38

MEKItANIKA ZHIDKOSTI I GAZA

C A L C U L A T I O N O F THE TRANSITION REGION O F A T U R B U L E N T J E T A. S. G i n e v s k i i Izv. AN SSSR. M e k h a n i k a Z h i d k o s t i i Gaza, No. 3, pp. 5 9 - 6 7 , 1966 The presently known methods for calculating plane and axisymmetric turbulent jets in a wake flow are based on dividing the flow region into two segments, initial and basic [ 1 - 3 ] . Here the matching of the parameters of the initial and basic segments is of an artificial nature, since it permits the existence of a physically impossible discontinuity of the curves of the velocity distribution and the jet width along the

=0,

~=0

for

y=5 r

(1.3)

Here t

dum

axis. The aerodynamic characteristics of the transition segment, extending from the point of convergence of the boundary layers at the end of the initial segment to the section corresponding to the point of i n flection of the curve ZXUm*(X), differ significantly from the characteristics of the initial and basic segments. This difference is due not only to the sharp increase of the velocity pulsations, but also the marked deformation of the average longitudinal velocity component profile. Consequently, the calculation of the transition segment, in contrast to the initial and basic segments, cannot be based on the single-parameter method. Generally speaking, the flow development in the transition segment m a y be calculated with the aid of the method [4], which reduces the solution of the problem to an equation of the heat conduction type and assumes the use of an experimental curve of the velocity distribution along the jet axis. Abramovich has carried out the c a l c u lation of the transition segment of a plane submerged jet on the basis of certain assumptions which are based on the results of experimental

With the a i d of (1.3) the p o l y n o m i a l (1. 2) m a y be r e p r e s e n t e d in the f o r m = aj8 ('1 - - 4'1~ + 3~1s) + x/#8 a (TI3 - - 2q ~ + ~1~) , (n = y/~).

T = pX~Ulm~

paul,,, (u p x u , ~ (us -

i. Velocity profiles in the transition segment of a jet. The equations of motion and continuity in plane 0 = O) and axisymmetric (j = i) turbulent boundary l a y e r s of an i n c o m p r e s s i b l e g a s h a v e the f o r m Ott

sou

.

"~

~_l_, ] v

(1.2)

ely + c~Y~ + . . 9 + cnyn.

O~

Oy---r

(1.7)

where

I1 ('1) = 1/~'1~ _~1.'1~ + 11~ '1e, I~ ('1) = 1/,'1~ _ 2/~'15 + 1/,'16.

(1. S)

F r o m (1.7) we o b t a i n the e x p r e s s i o n f o r the d i f ferential velocity profile

as~ --~y~ =~=/=0

(1.9)

To s i m p l i f y (1.9) we i n t r o d u c e the new p a r a m e t e r

(i. i) O~ -~y =cq,

u,,) = ai8 1Is + 1/s~83 i/so ,'

(1.1)

We d e t e r m i n e the c o e f f i c i e n t s of t h i s p o l y n o m i a l f r o m the c o n d i t i o n s on the j e t a x i s (y = 0, u = u m) and at the edge of t h e j e t (y = 6, u = u6) with a c c o u n t f o r

"r

u,~) - - a j ~ t l ('1) + 11#8~1~ ('1),

u - - u~ 11 (n) + (~82/6aj) 12 01) A u ~ = Urn-- Us = 5 l + {~8~172aj) "

H e r e x , y a r e t h e c o o r d i n a t e s of a C a r t e s i a n (j = 0) o r c y l i n d r i c a l (j = 1) c o o r d i n a t e s y s t e m , u , v a r e t h e c o m p o n e n t s of the a v e r a g e v e l o c i t y along t h e x , y a x e s , r is the R e y n o l d s s h e a r s t r e s s , and p is t h e gas density. To c o n s t r u c t the v e l o c i t y p r o f i l e s in the c r o s s s e c t i o n s of the j e t we r e p r e s e n t t h e p r o f i l e r(y) in the f o r m of t h e p o l y n o m i a l ~c = c o +

(1. 6)

O'~

p u ~ - + p v - g { - y - - 1 -~- ~- O--y-' o~ 0" 0--7 -}- - ~ - = -

(t~lm = q: (u~ -- u~)),

a f t e r s i m p l e t r a n s f o r m a t i o n s we have

studies [i]. Below is presented an approximate method of calculating the transition segment of plane and axisymmetric turbulent jets in a wake flow in which the velocity profiles obtained for the extreme sections of this segment are used for calculating the flow parameters in the initial and basic segments.

