A Monte Carlo method capable of predicting scalar (e.g., concentration) probability density functions (pdf's) in nonreacting and reacting elliptic flows was developed. A research combustor specifically designed to provide measurements of concentration and temperature pdf's in nonreacting and reacting flows is emphasized

Radhakrishnan, K.

Catalan 

NASA Technical Reports Server

MONTE CARLO MODELXNG I N  ELLIPTIC FLOWS 
K r  i shnan  Radha k r  i shnan  
Department of Mechanical Engineer ing and Applied Mechanics 
The U n i v e r s i t y  of Michigan 
Ann Arbor, Michigan 48104 
A Monte C a r l o  method capab le  of p r e d i c t i n g  s c a l a r  (e.g. concen t ra t ion)prob-  
a b i l i t y  d e n s i t y  f u n c t i o n s  ( p d f ' s )  i n  nonreac t ing  and r e a c t i n g  e l l i p t i c  f lows  
i s  being developed. I n  p a r t i c u l a r ,  a t t e n t i o n  i s  be ing  focused on a r e s e a r c h  
combustor s p e c i f i c a l l y  des igned t o  p rov ide  measurements of c o n c e n t r a t i o n  and 
t empera tu re  p d f ' s  i n  nonreac t ing  and r e a c t i n g  f lows.  (e.g.  8 ) .  
The modeling t echn ique  be ing  developed i n v o l v e s  t h e  coup l ing  of a Monte 
Car lo  c o a l e s c e n c e / d i s p e r s i o n  model (e.g.  2-7) w i t h  c u r r e n t l y  a v a i l a b l e  codes 
(such as t h e  f i n i t e - d i f f e r e n c e  scheme d e s c r i b e d  i n  ( 1 ) )  f o r  e l l i p t i c  flow- 
f i e l d  c a l c u l a t i o n s .  The la t ter  prov ides  s o l u t i o n s  f o r  t h e  mean v e l o c i t y ,  
t u r b u l e n t  k i n e t i c  energy and d i s s i p a t i o n  r a t e .  These s e r v e  a s  i n p u t s  t o  
t h e  Monte C a r l o  c a l c u l a t i o n  procedure  which i s  used t o  . solve  t h e  t r a n s p o r t  
e q u a t i o n s  f o r  s c a l a r  p d f ' s .  
The a t t r a c t i o n  of t h e  Monte C a r l o  m e t h o d ' i s  i t s  p r a c t i c a b i l i t y  f o r  
mul t i -d imensional  p d f ' s ,  w h i l e  conven t iona l  f i n i t e - d i f f e r e n c e  s o l u t i o n s  of 
t h e  pdf t r a n s p o r t  e q u a t i o n s  r e q u i r e  p r o h i b i t i v e  amounts of computer t ime  
(e .g .  2) .  T h i s  method a l s o  e l i m i n a t e s  the '  n e c e s s i t y  of assuming a shape 
f o r  t h e  pdf ;  i n s t e a d ,  g iven a n  i n i t i a l  d i s t r i b u t i o n  f u n c t i o n ,  t h e  model 
a l l o w s  f o r  t h e  temporal  e v o l u t i o n  of t h e  pdf .  Also,  i t  can account  f o r  
f i n i t e  r a t e  chemical  r e a c t i o n s ,  which i s  e s s e n t i a l  f o r  t h e  s u c c e s s f u l  
p r e d i c t i o n  of p o l l u t a n t  fo rmat ion  r a t e s  and f lame s t a b i l i t y .  
I n  t h e  Monte Car lo  c a l c u l a t i o n  procedure ,  t h e  composi t ion d i s t r i b u t i o n  
(due t o  c o n c e n t r a t i o n  f l u c t u a t i o n s )  a t  a p o i n t  i n  t h e  f l o w f i e l d  i s  d e s c r i b e d  
by a s t a t i s t i c a l  ensemble of computat ional  e lements  which have a d i s t r i b u t i o n  
i n  thermodynamic s t a t e s .  Each element h a s  s c a l a r  p r o p e r t i e s  a s s o c i a t e d  
w i t h  i t  s o  t h a t  i t s  thermodynamic s tate is w e l l  de f ined .  The number of such 
e lements  i n  a g iven  s tate can  v a r y  because  of t u r b u l e n t  mixing,  chemical  
r e a c t i o n s ,  e t c .  Given such a n  ensemble of e lements  a t  every g r i d  p o i n t  i n  
t h e  f l o w f i e l d ,  mean v a l u e s  of any s c a l a r ,  a s  w e l l  a s  v a r i a n c e  o r  h i g h e r  
