49810685 combustion physics by c k law

8/3/2019 49810685 Combustion Physics by C K Law

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A E R O D Y N A M I C S T U D Y O F T U R B U L E N T B U R N I N G F R E E

J E T S W I T H S W I R L

N. A. CHIGIER* AND A. CHERVINSKY

Department o f Aeronautica l Engineer ing , T echnio n- - I srae l Ins t i tu te o f Technology , Ha i fa , I srae l

Velocity and temperature measurements have been made in a series of turbulent swirling

free flames. Premixed butane-air jets issued from a round orifice with exit velocities of 60

m/see and the degree of swirl was varied by varying the ratio of the flow rates of air supplied

tangentially and axially to the burner. The flames were stabilized some 4 diameters from the

burner exit in the shape of an annular ring and were unconfined for a distance of 24 diameters.

The measurements made in this region for 3 different degrees of swirl show that the decay of

axial and swirl velocities is slower in a flame than in cold swirling jets. The turbulent burning

velocity was found to be 80 times greater than the laminar burning velocity and the velocity

gradient in the outer region of the jet where the fl~me was stabilized was 250 cm-L

In the theoretical analysis the turbulent equations of motion, energy and state have been

integrated to obta in expressions for the decay along the axis of the maximum values of axialvelocity, swirl velocity and temperature. Variations in flame shape are described in terms of

velocity decay constants and Gaussian error curve constants for radial spread of temperature

and velocity.

I n t r o d u c t i o n

The introdu ction of a swirling motion to air and

fuel streams is applied extensively in combustion

chambers and industrial furnaces in order toobtain flame stabilization in high velocity

streams and in order to control the size and shapeof the flame. The swirling motion is usually set

up by the introduction of vanes or by tangential

air entries in the burner. A number of experi-

mental investigations on flames with swirl

carried out at the Intern ationa l Flarae Re-

search Found ation in Holland *-4 together with

experimental and theoretical studies of swirlingair jets5-7 have shown that swirl lcads to an

increase in flame angle and results in wider and

shorter flames according to the degree of swirl.

These previous studies have also shown that

closed vortices are generated in the internal

region close to the burner exit and that the size

and strength of these vortices are dependentupon the axial and radial pressure gradients set

up in the jets. For the case of pressure-jet oil

flames, the need to adjust the degrce of swirl tothe fuel spray angle has been found and Kerr s has

suggested an optimum swirl number of 0.2.

Lee ~ has recently studied the swirling plume andshown the combined effects of swirl and buoy-

ancy on the behavior of vertical heated jets.

* Present address: University of Sheffield, De-

partment of Fuel Technology and Chemical Engi-neering, Sheffield, England

The stu dy reported in this paper deals with the

aerodynamic behavior of a series of free but an e-

propane-air flames with swirl. The flames were

of the general type encountered in industrial

practice and were stabilized some 20 cm fromthe burner exit. Air was entrained from the

stagnant surroundings having an ambient temper-ature of 30~ The average velocity at the orifice

was of the order of 60 m/see, and under these

conditions the buoyancy effect was small in the

region up to 24 orifice d iamete rs in which temper -

aturc and velocity measurements were nmde.

The theoretical analysis is based upon the inte-

grated forms of the Reynolds' equations of

motion for the case of a noni sothc rmal axi-symmetric swirling jet. Applying the assu mption

of similarity, theoretical expressions are obtained

for the decay of the maxima of axial velocity,

tangential velocity and temperature along the

axis of the jet. The expressions obtained include

the isothe rmal swirling je t as a special case. ~

T h e o r e t i c a l A n a l y s i s

Consider a cylindrical coordinate system withx, r, and r as the axial radial and azimuthal

coordinates. Let u, v, and w be the time mean

velocities in the x, r, an d r directions, respec-

tively, and let u' , v ' , and w' be the corresponding

turbulent fluctuating velocities. For the region

of the jet, where the b oundar y-laye r approxima-

tions are valid, and the molecular viscous termsare negligible in comparison with the tur bule nt

489

4 90 F I R E R E S E A R C H

v i sco u s t e rms , t h e g o v e rn in g eq u a t i o n s fo r t u rb u l en t a s i x y mm et r i ca l s t a t i o n a r y f lo w b eco m e:

Co n t in u i t y :

( O / O x ) ( r p u ) + ( O / O r ) ( r p v ) = O . (11

M o m e n t u m :

( O / O x ) ( p u 2 ) -[ - r - 1 ( O / O r ) ( r p u v ) = - - ( O p / O z ) - - ( O / O z ) ( p - ~ - r u ( p ' v ' I - ~ - r p ( u ' v ' ) l , (2 1

- - p w ~ / r = - - ( O p / O r l - - r - l ( O / O r l ( r p ( v' 2 > + 2 r v ( p ' v ' ) ) + P ( W ' 2 1 + 2w (p'w '___~ >, (3 )r r

( O / O x ) ( p u w ) - ~ - r - ' ( O / O r ) ( r p v w I + ( p v w / r ) = - - r - ' ( O / O r ) r ( p ( w ' v ' > + v v ~ + -[- v ( p ' T ' > ) ' ] . (5)

S t a t e :

p = p R T . ( 6 )

In Eq s . (1 ) - (6 ) p , p , an d T d en o t e t h e p re ssu re , d en s i t y , an d t emp era tu re , r e sp ec t i v e ly , an d t h especif ic he at c~ was assu me d to be cons tan t .

Th e fo l l o win g b o u n d a r y co n d i t io n s p rev a i l :

A t r = 0 :

v = w = O u / O r = O

A t r = o o :

u = w = O u / O r = O w / O r = O T / O r = ( p ' u ' ) = ( p ' v ' ) = ( p ' v ' T ' ) = ( u ' v ' )

= ( u ' 2 > = 0 . ( 7 )

M u l t i p l y E q . ( 2 ) b y r a n d E q s . ( 3 ) a n d ( 4 ) b y r . I n t e g r a t i n g w i t h r e s p e c t t o r f r o m r = 0 t o r =

o o, u s i n g b o u n d a r y c o n d i t i o n s ( 7 ) , y i e l d s

( d / d x ) r i o u 2 ~ - ( p - - p ~ ) J r- O - {- 2 u ( p ' u ' > ' ] d r = 0 (8)

f ~ / ~[ p w + p ( < w ' 2 > + < v ' 2 > ) - ]- 2 ( w ~ ' w ' > - -~ ( v p ' v '> ) ] r = - - 2 ( p - - p ~ ) r d r ( 9 )

" 0

( a / a x ) r ~ p ~ w a r= O. (10 )

By in t ro d u c in g Eq . (9 ) i n to (8 ) we o b t a in

( d / d x ) r { p ( u - 8 9 ) a L p [ ( - 9 + ( v " > ) ]

+ 2 u ( p ' u ' > - - < v p ' v ' ) - {- ( w p ' w ' > } dr = 0. (11)

M easu rem en t s o f tu rb u l en t v e lo c i t ie s i n j e t s an d wak es ~show th at (u '2>, (v '2>, and (w'2> are of thes a m e m a g n i t u d e . M e a s u r e m e n t s o f t e m p e r a t u r e v e l o c it y c o r re l a ti o n s b y C o r r si n a n d U b e r o iu in



T U R B U L E N T B U R N I N G F R E E J E T S W I T H S W I R L 491

h e a t e d j e t s in d i c a te t h a t (vp'v '> i s o f t h e s a m e m a g n i t u d e a s u ( p ' u ' > . Assum ing on t he bas i s o f t heava i l ab l e expe r imen ta l ev idence t ha t u ( p ~ u ' ) " ~ v (p ~ v ) ~ w ( p ' w ' ) , Equ a t ions (11 ) and (10 ) reduce t o

a n d

/ :d / d x ) r E p ( u 2 - - 89 d r = 0 ( 1 2 )

( d / d x ) r 2 p u w d r --- O . (13)

Equa t ion (12 ) i s an exp ress ion o f t he con se rva t i on o f t he ax i a l f lux o f l i nea r mom en tu m and E q .(13 ) exp resses t he conse rva t i on o f t he ax i a l f lux o f angu la r m om en tum .

