ^^2 T H ; * i r - r ^1 V^) ^ ^l(''2 - ^1^fm^

r „ „ N U ( x ) C ( x ) "1

Prom considerations of conservation of mass flow

1 ^

ƒ 00 |« eo /«CO

2»^ / ''u (1 - c)u r dr + 2ïr / p^o u r dr = P2 w r g u + 2?^ / P.u r

a

Thus, since P. aind Pp are independent

/ c u r dr = w r- Up (10)

7

-S u b s t i t u t i n g f o r c and u in terms of z from (3) and ( 4 ) , (IO) beoanes 2 - u. u (x) ^2^2 T- O ( x ) k m^ '' u^ u (_x) -] 1 m^ ^ I" "^ fi + 1 J or k rZ n^ ^ (ij + 1 )

°m^^^ - "~2" • ïï^uTTTTJ^rüTx)

The momentum equation for zero pressure gradient is

[^ 2 r^ 2 f"" 2

ƒ p u r dr +/ Pp u r dr = / pu r dr and using (2), (3) and (4) this reduces to

(p^u^ - P^ u2) ^ = ^i^p^ u j x ) [ V ^ + u j x ) ] + 2(P2- P ^ ) c j x ) ^H 2 u^ u j x ) u2 (x) (11) (12) (13) • • ^ /i + 1 ^ + 2 •-' (14)

Eliminating c (x) using (12) v^e have

, 2

P. u (jLi+ 2)(5/i+ 1) + 2 - ^ (p - p ) / i ( ; i + l )

X Jr Y»

+ u^(x) 4 P^ u^ ()ii + 1 ) ( M + 2) + 2/i (JU + 2) - ^ j pg u (2u^ - u^)* ^ X L

+

P^

u-^ (u^

- 2

u^)

J I

U^

k r ,

+ 2 — ^ (/i + ^)(^ + 2)u^(u^ - U2)( Pg Ug + P^ u^) = 0 (-15) I f we novif put u zero ( i . e . t h e j e t issuing i n t o a s t i l l gas) we obtain

u (x) m^ ' ^2 u 2 -J + 2 k •P ^p ^ 2 1 A hJ+j\ IM- 2J \x u (x) m^ ^ Urt 2 k '2 (-^2 P, v x = 0

T h i s i s t h e e q u a t i o n o b t a i n e d b y B l o t t n e r and h a s t h e s o l u t i o n u (x) m^ ' u^ = k

/!i

- P. \ P u t t i n g _{^"2 ^} 73- = cr 1 l"^ 2, + T -2 " 2 - - = U and "^1 " 2 - ' l '2 .

' i1]

Uih 1

1 ^ 2

I

(16) u (x) - ^ = U^ i n (15)

t h e r e l a t i o n betv/een c e n t r e - l i n e v e l o c i t y and d i s t a n c e downstream of t h e j e t e x i t f o r t h e g e n e r a l problem i s 'J 2 u-^ + U m ^ m c ^ 2 k r ^ /J x ^ ^ V • / /i + 2 J + u _m , ^•^. ^ 2 k r ^ 4 - - — + 2 1 - 2a fl - cr| - cru' 2 * - ^ - ^ 0 - U ) ( 1 +a-u) = 0 (17) X ^

which may b e s o l v e d f o r U i n terms of x and t h e p r o p e r t i e s of t h e j e t and s t r e a m . The v a l u e o r u ( x ) so o b t a i n e d irhen used i n (3) g i v e s t h e v e l o c i t y d i s t r i b u t i o n i n t h e j e t and v/ith (12) and (4) i t g i v e s t h e c o n c e n t r a t i o n d i s t r i b u t i o n of t h e j e t g a s ,

E q u a t i o n (l 7) i s n o t r e a d i l y s o l v a b l e i n p r e c i s e f onn. Numerical s o l u t i o r i s h a v e b e e n o b t a i n e d u s i n g t h e v a l u e s of /i and k g i v e n b y B l o t t n e r , I t i s assumed t h a t t h e s e v a l u e s c a n be a p p l i e d t o a j e t mixing w i t h a moving s t r e a m . The r e s u l t s f o r j e t s of heliimi, n i t r o g e n and carbon d i o x i d e

i s s u i n g i n t o a i r a r e g i v e n i n F i g s , 4 , 5 , 6. The e f f e c t of d e n s i t y changes i s shovm i n P i g . 7 .

