combustion and flame volume 48 issue none 1982 [doi 10.1016%2f0010-2180%2882%2990112-2] w.p. jones;...

8/10/2019 Combustion and Flame Volume 48 Issue None 1982 [Doi 10.1016%2F0010-2180%2882%2990112-2] W.P. Jones; J.

1/26

COMBUSTION ND FL ME

8 : 1 2 6 1982) 1

C a l c u l a t i o n M e t h o d s f o r R e a c t in g T u r b u l e n t F l ow s A R e v i e w

W . P . J O N E S

Department of Chemical Engineering and Chemical Technology Imperial College London SW7 2By England

and

J . H . W H I T E L A W

Department of Mechanical Engineering Imperial College London SW7 2By England

T h e p u r p o s e o f t h i s r e v i e w i s to d e s c r i b e a n d a p p r a i s e c o m p o n e n t s o f ca l c u la t i o n m e t h o d s , b a s e d o n t h e s o l u t i o n o f

conse rva t ion equa t ions in d i f fe ren t ia l fo rm, fo r the ve loc i ty , t empera tu re and concen t ra t ion f ie ld s in tu rbu len t

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

g a s e o u s - c o m b u s t i o n a p p l i c at i o n s .

T h e d i f f e r e n t i a l e q u a t i o n s a r e c o n s i d e r e d f i r s t a n d t h e i m p l i c a t i o n s o f c o n v e n t i o n a l i . e . , u n w e i g h t e d ) a n d d e n s i t y -

w e i g h t e d a v e r a g i n g d i s c u s s e d i n t h e c o n t e x t s o f s o l u t i o n m e t h o d s a n d p h y s i c a l i n t e rp r e t at i o n . I n g e n e r a l , i t i s c o n c l u d e d

t h a t e q u a t i o n s s h o u l d b e s o l v e d w i t h d e p e n d e n t v a r i a b l es i n d e n s i t y - w e i g h t e d f o r m a n d t h a t t h e i n t e r p r e ta t i o n o f

m e a s u r e m e n t s r e q u i r e s s p e c i a l c a r e t o d i s t i n g u i s h b e t w e e n c o n v e n t i o n a l l y a v e r ag e d a n d d e n s i t y - w e i g h t e d p r o p e r ti e s .

inite

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

t h r e e - d i m e n s i o n a l r e c i r c u l a ti n g f l o w s , s u c h a s a r e t o b e f o u n d i n c o m b u s t i o n c h a m b e r s , a r e c o n s i d e r e d b r i e f l y . I t i s

c o n c l u d e d t h a t c o m p u t e r s t o r a g e r e q u i r e m e n t s v e r y o f t e n p r e c lu d e t h e p o s s i b i li t y o f r e d u c i n g t h e n u m e r i c a l e r r o r t o

e n t i r e l y n e g l i g i b l e p r o p o r t i o n s e v e r y w h e r e . C o n s i d e r a b l e c a r e m u s t t h e r e f o r e b e t a k en i n b o t h s p e c i f y i n g a s u f f i c i e n t

n u m b e r a n d t h e d i s t r ib u t i o n o f m e s h p o i n t s t o b e u s e d a n d a l s o i n t h e i n t e r p r e ta t i on o f c o m p u t a t i o n a l r e s u lt s . T u r b u l e n c e

m o d e l s a r e a l s o d i s c u s s e d b r i e fl y a n d d e f i c i e n c i e s n o te d . S i n c e m a n y f l o w s a r e p ar t l y c o n tr o l l e d b y m e c h a n i s m s o t h e r

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

w h i c h a r e d i s c u s s e d .

T h e v a r i o u s r e c e n t m e t h o d s p r o p o s e d t o r e p r e s e n t r e a c ti o n i n t u r b u le n t f l a m e s a r e r e v i e w e d i n r e l a t io n t o d i f f u s i o n a n d

p r e m i x e d f l a m e s a n d t o f l a m e s i n w h i c h a n e l e m e n t o f b o t h i s pr e s e n t . T h e a p p l ic a t i on o f l a m i n a r fl a m e s h e e t m o d e l s ,

c h e m i c a l e q u i l i b r i u m a s s u m p t i o n s , p r o b a b i l i t y d e n s it y f u n c t i o n s o f d i f f e r e n t f o r m s , a n d t r u n c a t e d s e r ie s e x p a n s i o n o f

r e a c t i o n r a t e e x p r e s s i o n s a r e c o n s i d e r e d t o g e t h e r w i t h t h e u s e o f p r o b a b i li t y d e n s it y f u n c t i o n t r a n sp o r t e q u a t i o n s a n d

the i r M on te C a r lo ) so lu t ion . Th e app ra i sa l i s mad e in re la t ion to p re sen t ly ava i lab le re su l t s and fu tu re requ i reme n ts and

p o s s i b i l i t i e s .

1 . I N T R O D U C T O R Y R E M A R K S

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

h a s r e c e i v e d c o n s i d e r a b l e a t t e n t i o n i n r e c e n t y e a r s ,

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

r e v ie w s o f B r a c c o 1 9 7 6 ) , t h e A G A R D C o n f e r-

e n c e o n C o m b u s t i o n M o d e l li ng 1 9 8 0 ) , J o n e s

1 9 8 0 ) , a n d L i b b y a n d W i l li a ms 1 9 8 0 ) . A s a c o n

C o p y r i g h t 1 9 8 2 b y T h e C o m b u s t i o n I n s ti t u t e

P u b l i s h e d b y E l s e v i e r S c i e n c e P u b l i s h i n g C o . , I n c .

5 2 V a n d e r b i l f A v e n u e , N e w Y o r k , N Y 1 0 0 1 7

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

in t e res t in the ab i l i t i es and po ten t i a l ab i l i t i es o f

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

c r e a s e d , a n d t h e m a i n p u r p o s e o f t h is r e v ie w is

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

p o n e n t s a n d , a s a r e s u lt , t o p r o v i d e a b a s is f o r t h e

e v a l u a t i o n o f r e s u l t i n g c a l c u l a t i o n s . A t t e n t i o n i s

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

0010-2180182/10001 +26502.75


2/26


3/26

R E A C T I N G T U R B U L E N T F LO W S 3

b ine to f o rm spec ies C, i .e . ,

A + B - , c , 8 )

t he fo rw ard ra t e con s tan t i s, t yp ica l ly , k t = a

e x p ( - E / R T ) , a n d t h e n e t f o r ma t i o n r a t e s , w h i c h

a p p e a r i n t h e s p e c ie s ma s s f r a c t i o n e q u a t i o n s , a r e

P

r ~

= - - - k ~ C a C B

M B

P

i'B = - - - k f C a G ,

M A

P M c

r c = + k fC a CB . 9 )

M A M B

E q u a t i o n s 3 ) a n d 4 ) i mp l y t h a t th e d if f u s io n

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

t he l a w s o f F o u r ie r a n d F i c k . F o r m u l t i c o m p o n e n t

s y s t e ms t h e c o n s t i t u t i v e e q u a t i o n s f o r t h e d i f f u -

s io n o f h e a t a n d ma s s a re mo r e c o m p l i c a t e d s i n c e t h e

s p e c ie s ma s s f l u x d e p e n d s i n a c o mp l e x m a n n e r o n

t h e c o n c e n t r a t i o n g r a d i e n t s o f a ll s u b s ta n c e s p r e s -

e n t a n d t h e r ma l - d i f f u s i o n So r e t ) a n d d i f f u s i o n -

t h e r m o m e t r i c D u f o u r ) e f f e c ts c a n b e i m p o r t a n t .

T h u s , a l t h o u g h th e s e t o f e q u a t i o n s 1 ) - 4 ) m a y

b e a p p l i e d t o l a mi n a r f l a me s , a p p r e c i a b l e e r r o r i s

in th i s case l ike ly : Dixon-L ewis 1970 ) , fo r ex -

a mp l e , h a s d e mo n s t r a t e d t h a t t h e t h e r ma l d i f f u -

s iona l and o rd in ary d i f fus iona l f luxes a re o f the

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

a i r f l a me . So m e j u s t i f ic a t i o n f o r t h e u s e o f E q s .

3 ) a n d 4 ) i n t u r b u l e n t f l o w is t h e r e f o r e r e q u i r e d .

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

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

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

a n d t h e g r a d i e n t s o f me a n q u a n t i t i e s . T h e s e f l u c -

t u a t i o n s h a v e l e n g t h a n d t i me s c a le s o f t h e o r d e r

o f t h o s e o f t h e me a n f l o w a n d , p r o v i d i n g t h e Re y -

n o l d s n u m b e r i s l a rg e , a r e ma n y o r d e r s o f ma g n i -

tude l a rger than the f ine sca les a t wh ich the

mo l e c u l a r d i f fu s i o n b e c o m e s s ig n i f ic a n t . T h e t u r b u -

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

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

mo l e c u l a r d i f f u s i o n b e c o me s s i g n i f i c a n t b y a n i n -

er t i a l p rocess which i s ra t e con t ro l l ing . The impl i -

ca t ion i s t ha t , wh i l e molecu lar p rocesses a re re -

spons ib le fo r mix ing a t a micro leve l , t he ra t e a t

w h i c h t h e y o c c u r i s i n d e p e n d e n t o f t h e va l u e s o f

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

p r e c i s e me c h a n i s m w h e r e b y mo l e c u l a r d i f f u s i o n

o c c u r s. H e n c e t h e a s s u mp t i o n t h a t w e ma k e is n o t

tha t t he ac tua l molecu lar p rocesses a re as impl i ed

in Eqs . 3 ) and 4 ) bu t ra ther tha t t u rbu le nce i s

d o mi n a t e d b y i n e r t i a l e f f e c t s . T h u s , t h e mo d e l s

t o b e d e s c r i b e d t a k e n o e x p l ic i t a c c o u n t o f mo l e c -

u la r t rans por t a t a ll and Eqs . 1 ) , 3 ) and 4 ) a re

r e w r i t t e n i n th e m o r e g e n e r a l f o r ms

au~ au~ a p a r~j .

p ~ r pU j ax j - aX 1- aXj Pgt ,

10)

p - ~ - p U s - - = - - p S u ( p , ~) , 1 1 )

axj axz

where rep resen t s the sca la r se t Ca and h . In the

c a s e o f e n t h a l p y b p / a t i s o m i t t e d o n t h e b a si s t h a t

t h e M a c h n u mb e r i s l o w a n d t h e v o l u me t r i c s o u r c e

p S wi ll t hen be zero un les s rad ia t ion i s t o be con-

s idered .