At some d i s tance f ro m the bu rne r ex i t a fu l l y deve loped reg ion is e s t ab l ished i n wh ich t he ve loc it y ,t empera tu re , and dens i t y p ro f i l e s have s imi l a r o r a f f i ne fo rms . Our expe r imen ta l re su l t s show unde rwha t cond i t i ons and t o wha t ex t en t t h i s s imi l a r i t y i s ob t a ined . We may then choose t he fo l l owingsepa ra t i on o f va r i ab l e s:

u = u ~ ( x ) u ( ~ ) ,

w = ~ ( x ) w ( ~ ) ,

p - p ~ = ( p ~ - p o o ) p ( ~ ) .

T - To~ = ( T m - T = ) O ( ~ ), (14)

where ~ - - r / ( x ~ a ) and u~ , w in, T ,~ , pm a re t he m ax im um va lues o f ve loc i ty , t em pera tu re , anddens i t y a t each ax i a l s t a ti on . Th e cen t e r o f t he o r if ice a t t he bu rne r ex i t is t ake n a s t he o r ig in o f t hecoo rd ina t e sy s t em. Th e e f fec t i ve po in t o r ig in o f s imi l a r i t y i s a t d i s t ance 'a ' from the ori f ice and theva lue o f 'a ' i s dependen t up on the d i am e te r and fo rm o f t he o r if ice.

In t roduc ing the re l a t i ons (14 ) i n to Eqs . (12 ) an d (13 ) and rea r rang ing t e rm s we ob t a in

a n d

( d / d x ) [ ( x - l - a ) 2 { u ~ 2 ( K 1 - ~ - p - ~ ) ] } ] ( 1 5 )

( d / d x ) { ( x + a )3 [(p P --~ -- l + K 4 ) u ~ w m ] } = O , (16)

w h e r e/ ~ 1 , g~ , /~ 8 , a n d / ~ 4 a r e f o r m f a c to r s d e p e n d e n t u p o n t h e f o r m o f th e v e l o c it y t e m p e r a t u r e a n ddensi ty profi les and are defined as :

g~ = ~ fo~~ / fo~p(~)u~(~)d~;

= / / :

In t eg ra t i on o f Eq . (16 ) w i th re spec t to z y ie ld s ,

C1

~ w ~ = ( x + a ) ~ E g , + ( ~ / p ~ ) - 1 ]"(17)



4 9 2 F I R E R E S E A R C H

I n t e g r a t i o n o f E q . ( 1 5 ) a n d e l i m i n a t io n o f w ~ 2 w i t h t h e a i d o f E q . ( 1 7 ) y i e ld s a q u a d r a t i c e q u a t i o nf o r u ~ ~:

(z + a ) * [ ' R ~ + ( p ~ / p ~ ) - 1 ] u 2 - C ~ ( z + a )~ 'u ~ , ~ - - -C , 2 1 ~ 2 + I~3 [ ( p ~ / p ~ ) - - 1 ]

( x--[-

a ) 2 F I ~ ' ,+

( p m / p ~ ) - - l ' l 2 - - -- O,( i s )

w h e r e C 1 a n d C 2 a r e c o n s t a n t s o f i n t e ~ a t i o n w h o s e v a l u e s a r e d e p e n d e n t u p o n t h e d e g r e e o f s w i rl .A s o l u t i o n o f E q . ( 1 8 ) g i v e s

w h e r e

~21 '2 f l12

U m = ( X + a ) [ 1 ~ + ( p ~ / p = ) - - 1 11 1~ '

f = I + ~ [ 1 J r - \ ~ ] (2 C i /C '- '~ 2 [ ]~ 1 - ~ ( p m / p ~ ) ~ , -- ~ _ 1-]l~'2-1-(p~_p~{3F(p~/p~)_ - I ] ] 1 ' 2 "

S u b s t i t u t i n g E q . ( 1 9 ) i n to ( 1 7 ) g i v e s

c ~ [ ~ + ( p ~ / p ~ ) - 1 ] ' , ~ - f - ' ~

w m = ( x - ~ - a ) 2 E K 4 - I - ( p ~ / p ~ ) - 1 - ]

( 1 9 )

( 2 0 )

I n o r d e r t o c o m p a r e t h e t h e o r y w i t h e x p e r i m e n t i t is c o n v e n i e n t t o r e l a t e t h e v a l u e s o f u ~ a n d

w ,~ t o Urn0 a n d w ~ 0 a s m e a s u r e d a t t h e o r i fi c e . E q u a t i o n s ( 1 9 ) a n d ( 2 0 ) m a y b e r e w r i t t e n i n t h e n o n -d i m e n s i o n a l f o r m

a n d

d f~12~.,/~o = A ( 2 1 )

X -~- a F K I - { - ( P m / p ~ ) - - 1 ] 1/2

[ d _ ~ 2 [ g l - ' F ( P m / P o o ) - - i ] ' / 2 - 1 2Wm/Wmo ~- B

- k ~ - + ( p - T ~ p ~ ) - T " - ~ 'x+~]( 2 2 )

T h e t e m p e r a t u r e d e c a y c a n be d e t e r m i n e d i n a li ke m a n n e r b y m u l t i p ly i n g t h e e n e r g y e q u a t i o n ( 5 )

b y r a n d i n t e g r a t i n g w i t h r e s p e c t t o r f r o m r = 0 t o r = oo u s i n g b o u n d a r y c o n d i t i o n s ( 7 ) t o g i v e

( d / d x ) r p u ( T - - T ~ ) d r

I n t r o d u c i n g t h e s i m i l a r i ty re l a t i o n s ( 1 4 ) i n t o ( 2 3 ) y i e l d s

= 0 . (23 )

w h e r e

( d / d x ) {( x -4- a ) 2 [ /~5 - -~ ( P ,, /p o ~ ) - - 1 7 u ~ ( T m - - T = ) } = 0 ,

/ ~ 5 - - f 0 ~ u ( ~ )0 ( ~ ) d ~ / L ~ } p ( ~ ) u ( ~ ) 0 ( ~ ) d ~ .