(1, 5)

E q u a t i n g t h i s e x p r e s s i o n and the e x p r e s s i o n (here and in w h a t f o l l o w s the u p p e r s i g n c o r r e s p o n d s to t h e c a s e u m < u6 and t h e l o w e r to the c a s e Um > u6)

for

y=O,

- [ ~176

1

9

U s i n g (1.9) and (1.8) it is e a s y to show t h a t t h i s p a r a m e t e r i s v e r y s i m p l y r e l a t e d to t h e p a r a m e t e r fl 6 z / 6 a j [3@2 "~-1.

(i. io)

U s i n g (1.10) the v e l o c i t y p r o f i l e (1.9) m a y be w r i t t e n in t h e f i n a l f o r m A u ~ = i - - 601~ 0]) + ~ [/1 ('1) - - i2/~ ('1)]. (1.11)

FLUID

DYNAMICS

39

Now l e t us t u r n to the d e t e r m i n a t i o n of the r a n g e of p o s s i b l e v a l u e s of t h e p a r a m e t e r X. R e q u i r i n g t h a t at t h e b e g i m d n g of t h e t r a n s i t i o n (end of i n i t i a l ) s e g m e n t the v e l o c i t y v a r i a t i o n a l o n g t h e a x i s d o e s not u n d e r g o a d i s c o n t i n u i t y , we d e t e r m i n e t h e v a l u e of t h e p a r a m e t e r X = X0 at t h e b e g i n n i n g of t h e t r a n s i t i o n (end of i n i t i a l ) s e g m e n t . T o do t h i s we m a k e u s e of t h e s e c o n d of the r e l a t i o n s (1.7) and ( 1 . 1 0 ) , h e n c e t/ t Jn

Z)ltlrn 2 " 9

d (•

= ~ ~

~"

(1.12)

.~ ou g~ dy , 0

and a l s o Eq. ( 1 . 1 2 ) ,

w h i c h we w r i t e in the f o r m umdu m

d (•

= :F 5 2ik--~,n

(2.3)

.

S u b s t i t u t i n g (1, ii) in ( 2 . 1 ) , we find the e x p r e s s i o n for the jet half-width 1

+ ~oo [ /,' " d )

w h i c h c o r r e s p o n d s to X00 = - 1 0 . X v a r i e s in t h e l i m i t s --

10 ~

~

~<

d)l

Au ~

T h u s , the p a r a m e t e r

O.

= 1 - - 15~1~ -F 24,1~ - - t0~ ~ -- 5~t~ + 15~14 -- 16~1~ -~- 5~1a

(i.

i3)

~7

--

0.4

0.8

1.2

1.5

Fig. I

%/;

~0

With t h e a i d of (1.11) and (2.4) we t r a n s f o r m and {2.3) to t h e f o r m

B=0), (~ = - - ~0).(1.14)

d (nx ~ = - -

I T

(2.2)

1

[ T K / 2gJ601+JLt52]1+i • J

Figure 1 shows the velocity profiles Au~ k) in the transition segment for X = -10 to 0; also shown are the experimental points [5] corresponding to the initial and basic segments of an axisymmetric submerged turbulent jet, and also the velocity profile of the basic segment of an axisymmetric turbulent jet [3] calculated using the equation hu ~ = t

(2. ,~)

= 0,

The resulting expression for the velocity profile for t h i s r e g i o n of v a r i a t i o n of t h e p a r a m e t e r X w i l l d e s c r i b e t h e d e f o r m a t i o n of t h e v e l o c i t y p r o f i l e s w i t h i n t h e l i m i t s of t h e t r a n s i t i o n s e g m e n t i f t h i s e x p r e s s i o n f o r X = X0 and X = X00 is in a g r e e m e n t w i t h t h e v e l o c i t y p r o f i l e s in t h e i n i t i a l and b a s i c s e g m e n t s of t h e j e t . E q u a t i o n (1.11) t a k e s the f o r m