Some s p e c i a l c a s e s of e q u a t i o n 17 ca.n be s o l v e d e x a c t l y and t h e s e ha-.-e b e e n u s e d t o g u i d e t h e n \ m e r i c a l programme.

9

-2.1.1. Jet and Stream at the same speed (i.e, U = l)

Equation 17 is satisfied by U = 0 which means that the velocity in the

m

mixing region i s uniform. The c e n t r e - l i n e concentration i s 2

o j x ) = /ik f2 (,8)

X

2 . 1 . 2 , Zero J e t Veloca.ty ( i , e . U = O)

This case corresponds t o t h e flov/ behind a bluff base of r a d i u s Tp (Pig, 8 ) . YiTien U = 0 eqijation 17 f a c t o r i s e s t o

2 k r j •2 2

("„ V ^ ) ( « f ^ 4 U „ . ^ ^ ) = 0

w i t h solutions X 2 k r ? \ 2 - 2 The appropriate s o l u t i o n i s J. 2 k r f 2

i-"m = ( ^ - -J^) - 2 09)

which i s alwajTs negative and tends t o zero a s x i n c r e a s e s ,

The v e l o c i t y d i s t r i b u t i o n s given by equations 3 and 19 a r e compared in P i g . 8 v/ith some experimental measureaents made i n the wake of a bluff bodj'- by t h e present author. I t i s shovm t h a t t h e v e l o c i t y p r o f i l e

(equation 3) adequately r e p r e s e n t s the v e l o c i t y p r o f i l e i n the v/alce of a bluff afterbody and t h a t t h e v a r i a t i o n of c e n t r e l i n e v e l o c i t y p r e d i c t e d by equation 19 conforms reasonably t o t h e experimental r e s u l t s . A b e t t e r agreement could be obtained by displacing t h e o r i g i n for x i n equation 19

some small d i s t a n c e upstream of t h e base,

2 , 1 , 3 . J e t and stream a t same d e n s i t y ( i . e . cr = i ) Equation (l 7) can again be f a c t o r i s e d t o give

The appropriate s o l u t i o n i s ^ i

U^ = ( 4 - 2 k j l - U ^ i - | ) - 2 (20) X /

2i'2e The mas_s flCTv; .in t h e j e t

The mass flow across any section of t h e j e t i n c r e a s e s more r a p i d l y than accounted f o r by t h e n a t u r a l inflow i n t o the growing j e t from the e x t e r n a l streain; t h e j e t induces a flow tor/ards i t s e l f by a process of entx-ainment. To determine t h i s inflcrw v/e have t o consider the r a t e of increase of mass flow i n t h e j e t .

Pron (9) the mass flow a t any dovmstream positicai x i s given by 2

m (x) = m^ + m^

^2^2 k

P, ujx) W P 2 - P , ) O J 4 ^ +

^

J

o (x) can be eliminated, using ( I I ) t o give

/ X '"1 1

„i(x) = m^ m^ -rrpq +

k

[''1 ^m(^)

2 k u r

^--PiP^'P^)

(21) •vrfiioh reduces t o

^ =(-S7^)(irT2)\^'l (^){("V^A^

(21 a)

for the jet issuing into still air. The ratio m(x)/mp is plotted against /rp for helium, nitrogen and carbon dioxide jets in Figs. 9, 10, 11,

The rate of increase of mass flow is

7rp

dx

m(x) = -JL I- (x^ u (x) ) ^ ' k ds ^ m^ ' ' (22)

or expressed non-dimensionally in terms of the j e t ms.ss flow and the j e t r a d i u s d (m(x)/m2)

dCx/r^)

f

k P^ u^ . ^ I 2 u^(x) 4.