I n t r o d u c i n g u ~ f r o m E q . ( 1 9 ) i n t o E q . ( 2 4 ) a n d i n t e g r a t i n g w i t h r e s p e c t t o x y i e l d s ;

- - I ] 2

T ~ x - } - a K 5 -~ - ( p m / p ~ ) - - 1 "

( 2 4 )

( 2 5 )

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



T U R B U L E N T B U R N I N G F R E E J E T S W I T H S W I R L 493

Experimental Apparatus and Procedure

I n t h e s w i r l b u r n e r s h o w n i n F i g . 1 , a i r w a s

s u p p l i e d a x i a l ly a n d t a n g e n t i a l l y t h r o u g h f o u r

s l o t s . T h e d e g r e e o f s w i r l w a s v a r i e d b y i n d e -p e n d e n t l y v a r y i n g t h e a x i a l a n d t a n g e n t i a l

f lo w r a t e s o f a ir . L i q u i f ie d p e t r o l e u m g a s ( 5 1 %

p r o p a n e , 3 4 % b u t a n e , 1 5 % e t h a n e ) w a s

e v a p o r a t e d in a h e a t e x c h a n g e r a n d t h e n i n - Q t= n

j e c t e d r a d i a l l y i n t o t h e a x i a l a i r s t r e a m . T h e Qa=

g a s / a i r m i x t u r e i s s u e d f r o m t h e b u r n e r t h r o u g h Qt=./Qoa 5 - c m - d i a m o r i fi c e w i t h a v e l o c i t y o f t h e o r d e r u ,,0

o f 6 0 m / s e e . M a s s - f l o w r a t e s o f t h e a x i a l a i r , w ,,0

t a n g e n t i a l a i r , a n d g a s w e r e m e a s u r e d w i t h W,,o/U,,ot h e a i d of s h a r p - c d g e d o r if ic e p l a t e s . T h e m i x - G ~

t u r e w a s i g n i t e d b y a n o x y g e n - g a s p i l o t f la m e , G ~

a n d b y a d j u s t m e n t o f t h e g a s / a i r m i x t u r e r a t io S

t h e f la m e w a s s t a b i l i z e d s o m e 2 0 c m f r o m t h e

T A B L E I

I n p u t v a r i a b l e s .

F l a m e N o .

F 6 F 8 F 1 0

m 3/se c 0 . 0278 0 . 0556 0 . 0834

m 3/se e 0 . 0834 0 . 0556 0 . 0278

- - 0 . 5 0 0 . 7 5

m / s e e 6 2 . 0 6 5 . 6 6 8 . 6

m / s e e 5 . 1 2 1 . 3 2 2 . 8

0 . 0 8 0 . 3 2 5 0 . 3 3 2

kg 1 . 7 1 . 55 1 . 77

kg m 0 0 . 0045 0 . 0095

0 0 . 116 0 . 214

TANGENTIAL

AIR

AXIAL " ~ ~AR.~

t..EGAS

TANGENTIAL

SLOTS

FIG. 1 . Sw ir l bu rner .

b u r n e r e x i t . F o r a d i s t a n c e o f 1 5 0 c m t h e f l a m e

w a s u n c o n f in e d a n d a i r w a s e n t r a i n e d f r o m t h e

s t a g n a n t s u r r o u n d i n g s h a v i n g a n a m b i e n t t e m -

p e r a t u r e o f 3 0 ~ D o w n s t r e a m o f t h e f r ee - fl a m e

r e g i o n t h e c o m b u s t i o n g a s e s e n t e r e d a c y l i n d r i c a l

e x h a u s t c h a m b e r a n d a f t e r p as s i n g t h r o u g h a n

e j e c t o r w e re e x h a u s t e d t o t h e a t m o s p h e r e .

M e a s u r e m e n t s o f t h e t i m e - m e a n v a l u e s of

v e l o c i t y a n d s t a t i c p r e s s u r e w e r e m a d e w i t h a

5 - h o l e h e m i s p h e r i c a l w a t e r - c o o l e d i m p a c t p r o b e .

T h e p r o b e w a s c a l i b r a t e d i n a w i n d t u n n e l o v e r a

r a n g e o f y a w a n d p i t c h a n g le s . F r o m t h e 5

p r e s s u r e m e a s u r e m e n t s , r e c o r d e d o n a n i n c l i n e da l c o h o l m a n o m c t e r , t h e a x i a l , s w i r l, a n d r a d i a l

v e l o c i t y c o m p o n e n t s a s w e l l a s t h e s t a t i c p r e s s u r e

w e r e c o m p u t e d a t e a c h m e a s u r i n g p o i n t . T e m p e r -

a t u r e m e a s u r e m e n t s w e r e m a d e w i t h a b a r e

p l a t i n u m / p l a t i n u m r h o d i u m t h e rm o c o u p l e a n d

w i t h t h e a i d o f t h e v e l o c i t y m e a s u r e m e n t s , c o r -

r e c t i o n s w e r e m a d e f o r r a d i a t i o n l o s s e s a s

r e c o m m e n d e d b y F r i s t r o m . 12

M e a n v e l o ci ty a n d t e m p e r a t u r e t r a v e r se s w e re

m a d e a t 7 ax i a l s t a t i o n s x /d = 0, 4, 8, 12, 16, 20,

a n d 2 4 . T h e l i s t o f i n p u t v a r i a b l e s fo r t h e t h r e e

f l a m e s i s g i v e n i n T a b l e I .

Experimental Results and Discussion

T h e t y p e o f f l am e i n v e s t i g a t e d c a n b e s t b e

d e s c r i b e d w i t h t h e a i d o f t h e c o n s t a n t t e m p e r -

a t u r e l in e s sh o w n i n F i g . 2 . T h e v e l o c i t y g r a d i e n t s

a t t h e b u r n e r e x i t a r e t o o l a r g e t o p e r m i t f l a m e

s t a b i l i z a t i o n a n d i t i s o n l y a t s o m e 2 0 c m ( 4

d i a m e t e r s ) f r o m t h e o r if ic e t h a t t h e v e l o c i t y

m a g n i t u d e s a n d v e l o c i t y g r a d i e n t s h a v e b e e n

s u f fi c i en t l y r e d u c e d b y e n t r a i n m e n t t o a l l o w a na n n u l a r f l a m e fr o n t t o b e s t a b i l i z e d i n t h e

V l ~ ~iooo~- - ~ 1 2 o o

IJ W ~ ~ ~ i ~ o

"1000

0 50 I00~

F I G. 2 . Te m pe r a tu r e f i e ld in f l a m e wi th swi r l

( F 8 ) . I , bu r ne r ; I I , c o ld h igh - ve loc i ty c o r e ; I I I ,f l a m e f r on t ; I V , l um inous z one ; V , m a in r e a c t ion

z one ; VI , c o ld a m bie n t a i r .




1 . o ~ [ I l

u o 4 1 !I - -

0 0 .0 2 0 .0 4 0 .0 6 0 .0 8

E

Ix /d

i o - 1 2

o -1 6 - - ~ '

" - 2 0L ~o

i .

0 .02 0 .04 0 .06 0 .08

E.&

F r o . 3 . R a d ia l d i s t r i bu t ions o f a x i a l ve loc i ty .

0

I

i _ _0 .02 0 . 0 4 0 . 0 5 0 . 0 8

E

b o u n d a r i e s o f t h e j e t . T h e r e l a t i v e l y c o l d , h ig h -

v e l o c i t y c o r e p e r s i s t s o v e r a h n o s t t h e w h o l e

l e n g t h o f t h e f l a m e i n v e s t i g a t e d ( 2 4 d i a m e t e r s )

a n d t h e t e m p e r a t u r e d i s t ri b u t i o n s i n d i c a t e t h a t

r e a c t i o n r a t e s a r e l o w i n t h i s c e n t r a l c o r e . T h e

m a i n r e a c t i o n z o n e s u r r o u n d e d b y t h e l u m i n o u s

z o n e i s c o n f in e d t o t h e a n n u l a r s p a c e b e t w e e n t h e

c o l d c e n t r a l c o r e a n d t h e c o l d s u r r o u n d i n g a i r .