T h e f l o w i n p r a c t i c a l c o m b u s t i o n s y s t e ms i s i n-

v a r i b l y t u r b u l e n t , a n d t h e t e mp o r a l a n d s p a t i a l

v a r i a t i o n s i n t h e d e p e n d e n t v a r i a b l e s e n c o mp a s s

such a wide range o f t ime and l eng th sca les as to

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

g o v e r n i n g e q u a t i o n s . T h e c o mp u t e r s t o r a g e a n d

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

a n y e x i s t i n g o r p l a n n e d c o mp u t e r .

T o c i r c u mv e n t t h i s d i f f i c u l t y , t h e d e p e n d e n t

v a r i ab l e s ar e d e c o m p o s e d i n t o me a n a n d f l u c t u a t i n g

c o mp o n e n t s , a n d t h e r e s u l t i n g e q u a t i o n s a v e r a g e d

to conver t t hem in to s t a t i s t i ca l equa t ions descr ib -

i n g th e e v o l u t i o n o f me a n q u a n t it i e s . I n ge n e r a l,

t he averag ing p rocess shou ld invo lve ensem ble

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

here th i s i s i nd i s t ingu i shab le f rom t ime averag ing . )

H o w e v e r , a s a c o n s e q u e n c e o f t h e n o n l i n e a r i t y o f

the equa t ions , averag ing resu l t s i n a los s o f in fo r -

ma t i o n , s o t h a t t h e e q u a t i o n s a r e n o lo n g e r c lo s e d

a n d c l o s u r e a s s u m p t i o n s a r e n e ce s s a r y b e f o r e s ol u -

t ion is poss ible.

Fo r v a r i a b l e d e n s i t y f l o w s t w o t y p e s o f d e c o m-

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


4/26

4 W . P . J O N E S a n d J . H . W H I T E L A W

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

dens i ty -w e igh ted decom pos i t ion s ugges tedby F av re

(1969) . The unw e igh ted decom pos i t ion and averag -

ing are represented by

u = U + u ;

where

_ 1 [ to+r

U = l i m - U d t a n d ~ ' = 0 ,

-'--~ T 4tO

and dens i ty-w eighted averaging by

p U = - ~ + p u =- - fl U + p u ,

= + p c ~ - + p e y ,

where

1 f t o + l

-~J = l ira pU d t and f f f f = 0

,r.. oo T to

b ut ~ 4= O.

With dens i ty-w eighted averaging, the equ at ions of

c o n t i n u i t y a n d c o n s er v a ti o n o f m o m e n t u m a n d a

scalar may be wr i t ten , for h igh- turbulence Rey-

nolds numbers , as

a

- - ~ O ~ ) = 0

(12 )

axt

a ~ _ @ a , , -~ , , ,

u t u s ) = g i a x t a x s u s ) , (13)

a a

- - - -~-~x i(Pui (o ) P ,~a(~) . (14)

The use of unweighted averaging resul ts in s imilar

equa t ions bu t w i th tw o impor tan t d i f f e r ences .

F i r s t , t he dens i ty -w e igh ted quan t i t i e s r ep res en ted

by the t i lde (~ ) ove r the quan t i ty a r e r ep laced by

unweighted-averaged quant i t ies with the overbar .

Secon d, corre la t ion s involving f luctu at ing dens i ty

corre la t ions such as p'u ' and ~ appea r in the

equat ions .

The cho ice o f d ecom pos i t ion to be adop ted i s

to some degree arb i t rary though the dens i ty-

weighted decompos i t ion and averaging process is

to be prefer red on the phys ical grounds that i t

provides equ at ions descr ib ing the var ia t ion of the

mean values of those quant i t ies which are con-

served. Fo r ex amp le , the appl icat ion of dens i ty-

weighted averaging to Eq. (1) , w hich is a s ta tem ent

o f t h e c o n s e rv a t io n o f m o m e n t u m p e r u n i t v ol u m e

in the x l-d irect ion , g ives an equat ion which de-

s cr ibes the va r i a t ion o f me an mom en tu m in the x i-

d i r ec t ion , i . e ., pU i = pU i , no t ~ O . I n add i t ion the

resul t ing equat ions are of s impler form and have

te rms w h ich a r e more eas i ly in te rp re ted than thos e

which ar ise with unweighted averaging. More de-

ta i led d iscussions of the m er i ts of bo th ty pes of

averaging are given by Bilger (1976), Jones (1980),

and Lib by and W ill iams (1980) .

I t s hou ld be no te d tha t the s o lu t ion o f dens i ty -

weighted equa t ions y ie lds dens i ty-weigh ted proper-

t i e s , and in s t rumen ts may meas u re unw eigh ted

values, dens i ty-weig hted values or some o ther

value. An inf in i te ly smal l thermocouple , for ex-

ample , wi ll de tect an unwe ighted record of in-

s t an taneous t empera tu re , bu t s ome deg ree o f

dens i ty weight ing is associa ted with thermo-

couples o f f in i te s ize . The ex tent o f th is weight ing

with 40- /~m-diam thermocouples is unl ikely to ex-

ceed aro und 50C (see A tty a and W hite law, 1981)

comp ared w i th d i f fe r ences be tw een w e igh ted and

unw eigh ted va lues w h ich can exceed 300C . In

contras t , sampling probes measure values very

close to dens i ty-weighted concentra t ion , and i t i s

c l ea r tha t the ab i l i ty to ca lcu la te bo th un w eigh ted

and d ens i ty weighted is des irable . As wil l be sho wn,

th is be com es poss ib le i f a probab i l i ty-dens i ty func-

t ion fo rm ula t ion is adop ted .

I f unweighted values of the dependent var iables

are require d, th en th e corr elation s involving fluc-

tua t ing dens i ty m us t be eva lua ted : the co r r e la tion

p'ui ' i s needed to eva lua te the unw eigh ted mean

veloci ty , and s imilar ly ~ is required to obta in

the unw eigh ted m ean va lue o f ~ ,~ . The fo rm er

q u a n t i t y p'ui ' mu st be m odeled , i .e . , re la ted to

kno w n quan t i t ie s , w hereas the co r r e la t ion ~ can

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

fo r m ode l s w here the app rop r ia te p robab i l i ty den -


5/26


s i t y f u n c t i o n is k n o w n . A l t e r n a t i v e l y , t h e q u a n t i t y

c a n b e d e t e r m i n e d v i a d i r e c t mo d e l i n g a s-

s u m p t i o n . H o w e v e r , m o s t c o n c e n t r a t i o n m e a s u r e -

me n t s , i n c lu d i n g t h o s e w i t h w h i c h c o m p a r i s o n w ill

l a t e r b e d r a w n , a r e o b t a i n e d w i t h sa mp l i n g p r o b e s

w h i c h me a s u r e c l o s e a p p r o x i ma t i o n s t o d e n s i t y -

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

c o r r e l a t io n s o f t h e t y p e ~ is t h e r e f o r e n o t u s u a l ly

requ i red .

Cons iderab le p rogress has been made in dev i s ing

m o d e l s w h i c h a ll o w t h e c a l c u la t i o n o f t h e Re y -

nolds s t ress ,oui'u a n d t u r b u l e n t f l u x p u / - ~ r ap-

pear ing in Eqs . (13 ) and (14) and is d i scussed in

de ta i l i n the fo ur th sec t ion . L ike Eqs . (13) and

(14) the model s a re genera l ly dev i sed fo r h igh Rey-

n o l d s - n u m b e r f l o w s a n d s p e ci a l c a re mu s t b e e x e r -

c i sed in the in t e rp re t a t ion o f resu l t s i n s i tua t ions ,

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

n u m b e r o f t h e i s o th e r m a l f l o w u p s tr e a m o f t h e

f l a me m a y b e i n s u f f i c i e n t i n t h e p r e s e n c e o f a t e m-

pera tu re r ise and as soc ia t ed changes in v i scos i ty and

d e n s i t y t o m a i n t a i n a f u l l y t u r b u l e n t c o n d i t i o n .

3 N U M E R I C A L S O L U T I O N M E T H O D S

The uncer t a in t i es as soc ia t ed wi th f in i t e -d i f fe rence

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

j e c t o f s o me c o n t r o v e r s y . Sw i t h e n b a n k e t a l.

( 1 9 7 9 ) a n d D r u m m o n d e t a l. ( 1 9 7 9 ) , f o r e x a m p l e ,

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

f l o w s a n d ma k e n o me n t i o n o f p o s s ib l e u n c e r t a i n -

t i es as soc ia t ed wi th numer ica l as sumpt ions . In

c o n t r a s t , Mc D o n a l d ( 1 9 7 9 ) a n d J o n e s a n d Mc -

G u i r k ( 1 9 7 9 ) r a i se d o u b t s a b o u t t h e s e a s s u mp t i o n s ,

even fo r two-d imens iona l f lows .

N u me r i c a l p r o c e d u r e s r e p r e s e n t d i f f e r e n t i a l

e q u a t i o n s b y a p p r o x i ma t e a l g e b r a i c e q u a t i o n s

w h i c h b e c o me l e s s a p p r o x i ma t e w i t h d e c r e a s i n g

d i s ta n c e b e t w e e n t h e n o d e p o i n t s u s e d t o l i n k t h e

a p p r o x i ma t e e q u a t i o n s . I n p r i n c i p l e , t h e n u mb e r

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

the so lu t ion to the d i f fe ren t i a l equa t ions i s

ach ieved . In p rac t i ce , t h i s p roced ure i s l imi t ed by

c o m p u t e r s to r a ge a n d t h e co s t o f c o m p u t e r r u n

t i me . O n e o b v i o u s me t h o d o f o v e r c o m i n g t h is d if -

f i cu l ty i s t o use more accura te (o r h igher -o rder )

f i n i t e - d i f f e r e n c e a p p r o x i ma t i o n s . Su c h me t h o d s

a p p e a r n o t t o h a v e p r o g r e s s e d b e y o n d t h e f o r ma -

t ive s t a t e , and an app l i ca t ion o f h igher -o rde r

s c h e me s t o a s e t o f c o u p l e d n o n l i n e a r ( e l l i p t i c )

c o n s e r v a t i o n e q u a t i o n s r e ma i n s t o b e d e mo n -

s t ra t ed .

