The work in numerical modeling is focused on the use of the random vortex method to treat turbulent flow fields associated with combustion while flame fronts are considered as interfaces between reactants and products, propagating with the flow and at the same time advancing in the direction normal to themselves at a prescribed burning speed. The latter is associated with the generation of specific volume (the flame front acting, in effect, as the locus of volumetric sources) to account for the expansion of the flow field due to the exothermicity of the combustion process. The model was applied to the flow in a channel equipped with a rearward facing step. The results obtained revealed the mechanism of the formation of large scale turbulent structure in the wake of the step, while it showed the flame to stabilize on the outer edges of these eddies

Chorin, A. J.

Ghoneim, A. F.

Oppenheim, A. K.

English

NASA Technical Reports Server

NUMERICAL MODELING OF TUPBULENT COMBUSTION 
A. F. Ghoneim, A. J. Chorin, and A .  K. Oppenheim 
U n i v e r s i t y  o f  C a l i f o r n i a ,  Berkeley 
Berkeley , Cal i f  o r n i  a 94720 
Our work i n  numerical modeling i s  focused on the  use o f  the 
random vor tex  method t o  t r e a t  t u rbu len t  f low f i e l d s  associated w i t h  
combus t i o n  wh i l e  f lame f ron ts  are considered as in te r faces  between 
reactants and products, propagat ing w i t h  t h e  f l o w  and a t  t he  same 
t ime advancing i n  t he  d i r e c t i o n  normal t o  themselves a t  a  p rescr ibed 
burn ing speed. The l a t t e r  i s  associated w i t h  the generat ion o f  
s p e c i f i c  volume ( t h e  flame f r o n t  ac t ing ,  i n  e f f e c t ,  as t he  locus o f  
vo lumet r ic  sources) t o  account Fcr t h e  expansion of the f l o w  f i e l d  
due t o  the exo thermic i ty  o f  t h e  combustion process. The model was 
app l ied  t o  t he  f l o w  i n  a  channel equipped w i t h  a rearward f a c i n g  
j t e p .  The r e s u l t s  we obta ined revealed the  mechanism o f  the  forma- 
t i o n  o f  l a r g e  scale t u r b u l e n t  s t r u c t u r e  i n  t h e  wake of the  step, 
wh i l e  i t  showed the  f lame t o  s t a b i l i z e  on the  ou te r  edges o f  these 
(1) eddies . 
I n  the course o f  t h i s  year  we concentrated upon th ree  top i cs :  
1. Study o f  the  fundamental aspects o f  t h e  modeling technique 
2. A p p l i c a t i o n  o f  the  model t o  t he  fo rmat ion  o f  a  t u r b u l e n t  j e t  
3. Development o f  novel concepts b r i n g i n g  f o r t h  the aerodynamic 
p rope r t i es  o f  t u r b u l e n t  flames 
On the  f i r s t  subject ,  th ree  e f fo r t s  have besn successfu l ly  pur- 
sued. F i r s t  i s  t h e  development o f  t h e  random element method, pre- 
sen t ing  the  a p p l i c a t i o n  o f  t he  random walk technique t o  model d i f -  
fusion o f  energy. The concept o f  vor tex  sheets and vor tex  b lobs i s  
extended f o r  t h i s  purpose t o  heat  t r a n s f e r  sheets o r  temperature jump 
elements, and blobs o f  i n t e r n a l  energy, t h a t  c a r r y  thermal g rad ien ts  
from heat  conducting w a l l s  t o  t he  i n t e r i o r  o f  t he  f i e l d  and r e d i s t r i b u t e  
them by d i f f u s i o n .  Table 1 presents t he  fundmental idea of random 
https://ntrs.nasa.gov/search.jsp?R=19830020962 2020-03-22T00:48:38+00:00Z
walk modeling of diffusion, exploiting the analogy between the 
Green junction of the diffusion equation and the probability density 
junction of a Gaussian random variable. Figures 1 and 2 show a 
comparison between the exact solution and the numerical solution for  