T h e r e i s n o s i g n i f i c a n t v a r i a t i o n i n t h e m a x i m u m

t e m p e r a t u r e a l o n g t h e le n g t h o f t h e f l am e

E F i g . 6 ( e ) - ] s i n c e a t e a c h a x i a l s t a t i o n t h e

m a x i m u m t e m p e r a t u r e i s d e t e r m i n e d b y t h e

m a x i m u m f l a m e t e m p e r a t u r e . I n t h e c o l d c o r e ,

t e m p e r a t u r e s r i se in t h e d o w n s t r e a m d i r e c t i o n

l a r g e l y d u e t o t u r b u l e n t m i x i n g w i t h h o t c o m -

b u s t i o n g a s e s fr o m t h e r e a c t i o n z o n e .

F i g u r e 3 s h o w s t h e r a d i a l d i s t r i b u t i o n s o f u/u,,p l o t t e d a g a i n s t r / ( x + a ) . T h e p o s i t i o n o f t h e

a p p a r e n t o r i g in % ' w a s d e t e r m i n e d b y e x t r a p o l a t -

i n g t h e l i n e s o f 1 / u ~ a g a i n s t x t o t h e v a l u e o f

l / u , , = 0 . I t c a n b e s e e n t h a t , d e s p i t e th e

l a r g e t e m p e r a t u r e c h a n g e s w h i c h i n f l u e n c e t h e

v e l o c i t y p r o f i l e s , t h e d i s t r i b u t i o n s a t x /d = 12,

1 6, 2 0 , a n d 2 4 h a v e a s i m i l a r f o r m w h i c h m a y b e

d e s c r i b e d b y t h e e q u a t i o n

~ / ~ . , = e xp ( - k . ~ ) , (2 6)

w h e r e t h e v e l o c i t y e r r o r c u r v e r k u g i v e s

a m e a s u r e o f t h e d e g r e e o f r a d i a l s p re a d .

T h e n o r m a l i z e d r a d i a l d i s t r i b u t i o n s o f s w i r l

v e l o c i t y a r e s h o w n i n F i g . 4 a n d h a v e t h e f o r m o f

a c o m b i n e d v o r t e x w i t h a n a l m o s t l i n e a r i n n e r

r e g i o n c o r r e s p o n d i n g t o s o l i d b o d y r o t a t i o n a n d

a n o u t e r r e g i o n c o r r e s p o n d i n g t o f r e e v o r t e x

f l o w . T h e s i m i l a r f o r m o f t h e n o r m a l i z e d s w i r l

v e l o c i t y c a n b e d e s c r i b e d a s a f u n c t i o n o f ~ b y

a t h i r d - o r d e r p o l y n o m i a l ( 7 ) .

F o r t h e r a d i a l te m p e r a t u r e d i s t ri b u t i o n s

( F i g . 5 ) , t h e t e m p e r a t u r e m a x i m a w e r e fo u n d t o

b e in t h e m a i n r e a c t i o n z o ne a n d n o t o n t h e i e t

a x i s . T h e t e m p e r a t u r e i n t h e c e n t r a l c o r e i n -

wW .

1 . 0 '

G 8

0 .6

0. 4

O.2

m

0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8

)

F- IO_

/ -

0 0 . 0 2 0 . 0 4 0 . 0 5 0 . 0 8

F I G. 4 . R a d ia l d i s t r i bu t ions o f sw i r l ve loc i ty .



495

1.0 a I

9 ' o

0 2 T ~ - ~ , ~ - -

0 0.02 0.04 0,06 0.0 8

9 o l

| / o - 1 2 I ~ ~ Io e-15

0 0.02 0.04 0.05 0.08

E - x r 'a

0 0.04

FIG. 5. Radial distributions of temperature.

0.02

T U R B U L E N T B U R N I N G F R E E J E T S W I T H S W I R L

0 . 0 6 0.08

creases, due to mixing with ho t combustion gases,and full similarity cannot be obtained underthese conditions until farther downstream wherethe position of the temperature maximum willconverge onto the jet axis. In the outer region ofthe flow, as shown in Fig. 5, the temperatureprofiles have similar forms and may be describedby the equation

( T - T ~ ) / ( T . , - T ~ )

- - e x p E--kT(~- ~T,~)~-], ( 2 7 )

where k T gives a measure of the degree of radialspread of temperature and ~T, is the radial dist-ance to the position of temperature maximum.

The form factors defined in the theoreticalanalysis can now be computed by integrating thenondimensional radial distribution and the valuesare shown in Table II.

The decay along the axis of the maximumvalues u~, win, and T,~ -- T~ is shown in Fig. 6.

TABLE II

Form factors

Flame No.

F6 F8 F10

/~ 1.087 1.087 1.053/~2 0 0.272 0.800/s 0 0.260 0.788

/ ( 4 - - 1.031 1.021f* 1.0 0.99 0.97

* For x / d = 10.

The decay of axial velocity increases with thedegree of swirl and the lines in Fig. 6(a) followthe hyperbolic decay predicted by Eq. (21). I tis, however, to be noted that the decay constantsare larger than in cold swirling jets, thus in-dicating a slower decay in the axial velocity.This is largely due to the temperature changesin the jet. The decay of swirl velocity does notshow any significant variation with the degree ofswirl and this is in agreement with the resultsobtained previously in cold jets7 The swirl

velocity decays in the flame as 1 / x 2 and thedecay constant is considerably larger than in thecold swirling jet, indicating again a slower decayof swirl velocity in the flame. The temperaturemaxima shown in Fig. 6(c) remain essentiallyconstant and the temperature decay Eq. (25)does not apply in this region of the flame. Thedecay and distribution constants computed fromthe results are given in Table III.

Concerning the stabilization of the flame, it wasnot possible for the flame to stabilize at the burnerexit, where velocities were of the order of 60

m/see and velocity gradients of the order of6000 sec 1. At distances varying from 20 to 30cm (4 to 6 diameters) in the low velocity regionnear the jet boundary the turbulent burningvelocity exceeds the forward velocity and holdingpoints (flame front) are set up. Because of theturbulent nature of the flow, the flame front is notstationary but fluctuates about a certain timemean average position. It was noted that themaximum temperature was found at each axialstation where the velocity was 24 m/see and thevelocity gradient was 250 sec 1. Since the laminarburning velocity of a stoichiometrie butane-

propane-air mixture is 0.3 m/sec, the turbulentburning velocity for this type of flame is nearly80 times that of the laminar burning velocity.Since the air and fuel were fully mixed before

(a )1.0 p,~ : , 1,0 It

. 8 ~ - " ~ -~ - - i o

u : i i % t - \

0 5 10 15 20 25 0

X /d

( b ) ( c )

1 o - I 1

/ < '?;--T ''~10 15 20 25 0 10 15 20 25

X / d X / d

FIG. 6 . Dec ay a long the axis of ma xim um va lues of ( a) axia l ve loc i ty ; (b) swirl ve loc i ty ; and (c)

temperature difference.

l e a v i ng t he bu r ne r , mi x i ng r a t e p l a ys no r o l e i nd e t e r m i n i n g t h e n a t u r e o f t h e f l am e a n d t h e

pos i t i on o f t he f l a me f r on t , pos i t i on o f t e mpe r -

a t u r e m a x i m a , a n d t h e g e n e r a l fo r m o f t h e f l a m e

a r e g o v e r n e d b y t h e v e l o c i t y m a g n i t u d e s a n dv e l o c i t y g r a d i e n t .