Au ~ = i

(~ = u~ / ui,.) 9

60//" (9 +

-

~2//"

-

1

5 = [ ~ K l 2 n i l i ~ { u ~ m [~A~ (~,) :F :'l~ (~)1} ~+~ '

S i n c e d u m / d x = 0 at t h e end of the i n i t i a l s e g m e n t , t h e r e f o l l o w s Xo = 0. T h e u p p e r l i m i t of t h e p o s s i b l e v a l u e s of X is t h e r e s u l t of the f a c t t h a t f o r l a r g e n e g a t i v e v a l u e s of t h i s p a r a m e t e r t h e e x p r e s s i o n (1.11) f o r t h e v e l o c i t y p r o f i l e g i v e s p h y s i c a l l y i m p o s s i b l e v a l u e s of t h e v e l o c i t y n e a r t h e u p p e r e d g e of t h e j e t . T h i s v a l u e X = X00 m a y be found f r o m the r e l a t i o n

(O'~Au~ / 0~f)~=, =

(2.2)

o

X A4 -~ (~,) ~3-i [~A~ (~,) ~: A~ (~)1 ~+j dF (~, ~) ;

(2.5)

1

6~]2 + 8~1~ -- 3~1~ ,

0~ = -- i2). It follows from this figure that the family (1.11) of velocity profiles for ). = 0 and X = -10 satisfactorily describes the velocity profiles in the initial and basic segments of the turbulent jet. Thus, within the framework of the approximation used here, it is shown that the velocity profiles in the transition segment are the same for both the plane and axisymmetrie jets and also for the corresponding jets in a wake flow.

F (~, k) = [3A2 (~) -T 2~A~ (X) =F ~-IAs

()~)1 [~Ai

(~') 2}- A2

(~')]-~"

Here 1

& (~) =- ! Au~

: kl, + ka~k,

1

2. V a r i a t i o n of t h e j e t p a r a m e t e r s a l o n g t h e flow. To d e t e r m i n e t h e law of v a r i a t i o n of t h e j e t p a r a m e t e r s a l o n g i t s a x i s w e m a k e u s e of t h e s e t of i n t e g r a l r e l a t i o n s f o r m o m e n t u m and e n e r g y (the G o l u b e v integral relations)

A ~ ()~) =

i

(Au~ nj d~l

1

A:; (~) - f (Att~

2 ~ I u (u - - u~) g~ dg : cons~, 0

(2.1)

-= k:,, + :~k~,)~ -t :~k,.y ',- ~,,:A:~ ,

{I

A4 ()~) := i (Au~ K

k,.,a -t- 2kuk § k...QC-,

0

It

~1; d~l = nl -~- t~e)~q lh~M,

(2.7)

40

MEKHANIKA ZHIDKOSTI I GAZA

where

f r o m unity at the n o z z l e exit to z e r o at infinity. T h u s , 1

ki~ = I (t - - 60/2) ~(/1 - - t2/~) ~~1~d~l

m

(i, k = O, 1, 2, 3),

1

m

~:Tl__---mA~mo,

u~

~ 0 = - T l - ~ - -~,

u~"

m=

0 1

1

n, = 36001)r

~l~d~l,

n~ = t20 1 (121(* -- I,'1.') n ~dn.

o

0 1

n~ = l (/(~ + t44/~'~ -- 2 4 t ( / ( ) ~ d~.

E x p r e s s i o n (2.8) is s i m p l i f i e d c o n s i d e r a b l y in the p a r t i c u l a r c a s e of the s u b m e r g e d j e t (u 6 = 0, ~ = m = = 0) lk

o

Au~o - -

= oxp[-- ~

Um

-- ~

~A~(D

dA],

d (2--/)A4(k)+

~As(Z)

Ao

(~) A(~) = As A~--~"

The v a l u e s of the c o e f f i c i e n t s k i k and n i f o r the p l a n e and a x i s y m m e t r i c p r o b l e m s a r e g i v e n in the t a b l e .