L

_m* X ^2 m u ' ( x ) (22a)

For t h e j e t i s s u i n g i n t o s t i l l a i r we may s u b s t i t u t e in (24) for u (x) frcan (16) and obtain t h e approximate r e l a t i o n

d(m(x)/m2)

dCv^P

k P, (23)

indicating that the mass flew increases linearly with distance downstream. Pigs, 9, 10 and 11 show that this is also almost exactly true for the jet

- 11

2.3. The inflow velocity

The rate at v/hich the jet mass flow is increasing is related to the strength of a line sink positioned along the jet axis. If w (x) is the sink strength at x, then

1 o r , using (22),

<-'> - i h [:=' ujx)] (24)

Now theinduced v e l o c i t y v ( x , r ) can be w r i t t e n dovm d i r e c t l y frcm t h e

sink s t r e n g t h as / X - 2 w(x) v ( x , r J = , • •J' '• ^ » ^ li. ir r or from (24) .

^==.=^) = - W - ^ [^' "„(:=)) (25)

¥e may arrive at the same result for the induced velocity by

considering the Stokes stream function 5* (x,R) for the flow do^vnstream of the potential core,

-kz

/ 2.f r

* (x,R) = / ur dr = X / u 4. u (x)e 0 r= 0 L z dz v/here z = r / and R i s l a r g e X Using (6) R 2 / X * (x,R) = u r dr 4. ^ ^ J ^ ) •'0 ^ 2 k

The induced v e l o c i t y i s given by - r v = ^ and hence

v ( x , r ) = - ^ ^ ^ [ - ' H n ( - ) ] <2«) which i s t h e r e s u l t of equation 25.

Por t h e j e t i s s u i n g i n t o s t i l l a i r t h i s reduces t o a form independent of X

vüi ^ . a . (_f2 f (2fe)

Ug r • \2 kp^y ^ '

Equation 26a shows immediately that the inflow velocity is increased by increase of jet density. Substitution of values of u (x) into (26)

shows that this restilt holds for a jet issuing into a free stream, Furthermore the induced velocity increases with increase of the ratio

cf jet to free stream speed.

3» The T\70-dimensional Jet

Consider a jet of gas density p^ issuing from a slot of v/idth 2a with speed Up into a free stream of speed u and density p.. At any point X downstream of the potential core we may again write the speed u

of the jet and the density p as

u = u^ + u j x ) f(z) (28) p = p^ + (p^- p^)o (29)

vï^ere z = y, and c i s a concentration parameter. / x

As before we assume t h a t t h e v e l o c i t y and concentration d i s t r i b u t i o n s are similar so t h a t

f ( z ) = e"^^^ (30) and c = oJ,x) g(z) = c^(x) e " (31)

3 . 1 . The flov/ downstream of t h e j e t

Considering tlie conservation of j e t mass flov/ per u n i t length of s l o t we have CO o u dy = 2a u^ (32) - C O 0 0 p 00 2 2 or / ^ ojx) e- ^ [ u ^ + u j x ) e " ^ ] x dz = 2a U2 (53) But (34)

13 -Thus (32) beccmes m^ ' u., ^ (x) 1 m^ "^ + L^/i

^"+r J

2a u _{__2 k} X ^ M r

(35)

From momentum consideraticjns

pu^ dy = 2a Pp Up + 2 P. u dy

2 2 1 1 (36)

Siobstituting from(^), (29),(30) and C5l)in(36) and performing the i n t e g r a t i o n we obtain 2 p^ u^ u j x ) + p^ < / l + 2(P2 - P^)u^ °m(j_;^n(^) ^2 ^i _{ti + 1} ( ^ 0 - ^ , ) 2 ? c (x)u. c (x) u (x) m'- ^ 1 / s m^ '^ m^ ' — p — + KPr, - P.) —rrrr: Ai + 2 2a fk /„ 2 „ 2N

= r- J^(^2^2-''l ^ )

S u b s t i t u t i n g for c (x) from (35) we have a cubic equation for t h e j e t c e n t r e l i n e v e l o c i t y in terms of the distance dovmstream of the j e t e x i t

m' ra

u„(x) [./i u^i^ . ^ / I (u, - up(.,u, . »,u,)]

2 ^ 2 _a

^-1 x^, i ^ ^ t J l u^ (u^ . ug) (P2U2+P^u^) = 0 (37) l^Tien the j e t i s issuing i n t o a gas a t r e s t ( i . e . u = O) equation (37)

which can be solved in the form u (x) m^ ' u„ a X ^ (^ + 2)

r^;^

I

'i

mj1\

^-'