T h e e x p r e s s i o n s g i v e n i n t h e t h e o r e t i c a l a n -

a l y s is fo r th e d e c a y o f th e v a r i o u s c o m p o n e n t s o f

ve l oc i t y inc l ude t he c a se o f t he t u r bu l e n t sw i r li ng

i so t he r ma l j e t a s a spe c i a l c a se . I t shou l d be

n o t e d t h a t f a r d o w n s t r e a m f r o m t h e o ri fic e

E q . ( 21 ) f o r t he d e c a y o f t he a x i a l ve l oc i t y w il l

b e s i m i l a r t o t h a t o b t a i n e d b y T h r i n g a n dN e w b y 3 a n d r e d u ce s t o

t ~ / t ~ o = A E d / ( x + a ) - ] ( p ~ / p , , ) 1 12 . ( 2 8 )

I t i s t o b e e x p e c t e d t h a t i n t h e f a r - d o w n s t r e a m

r e g i on t he e xpr e s s i ons f o r sw i r l ve l oc i t y a nd

t e m p e r a t u r e w i l l b e c o m e

Wm//Wmo= B E d/ (x 4- a) ] '2 (poo/pm)u2 ( 29 )

T A B L E I I I

D e c a y a nd d i s tr i bu ti on c ons t a n t s

F l a m e N o .

F 6 F 8 F 1 0

a /d 35 45 60

A 18 .3 19 .6 20 .4B - - 211 345

k, 360 435 770kr 1440 1210 1580

~T~ 0.03 5 0 .033 0 .03 0

a n d

( T - T ~ ) / ( T m - Too)

= C[ d/( z + a ) 3 ( p ~ / p = ) ' / ' ( 3 0 )

C o n c l u s i o n s

F l a m e s t a b i l iz a t i o n o f a p r e m i x e d p r o p a n e -

b u t a n e - a i r m i x t u r e i n a f r e e t u r b u l e n t j e t w i t h

l ow de gr e e s o f sw i r l t a ke s p l a c e a t a d i s t a nc e o f

b e t w e e n 4 a n d 6 d i a m e t e r s f r o m t h e b u r n e r e x it .

T h e f l a m e f r o n t i s c o n f i n e d t o t h e a n n u l a r s p a c ebe t w e e n a c o l d h i gh - ve l oc i t y c o r e a nd t he c o l d

a m b i e n t a i r . T u r b u l e n t b u r n i n g v e l o c i t i e s a r e 8 0

t i m e s l a r g e r t h a n l a m i n a r b u r n i n g v e l o c i t i e s

n e a r t h e j e t b o u n d a r i e s w h e r e v e l o c i t y g r a d i e n t sa re 250 see 1 .

M e a s u r e m e n t s o f v e l o c i ty a n d t e m p e r a t u r ei n f l a m e s w i t h s w i r l s h o w t h a t r e a s o n a b l y g o o d

s i mi l a r i t y i s f ou nd f o r t he p r o f il e s o f a x i a l a n d

s w ir l v e l o c i t y a n d o n l y i n t h e o u t e r r e g io n o f t h e

j e t f o r t he t e m pe r a t u r e p ro fi le s. T h e l a r ge e f fe c t

o f t e m p e r a t u r e c h a n g e s in t h e j e t c a u s e a s l o w e r

d e c a y o f v e l o c i t i e s t h a n f o u n d p r e v i o u s l y i n t h e

ease of a cold swir l ing je t .T he or e t i c a l e xpr e s s i ons de sc r i b i ng t he de c a y

a l o n g t h e a x i s o f m a x i m u m v a l u e s o f a x i a l

v e l o c it y , s w i rl v e l o c i t y a n d t e m p e r a t u r e h a v eb e e n o b t a i n e d b y i n t e g r a t i o n o f t h e t u r b u l e n t

e q u a t i o n s o f m o t i o n , e n e r g y a n d s t a t e . T h e s e

e xpr e s s i ons t oge t he r w i t h t he s e mi - e mpi r i c a l

e xpr e s s ions ob t a i ne d f o r t he r a d i a l sp r e a d o f

v e l o c i ty a n d t e m p e r a t u r e p r o v i d e a m e a n s o f d e -t e r m i n g t h e s e q u a n t i t i e s a t e a c h p o i n t i n t h e

f l a me a s a f unc t i on o f t he de gr e e o f sw i r l .

N o m e n c l a t u r e

a d i s t a n c e o f po i n t o r i g i n o f j e t f r om o r if ic e

A e m p i r i c a l c o n s t a n t d e f in e d b y E q . ( 2 1 )



T U R B U L E N T B U R N I N G F R E E J E T S W I T H S W I R L 49 7

B e m p i r i c a l c o n s t a n t d e f i n e d b y E q . (2 2 )

c , s p e c if ic h e a t a t c o n s t a n t p r e s s u r e

d d i a m e t e r o f o r if i c e

f f u n c t i o n d e f i n e d b y E q . ( 1 9 )

G ~ a x i a l f l ux o f l i n e a r m o m e n t u m

G , a x ia l f lu x o f a n g u l a r m o m e n t u m

k t h e r m a l c o n d u c t i v i t y

k ~ v e l o c i t y G a u s s i a n e r r o r c u r v e c o n s t a n t

E q . ( 2 6 )

kT t e m p e r a t u r e G a u s s i a n e r r or c u r v e c o n s t a n t

E q . ( 2 7 )

/ ~ f o r m f a c t o r s d e f i n e d i n E q s . ( 1 5 ) , ( 1 6 ) ,

a n d ( 2 4 )

p s t a t i c p r e s s u r e

Q v o l u m e t r i c f lo w r a t e

r r a d i a l c o o r d i n a t e

R r a d i u s o f o r i f i c e

S ---- G , /G ~RT t e m p e r a t u r e

u a x i a l c o m p o n e n t o f v e l o c i t y v e c t o r

v r a d ia l c o m p o n e n t o f v e l o c i t y v e c t o r

w t a n g e n t i a l c o m p o n e n t of v e l o c i t y v e c t o r

x a x i a l c o o r d i n a t e

= r / ( x - -~ a )

p d e n s i t y

S u p e rsc r i p t s

' t u r b u l e n t f l u c t u a t i n g c o m p o n e n t

S u b sc r i p t s

m m a x i m u m v a l u e

0 v a l u e a t o r i f ic e x = 0

v a l u e i n a m b i e n t a i r b e y o n d f l a m e b o u n d a r y

ACKNOWLEDGMENT

T h e a u t h o r s t h a n k J . A v n y f o r a s s i s t a n c e w i t h

t h e e x p e r i m e n t s a n d c o m p u t a t i o n s .

R E F E R E N C E S

1. BE]~R, J . M. AND CHIGIER, N . A . : S w i r l i n g

J e t F l a m e s I s s u i n g f r o m a n A n n u l a r B u r n e r ,

5 m e J o u r n e e d ' E t u d e s s u r le s F l a m e s , P a r i s ,

1 96 3; a ls o a v a i l a b l e a s D o c . N o . K 2 0 / a / 9 ,

I n t e r n a t i o n a l F l a m e R e s e a r c h F o u n d a t i o n ,

I J m u i d e n , H o l l a nd .

2 . B E ~ R , J . M . : O n t h e S t a b i l i t y a n d I n t e n s i t y

o f P r e s s u re J e t O i l F l a m e s , A m e r i c a n P e t r o l e u m

I n s t i t u t e R e s e a r c h C o n f e r e n c e , J u n e 1 9 6 4 ,

p a p e r C P 6 4 - 9 .