T h e a c c u r a c y o f c a lc u l a t io n s is d e p e n d e n t o n

t h e n u me r i c a l s c h e m e a n d a l so o n t h e f l o w a n d i ts

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

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

o f V l a c h o s a n d W h i t e l a w ( 1 9 7 9 ) o b t a i n e d f o r t h e

l a mi n a r f l o w a r o u n d a n a x i s y m me t r i c c o n s t r i c t i o n

i n a p i p e w i t h t h o s e o f G r e e n a n d W h i t el a w ( 1 9 8 0 )

f o r a n a x i s y mm e t r i c r e c ir c u l a ti o n o f a c o m b u s t o r .

In the p ipe f low , ca l cu la t ions w i th 12 7 and

2 2 1 7 n o d e s s h o w e d a ma x i m u m d i f f e r e n c e o f

a r o u n d 2 0 % o f t h e m a x i m u m v e l o c it y va l ue ; t h e

v e l o c i t y v a r i a t io n s w e r e m o n o t o n i c w i t h i n c r ea s e

i n n u mb e r o f n o d e s ; a n d t h e r e s u l t s w i t h 2 2 1 7

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

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

s c h e me w a s u s e d b y G r e e n a n d W h i t e l a w a n d t h e

use o f 22 16 nodes l ed to resu l ts which w ere fa r

f r o m c o r r e c t d u e ma i n l y t o t h e i n a b i li t y o f th i s

me sh to reso lve p rop er ly a smal l reg ion o f rec i rcu -

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

prof i l e . Calcu la t ions wi th 28 X 30 nodes were

s t i l l i nadequate un t i l t he pos i t ions o f the mesh

l ines were a r ranged so as to p rov ide a p rop er reso -

lu t ion o f the p ressu re f ie ld . Resu l t s ob ta ine d wi th

50 50 nodes were s imi l a r , i . e . , max imum veloc-

i t ies wi th in 4%, to those ob ta ined wi th 28 X 30

nodes loca ted to reso lve the p ressu re f i e ld . In ad -

d i t ion to p rob lems as soc ia t ed wi th loca t ing g r id

nodes to ensure tha t smal l reg ions which exer t a

l a rge in f luence on the overa l l f low are adequate ly

reso lved , e r ro rs can a r i se d i rec t ly f rom the f in i t e -

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

c a se o f e l l i p t i c f l o w p r o b l e m s , s o me t i me s f r o m

a fa i lu re to p rocure a su f f i c i en t ly converged so lu -

t i o n . L e o n a r d ( 1 9 7 9 ) h a s d i s c u s s e d s o me o f t h e s e

p r o b l e ms a n d p r o v i d e d a r e mi n d e r t h a t l a c k o f

s k il l i n t h e a p p l i c a t i o n o f n u me r i c a l me t h o d s c a n

lead to se r ious e r ro rs .

T h e n u me r i c a l s c h e me s u s e d , f o r e x a mp l e , b y

Sw i t h e n b a n k e t a l . , D r u mmo n d e t a l . , J o n e s a n d

Mc G u i r k , V l a c h o s a n d W h i t e l a w , a n d G r e e n a n d

W h i t e l aw , ma k e u s e o f c e n t e r e d i m p l i c it d i f fe r -


6/26


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

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

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

t o r y ( i n s p a c e ) s o l u t i o n s a l mo s t a l w a y s a r i s e f o r

ce l l Pec le t num ber s P l Us I Axe,[~at g r e a t e r t h a n

t w o . Fo r n o n l i n e a r p r o b l e m s th e s e i n e v i t a b l y p r e -

v e n t c o n v e r g e n c e f o r e l li p t ic f l o w p r o b l e ms a n d

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

mu s t t h e r e f o r e b e a d o p t e d t o e n s u r e t h a t t h e

f in i t e -d i f fe rence equat ions re t a in the p roper t i es

o f t h e p a r t ia l d i f f e r e n t ia l e q u a t i o n s f r o m w h i c h

t h e y w e r e d e r i v ed , e . g. , w i g g l e s d o n o t d e v e l o p

i n t h e n u me r i c a l s o l u t i o n . T h e p r o c e d u r e mo s t

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

f e r e n c i n g 1 f o r t h e c o n v e c t i o n t e r ms f o r c e ll Pe c -

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

bo th loca l ly and on a d i rec t iona l bas i s , e .g ., i f i n

one f in i t e -d i f fe rence ce l l

p lUI Ax//at

i s g rea te r

than two bu t p l V I

Ay / /a t

i s l es s than two , then

the don or ce l l d i f fe renc ing i s used fo r the x -d i -

r e c t i o n b u t c e n t e r e d d i f f e re n c i n g re t a i n e d f o r th e

y-d i r ec t ion . In p rac t i ce th i s swi tch ing is imple-

m e n t e d b y w h a t a m o u n t s t o a d d i n g a p s e u d o v is -

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

p l f , ~ l A x , ~ / 2

t o the ac tua l

v i s c o s i t y ( mo l e c u l a r p l u s t u r b u l e n t ) a n d a s a c o n -

s e q u e n c e m a y r e s u lt i n a f a l s e o r n u m e r i c a l

d i f fus ion . Whether o r no t s ign i f i can t e r ro r i s there-

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

t h e o t h e r t e r m s in t h e e q u a t i o n : i t is i mp o r t a n t i f

the to ta l , i . e . , numer ica l p lus phys ica l d i f fus ion in

the d i rec t ion in which donor ce l l d i f fe renc ing i s

u s e d ma k e s a s i g n i f i c a n t c o n t r i b u t i o n t o t h e c o n -

s e r v a t io n e q u a t i o n .

N u m e r i c a l d i f f u si o n e r r o r s a r e u s u a l l y mo s t

ser ious whe n don or ce l l d i f fe renc ing i s used in s i tu -

a t ions where the ve loc i ty vec to r i s a l igned c lose to

the d iagonal s o f a f in i t e -d i f fe rence g r id . An es t i -

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

e x p r e s s i o n s u g g e s t e d b y Ru n c h a l a n d W o l f st e i n

( 1 9 6 9 ) ,

P 'numer iea l = 0 .36p I U t l A x s in 20 ,

w h e r e I Ui [ i s th e ma g n i t u d e o f t h e v e l o c i t y a n d

w h e r e 0 i s t h e a n g l e b e t w e e n t h e v e l o c i t y v e c t o r

1 D onor cell differencing involves the use of backward or

forward space differences; which is to be u sed depends

on the direction of the velocity; e.g., see Roach e (1976,

p. 73).

and the g r id . In the case o f the resu l t s o f Gre en

a n d W h i te l a w , f o r e x a mp l e , Pe c l e t n u m b e r s u p

t o 5 0 0 w e r e e n c o u n t e r e d a n d , w i t h t h e 2 8 X 3 0

node s re fer red to ear l i e r, cou ld no t be avo ided ,

espec ia l ly s ince spec ia l a t t en t ion and a concen t ra -

t i o n o f n o d e s w e r e r e q u i r e d t o r e p r e s e n t a s ma l l

b u t i mp o r t a n t r e g io n o f t h e f l o w . T h e ma g n i t u d e

o f / a n u m e t i e a l r a n g e d f r o m / a t X 10 3 to /a t in the

r e g io n o f h ig h e s t Pe c l e t n u mb e r . Fo r t u n a t e l y , i n

the reg ions o f h igh numer ica l d i f fus ion , the mag-

n i t u d e o f t h e p r e s s u re a n d c o n v e c t iv e t e r ms i n t h e

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

d i f fus ion t e rms .

T h e p r e c e d i n g p a r a g r a p h s i n d i c a t e t h a t e r r o r s

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

d i f f e r e n c e a p p r o x i ma t i o n s a n d t o t h e l i m i t e d n u m-

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

Usual ly , t he e r ro rs can be a r ranged to be smal l ,

bu t care i s needed to ensure tha t th i s is so . Wi th

ca lcu la t ions invo lv ing th ree independen t var i ab les ,

t h e i n c r e a s e d c o mp u t e r s t o r a g e r e q u i r e me n t s

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

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

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

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

n u m b e r o f e q u a t i o n s h a v e t o b e s o l v e d . I n s o me

f lows , i t i s l i ke ly tha t the numer ica l so lu t ion

me t h o d w i ll li m i t t h e a c c u r a c y o f c a l cu l a t io n s .

4 . T U R B U L E N C E M O D E L

T h e u n k n o w n v e l o c it y c o va r ia n c e o f E q . ( 1 3 ) - t h e

R e y n o l d s s t r e s s t e n s o r - m a y b e e x p r e s s e d a s t h e

d e p e n d e n t v a r i a b le o f t h e e x a c t c o n s e r v a t i o n e q u a-

t ion:

P

a t u , u / ' + - ~ u ~ - - u i u / '

a x k

x ~ { x j

-u j ax~

a

a p [ a a ,

l u j o p + u t

I o x , J - p u , u

-- P-us uk ax k - -~eu

( 1 5 )


7/26

R E A C T I N G T U R B U L E N T F L O W S 7

A re l a t ed e qua t i on can a l so be de r i ved fo r the un -

we i gh t ed -ave raged co r re l a t i on , bu t , i n bo t h cas es ,

unk now n co r re l a t i ons a ri s e. Th i s is a m a n i fes t a t i on

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

t h e c l o s u r e p r o b l e m - t o w h i c h a s o l u t i o n c a n b e

a t t em p t ed by r e l a t i ng h i ghe r -o rde r co r re l a t i ons t o

t e rm s o f a l ower o rde r and , t he reby , c l o s i ng t he

equ a t i on s e t . I n gene ra l t he t r i p le -ve l oc i t y co r re l a-

t i on , co r r e l a t i ons i nvo l v i ng f l uc t ua t i ng p res s u re ,

and t he v i s cous d i s s i pa t i on ei~ must a l l be ap-

p r o x i m a t e d i n t e r m s o f l o w e r - o r d e r k n o w n

quant i t i es .