moments can b e  o b t a i n e d  from i n s t a n t a n e o u s  averag ing  over  t h e  ensemble pop- 
u l a t i o n .  
The e f f e c t s  of mixing,  convec t ion ,  d i f f u s i o n ,  and chemical  r e a c t i o n  on 
t h e  s c a l a r  p d f ' s  a r e  i n c o r p o r a t e d  i n t o  t h e  c a l c u l a t i o n  procedure  a s  fo l lows .  
The t u r b u l e n t  and moecular mixing p r o c e s s e s  are r e p r e s e n t e d  by a l l o w i n g  
e lements  w i t h i n  each ensemble t o  m i x  w i t h  one a n o t h e r .  These mixing i n t e r -  
a c t i o n s  occur  a t  d i s c r e t e  time i n t e r v a l s  g iven  by a mixing f requency which 
can  b e  r e l a t e d  t o  t h e  l o c a l  t u r b u l e n t  k i n e t i c  energy and d i s s i p a t i o n  r a t e  
( s e e  r e f e r e n c e  2 f o r  d e t a i l s ) .  The e f f e c t s  of t u r b u l e n c e  on mixing and 
chemical  r e a c t i o n  a r e  t h u s  in t roduced  through t h e  mixing f requency.  During 
t h e  t i m e  between mixing i n t e r a c t i o n s ,  each element undergoes chemical  
https://ntrs.nasa.gov/search.jsp?R=19830020954 2020-03-22T00:49:37+00:00Z
r e a c t i o n  a t  a r a t e  s p e c i f i e d  by t h e  u s e r ,  E i t h e r  a n  i n f i n i t e  r a t e  assumption 
can  be made o r  a chemical  r e a c t i o n  mechanism and r a t e  d a t a  s p e c i f i e d ,  S i n c e  
t h e  p r o c e s s e s  of mixing and r e a c t i o n  a r e  t r e a t e d  s e p a r a t e l y ,  t h e  u s e  of 
m u l t i p l e  f i n i t e  r a t e  r e a c t i o n s  d a e s . n o t  i n c r e a s e  t h e  complexi ty  of t h e  form- 
u l a t i o n .  Also,  t h e  i n s t a n t a n e o u s  r e a c t i o n  r a t e  f o r  each element i s  c a l c u l a t e d ,  
so  t h e  mean r e a c t i o n  r a t e  can b e  ob ta ined  by s imply averag ing  over  t h e  ensem- 
b l e  popula t ion .  The t r a n s p o r t  p r o c e s s e s  of convec t ion  and d i f f u s i o n  a r e  a l s o  
t r e a t e d  s e p a r a t e l y ;  t h e  former by t r a n s p o r t i n g  e lements  from one ensemble 
( i . e .  g r i d  p o i n t )  t o  a n o t h e r  a t  a r a t e  governed by t h e  l o c a l  f l u i d  v e l o c i t y ,  
and t h e  la t ter  by exchanging elements between ensembles a t  a d j a c e n t  nodes a t  
a r a t e  governed by t h e  l o c a l  t u r b u l e n t  d i f f u s i v i t y .  
The s p e c i f i c a t i o n  of i n i t i a l  ensembles a t  t h e  g r i d  p o i n t s  is  as fo l lows .  
For e l l i p t i c  f lows ,  p r o p e r t i e s  a t  a p o i n t  can  b e  a f f e c t e d  by v a l u e s  down- 
s t ream.  The c a l c u l a t i o n  procedure  w i l l  t h e r e f o r e  i n v o l v e  i t e r a t i o n  on some 
assumed i n i t i a l  d i s t r i b u t i o n s .  The p r o c e s s e s  of convec t ion ,  d i f f u s i o n ,  
mixing,  and r e a c t i o n  w i l l  be  s imula ted  as d e s c r i b e d  above. Also,  t h e  i n l e t  
f low c o n d i t i o n s  w i l l  s p e c i f y  ensembles a long one a x i a l  l o c a t i o n .  The itera- 
t i o n  w i l l  c o n t i n u e  u n t i l  t h e  ensembles have become time s t a t i o n a r y ;  a 
c o r r e c t  ensemble w i l l  t h e n  have been c r e a t e d  a t  a l l  s p a t i a l  l o c a t i o n s  i n  
t h e  f l o w f i e l d .  
The a b i l i t y  of t h e  Monte Car lo  method t o  p rov ide  t h e  coup l ing  of f i n i t e  
r a t e  f l u i d  mechanic mixing and f i n i t e  r a t e  combustion chemis t ry  i s  i l l u s t r a t e d  