3 . C HIGmR , N . A . : I s r ae l J . Tec h n o l . 3 , 3 2 (1 9 65 ).

4. MON NOT, G. , CH EDAILLE,J. , AND LUECKEL, W .:

F l a m e s d e C o m b u s t i b l e L i q u i d e o b t e n u e s a v e c

d e s B r u l e u r s a P u l v e r i s a t i o n M e c a n i q u e e t a

R o t a t i o n d ' A i r , 6 m e J o u r n e e d ' E t u d e s s u r le s

F l a m m e s , P a r i s N o v . 1 9 65 ; al s o a v a i l a b l e a sD o c . N o . K 2 0 / a / 2 7 , I n t e r n a t i o n a l F l a m e

R e s e a r c h F o u n d a t i o n , I J m u i d e n , H o l l a n d .

5 . CHIGIER, N . A . AND BE~R, J . M . : J . Bas i c

Eng. 4 , 788 (1964) .

6 . LEE, SHAo -LIN: J . Ap p l . M ech . 32 , 258 (1965).

7 . CHIG IER, N. A. AND CHERVINSKY, A. : I s r a e l

J . Tech n o l . 4 , 4 4 (1 9 6 6 ) .

8 . KERR, N. M . AND FRASER, D. : J . I n s t . Fu e l

39 , 5 1 9 -5 3 8 (1 9 6 5 ) .

9 . L E E , S H A o -L IN : A x i s y m m e t r i c a l T u r b u l e n t

S w i rl in g N a t u r a l C o n v e c t i o n P lu m e , R e p o r t

N o . 5 8 , C o l l e g e o f E n g i n e e r i n g , S t a t e U n i v e r -

s i t y o f N e w Y o r k , N o v . 1 96 5.

1 0. H I N ZE , J . O . : T u r b u l e n c e , M c G r a w - H i l l , 1 95 9.

11. CORRSlN, S. AND UBEROI, M . S . : F u r t h e r E x -

p e r i m e n t s o n t h e F l o w a n d H e a t T r a n s f e r i n

a H e a t e d T u r b u l e n t J e t , N A C A T N - 1 8 6 5 ,

1949.

12 . FRISTROM, R . M . : E x p e r i m e n t a l T e c h n i q u e s

f o r t h e S t u d y o f F l a m e S t r u ct u r e, R e p o r t N o .

3 00 , A p p l i e d P h y s i c s L a b . , T h e J o h n s H o p k i n s

U n i v e r s i t y , J a n . 1 9 6 3 .

13 . THRING, M . W . A ND N E W B Y , M . P . : Fo u r t h

Sy m po siu m (International) on Combustion, p.

7 89 , W i l l i a m s a n d W i l k i n s , 1 9 53 .

COMMENTS

P r o f . H . W . E m m o n s ( H a r v a r d U n i v e r s i t y ) :

T h e a i r r o t a t i o n i n t r o d u c e d i n t h e f i r e w h i r l c a u s e s

a s m a l l t u r b u l e n t f i re to b e c o m e l o n g a n d o f l o w

t u r b u l e n c e . T h e f u e l - a i r m i x t u r e r o t a t i o n i n t ro d u c e d

i n t o t h e j e t f l a m e o f t h e p a p e r ' s e x p e r i m e n t s c a u s e s

a s m a l l t u r b u l e n t f i re to b e c o m e s t i l l s m a l l e r a n d

m o r e t u r b u l e n t . I b e l i e v e t h e s e i n t e r e s t in g d i f f er -

e n c e s a r e t h e r e s u l t o f t h e d i f f e r e n c e s i n r a d i a l

s t a b i l i t y r a t h e r t h a n d i ff e r en c e s i n b u o y a n c y , a x i a l

M a c h n u m b e r , o r o t h e r p h e n o m e n a p r e s e n t i n t h e

t w o e x p e r i m e n t s .

M a n y y e a rs a g o, L o r d R a y l e i g h p r o v e d t h a t a

r o t a t i n g f lu i d i s s t a b l e i f t h e a n g u l a r m o m e n t u m

i n c re a s e s w i t h r a d i u s , i s u n s t a b l e i f t h e a n g u l a r

m o m e n t u m d e c r e a s e s w i t h r a d i u s , a n d i s n e u t r a l l y

s t a b l e if t h e a n g u l a r m o m e n t u m i s i n d e p e n d e n t o f

r a d i u s .

I n t h e f i r e w h i r l, t h e a n g u l a r m o m e n t u m i n c re a s e s

t h r o u g h t h e c o r e a n d t h e n r e m a i n s c o n s t a n t i n t h e

f r e e v o r t e x . T h e s t a b l e c o r e s u p p re s s e s t u r b u l e n c e

t h e r e ; a n d t h e r e i s n o r a d i u s r a n g e w h i c h is u n s ta b l e .

I n t h e r o t a t i n g j e t b u r n e r , t h e j e t b u r n s i n a n o n -

r o t a t i n g a t m o s p h e r e a n d , h e n c e , t h e a n g u l a r m o -

m e n t u m i n c r e a s e s f r o m z e r o t o a m a x i m u m n e a r



498 F I R E R E S E A R C H

t h e e d g e o f t h e j e t a n d t h e n f a l l s t o z e r o t h r o u g h a

j e t b o u n d a r y r e g i o n , a s s h o w n i n t h e p a p e r . T h i s

b o u n d a r y r e g i o n i s h i g h l y u ns t a b l e . I t p r o m o t e s

t u r b u l e n c e , w h i c h f u r t h e r b r o a d e n s t h e b o u n d a r y

r e g i o n b y d e c a y o f t h e c o r e a n d m i x i n g w i t h a m b i e n t

a i r. T h e t h u s - e n i m n c e d m i x i n g p r o ce s s i n c re a s e s t h e

e f f e c t i v e f l a m e s p e e d i n a p r e m i x e d i e t , o r m o r e

r a p i d l y s u p p l i e s o x i d i z er in a n o n p r e m i x e d j e t . I n

e i t h e r c a se , t h e f l a m e is m o r e t u r b u l e n t , a n d s h o r t e r .

P r o f . M . W . T h r i n g ( Q u e e n M a r y C o l l e g e , L o n -

d o n U n i v e r s i t y ) : I th i n k t h a t i n m a n y c a s es t h e

p r i m a r y e f fe c t o f s w i r l i s n o t t h e a n g u l a r m o m e n t u m

b u t t h e r a d i a l c o m p o n e n t o f v e l o c i t y i n t r o d u c e d b y

s w i r l w h e n t h e f l a m e s p r e a ds o u t f r o m a s m a l l i n l e t

i n t o a l a r g e c o m b u s t i o n c h a m b e r . I w o u l d l i k e t os e e a n e x p e r i m e n t o n a f l a m e w i t h a l a r g e r a d i a l l y

o u t w a r d v e l o c i t y c o m p o n e n t b u t n o s w ir l .

I n t h e p a p e r, t h e f l a m e d i d n o t h a v e a n y e n -

c l o si n g w a l l . A t I J m u i d e n , w e f o u n d t h a t v e r y

m u c h h i g h e r r a t e s o f c o m b u s t i o n c a n b e o b t a i n e d

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

u s i n g a d i v e r g e n t c o n e w a l l w h i c h g i v e s a v e r y

h i g h r a t e o f s h e a r a n d , s o , f u e l - a i r m i x i n g . F u e l - a i r

m i x i n g i s j u s t a s i m p o r t a n t a s r e c i r c u l a t i o n i n s u c h

f l a m e s .