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

t h e e q u a t i o n s f o r t h e c o m p o n e n t s o f t h e s t r e s s

t ens o r have p rev i ous l y been p ropos ed i n t he con -

t ex t o f i s o t he rm al f lows s ee , fo r exam pl e , Lau nder ,

1975 ; B rads haw, 1976 ; Lum l ey , 197 8 ; and B rad -

s haw e t a l . , 1981 , and t he i r ex t ens i on t o r eac t i ng

f l ows wi t h a s s oc i a t ed l a rge dens i t y va r i a t i ons has

b e e n c o n s i d e r e d b y J o n e s ( 1 9 8 0 ) . H o w e v e r , s u c h

m o d e l s a r e o f c o n s id e r a b le c o m p l e x i t y , an d f o r

reci rcu la t ing f lows there are subs tant ia l d i f f i cu l t i es

i n ob t a i n i ng e r ro r f r ee num er i ca l s o l u t i ons . As a

cons equence , s econd -o rde r s t r e s s c l o s u re m ode l s

m u s t be co ns i de red t o be la rge l y un t es t e d i n e l-

l i p ti c f l ows , and , t he re fo re , a t t he p res en t t im e

s i m p l e r e ddy v i s cos i t y - t ype m ode l s , s uch as t ha t

des c r i bed be l ow , a re t o b e p re fe r r ed . An a l t e rna -

t ive ap p roac h wou l d be t ha t o f s ubg ri d sca le

m od e l i ng ( see , fo r exam pl e , R eyn o l ds , 1976 ; and

S chu m a nn e t a l ., 1980 , bu t i n v i ew o f bo t h it s

c o m p l e x i t y a n d e a r ly s t a te o f d e v e l o p m e n t t h is is

a ls o inapp rop r i a t e a t t he p res en t t i m e .

T h e a p p r o a c h t o t h e t u r b u l e n c e m o d e l s c o m -

m o n l y u s ed fo r r ec i r cu l a t i ng f lows i s a fo rm o f

t h e t w o - e q u a t i o n m o d e l i n tr o d u c e d b y J o n e s a n d

Launder (1972 ) . Th i s i nvo l ves an a s s um ed l i nea r

r e l a t i on be t ween t he R eyno l ds s t r e s s and r a t e o f

strain:

- - ~ t t

pU i Uj

= ~ i J l -P k + t a t ~ X k I

( 1 6 )

F o r t he t u rbu l en t f l ux o f s ca l a r quan t i t i e s a g rad i -

ent d i f fus ion model i s used , v iz . ,

t a t a ~ ~

~ u y ~ -

ot axj

( 1 7 )

The t u rbu l en t (o r edd y ) v i s cos i ty is gi ven by

ta t = G ~ k 2 / e ,

where k(---- Ui Ui / 2 ) and e a re t he t u rbu l ence k i -

ne t i c e ne rgy and d i s s ipa t i on r a t e , r e s pec t i ve l y , t he

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

t he t r ans po r t equa t i ons :

~ k

Oxj

= . . . .tat + 12 _ p._Ui,,Uj,,

a x j a x ~ a x j

tat O~ O~

- U0x

( 1 8 )

~ U j b

Oxj

tat + ta

ax~ axj

e [ _ - 7 7 ,, a 0 , +

C l k pui uj ~xj

e 2

- C z - o k

u t a ~ a ~ ]

d

~2 ax i Ox~

( 1 9 )

The as s um pt i ons i nvo l ved i n t hes e equa t i ons have

been ex t ens i ve l y d i scus s ed by J ones and L aunde r

( 1 9 7 2 ) a n d b y J o n e s a n d M c G u i r k ( 1 9 8 0 a ) a n d

e l s e w h e r e . I t i s r e c o m m e n d e d t h a t t h e c o n s t a n t s

be ass igned the fo l lowin g values :

Cu = 0 .0 9 , C 1 = 1 . 4 4 , 6 '2 = 1 . 9 2 ;

o k = 1 . 0 , a e = 1 . 3 0 , a t = 0 . 7 .

Equa t i ons (16 ) - (19 ) can a l s o be wr i t t en i n un -

we i gh t ed -ave rage fo rm and , a s i nd i ca t ed by B ray


8/26


( 1 9 7 3 ) i n c l u d e a d d i t i o n a l t e r m s w h i c h s t e m f r o m

t h e f l u c t u a ti n g d e n s i ty a n d w h i c h m u s t b e m o d e l e d .

The u s e o f Eqs . (16 ) - (1 9 ) i nvo l ves t he a s s um pt i on

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

fo r con s t an t dens i t y f l ows and wh i ch inc l ude t he

g rad i en t d i f fu s i on m ode l can s i m p l y be r ewr i t t en

i n dens i t y -we i gh t ed fo rm wi t hou t any exp l i c i t ac -

cou n t be i ng t ake n o f dens i t y f l uc t ua t i ons , t he in -

f l uence o f t hes e be i ng as s um ed t o be en t i r e l y de -

scr ibed b y the use o f den s i ty-we ighted averaging .

Thes e a s s um pt i ons have no t b een and a re un l i ke l y

t o be fu l l y t e s t ed i n a d i r ec t sens e , bu t t he p res en t

ev i dence wi l l s uppo r t t he i r va l i d i t y i n m an y cases:

t he coun t e rg rad i en t d i f fu s i on obs e rved by Mos s

(19 80 ) i s an excep t i on . In the m a j o r i t y o f cas es

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

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

s i gn i f i can t l y d i f f e ren t t o t ha t o f t he cons t an t den -

s i t y ve r s i on app rop r i a t e t o i s o t he rm al f l ows . The

m e r i t s o f t h e t w o - e q u a t i o n m o d e l h a ve b e e n a p -

prai sed for i so thermal f lows in several papers , for

e x a m p l e , L a u n d e r a n d S p a ld i n g ( 1 9 7 4 ) , J o n e s a n d

M c G u i r k ( 1 9 8 0 b ) , a n d R i b e i ro a n d W h i te la w

( 1 9 8 0 a ) , a n d a p p e a r s t o r e p r e s e n t a g o o d c o m -

p rom i s e .

T h e c o n t r a s t b e t w e e n t h e f lo w s o f J o n e s a n d

M c G u i r k ( 1 9 8 0 b ) a n d R i b e i r o a n d W h i t e l a w

(1980a) i s i n s t ruc t i ve . The t r a j ec t o ry o f t he j e t i n

c ro s s f l ow i s con t ro l l ed l a rge l y by p res s u re fo rces ,

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

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

In t he coax i a l j e t s o f R i be i ro and Whi t e l aw , how-

ever , t u rbu l enc e a s s um pt i ons a re i m p or t an t , and

t hos e a s s oc i a t ed w i t h t w o-equ a t i on m ode l s and i n-

deed ex i s t i ng R eyno l ds s t r e s s c l o s u res were s hown

to be incorrect in several respect s . That such d i s -

c repanc i es s om et i m es o ccu r i s no t en t i r e l y s urp r is -

i ng , fo r bo t h R eyno l ds s t r e s s c l o s u res and t wo-

e q u a t i o n m o d e l s i n v o l v e a p p r o x i m a t i n g u n k n o w n

t e rm s p u re l y i n t e rm s o f l oca l t u rbu l enc e quan -

t i t i e s . Im p l i c i t l y con t a i ned wi t h i n t h i s p rocedu re

is t he a s s um pt i on t ha t d epa r t u res o f the t u rbu -

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

s ense s m a ll ; an app ro x i m a t i on wh i ch is un dou b t ed l y

i n con f l i c t w i t h obs e rva t i on fo r s om e f l ows . In

s p i t e o f t h i s l i m i t a t i on and t hos e a s s oc i a t ed wi t h

c l o s u re o f t he t u rbu l ence ene rgy d i s s i pa t i on r a t e

equa t i on , wh i ch can no t be s ep a ra t e l y eva l ua t ed ,

t h e m a n y c o m p a r i s o n s w i t h e x p e r i m e n t s w h i c h

have been r epo r t ed s ugges t t ha t a t p res en t t he t w o .

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

com bus t i ng f l ows and wil l l i m i t ob t a i nab l e accu racy

i n a r e s t r i c t ed num ber o f c ir cum s t ances wh i ch can

i nc l ude , fo r exam pl e , nea r wakes beh i nd b l u f f

bodies and some swi r l ing f lows . I t should be re-

m e m b ered t ha t a ll m ode l s neces s a r i l y have l i m i ts

t o t he i r r ange o f app l i cab i l i t y , and t hus accu ra t e

res u lt s ca nno t be expec t ed unde r a ll c i r cum -

s t ances ; fo r exam pl e , and as d i s cus s ed by R i be i ro

and Whi t e l aw (1980b ) , ro und j e t s is su ing i n t o

s t agnan t s u r rounds a re am ong s eve ra l t u rbu l ence -

con t ro l l ed f l ows wh i ch a re no t we ll r ep res en t ed by

ex i s t ing m ode l s .