If the r e l a t i o n h(~), defined by Eq. (2.8), is known, the d e p e n d e n c e ~ (~x ~ m a y b e d e t e r m i n e d b y i n t e g r a t ing the differential relation (2.5) 1

1

2~J60i+Ju 2 F(~,X)

• ~/

d2

•

j

(~) ~-J [~A~ (~) -T A, (~)l~+~--dF(~, U. (2.9)

H e r e X 0 i s the leng"~h of the i n i t i a l s e g m e n t of the j e t . A s s u m i n g t h a t the v e l o c i t y is c o n s t a n t and equal to u 0 at the n o z z l e e x i t , we o b t a i n

,vx ~

g

i--r~

T 2~J~ol+Ju~2

(1 + / ) mS

9

E x p r e s s i o n (2.9) is s i m p l i f i e d c o n s i d e r a b l y in the p a r t i c u l a r c a s e of a s u b m e r g e d j e t Z

~I

.vx ~ ~2

(5~ - x~176= -

l -

2_11+I/.# ~ A~/2J00 ~0 ( A ~ ~ ~) dx,

Fig. 2 = ~As (~) hu~ ~ Equating (2.5) and (2.6), we o b t a i n the e x p r e s s i o n w h i c h e s t a b l i s h e s the c o n n e c t i o n b e t w e e n the p a r a m e t e r s ~ and k FIX,~) --

~o =

~(~!,~o~~E (~, ~) df (~,, ~),

E ( ~ , ~) =

21_~ (~ ~_ 1) Aa (~,)

9

(2.8)

The function F(X, ~) is defined by Eq. (2.7). The initial values k = k 0 and ~ = ~0 correspond to the beginning of the transition segment. The parameter ~ is expressed very simply in terms of the dimensionless velocity AUm ~ = (Um -- U6)/(u 0 -- U6), which varies

i=o kl0

kol k2o kil kso

0.57143 0.01905 0.47180 0.02520 0.569.10 -8 0.41980

H e r e • 0 c o r r e s p o n d s to the i n i t i a l v a l u e s X = 0 and Aura ~ = 1, and the r e l a t i o n X(AUm~ i s found with the a i d of (2.8). T h u s , the f o r m u l a s p r e s e n t e d above allow the c a l c u l a t i o n of a l l the c h a r a c t e r i s t i c s of the t r a n s i t i o n s e g m e n t of a j e t in a wake flow, AUm~176 6(~X~ and the v e l o c i t y p r o f i l e , f r o m the s e c t i o n x = x0 c o r r e s p o n d i n g to X = 0, to the s e c t i o n x = x . c o r r e s p o n d i n g to X = - 1 0 . 3. C a l c u l a t i o n of the p a r a m e t e r s of the b a s i c s e g m e n t of a j e t . As i n d i c a t e d a b o v e , the v e l o c i t y p r o f i l e (1.11) f o r the v a l u e of the p a r a m e t e r ~ = 210 quite s a t i s f a c t o r i l y d e s c r i b e s the c o r r e s p o n d i n g v e l o c i t y p r o f i l e of the b a s i c s e g m e n t of the t u r b u l e n t j e t

j=t 0.17857 0.00893 0.12238 0.01007 0.264.t0-s 0.09706

(2.10)

i=o k21 k12 ko3 nl n2 n3

0.02896 0.116.10 -~ 0.188.10 -4 t.55844 0.08750 0.00810

i=l 0.0i047 0. 495.10 -3 0 . 8 6 2 . i 0 -~ 0. 90909 0.07360 0. 003(18

FLUID DYNAMICS

41

in a wake flow. T h u s , s e t t i n g k = c o n s t = - 1 0 , we can with the aid of (2.9) and (2.10) c a l c u l a t e the v e l o c i t y v a r i a t i o n along the jet axis for v a l u e s of x >- x , . F o r X = c o n s t the i n t e g r a t i o n in (2.9) and (2.10) m a y be p e r f o r m e d in closed f o r m , as a r e s u l t of which we obtain the c o r r e s p o n d i n g c o m p u t a t i o n a l f o r m u l a s for the p l a n e and a x i s y m m e t r i c j e t in a wake flow: in the p l a n e jet c a s e (j = 0) •

L,, (.~) =

- -

~176

~_ 1~

= ~i--m[L0(~)__L0(~,)l;,n

a21~

+

t~21

2a21

(• a~l

- -

+'3a~te(a~tTa~.)ln(ai~q:l)!,

a~)