Pro f . J . M . Be e r ( U n i v e r s i t y o f S h e f f i e l d ) :

1 . S w i r l i n g f l o w s a r e d i s c u s s e d i n t h e p a p e r s o f

P r o f . E m m o n s a n d o f D r . C h i g i e r. T h e r e i s , h o w -

e v e r , a n a p p a r e n t d i s c r ep a n c y i n t h e e f f ec t s w i rl

h a s o n t h e d e v e l o p m e n t o f t h e f l a m e i n t h e s e t w o

s y s t e m s . I n t h e c a s e of t h e f i r e w h i r l , t h e f l a m e

l e n g t h i n c r e a s e s a n d t u r b u l e n t m i x i n g b e t w e e n t h e

f l a m e g a s e s a n d t h e r o t a t i n g f l o w b e c o m e s l e s s

i n t e n s i v e w i t h i n c r e a s i n g r o t a t i o n o f t h e a i r f l o w

( i nc r e a si n g r a t i o o f a n g u l a r t o f o r w a r d m o m e n t a

r a t i o o r R o s s b y n u m b e r ) w h i l e t h e f l a m e l e n g t h

d e c r e a s e s w i t h i n c r e a s i n g i n t e n s i t y o f t u r b u l e n tm i x i n g i n s w i rl i n g j e t f l am e s . I t h i n k t h a t t h i s

d i s c r e p a n c y c a n b e r e s o l v e d b y c o n s i d e r in g t h e

s p a t i a l d i s t r i b u t i o n s o f s h e a r a n d o f f u e l , r e s p e c -

t i v e l y , i n b o t h s y s t e m s . I n t h e c a s e o f t h e f i re w h i r l ,

t h e r a d i a l p r e s s u r e g r a d i e n t s e t u p b y t h e s w i r l i n g

f lo w i n c r e as e s t h e f lo w r a t e o f a i r d r a w n i n t o t h e

c e n t e r a t g r o u n d l e v e l w h i c h i n t u r n e n a b l e s m o r e

o f t h e p o o l o f li q u i d f u e l t o b e b u r n e d . T h e s w i r li n g

f lo w d oe s n o t c o n t r i b u t e , h o w e v e r , t o t h e f a s t e r

m i x i n g b e t w e e n f u e l a n d a i r b e c a u s e t h e r e g i o n s o f

h i g h s h e a r i n t h e r o t a t i n g f l o w a r e fa r o u t s i d e t h e

f l a m e b o u n d a r ie s . T h e s w i r l i s t o o w e a k t o p r o d u c e

a r e c i r c u l a t i o n v o r t e x i n t h e c o r e r e g i o n a n d i t se f f e c t i n t h i s e a s e i s t h a t i t s e p a r a t e s f u e l a n d a i r

w i t h t h e c o n s e q u e n c e o f i n c r e a s i n g f l a m e l e n g t h .

I t c a n b e e x p e c t e d t h a t , b y i n c r e a s i n g t h e i n t e n s i t y

o f s w i r l b e y o n d a c e r t a i n v a l u e t h a t i s s tr o n g

e n o u g h t o p r o d u c e t h e r e c i r c u l a t i o n v o r t e x i n t h e

c e n t r a l r e g i o n o f t h e r o t a t i n g f lo w , t h e f l a m e l e n g t h

w i l l d e c r e a s e w i t h i n c r e a s i n g s w i r l .

I n t h e c a s e of t h e t u r b u l e n t j e t f l a m e w i t h a

h i g h d e g r e e o f s w i r l , t h e f u e l c o n c e n t r a t i o n i s h i g h

i n t h e r e g i o n o f h i g h s h e a r s t r e s s e s i n t h e f l o w .

T h i s i s b e c a u s e o f t h e v e l o c i t y d i s tr i b u t i o n i n

s o l i d - b o d y - t y p e s w i r l s t h a t p r o d u c e s p e a k s o f b o t h

t h e a x i a l a n d t a n g e n t i a l v e l o c i t y p ro f il e s in t h e

s a m e r e g i o n a t s o m e r a d i a l d i s t a n c e f r o m t h e a x i s

o f t h e r o t a t i o n . I n t h i s l a t t e r c a s e , s w i r l e f f e c t i v e l y

p r o m o t e s t u r b u l e n t m i x i n g a n d r e d u c e s f l a m e le n g t h .

2 . C o n c e r n i n g P r of e s s or E m m o n s ' o b s e r v a t i o n o n

t h e r a d i a l s p r e a d o f t h e s w i r l i n g f l o w , I w o u l d l i k e

t o m e n t i o n a s i m i l a r o b s e r v a t i o n i n f u l l y s e p a r a t e d

f l ow . 1 I t w a s f o u n d t h a t t h e a n g l e o f s p r e a d o f a

s w i r l i n g j e t i n c r e a s e d w h e n h e a t w a s r e l e a s e d ,c o m p a r e d w i t h a c o l d i s o t h e r m a l s w i r l i n g j e t o f

t h e s a m e a n g u l a r t o l i n e a r m o m e n t a r a t i o . T h i s

c o u l d b e e x p l a i n e d b y t h e i n c r e a s e o f t h e v a l u e o f

t h e p r e s s u r e t e r m i n t h e e x p r e s s i o n o f t h e a x i a l

f lu x o f t h e l i n e a r m o m e n t u m d u e t o t h e r m a l e x -

p a n s i o n i n t h e c o r e o f t h e j e t . T h e s w i r l i n g j e t c a n

b e c h a r a c t e r i z e d b y t h e r a t i o :

S = f p U W r 2 d r / (f p U ~ r d r q - f P r d r),

w h e r e b o t h t h e a n g u l a r a n d l in e a r m o m e n t a a r e

c o n s e r v e d . 1 A s a r e s u l t o f t h e h e a t - r e l e a s e a n d

b u o y a n t e f fe c ts , t h e p r e s s u r e w i l l r i se w i t h h e i g h t

r a p i d l y o n t h e a x i s a n d t h i s , in t u r n , w i l l i n c r e a s e

t h e r a d i a l s e p a r a t i o n b e t w e e n t h e f l a m e a n d p e a k s

o f t h e a x i a l - v e l o c i t y p r o fi l e. T h e i n c r e a s i n g r a d i a l

s e p a r a t i o n b e t w e e n t h e f la m e a n d t h e h i g h - s h e a r

z o n e w i t h i n c r e a s i n g h e i g h t o f t h e f l a m e c a n t h u s

b e e x p l a i n e d .

Reference

1 . B e 6 r , J . M . a n d C h i g i e r , N . A . : 5 t h J o u r n e e

d ' E t u d e s s u r l es F l a m m e s , P a r i s , N o v . 1 9 63 .