by p r e s e n t i n g  a c o a l e s c e n c e / d i s p e r s i o n  model developed f o r  s t u d i e s  of i g n i t i o n  
and blowout i n  a combustor primary zone. The u s e  of t h e  f lame s t a b i l i t y  model 
(desc r ibed  i n  r e f e r e n c e s  5 -7) i s  demonstrated by g e n e r a t i n g  l e a n  i g n i t i o n  
l i m i t s  of premixed t u r b u l e n t  f lames.  The v a r i a t i o n s  of t h e  p r e d i c t e d  l e a n  
i g n i t i o n  l i m i t  w i t h  r e f e r e n c e  v e l o c i t y ,  i n l e t  t empera tu re ,  and p r e s s u r e  
compare f a v o r a b l y  w i t h  exper imental  t r e n d s  and magnitudes.  
T h i s  work i s  suppor ted  by NASA Gran t  NAG 3-294. 
References  
L i l l e y ,  D.G. and Rhode, D.L. (1982). NASA CR-3442. 
Pope, S.B. (19811.. Combustion Sc ience  & Technology, 25, p. 159.  
P r a t t ,  D.T. (1976). P r o g r e s s  i n  Energy and Combustion Sc ience ,  1, p. 73. 
Radhakrishnan, K. and Heywood, J . B .  (1979). NASA CP-2078, p.179. 
Radhakrishnan, K. e t  a l .  (1979). NASA CR-3216. 
Radhakrishnan, K. and Heywood, J . B .  (1981). Combustion Sc ience  & Technology, 
24, p. 165. 
-
Radhakrishnan, K. and ~ r a t t ,  D.T. (1982). A I M  Paper 82-1158. 
Roquemore, W.M. e t  a l .  (1982) AIAA Paper 82-0240. 
Westbrook, C.K. and Dryer ,  F.L. (1981). Combustion Sc ience  & Technology, 
27, p.31. 
Method 
Combine e x i s t i n g  computer codes f o r  e l l i p t i c  flow f i e l d  calcu-  
l a t i o n s  with a  Monte Car10 meihod t o  s imulate  t h e  t r a n s p o r t  
equat ions f o r  s c a l a r  probabil i t y  dens i ty  func t ions .  
Computer Code 
For Flow Field 
Calcula t ions  
. 
Monte Carlo 
Calculat ion 
Procedure 
163 
Sca la r  Proper t ies  
cp,,  <T, , 4. > 
1 
Fl ow Fi e l  d  
- U ,  I(,  E e t c .  
S o l u t i o n  Technique For E l l i p t i c  Flows 
I n t e r a t i v e  as f o l l o w s :  
Assume i n i t i a l  ensemble (and hence, p d f ' s )  a t  a l l  g r i d  p o i n t s .  
C a l c u l a t e  < p >, < T >, e tc .  
So lve f o r  f l o w  v e l o c i t y ,  t u r b u l e n t  k i n e t i c  energy and d i s s i p a -  
t i o n  r a t e s  a t  a l l  g r i d  p o i n t s .  
Generate new p d f ' s  a t  a l l  g r i d  p o i n t s .  
Check i f  new p d f ' s  agree w i t h  o l d  va lues .  
Schematic o f  B l u f f  Body Stab i l i zed  Flame 
Rec i rcu lo t  ion Zone 
Simulation of 'Recirculat ion Zone 
Partial ly Stirred Reactor 
Unburnt 
Mixture 0 
0 Average 
@ Mixture 
Monte Carlo Mix ing Interact ion 
Element I j I j 
Burnt i i I B i  + B j l / 2  Fraction 
Model Parameters and R e l a t i o n s  
M i x i n g  Frequency 
I n f l o w  Time 
Residence Time 
Chemical K i n e t i c s  
Two Step Mechanism 
C3H8 + 3.5 O2 + 3 CO + 4 H20 
CO + 0.5 O2 = CO, 
Reac t ion  Rates f rom Westbroo k and Drye r  ( 1  981 ) 
Preexponent ia l  Constants Ad jus ted  
Ensemble P r o p e r t y  
Example : Ignition 
Development of 
f (8) Distr ibut ion Function 
Due to  Mix ing 
0 
Igni t ion 
- 
B lnc~eases 
No Ignit ion 
- 
B Decrecses 
EXAMPLE:LEAN IGNITION LIMIT  
Variations of Limits 
with Inlet Temperature 
Variations o f  Limits with Velocity 
I U=10 m/s.  P =  1 a t m  Lean Ignition Limits 
1.0 
$ 
0 .5  
0 . 3  
I 0 D a t a  (Ref .6)  - M o d e l  1.0 
TU= 5 0 0  K , P =  1 a t m  
Lean Ignition Limits 
0 Data ( R e f .  6 ) 
- Model 1 ~3 ~8 
r 
- 
! - A 
I I  I l  I l l  I  I 1  
3 5 10 15 20 25 
U :m/s)  
Variation of Predicted Limit with Pressure 
I Lean Ignition Limit Model Predictions 
0.3 
1 5 1 0  
P (at m) 


Monte Carlo Modeling in Elliptic Flows

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