In pas s i ng i t s hou l d a l s o be no t ed t ha t t he u s e

o f g rad i en t d i f fu s i on m ode l s c l ea r l y p rec l udes t he

pos s i b i l i t y o f r ep res en t i ng s o -ca l l ed coun t e r -

g r a d ie n t d i f f u s i o n - t h a t is , t u r b u l e n t d i f f u s io n o f ,

s ay , s pec ies up i ts m e an g rad i en t . L i bb y and B ray

(1980a , b ) have p red i c t ed s uch behav i o r t o be i m -

po r t an t i n p rem i xed p l ana r f l am es , and Mos s

(198 0 ) has obs e rved cou n t e r g rad i en t d i f fu s i on i n

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

t i m e i t i s d i f f icu l t t o a s s es s whe t he r o r no t c oun t e r -

g rad i en t d i f fu s i on e f f ec t s h ave s ign i f ican t i n f lu -

ence i n p rac t i ca l s y s t em s , t hough p res en t i nd i ca -

t i ons a re t ha t t hey do no t . C oun t e rg rad i en t d i f fu -

s i on can be a t t r i bu t ed t o t he p re fe ren t i a l i n f l uence

o f m ea n p res s u re g rad i en t on l ow- and h i gh -dens i ty

gas wh i ch i s m an i fes t t h rough t he appea rance o f

t e rm s l i ke

p ' , / ' a n d

i n t he exac t F av re -ave raged R eyno l ds s t r e s s and

t u rbu l en t s ca l a r f l ux t r ans po r t equa t i ons . In p re -

m i xed f l ows t h i s p roces s can be des c r i bed by t he

L i b b y - B r a y m o d e l . A l t e r n a t i v e l y , a n d p o t e n t i a l l y

m ore gene ra l l y , it c an be r ep res en t ed v ia a s econd -

o rde r c l o s u re m o de l , s uch as t ha t o u t l i ned i n J ones

(1980 ) , i n wh i ch t r ans po r t equa t i ons a re t o be

s o l ved fo r a ll t h e i ndep ende n t s econd -o rd e r m o-

i n U ~ ~ . - - r - - w ~ .

m e n t s c i a m g

u i u j , u i

q~,~ , a n a p u i . n o w -

ever , fo r bo t h app roaches fu r t he r t e s t i ng and i n -

ves t igat ion i s needed.


9/26


5 . C O M B U S T I O N M O D E L S

T h e c o m b u s t i o n mo d e l in e ss e n c e mu s t p r o v i d e a

m e t h o d o f e v a l u a ti n g t h e m e a n f o r m a t i o n r at e o f

each spec ies p resen t , and in add i t ion a l low the ca l -

c u l a t i o n o f me a n f l ui d t e mp e r a t u r e a n d t h e d e n -

s i ty o f t h e mi x t u r e . U n d e r i d e a l c ir c u ms t a n c e s w e

shou ld l ike to be ab le to han d le rea c t ing f lows

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

t i o n ma y o c c u r v i a a l a r g e n u mb e r o f f i n i t e - r a t e

r e a c t i o n s t e p s . [ Fo r e x a m p l e , t h e s i mp l if i e d me c h -

a n i s m f o r t h e h i g h t e mp e r a t u r e o x i d i z a t i o n o f

me t h a n e o f Bo w ma n ( 1 9 7 0 ) i n v ol v es 1 1 s p e c ie s

and 14 reac t ion s teps . ] In genera l t h i s wo u ld neces -

s i ta t e t h e s o l u t i o n o f c o n s e r v a t i o n e q u a t i o n s f o r

t h e me a n v a l u e o f e a c h o f t h e i n d e p e n d e n t s p e ci e s,

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

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

t ion ra t es a re inev i t ab ly h igh ly non l inear func t ions

o f t e m p e r a t u r e a n d s p e c ie s c o n c e n t r a t i o n s , a n d

t h u s k n o w l e d g e o f t h e me a n v a l u e s o f t h e s e l a t t e r

quan t i t i es i s i n su f f i c i en t t o a l low the eva lua t ion o f

me a n f o r ma t i o n r a t e s . I n f a c t t h e d e t e r mi n a t i o n o f

me a n f o r m a t i o n r a t e s r e p r e s e n t s a ma j o r d i f f ic u l t y

i n th e d e v e l o p m e n t o f p r e d i c t i o n m e t h o d s f o r c o rn -

b u s t i n g f l o w s . T h e i mp o r t a n c e o f f l u c t u a t i o n s i n

t u r b u l e n t f l o w s is w e l l k n o w n a n d mu s t b e r e p r e -

s e n t e d i n th e e v a l u a t i o n o f me a n f o r m a t i o n r a t e s ,

d e n s i t y , a n d t e mp e r a t u r e . T h u s t h e r e a c t i o n r a t e

f o r t h e o n e - s t e p r e a c t io n o f Se c t i o n 1 c a n n o t b e

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

a n d , f o r e x a mp l e , t h e e q u a t i o n

f A = - - - p k , ( T ) C A C B

wi ll l ead to e r ro rs up to th ree o rders o f magn i -

t u d e . T h e mo s t c o n v e n i e n t w a y t o r e p r e s e n t

the necessary sca la r f luc tua t ions i s wi th a p roba-

b i l i t y d e n s i t y f u n c t i o n ( p . d . f . ) . O t h e r a p p r o a c h e s

are poss ib le and can be descr ibed in t e rms o f p .d . f .

p r o c e d u r e s . A t t h e p r e s e n t t ime it a p p e a r s t h a t

o n l y p . d . f , t r a n s p o r t e q u a t i o n f o r mu l a t i o n s o f f e r

t h e p o s s i b i l it y o f h a n d l in g l a rg e n u m b e r s o f r e a c t in g

s p e c i e s , b u t e v e n h e r e i n v i e w o f t h e mu l t i d i me n -

s i o n a l i t y o f t h e a p p r o a c h , c o mp u t e r s t o r a g e r e -

q u i r e me n t s a n d r u n t i me s c a n b e e x p e c t e d t o b e

v e r y l a rg e , a n d t h e i r v i a b i l it y t h e r e f o r e r e ma i n s t o

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

a n a d d i t i o n a l p o t e n t i a l p r o b l e m, f o r t h e d e t a i l e d

r e a c t i o n me c h a n i s m w h e r e b y o x i d a t i o n o c c u r s i s

k n o w n o n l y f o r a fe w o f t h e s i mp l e r h y d r o c a r b o n s .

I n p r a c t i c e , h o w e v e r , t h i s ma y n o t p r e s e n t a s i g -

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

e v a l u a t i n g me a n f o r ma t i o n r a t e s a r e s u c h t h a t , f o r

the fo reseeab le fu tu re , t he use o f on ly a smal l

n u m b e r o f f i ni te - r at e r e a c t i o n m e c h a n i s m s m a y b e

t h e o n l y p r a c t i c a b l e r e c o u r s e .

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

a s ma l l n u m b e r o f r e a c t i o n s d o e s n o t a p p e a r t o b e

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

reac t ions as soc ia t ed wi th hea t re l ease in the h igh-

t e m p e r a t u r e o x i d a t io n o f h y d r o c a r b o n f u e ls ( a n d

h y d r o g e n ) u s u a l ly h a v e t i me s c a le s v e r y s h o r t c o m -

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

p r o c e s se s a n d t h e a s s u mp t i o n o f f a s t c h e m i s t r y

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

c i r c u ms t a n c e s . O f c o u r s e , i t i s n o t a p p r o p r i a t e u n -

der a l l c i rcums tances . In par t i cu la r , t he ca l cu la t ion

o f th e f o r m a t i o n a n d e mi s s i o n o f p o l l u t a n t s s u c h

a s c a r b o n mo n o x i d e , u n b u r n t f u e l , a n d n i t r i c

ox ide a l l requ i re cons idera t ion o f f in i t e - ra t e chemi -

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

p h e n o me n a . N e v e r t h e l e ss t h e f a s t c h e m i s t r y a s-

sumpt ion resu l t s i n a cons iderab le s impl i ca t ion as

fa r as model ing i s concerned and does a l low a rea-

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

c l u d in g t h e f i el d s o f t e mp e r a t u r e a n d c o n c e n t r a -

t i o n s o f ma j o r s p e c i e s in a w i d e r a n g e o f c o n t i n u o u s

f l o w c o mb u s t i o n s y s t e ms .

D i f f u s i o n o r U n p r e m i x e d F l a m e s

T h e f a s t c h e m i s t r y a s s u m p t i o n is i n v o k e d m o s t

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

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

s t r e a ms . T h e n , i f t h e c h e mi s t r y i s a s s u me d s u f -

f i c i en t ly fas t fo r a l l reac t ions to go to comple t ion

( o r e q u i l i b r i u m) a s so o n a s th e r e a c t a n t s a r e mi x e d ,

t h e t h e r mo c h e mi c a l s t a t e o f t h e r e s u l t i n g mi x t u r e

ma y b e d e t e r mi n e d p u r e l y i n t e r ms o f s t r i c t l y c o n -

served sca la r var i ab les . The need to eva lua te mean

r e a c t i o n r a t e s is t h e r e b y r e mo v e d . I n p ri n c i p le

t h e e q u i l ib r i u m c o m p o s i t i o n , t e mp e r a t u r e , a n d

f l u id d e n s i t y o f a g a s mi x t u r e c a n b e d e t e r mi n e d

i f th e e l e me n t a l m a s s f r a c t i o n s o f a ll e l e me n t s p res -

e n t t h e p r e s s u r e , a n d e n t h a l p y a r e k n o w n . H o w -


10/26

1 0 W . P . J O N E S a n d J . H . W H I T E L A W

e v e r , in t h e c a se o f tu r b u l e n t f l a me s a n u m b e r o f

s impl i f i ca t ions a re invar i ab ly made . Wi th reason-

ab le accuracy i t can be as sumed tha t a l l spec ies

and hea t have equal d i f fus ion coef f i c i en t s , i . e . t he

t u r b u l e n t P r a n d t l / Sc h mi d t n u mb e r s a r e a l l e q u a l ,

a n d i n ma n y c i r c u ms t a n c e s t h a t t h e h e a t l o s s t o

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

r e le a s e. I n t h i s s it u a t i o n t h e i n s t a n t a n e o u s t h e r m o -

chemica l s t a t e o f the gas i s de te rminab le as a non-

l inear func t ion o f a s ing le s t r i c t ly conse rved ( i .e . ,

ze ro source) sca la r var i ab le . Al l t he re l evan t s t r i c t ly

conserved sca la r var i ab les a re l i near ly re l a t ed , so

t h e a c t u a l o n e t o b e u s e d b e c o me s a ma t t e r o f

t a s te . T y p i c a l c h o i c e s a re t h e mi x t u r e f r a c t i o n f ,

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

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

a n y e l e me n t p r e s e n t , a n d t h e e n t h a l p y o r a n y c o m-

p o s i t e v a r i a b l e ma d e u p o f t h e s e q u a n t i t i e s . I n t h e

p r e s e n t c o n t e x t t h e mi x t u r e f r a c t i o n f w il l b e

t aken as the sca la r var i ab le to be used ; a l l o ther

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

h = ( h ~u - - h a ) f + h a ,

Y a = ( Y a t u - - Y a a ) f + Y ,

w h e r e Y a is th e ma s s f r a c t i o n o f e l e me n t a a n d t h e

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

to the fue l and a i r s t reams , respec t ive ly . The in -

s t a n t a n e o u s c o mp o s i t i o n , t e mp e r a t u r e , a n d d e n s i t y

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

f . T h e s i mp l e s t me t h o d o f d o i n g t h i s i s t o a s s u me

tha t the r eac t ion p roce eds v ia a s ing le-s t ep i r revers -

i b le r e a c t i o n o f t h e f o r m

f u e l + o x i d a n t ~ p r o d u c t s .