+

~T~:v-~

ai~=a~/a~;

and in the case of the axisymmetric jet the variation of the velocity profile within the limits of the initial segment may also be neglected [I], this profile (Eq. (i. ii)) may be used for calculating the initial segment.

i

<%

(3.1) ~6

1)

+

4

(3,2)

84

1 in the a x i s y m m e t r i c jet c a s e (j = 1)

I

O

0.08

0./~

0.20

m.s

"

0.32

~

~

i 3r

i ~

I

048

Fig. 4

[ - i~--J m F

•176176

G'Iz

[L~(~)--L~(~,)I;

(3.3) To do this we need only introduce in place of 77 = y/6 another dimensionless transverse coordinate

2 ~ ] [bi~ :7: b~]-'i: , + 4 ( @ b , , - - b~,) ~ :t: ~-

bi~ = bi f G.

(3.4) Y - - Y0 __ Y - - Y0

The v a l u e s of the c o e f f i c i e n t s a i and b i a p p e a r i n g h e r e a r e equal for j = 0 and j = 1 to the c o r r e s p o n d i n g v a l u e s of A i d e t e r m i n e d u s i n g (2.7) for k = - 1 0 . E q u a t i o n s ( 3 . 1 ) - ( 3 . 4 ) a r e s i m p l i f i e d in the c a s e of a s u b m e r g e d j e t , when the p a r a m e t e r ~ = 0 1

• (x~ - - x * ~

--

(2 - - / ) 2 ~+V-4 •

w h e r e Y0 and Y5 a r e , r e s p e c t i v e l y , the d i s t a n c e s f r o m the j e t axis to its i n n e r and o u t e r edges. The v e l o c i t y p r o f i l e (1.11), t o g e t h e r with the i n t e g r a l r e l a t i o n s (2.1) and (2.2), p e r m i t s c o m p l e t e c a l c u l a t i o n of the p a r a m e t e r s of the m i x i n g zone within the l i m i t s of the i n i t i a l s e g m e n t , and also its length 34 X0.

x

A~(-- 10)

A4 ( - - t0) A~ ~-'1~ ( - - t0)

[(Au.~~ ~-~"- - (Au~.) ~-~1

(3.5)

We write the corresponding for the plane and axisymmetric plane jet

It m u s t be noted that the c u r v e s Aura ~ (x) defined by Eqs. (2.9) f o r x - < x , and by Eq. (3.1) o r (3.3) for x -> x , a r e s m o o t h at the s e c t i o n x = x , ; the c u r v e s of 6 (x) have a d i s c o n t i n u i t y at x = x , , which is due to the d i s c o n t i n u i t y of the c u r v e Mx) at x = x , .

!-- !I

formulas

~pl+ ~2m + q~sm~

(4.

~4X0~ = ~ 2a 4 (t - - rn)'-' [(at - - a2) m @ a2] ; T1 =

a3 - - a2,

%= Yo -= i - -

I // ..........!t, i -'fi -%/+

computational jets:

ux~

% =

I)

- - a l -+- 3 a 2 - - 2 a a ,

al--2a2+a3,

~

5

2aa(i--m)"

a x i s y m m e t r i c jet xz ~ =

~ (a~ d~ (8~ +(i ~ o

(p (5~ = [2 (b4 - - a,~l) 6~ + 2a~ gT + r (5~

0

~9

~8

ZZ

l~

,

Zd , ( a o) = 7~(~o? + ~ a ~ Vt + ~(a~ ~- ,

Fig. 3

TI ~

4. C a l c u l a t i o n of the p a r a m e t e r s of the i n i t i a l s e g m e n t of a jet. F o r the v a l u e of the p a r a m e t e r X = 0, the v e l o c i t y p r o f i l e (1.11) d e s c r i b e s quite s a t i s f a c t o r i l y the c o r r e s p o n d i n g v e l o c i t y p r o f i l e of the i n i t i a l s e g m e n t of p l a n e and a x i s y m m e t r i c t u r b u lent j e t s . Since within the l i m i t s of the i n i t i a l s e g m e n t of the p l a n e jet the v e l o c i t y p r o f i l e is s t r i c t l y c o n s t a n t as a r e s u l t of the s e l f - s i m i l a r i t y of the flow,

i

-~%r @ I-~(~i~12 -- (~2~I -~

(4.3)

~3,

Here a,=(~o-~ t)(+2G--I),

~, = (a,~o :4- a,D / (G :4: l),

~..= §

2-3a~G

~-.~,.