Dr. N. A. Chigier: T h e e x p e r i m e n t s c a r r i e d o u t

b y P r o f e s s o r E m m o n s a n d D r . S h u h - J i n g Y i n g

h a v e s h o w n t h a t t h e i n t r o d u c t i o n o f s w i r l t o t h e

s u r r o u n d i n g a t m o s p h e r e l e a d s t o a l a r g e i n c r e a s e

i n f la m e l e n g t h a n d a r e d u c t i o n i n t h e r a t e o f e n -

t r a i n m e n t . I n o u r w o rk , w e h a v e f o u n d t h a t t h e

i n t r o d u c t i o n o f a s w i rl i n g m o t i o n t o t h e a i r s u p p l y

l e a d s to a r e d u c t i o n i n t h e f l a m e l e n g t h a n d a n

i n c r e a s e i n t h e r a t e o f e n t r a i n m e n t . T h e s e a p -

p a r e n t l y c o n t r a d i c t o r y e f f e ct s c a n o n l y b e e x -

p l a i n e d b y t h e d i f fe r e n c e s i n t h e f l o w f i e l ds s e t u p

i n th e t w o s y s te m s . W e e x p e c t t h a t t h e i n c re a s e d

r a t e o f e n t r a i n m e n t i n o u r c a se o f t h e s w i r li n g j e t

is a s s o c i a t e d w i t h a n i n c r e a s e i n t h e i n t e n s i t i e s o f



T U R B U L E N T B U R N I N G F R E E J E T S W I T H S W I R L 4 99

t u r b u le n c e . C o r r e s p o n d i n g ly , w e m i g h t e x p e c t a

d a m p i n g o f t h e t u r b u l e n c e i n E m m o n ' s c a se o f t h e

f i r e w h i r l a s s o c i a t e d w i t h t h e r e d u c e d r a t e o f e n -

t r a i n m e n t . T h e e x p l a n a t i o n o f t h e r e a s on s f o r

o b t a i n i n g t h e s e d i f f e r e n t e f f e c t s , a s p u t f o r w a r d b y

E m m o n s , i s e x t r e m e l y i n t e re s t i n g , a n d a p p e a r s t o

b e a p l a u s i b l e e x p l a n a t i o n . T h e d i f f e r e n t e f f e c t s

a r e s u f f i c ie n t l y i n t e r e s t i n g t o m e r i t a d e t a i l e d

c o m p a r i s o n o f t h e f lo w f ie l ds a n d a e r o d y n a m i c

b e h a v i o r o f t h e t w o s y s t e m s i n o r d e r t o v e r i f y

E m m o n s ' h y p o t h es i s.

I n r e p l y to P r o f e s s o r T h r i n g ' s c o m m e n t , I c a n n o t

a g r e e w i th h i s c o n t e n t i o n t h a t t h e p r i m a r y e f f ec t

o f s w i r l i s n o t t h e a n g u l a r m o m e n t u m b u t t h e

r a d i a l c o m p o n e n t o f v e l o c i t y i n t r o d u c e d b y t h e

s w i r l . I n t h e f i r s t i n s t a n c e , t h e i n t r o d u c t i o n o f a

s w i r li n g m o t i o n w i t h t a n g e n t i a l c o m p o n e n t s o f

v e l o c i t y a lt e r s t h e n a t u r e , a s c a n b e s e e n f r o mt h e e q u a t i o n s o f m o t i o n . I n t h e e x p e r i m e n t s u g -

g e s t e d b y P r o f e s s o r T h r i n g , i n w h i c h t h e r e w o u l d

b e o n l y a x i a l a n d r a d i a l v e l o c i t y c o m p o n e n t s , i t

w o u l d o n l y b e n e c e s s a r y t o c o n s i d e r t h e f i r s t t w o

e q u a t i o n s o f m o t i o n i n c y l i n d r i c a l c o o r d i n a t e s , a n d

n o c o n s i d e r a ti o n n e e d b e m a d e o f th e t h i r d e q u a -

t i o n o f m o t i o n .

A f l a m e w i t h a l a r g e r a d i a l ly o u t w a r d v e l o c i t y

c o m p o n e n t w o u l d h a v e a h i g h e r r a t e o f e n t r a i n m e n t

t h a n a n o n s w i r l i n g j e t , d u e t o t h e l a r g e r " e x t e r n a l -

s u r f a c e a r e a . " H o w e v e r , t h e i n c r e a s e d r a t e o f

s h e a r , a s w e l l a s t h e a d d i t i o n a l R e y n o l d s s t r e ss e s

s e t u p b y t h e s w i r l i n g m o t i o n , w i l l h a v e a s i g n if i -

c a n t e f fe c t u p o n t h e r a t e o f e n t ra i n m e n t .

Prof. H. W. Emmons: T h e r e a r e m a n y i n t e ra c t i n g

p h e n o m e n a i n b o t h t h e f i r e w h i r l a n d t h e s w i r l i n g

j e t b u r n e r . T h e r e a r e i n f a c t s o m a n y e f fe c ts p r e s e n t

t h a t i t i s i m p o s s i b l e t o s o l v e t h e N a v i e r - S t o k e s

e q u a t i o n s f o r e i t h e r c a s e , e x c e p t i n a r a t h e r c r u d e

a p p r o x i m a t e f o rm . T h u s , a l l " e x p l a n a t i o n s " o f a

v e r b a l c h a r a c t e r , s u c h a s t h o s e o f D r . B e e r a n d

t h e s u c c e e d i n g o n e s b y t h e a u t h o r s , a r e a t t e m p t s

t o p i c k o u t o f t h e c o m p l e x i t y a f e w e f f e c t s w h i c h

l e a d t o c o r r e c t q u a l i t a t i v e p r e d i c t i o n s a n d t h u s

s a t is f y o u r d e s ir e t o f ee l w e " u n d e r s t a n d " w h a t i s

h a p p e n i n g .

I d o n o t b e li e v e t h a t t h e d o m i n a t i n g m e c h a n i s m

c o n t r o l l i n g t h e d i f f e r e n c e b e t w e e n t h e w h i r l a n d

t h e j e t i s a s s o c i a te d w i t h t h e c o m b u s t i o n , b u o y a n c y ,

i n i ti a l m o m e n t u m , e t c . I b e l i e v e t h e d o m i n a t i n g

d i f f e re n c e is t h e d i f f e r e n c e i n r a d i a l s t a b i l i t y o f t h e

t w o c a se s . T h e w h i r l h a s a f r e e v o r t e x s u r ro u n d i n ga r o t a t i n g c o r e . T h e a n g u l a r m o m e n t u m h a s a

p o s i t i v e o r z e r o r a d i a l g r a d i e n t e v e r y w h e r e . T h e

f lo w is t h e r e f o r e s t a b l e o r n e u t r a l e v e r y w h e r e .

T h e s w i rl in g je t , o n t h e o t h e r h a n d , h a s s t a t i o n a r y

a i r o u t s i d e o f t h e j e t . T h u s , a t t h e j e t b o u n d a r y ,

t h e r e i s a s t r o n g n e g a t i v e a n g u l a r m o m e n t u m

r a d i a l g r a d i e n t . T h e j e t b o u n d a r y i s t h e r e f o r e

u n s t a b l e .

O f c o u r se , t h e r e w i l l b e m o d i f i c a t io n s o f th e a b o v e

b a s i c d i f f e re n c e b y c o m b u s t i o n , b u o y a n c y , e t c . ,

w h i c h e f f e c t s w i l l b e i m p o r t a n t , e v e n d o m i n a t i n g ,

i n s o m e r a n g e s o f t h e a p p r o p r i a t e d i m e n s io n l e ss

p a r a m e t e r s . T h e r e i s t o o l i t t l e c u r r e n t l y k n o w n

a b o u t t h e s e e ff e c ts t o s t a t e r i go r o u s ly t h a t t h e y

a r e u n i m p o r t a n t h e r e , a l th o u g h I b e l i e v e t h i s t o

b e t h e c a s e .

49810685 combustion physics by c k law

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