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

mi x t u r e ( a t a n y i n s t a n t ) c o n s i st s o f e i th e r f u e l p lu s

p r o d u c t s o r o x i d a n t p l u s p r o d u c t s ; i . e . , f u e l a n d

o x i d a n t c a n n o t c o e x i s t o n a n i n s t a n t a n e o u s b a s i s .

A k n o w l e d g e o f f i s th e n s u f f i c i e n t t o d e t e r m i n e

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

mass f rac t ions , and wi th the ad iaba t i c f low as -

s u m p t i o n t h e t e mp e r a t u r e m a y a l s o b e c a l c u l a te d .

I n t h e p a s t t h i s s i mp l i f i e d mo d e l h a s o f t e n b e e n

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

da ta , e .g ., con s tan t spec i f i c hea t s . Th i s i s an en -

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

l ig ib le s impl i f i ca t ion , v i r tua l ly no sav ings in com-

pu ta t iona l e f fo r t , and can resu l t i n l a rge e r ro rs in

t h e c a l c u l a t e d t e mp e r a t u r e . A n a l t e r n a t i v e me a n s

o f r e la t i n g t h e t h e r m o c h e m i c a l s t a te t o t h e m i x -

tu re f rac t ion i s v i a a chemica l equ i l ib r ium assump-

t ion , i . e . , reac t ion ra t es a re su f f i c i en t ly fas t fo r the

mi x t u r e t o b e i n a s t a t e o f c h e mi c a l e q u i li b r iu m.

Wi th th i s as sumpt ion i t i s no t necessary to spec i fy

a p r e c i s e r e a c t i o n me c h a n i s m, a n d t h e e q u i l i b r i u m

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

s i ty c a n b e o b t a i n e d b y m i n i mi z i n g t h e f r e e e n e r g y .

A w e l l - te s t e d a n d r e li a b le c o m p u t e r p r o g r a m b a s e d

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

Br i d e ( 1 9 7 1 ) . I n s o f a r a s ma j o r s p e c ie s a n d t e m-

p e r a t u r e a r e c o n c e r n e d b o t h f o r th e H 2 / a i r s y s t e m

a n d l e a n h y d r o c a r b o n / a i r m i x t u r e s , t h e r e i s l i tt l e

to c hoose be tw een the s ing le-st ep i r revers ib l e reac-

t i o n ( w i t h a c c u r a t e t h e r mo d y n a mi c d a t a ) a n d t h e

f u l l e q u i l i b r i u m c a l c u l a t i o n . H o w e v e r , f o r h y d r o -

c a r b o n / a i r m i x t u r e s r i c h e r t h a n s t o i c h i o me t r i c

w h e r e la rg e a m o u n t s o f c a r b o n m o n o x i d e m a y b e

f o r me d , d i f f e r e n c e s b e t w e e n t h e t w o me t h o d s b e -

com es s ign i f ican t : t he s ing le-s t ep reac t ion mode l

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

m a y g e n e r a t e t e mp e r a t u r e s u p t o 2 0 0 K h i g he r

t h a n t h e e q u i l i b r i u m t e mp e r a t u r e s .

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

w h e r e b y t h e i n s t a n t a n e o u s t h e r mo c h e mi c a l s t a t e

a t a n y p o i n t ma y b e r e l a t e d t o t h e i n s t a n t a n e o u s

v a l u e o f m i x t u r e f r a c t i o n a t t h a t p o i n t . Be c a u s e o f

the s t rong non l inea r i ty o f these re l a t ions i t i s neces -

s a r y t o t a k e a c c o u n t o f t h e f l u c t u a t i o n s i n mi x t u r e

f rac t ion which a r i se in a l l t u rbu len t d i f fus ion

f l a me s . T h e mo s t c o n v e n i e n t w a y o f a c h ie v i ng th i s

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

f u n c t io n f o r m i x t u r e f r a c t i o n , P ( f , xi).

A b o u t t h e s i m p l e st a n d m o s t p o p u l a r a p p ro a c h

is t o s p e c i f y a t w o - p a r a m e t e r f o r m o f t h e p r o b a -

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

var i ance o f f . I f Favre averag ing i s used , t hen the

v a l u e s o f t h e s e q u a n t i t i e s a r e o b t a i n e d f r o m t h e

s o l u ti o n o f t h e e q u a t i o n s

xj axj at Oxj

axj Oxi o t OXj l

2o , \ a j ?


11/26


where CD has a value of 2 .0 .

The p .d . f , t o be c ons t ruc ted i s now a dens i ty -

w e igh ted fu nc t ion w h ich a l low s eva lua tion o f bo th

dens i ty -w e igh ted and unw eigh ted mean va lues .

Dens i ty-w eighted mean values are g iven by

fO

*f f ) e f , xi ) a f

and unw eigh ted va lues by

= P [ 1 ~ ( f ) P ( f , x i ) dr .

~

4

o ( f )

The dens i ty can be ob ta ined f rom

= o f f )

(H ere s t ands fo r any qua n t i ty w h ich m ay be

un ique ly r e l a t ed to f , i nc lud ing t empe ra tu re and

species mass fraction.)

The use o f Favre averaging is no t a lways recog-

nized; of ten unweighted averaging is os tens ib ly

used, but the corre la t ions involving f luctuat ing

dens i ty are ignored . However , the neglect of den-

s i ty f luc tua t ions s imp ly r educes the unw eigh ted

averaged equat ions to a form ident ical to the

dens i ty-we ighted averaged equat ion s . The so lu-

t ion o f thes e equa t ions w ou ld then y ie ld dens i ty -

weighted values, exce pt tha t the de ns i ty is evalu-

ated in a m ann er incon s is tent wi th F avre averaging,

so that er roneous resul ts are obta ined.

Compu ta t ions have been made us ing a number

o f a s s umed fo rm s fo r the p robab i l i ty dens i ty func -

t ions . For exam ple , the rectangular wave var ia t ion

of f wi th t im e sugges ted by Spald ing (1971) and

u s ed b y G o s m a n a n d L o c k w o o d (1 9 7 3 ) a n d K h a-

l il e t a l . (1975) corresponds to tw o 6- funct ions lo-

c a t e d a t f + a n d f - :

e l f , x O = a~ ( f - f + + 1 - a ) 8 f - f - ) ,

w here a , f , and f - a re de te rmined f rom the

values of j r and f ' ~ . More realist ic specif icat ions of

the p robab i l i ty dens i ty func t ions have been p ro -

pos ed and us ed . F o r example , N agu ib and Lock-

w oo d (1975) p ropos ed and us ed a c l ipped G aus sian

dis t r ibut ion:

P(.f , xi)

= L e r f c { ~ % ]

2 8 ( . t )

1 - - f / 6 ( 1 - - f )

+ l e r f c X /~ o ]

+ [ H f ) - / ~ f - 1 ) ]

where H ( ) i s the Heaviside funct io n and again the

param eters ro and Oo are obta ined f rom the values

o f f and f f ' z . In th is case expl ic i t express ions for

fo and O o canno t be ob ta ined , and the ir va lues

mus t be ob ta ined i l e r a t ive ly ; f o r compu ta t iona l

pu rpos es th i s i s o f t en done once and fo r a l l

and the resul ts s tored in tab ular fo rm. An al terna-

t ive f orm ulat i on o f the c l ipped Gauss ian p .d .f , was

adop ted by K en t and B i lge r (1977) w hereby the

in te rm i t t ency w as u t il i zed . They w r i t e fo r the p .d . f .

P ( f , x i ) = [1 - I ( x i ) ] 8 ( f ) + l ( x i ) e 1 f , xi ) .

F or a j e t d i f fu s ion f l ame an empi r i ca l co r r e l a t ion

w as u s ed to e s t ima te the in t e rm i t t ancy l (x i ) : P I ( )

i s the p robab i l i ty func t ion fo r the tu rbu len t f lu id

for which a c l ipped Gauss ian d is t r ibut ion was

used.

The c l ipp ing o f a G aus s ian d i s t r ibu tion s o

that the probabi l i ty is f in i te only in the a l lowable

range of f i s arb i t rary , and for th is reason unappeal-

ing . A d is t r ibut ion f ree f rom th is procedure is the

/3-probabil ity dens i ty fun ct ion u t i l ized by R ichard-

son e t a l . (1953) , Rhodes , e t a l . (1974) , Jones and

P r idd in (1978) , and J ones and M cG ui rk (1980) . I t

can be w r i t t en

p - l 1

- j O b - x

P( ' f 'x i ) 1 , 0 < f < 1 ,

fo f - l ( 1 J O 1 d f


12/26


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

t h e v a l u e s o f j T a n d ]7 2 .

A c o m p a r i s o n o f c a l c u la t i on s , p e r f o r m e d w i t h

d e n s i t y - w e i g h t e d a v e r a g e d e q u a t i o n s a n d t h e

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

w i t h t h e m e a s u r e m e n t s o f K e n t a n d B i l ge r ( 1 9 7 2 )

i s g iven in Jones (1980) . In summary , the resu l t s

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

sa t i s fac to ry in tha t i t gen era t ed rad ia l p ro f i l es o f

u n w e i g h te d t e m p e r a t u r e w i t h t w o m a x i m a a n d

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

pro f i l es . There was l i t t l e t o choose be tween the

c l ip pe d Gauss i an and /3 -p .d . f . s , and in bo th ca l-

c u l a t i o n s t h e me a s u r e d t e mp e r a t u r e a n d s p e c i e s

ma s s f r a c t i o n p r o f i l e s w e r e a c c u r a t e l y p r e d i c t e d .