G = (lqG =i G) / (G -i 1).

42

MEKHANIKA

kO

au;

-'\

-

'

, ~*"

08 -5 -

~ r

x x" 0

8.04,

g,08

0.12,

0./6

I GAZA

plane and circular jet satisfactory agreement of theory and experiment in all three segments, initial, transition, and basic, may be obtained by specifying only the single constant x. The isotachs shown in Fig. G for the flow in the initial, transition, and basic segments of the plane jet (solid lines) indicate that in the first approximation the outer boundary of the transition segment is a straight line and is a continuation of the outer boundary of the initial segment (dashed lines). In the same way the isotachs for the flow in the transition segment for Au~ -< 0.75 are quite close to straight lines which are the continuation of the rectilinear isotachs of the initial segment. It was this particular type of assumption about the nature of the flow in the transition segment that was made by Abramovich [1] in calculating the transition segment of the turbulent jet.

/f!

08

ZHIDKOSTI

020

Fig. 5

Equations

(4.3) permit

calculation

5~ ~ up to the maximal the initial segment

value 5~

6~a~-

of the relation

9(0

i

)

)

-

i

"

x at the end of

( 2 ~ ) "~/'.

Then r176 ~XO ~

=

q) (5~ d*,p (~~

f o

y0 ~ -----] / i

-J- r (50) ` - - ~z(5 ~

(4.4) ~/5

0 The values i n (4. i ) - ( 4 . 4 ) tively,

of the coefficients a i and b i appearing are equal for j = 0 and j = 1, respec-

to the values

of A i determined

QY2

O48

Fig. 7

for X = 0 using

Eqs, (2.7).

Ill

In conclusion, we present the resuks of the calculation of a plane turbulent jet in a wake flow. These results are shown in Fig. 7 for values of m = 0, 0.25, and 0.5. Figure 8 shows the variation of the parameter k versus Aura ~

.....

m

ZO

002

~Y~-~

008

0.08

050

= 75

OZJ

m=O

O.IO .

Fig. 6 og

O~

Fig. Figure 2 shows the boundaries of the mixing zone of the initial segment of plane (solid lines) and circular (dashed lines) jets for various values of the parameter m . Figure 3 shows the relative length of the initial segment x0o = x0/(X0)m=0. 8. Analysis of results. Comparison of theozy and experiment. Figures 4 and 5 preseut the curves of aUm~ ~ and 8~ ~ for plane and axisymmetric submerged jets, and the points corresponding to the extreme values of the parameter k. We see that the transition segment in which the restructuring of the basic velocity profile t e r m i nates has an extent commensurate with the length of the initial segment. These same figures show the data from measurements of the velocity at the jet axis [1]. The measurements of the velocity along the axis of the plane turbulent jet in the initial, transition, and basic segments (open circles in Fig. 4) were made by the author and K. A. Pochldna. We see from these figures, in particular, that for both the

lO

8

Thus, with the aid of the method discussed here, we can calculate the continuous deformation of the velocity profile in the transition segment and can also determine the flow characteristics in the initial and basic segments of a turbulent jet.

REFERENCES i. G. N. Abramovich, Theory of Turbulent Jets [in Russian], Fizmatgiz, 1960.

FLUID DYNAMICS 2. H. B. Squire and J. Trouncer, Round jets in a general s t r e a m , AR R. a. M. 1974, 1944. 3. A. S. Ginevskii, "Turbulent nonisothermal jet flows of a compressible gas, " in collection: Industrial Aerodynamics [in Russian], Oborongiz, no. 23, 1962. 4. L. A. Vulis, "On the calculation of free t u r bulent flows with the aid of the equivalent problem of

43 heat conduction t h e o r y , " Izv. AN KazSSR, ser. energeticheskaya, no. 2(18), 1960. 5. T. F. Taylor, H. L. Grimmer, and E. W. Comings, Isothermal free jets of air mixing with air, Chem. and P r o g r e s s . Engng. 74, no. 4, 1951. 1 March 1965

Moscow