I f T/e vise t h e non-dimensional forms defined previously, t h e general equation of tv/o-dimensional problem (37) beccmes t h e cubic

m

<{-^-*r-^* = ^=" f <-0 ^ji^fHi ]

+ U 2 ^ " _m

| V * i ƒ ! 0 - U)(1 *cru)] . 2 ^2 I ,||^^^(1 - U)(1 .cru) = 0

ËSZUi

(40)

which may be solved for U i n teims of x and the p r o p e r t i e s of the j e t and stream. On the assumption t h a t the constants/J and k given by B l o t t n e r for t h e ajci-symmetric problem a r e applicable t o the two-dimensional problem equation (Z^O) has been solved ntimerically. The r e s u l t s a r e given i n

F i g s , 12, 13, 14. The e f f e c t of d e n s i t y i s shown i n Pig. 15,

As for t h e axi-symnetrio j e t , c e r t a i n s p e c i a l cases of the tv/o-dimensional problem nan be solved.

3*1 . 1 * J e t and Stream a t saane speed ( i . e . U = l )

I S

Eouation 40 i s s a t i s f i e d by U = 0 . The c e n t r e - l i n e concentration m c (x) = _{m^ '}

is

X TT (41) 3 . 1 , 2 , Zero J e t Velocity ( i . e . U = O) When U = 0, equation (40) f a c t o r i s e s t o ( ^m • y + 1 /^ /\ 'm

giving the appropriate solution

Uf + 2^2 U + 2 . p g I j = 0

m

IT X

U

m

2 - 2 1-^ ^ - T/2' iTT X (42)

15

-3 . 1 . -3 . J e t cjid Stream, of t h e same density ( i . 0 . 0 &#-34; = l ) IThen c = 1, equation (40) again f a c t o r i s e s t o

giving t h e solution

r i T

3 . 2 . The inflow v e l o c i t y

The stream function f for t h e flow downstream of the p o t e n t i a l core i s

^ (x, Y ) = / u dy and Y i s l a r g e

° Y

rl 2

= ^ X j (u + ujx) e-^'^ ) ÖZ

The induced v e l o c i t y v i s then given by

Thus, frcm (4^1-),

V = - r

-Ox

V = - -S".

iJk fe (== \(-^ ) C^'S),

T-I f t h e induced v e l o c i t y i s regarded as being dxie t o a d i s t r i b u t i o n of sinlcs along the j e t c e n t r e l i n e , then t h e sinlc s t r e n g t h u expressed a s a function of x i s

4 . Restriotictns on t h e Analysis

Examination of the v a r i a t i o n of c e n t r e - l i n e velcxjity with d i s t a n c e shows t h a t t h e s o l u t i o n becomes w i l d l y inadequate for low v a l u e s of

^ and — , This i s t o be expected since c l o s e t o t h e j e t e x i t t h e p o t e n t i a l oore e x i s t s and t h e assumed v e l o c i t y p r o f i l e does not apply. The equations (l 7) and (40) g i v e , f o r x and x , no r e a l r o o t s for any value cf d e n s i t y

^2 ^

r a t i o and velcxsity r a t i o . As o" and U i n c r e a s e so does t h e r e g i o n of inadequacy of t h e assumed velcxjity p r o f i l e ,

5. Conclusions

1, Exponential velocity and density profiles across a jet of one gas issuing into a stream of a second gas are enployed in a momentum integral approach to determine the centreline velocity and density distributions downstream of tlie potential core. The same form for the velocity profile describes the flow behind a bluff base.

2, Decrease of jet density ratio decreases the centre-line velocity in the jet dovmstream of the potential core.

3. At any distance downstream of the jet exit, decreasing the jet density increases the ratio of mass flux in the jet to jet mass flux at exit, This result becomes more pronoimced at the lower jet densities and the higher jet speeds,

4. The indixsed velocity into the jet is increased by increase of jet density and by increase of the ratio of jet exit velocity to free stream speed.

6. References

1, Keagy, IT.R, and A study of freely expanding inhomogeneoxis Weller, A.E, jets. Proc. Heat Transfer and Fluid

Mechanics Institute, 1949.