T h e ma j o r s p e c i e s a n d t e mp e r a t u r e t h u s a p p e a r t o

be re l a t ive ly insens i t ive to the p rec i se shape o f

t h e p .d . f , o n c e t h e me a n a n d va r i a n ce ( i ts w i d t h )

a r e d e t e r mi n e d , w i t h t h e p r o v i s i o n t h a t i t b e c o n -

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

D i r a c f u n c t i o n s a t t h e b o u n d s .

P o l l u t a n t F o r m a t i o n i n i f f u s io n F l a me s

Ni t r i c ox ide i s no rmal ly p resen t in t race quan t i t i es

and has neg l ig ib le in f luence on the reac t ions as -

s o c i a t e d w i t h h e a t r e l e a s e , t h e t h e r mo d y n a mi c

s t a t e o f the gas , o r the f low. I t i s genera l ly ac-

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

m e t r ic c o n d i t i o n s - t h a t f o r m a t i o n o f t h e r m a l N O

i s d e s c r ib e d b y t h e e x t e n d e d Z e l d o v i c h me c h a n i s m:

O + N z ~ N O + N ( i)

N + O 2 ~ N O + O , ( ii)

N + OH ~ NO + H. ( i ii )

I n m o s t c o m b u s t i o n s y s t em s t h e N O c o n c e n tr a -

t ions wi l l be wel l be low the i r equ i l ib r ium leve l s ,

a n d i f a s t e a d y - s t a t e a p p r o x i ma t i o n i s t h e n i n -

v o k e d f o r t h e N a t o m c o n c e n t r a t i o n , t h e i n s t a n -

t a n e o u s N O f o r ma t i o n r a t e c a n b e e x p r e s s e d a s

M N O

S N o = 2 p k~ ( O - - C o CN 2 .

M o M N 2

T h e r a t e c o n s t a n t k t c i) i s a f u n c t i o n o f t e mp e r a -

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

t ions a re as sum ed to have equ i l ib r ium va lues , t hen

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

a s a u n i q u e f u n c t i o n o f th e m i x t u r e f r a c t i o n . T h e

me a n f o r ma t i o n r a t e r e q u i r e d i n t h e me a n N O ma s s

f r a c t i o n e q u a t i o n c a n t h e n b e d e t e r mi n e d f r o m

3 N o x ~ . ) = SNo (f )Pff , x j)d f .

W h e r e a s t h e r e i s s o me e v i d e n c e t h a t t h e a b o v e

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

c o mb u s t o r o p e r a t i n g p r e s s u r e s a n d i n l e t t e mp e r a -

tu res (e .g . , Jones and Pr idd in , 1978) , i t does no t

p r o d u c e a c c u r a t e r e s u lt s a t a t mo s p h e r i c c o n d i t io n s .

Fo r e x a m p l e , w h e n a p p l i e d t o H 2 / a i r f la me s it

d o e s n o t r e p r o d u c e t h e r i c h s h i f t o f t h e ma x i -

mu m N O c o n c e n t r a t i o n o b s e r v e d b y B i l g e r a n d

Beck (19 75) and g ives NO leve ls subs tan t i a l ly d i f -

f e r e n t f r o m t h o s e me a s u r e d . I n v i e w o f t h e v e r y

s t ro n g d e p e n d e n c e o f N O f o r m a t i o n r a te o n t e m -

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

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

d i s c r e p a n c y , b u t a mo r e p o p u l a r v i e w is t h a t t h e

er ro r i s as soc ia t ed wi th the equ i l ib r ium oxygen

a t o m a s s u mp t i o n . T h e a p p r o a c h o f f r e e r a d i c a l s ,

such as oxyg en a to m s , to e qu i l ib r ium i s v i a re l a t ive ly

s l o w t h r e e - b o d y r e c o mb i n a t i o n r e a c t i o n s . T y p i -

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

o f m a g n i t u d e g r e a t e r t h a n e q u i l ib r i u m v a lu e s c a n

ar i se ; as soc ia t ed wi th th i s wi l l be some dev ia t ion

f r o m t h e e q u i l i b r i u m t e mp e r a t u r e . A d e s c r i p t i o n

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

f o r m a l i s m a d o p t e d b y J a n i c k a a n d K o l l m a n n

( 1 9 8 0 ) f o r t h e h y d r o g e n - a i r f l a me . T h e b i mo l e c u -

l a r r e a c t i o n s o f t h e H 2 / a i r s y s t e m w e r e e q u i l i b r a te d

a n d , f o l l o w i n g t h e w o r k o f D i x o n - L e w i s ( 1 9 7 5 ) , a

s ing le c om bine d reac t ion p rogress var i ab le r (which

i s re l a t ed to a com pos i t e mass f r ac t ion va r i ab le ) was

i n t r o d u c e d t o d e s c r ib e t h e t h r e e - b o d y r e a c t io n s .

T h e t h e r mo c h e m i c a l s t a t e o f t h e g a s c a n th e n b e

d e t e r mi n e d i n t e r ms o f t w o v a r i a b l e s s o t h a t t h e

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

progress var i ab le r i s needed . Jan icka and Kol l -

mann assumed f and r t o be s t a t i s t i ca l ly indepen-

d e n t a n d c o n s t r u c t e d a p . d . f, i n t e r ms o f t h e me a n s

a n d v a r i a n ce s o f f a n d r u s i n g a b e t a f u n c t i o n f o r

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

f o r t h e r e a c t i o n p r o g r e s s v a r i a b l e . U p t o a b o u t 8 0

n o z z l e d i a m e t e r s t h e r e s u l ti n g N O p r e d i c t i o n s w e r e

i n g o o d a g r e e me n t w i t h m e a s u r e m e n t s , a s w e r e t h e


13/26


ma j o r s p e c i e s a n d t e mp e r a t u r e s , t h o u g h t h e l a t t e r

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

t a i n e d w i t h t h e f a s t c h e mi s t r y - f u l l e q u i l i b r i u m

m o d e l . T h e e x t e n si o n o f t h e J a n i c k a - K o l l m a n n

f o r m a l i sm t o h y d r o c a r b o n - a i r s y s t e m s ca n b e

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

def in ing a re ac t ion p rogress var i ab le in t e rms o f

t h e t o t a l n u mb e r o f mo l e s / u n i t ma s s: o n l y th e

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

i . e ., p r odu ct io n ra t e , o f th is quan t i ty . I t i s l i ke ly ,

however , t ha t t he spec i f i ca t ion o f a more rea l i s t i c

f o r m f o r t h e j o i n t p . d . f , w o u l d b e b e n e f i c i a l a n d

lead to more re l i ab le resu l t s .

A n a l t e r n a t i v e me t h o d o f d e s c r i b i n g f i n i t e r a t e

ef fec t s in d i f fus ion f l ames i s t he per tu rba t ion

me t h o d f o r mu l a t e d b y B i l g e r ( 1 9 8 0 ) . I n t h i s a p -

p r o a c h ma s s f r a c t i o n s a r e s e p a r a t e d i n t o t h e i r

e q u i l i b r i u m v a l u e s d e p e n d e n t o n t h e mi x t u r e f r a c -

t i o n p l u s a p e r t u r b a t i o n ( o r d e p a r t u r e ) f r o m t h is

va lue . Exac t equa t ions a re then der ived fo r the per -

t u r b e d ma s s f r a c ti o n s . H o w e v e r , t h e a v e r a g e d f o r m s

o f t h es e e q u a t io n s c o n t a i n u n k n o w n t e r m s c o m -

p r is i ng t h e me a n f o r m a t i o n r a t e o f t h e ma s s f r ac -

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

wi th the f ine sca le f luc tua t ions and re l a t ed to the

sca la r d i s s ip a t ion ra t e . A t th i s s tage i t i s no t ob-

v ious tha t any th ing has been ga ined , and the use-

f u ln e s s o f t h e a p p r o a c h d e p e n d s b o t h o n t h e p r o -

v i s i o n o f a n a d e q u a t e mo d e l f o r t h e mi x i n g t e r m

a n d t h e f o r ma t i o n r a t e s o f th e p e r t u r b a t i o n s b e i n g

wel l -con d i t ioned , i . e. ap pro x im ate ly l inear , func-

t ions o f the independen t var i ab les . Bi lger has a r -

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

s mo g r e a c t i o n s a n d f o r t h e t w o v a ri a b le f o r mu l a -

t io n o f t h e H 2 / a i r s y s t e m a d o p t e d b y J a n i c k a an d

K o l l ma n n , a n d i n t h e l a t t e r s it u a t i o n th a t i t m a y

be poss ib le to ca l cu la t e n i t r i c ox ide concen t ra -

t i o ns w i t h o u t r e s o r t t o a j o i n t p . d . f . H o w e v e r ,

bo th s i tua t ions invo lve on ly t race spec ies and

s ma l l d e p a r t u r e s f r o m e q u i l i b r i u m t e mp e r a t u r e s .

I t i s as ye t unc lear whether the approach wi l l y i e ld

u s e f u l r e s u lt s f o r a p p r e c i a b l e d e p a r t u r e s o f ma j o r

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

f r o m e q u i l i b r i u m.

P r e m i x e d F l a m e s

Pr e mi x e d t u r b u l e n t f l a me s , i n c o n t r a s t t o d i f f u s i o n

f l a me s , r e q u i r e t h e e v a l u a t i o n o f me a n r e a c t i o n r a t e .

Fo r t h is r e a s o n , a n d p e r h a p s a l so b e c a u s e p e r f e c t l y

p r e mi x e d f l a me s o c c u r l e s s f r e q u e n t l y i n p r a c t i c a l

s y s t e ms , t h e y a p p e a r t o h a v e r e c e i v e d l e s s a t t e n -

t i o n t h a n h a v e d i f f u s i o n f l a me s . So f a r mo s t w o r k

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

can b e char ac te r i zed by a g loba l s ing le s t ep rea c t ion

o f t h e t y p e

f u e l + o x i d a n t ~ p r o d u c t

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

g i ve n b y a n e x p r e s s i o n o f t h e f o r m

S p, = A C ~ C o ~ e x p ( - T a / T ) ,

w h e r e b o t h A a n d

TA

a r e ( k n o w n ) c o n s t a n t s . I t i s

t h e e v a l u a t io n o f t h e m e a n ( t i me a v e r a g e d ) v a lu e

o f t h i s f o r ma t i o n r a t e w h i c h p r e s e n t s p r o b l e ms . I f

f luc tua t ions a re neg lec ted and the ra t e eva lua ted in

t e r m s o f th e m e a n v a lu e s o f t e m p e r a t u r e a n d m a s s

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

order o f ma gn i tude and wi ll exh ib i t a s t rong de-

p e n d e n c e o n t e mp e r a t u r e , p r e s s u r e , a n d mi x t u r e

s t r e n g t h . I n c o n t r a s t , e x p e r i m e n t a l r e s u l ts f o r p re -

mi x e d t u r b u l e n t f l a me s a r e o n l y w e a k l y d e p e n d e n t

o n t e mp e r a t u r e , p r e s s u r e , a n d mi x t u r e s t r e n g t h .

T h i s f a c t l e d Sp a l d in g ( 1 9 7 1 ) t o p r o p o s e t h e e d d y -

b r e a k u p mo d e l , w h i c h , f o r t h e me a n f o r ma t i o n

ra te o f p rod uct , g ives

and which i s based on the idea tha t t he mean reac-

t ion ra t e i s de te r m ined so le ly by the ra t e o f sca le

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

i ng . T h u s t h e m o d e l t a k e s n o e x p l i ci t a c c o u n t o f

c h e mi c a l k i n e t i c s a n d r e l a t e s t o c o mb u s t i o n w h i c h

i s en t i re ly mixed con t ro l l ed . In th i s s i tua t ion i t has

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

e v i d e n c e f o r p r e mi x e d f l a me s , t h o u g h i t mu s t b e

n o t e d t h a t , i n t h e o r i g i n a l f o r mu l a t i o n a n d ma n y

subsequen t app l i ca t ions , t he fue l mass f rac t ion

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

equa t ion in which a po ten t i a l ly l a rge t e rm invo lv ing

the fue l bu rnup ra t e was un jus t i f i ab ly ignored . The

mo d e l h a s a l s o b e e n u s e d o c c a s i o n a ll y f o r n o n p r e -

m ixed f l ames : i t is i nap prop r i a t e in th i s s i tua t ion

a n d Po p e ( 1 9 7 7 ) h a s s h o w n t h a t i t d o e s n o t t h e n

necessar i ly p rov ide un ique so lu t ions .


14/26


A f i rm er basi s fo r t he edd y -b reak up m od e l and

i ts l i m i t a t ions i s p rov i ded by t he B ray ,Mos s (197 7 )

m ode l ( s ee a l s o B ray , 1980 ) fo r p rem i xed f l am es .

For react ion v ia a s ingle g lobal s tep and adiabat ic

f low a s ingle re act io n progress var iab le , def ine d as

t he r a t i o o f p ro duc t m ass f r a c t i on t o i t s fu l l y bu rn t

va l ue , i s i n t roduced and i s s u f f i c i en t t o de t e rm i ne

t h e ( i n s t a n t a n e o u s ) c o m p o s i t i o n , t e m p e r a t u r e , a n d

d e n s i t y o f t h e m i x t u r e .

The

probab i l i t y dens i t y

func t i on fo r t he r e ac t i on p rog res s va r i ab l e r is t hen

a s su m e d t o h a v e t h e f o r m

P(r,

x j ) = o t (x j ~ ( r ) + ~ (x i )5 (1 - - r )

+ [ H ( r ) - - H ( r - -

1)]

7(x j )F (r , xy ) .

I n t h e li m i t o f l a rg e D a m k S h l e r a n d R e y n o l d s n u m -

ber s , i .e . , y ,< 1 , t he m ean p ro duc t fo rm a t i on r a t e i s

d e t e r m i n e d t o b e

S p = C a r-) - CR e e '

where C R i s a cons t an t dependen t on t he con t i nu -

o u s p a r t F ( ) o f t h e p . d . f. T h e b u r n i ng m o d e p a r t

o f t h e p . d . f . F ( ) c a n b e d e t e r m i n e d f r o m a la m i n a r

f l am el e t des c r i p t i on t houg h r e s u l ts a re foun d t o be

rela t ively insens i t ive to the shape chose (see , e .g . ,

B r a y 1 9 7 8 ) . F o r s m a ll a n d i n te r m e d i a t e D a m k o h l e r

num bers t h i s i s un l i ke l y t o be t he cas e and t he

s hape o f t he bu rn i ng m od e pa r t o f t he p .d .f , w il l

be o f g rea t e r i m por t ance , t hough i t s fo rm can s t i l l

b e d e t e r m i n e d f r o m a la m i n a r f la m e l e t m o d e l . T h e

m o d e l h a s a ls o b e e n e x t e n d e d t o c o v e r m o r e c o m -

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

i nvoked he re i s t ha t t he r eac t i on s t eps p roceed s e-

q u e n t i a ll y . C h a m p i o n e t a l. ( 1 9 7 8 ) h a v e a d o p t e d

t h is l a t t e r a pp roac h t o ca l cu l a te a one -d i m ens i ona l

p rem i xed p ro pane / a i r f l am e . C on s t an t s pec if i c

hea t s were a s s um ed and dens i t y -we i gh t ed ave rag ing

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

involv ing tw o seq uent ia l s t ages . In th e f i rs t s t a ge -

t h e d e la y z o n e - p r o p a n e w as t a k e n t o c o m b in e

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

oxide v ia a s ingle g lobal react ion , the ra te expres -

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

(1969 ) . The s econd s tage o f t he r eac t i on invo l v ing

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

o x i d e - t h e c o m b u s t i o n z o n e - w a s t h e a ss u m ed t o

c o m m e n c e o n l y a f t e r t he com pl e t e d i s appea rance

o f p r o p a n e . I n t h e c o m b u s t i o n z o n e t o s i m p l i f y

t he r eac t i on m echan i s m a pa r t i a l equ i l i b r i um as -

s um pt i on fo r t he f a s t b i m o l ecu l a r r eac t i ons was

i n v o k e d a n d t h e r a t i o o f o x y g e n t o h y d r o g e n a t o m

rad i ca l concen t r a t i ons was a s s um ed cons t an t . Th i s

l a t t e r a s s um p t i on was neces s it a t ed by t he des i r e t o

des c r i be t he s t a t e o f t he r eac t i ng m i x t u re i n t e rm s

o f a

single

r eac t i on p rog ress va r iab l e , nam el y , t he

t em pe ra t u re . F o r t he de l ay zone , ca l cu l a t ions were

p res en t ed u s i ng t wo m ode l s : t he m e t hod s ugges t ed

by B orgh i (1974 ) whereby t he ave raged s pec i es

m as s fo rm a t i on r a t e i s eva l ua t ed by an ave raged

Tay l o r s e r i e s expans i on t runca t ed a t s econd o rde r ;

and a m ode l bas ed on an a s s um ed p .d . f , f o r t he

reac t i on p rog res s va r i ab l e . The p .d . f , u s ed com -

p r i sed a D i rac de l t a fu nc t i on a t t he equ i l ib r i um

t e m p e r a t u r e p l u s a c o n t i n u o u s c o n t r i b u t i o n f o r

wh i ch bo t h r ec t angu l a r and t r i angu l a r fo rm s were

u t i l i zed . As pe rhaps m i gh t be an t i c i pa t ed - t he de -

lay zon e i s a reg ion of smal l hea t re lease and there-

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

s m al l d i f f e rences be t w een t he t w o m et hods , and

the resu l t s app ear a l so to be re la t ively insens i tive

t o t he p .d . f , s hape . The com bus t i on zone i s a r e -

g i on w i t h po t en t i a l l y l a rge f l uc t ua t i ons i n tem -

pera t u re , and t he t runca t ed s e r i e s expans i on t ech -

n i que i s no t app l i cab l e . The p res um ed p .d . f , f o r -

m a l i s m i s no t s o r e s t r i c ted , and the r e s u l t s ob t a i ne d

wi t h t h i s m ode l s how a l a rge and p l aus i b l e i n f l u -

e n c e o f t e m p e r a t u r e f l u c t u a t io n s o n t h e m e a n t e m -

p e r a t u r e i n t h e c o m b u s t i o n z on e . H o w e v e r , n o

com par i s on wi t h exper i m en t i s p res en t ed p re -

s um a b l y becaus e o f t he nonava i l ab i l i ty o f su i t ab le

m e a s u r e m e n t s .

T h e B r a y - M o s s m o d e l f o r p r e m i x e d c o m b u s -

t i o n h as b e e n e x t e n d e d b y L i b b y a n d B r a y ( 1 9 8 0 a ,

b ) , w ho u t i li ze t he con cep t o f l am i na r f l am e l e t s i n

o r d e r t o d e r iv e m o d e l s f o r t u r b u l e n t t r a n s p o r t a n d

diss ipat ive processes in one-d imens ional p lanar

f lam es . Th i s i s ach ieved th roug h t he i n t rod uc t i o n

o f a j o i n t p . d . f . P (u , r ) fo r t he r eac t i on p rog ress

v a ri a bl e a n d t h e c o m p o n e n t o f v e l o c it y n o r m a l t o

t he ave raged pos i ti on o f t he t u rbu l en t f l am e f ron t .

The f a s t r eac t i on a s s um pt i on i s i nvoked and t he

j o i n t p . d . f , can t hen be s epa ra t ed i n t o t wo con -

d i t i oned p .d , f . s co r r es pond i ng t o r e ac t an t and


15/26


p roduc t nodes , t he p robab i l i t y o f even t s i n t he

bu rn i ng m o de be i ng negl ig ib le . T he m ode l l eads to

a d e s c r i p ti o n o f p r e m i x e d c o m b u s t i o n i n t e rm s o f

fou r q uan t i t ie s , nam el y , t he m ea ns and va ri ances

o f t he t wo c ond i t i oned ve loc i t ie s . R ap i d d i s t o r t i on

t he o ry i s u s ed t o e l i m i na t e one o f thes e quan t i t i e s,

a n d t h e c o m p l e t e m o

combustion and flame volume 48 issue none 1982 [doi 10.1016%2f0010-2180%2882%2990112-2] w.p. jones;...

Documents