2, Blottner, P,G, Effect of Jet Density on Induced Flow around an Axially Syinnetric Jet.

Sandia Corporation Research and Develop-ment Report S C - 4 1 2 7 ( T R ) , October 1957. 3, Squire, H.B,, and Round Jets in a General Stream.

u, P,

FIG. I. DEFINITION OF SYMBOLS IN THE AXISYMMETRIC JET PROBLEM

l o k x l O ' ^ •5 > / HE Lll / y\^ M

y'

^ y ^

y

y ^ ^ ^^ NITROC >

1

;E ^ ' N ^ y ^ ^ ^ CARI DlOX ^ ^ ION IDE -2 •8 I O 1 2 1-4 1 6

FIG. 3. VARIATION OF PROFILE ODNSTANTS WITH DENSITY RATIO REFERENCE 2)

I - o 0 - 9 0 - 6 O - 4 O - 2 \ \ ft \

V

\ - k i *

Ï

\ \ 1 EXPERIMENTAL POINTS O ? = 4 8 V ' ^ ' O + A Q :

V

_{Q >} 5 = 4 8 i l 2 = 2 ^2 " 1 - = 29 - 2 s IO '"2 " 1 5 = 2 9 i 2 » 2 r j u , • ^ . ^ , . , - A O l 0 - 2 , , 1 0 - 3

FIG. 2. VELOCITY PROFILES ACROSS A JET OF AIR MIXING WITH A FREE AR STREAM

FIG. 4. CIRCULAR HELIUM JET VARIATION OF CENTRE LINE VELOCITY WITH DISTANCE

DOWNSTREAM OF JET EXIT. ^/u,

FIG. 6. CIRCULAR CARBON DIOXIDE J E T . VARIATION OF CENTRE LINE VELOCITY WITH

DISTANCE DOWNSTREAM OF JET EXIT

o 3 UJ oc I-z u o - 0 4 -O 8

FIG. 5. CIRCULAR NITROGEN JET. VARIATION OF CENTRE LINE VELOCITY WITH DISTANCE

DOWNSTREAM OF JET EXIT. 'J-Zu,

1 0

FIG.7. EFFECT OF JET DENSITY ON THL CENTRE - LINE VELOCITY DISTRIBUTION

Ol 0 2 0-3 0-4

I O , — »

- O 8

- 0 <

- O 4

FIG. 8. CENTRE LINE VELOCITY BEHIND A BLUFF CYLINDER.

FIG. 9. RATIO OF MASS FLUX IN JET TO EXIT MASS FLUX-HELIUM JET.

" ' " | 4 2 0 8 6 2 ^ ^ ^ VELOCITY RATIO yA ^ - - " ^

"CA

2 0 4 0 60 8 0 £ lOO

FIG IO RATIO OF MASS FLUX IN JET TO EXIT MASS FLUX - NITROJEN JET.

12 U.IO U « 8 U . 6 U>4 U . 2 " j t l " ' I O 8 6 4 2 -VELOCITY RATIO U«ooJ

FIG. I I . RATIO OF MASS FLUX IN JET TO EXIT MASS FLUX-CARBON DIOXIDE JET

0-2 0-4 0-6

0-5 I O 1-5 2 0 2-5 3 0 3-5 4 0 4-5 5 0

FIG. 12. TWO DIMENSIONAL HELIUM JET CENTRE LINE VELOCITY VARIATION.

10»^

u-os

FIG. 14. TWO DIMENSIONAL CARBON DIOXIDE JET CENTRE LINE VELOCITY DISTRIBUTION

O5 10 1-5 2 0 2 5 3 0 3 5 4 0 - 4 5 x 5 0

FIG. 13. TWO DIMENSIONAL NITROGEN J E T CENTRE LINE VELOCITY VARIATION

Uin 0 9 0 8 0-7 0 6 0 5 0 4 0 3 0 2 0 1 A1

V

^ ^ ^ 1 2 =0^6 .0-8 N S < I 2 ^ •6

w

^ ^ 2 0 4 0 6 0 8 0 ac 100 a

FIG. 15. EFFECT OF JET DENSITY ON THE CENTRE LINE VELOCITY DISTRIBUTION.