the problem of tieating an inf in i te  solid by an isothermal wall a t  
y = 0, i n  terms of the heat f l u x  profiles and the- temperature 
profiles.  Figures 3 and 4 depict the temperature profiles for  a 
f i n i t e  solid bounded by two isothermal walls, and an isothermal and 
an adiabatic wall, respectively. The extention of the procedure t o  
handle two-dimensional diffusion i s  presented in  Table 2.  A one- 
dimensional approximation close to  the walls i s  employed t o  imple- 
ment isothermal boundary conditions , whi l e  energy elements are used 
-. i n  the in te r ior  to  diffuse the energy there. t-igures 5 and 6 display 
the temperature profiles along the diagonal of a corner and a square, 
respectively, as compared to  the corresponding exact profiles.  
The second fundmental problem we have solved i s  concerned with the 
generation of vort ic i ty  by the interaction between the pressure f i e ld  and 
densi tjl gradients -- a mechanism which i s  of importance to  both flames 
and buoyancy effects.  The motion of the temperature jump elements, 
governed by. the a1 gori t h m  of the random element method, generates el e- 
ments of vort ic i ty  i n  the in te r ior  of the f i e ld  while additional 
elements are generated by the no-sli p condition ' a t  the walls. Table 3 
describes how random walk i s  used i n  conjunction w i t h  the principle of 
time sp l i t t i ng  t o  solve the system of equations that  describes natural 
convection flow over a vertical  inf in i te  isothermal wall. Figure 7 
shows a comparison between our solution (thick l ines)  and a f i n i t e  
(3  difference solution for  two values of the Prandtl number . 
The third problem i n  th i s  category i s  that  of flame propagation. 
In our original model an interface advection and propagation algorithm, 
based on the assumption that  the flame i s  a jump in density, was used t o  
propagate the flame. In order to  t r e a t  flame propagation as that  of 
a reacting surface governed by a f i n i t e  rate  of reaction, the problem 
was recast i n  terms of a reaction-diffusion equation in temperature. 
The algorithm of the random element method was extended t o  solve th i s  
problem by adjusting the strength of the temperature jump elements as 
they move according to  the rate of reaction as described i n  Table 4. 
The a l g o r i t h m  was t e s t e d  by  s o l v i n g  problems by t he  use o f  f i n i t e  
d i f f e r e n c e  and f i n i t e  element methods, demonstrat ing i t s  c a p a b i l i t y  
of c a l c u l a t i n g  flame propagat ion w i t h  proper  accuracy. F igures 8 and 
9 show a  comparison between the  exact  s o l u t i o n ,  eva lua ted  f o r  
f ( T )  = T ( l -T ) ,  and t h e  numerical  computations when t h e  i n t e g r a t i o n  
of t h e  r e a c t i o n  p a r t  o f  t h e  equa t ion  i s  done us ing  a  f i r s t - o r d e r  
Eu le r  scheme i n  t he  f i r s t  case, and an exac t  i n t e g r a l  i n  t h e  second 
( 4 )  case, r e s p e c t i v e l y  . 
On t h e  second sub jec t ,  t h e  random vo r tex  method was a p p l i e d  t o  
t he  problem o f  t h e  f o rma t i on  o f  p lanar ,  two-dimensional t u r b u l e n t  j e t  
a t  h i g h  Reynolds numbers. F igure  10 descr ibes schemat ica l l y  t he  
elementary processes o f  t h e  random vo r tex  rneth0.d and how they  a re  
implemented t o  so l ve  t h e  convec t i on -d i f f us i on  equat ion.  The r e s u l t s  , 
expressed i n  terms o f  t h e  development o f  t h e  v o r t i c i t y  f i e l d ,  a re  
presented i n  Fig. 11. They revea l  the  format ion o f  l a r g e  sca le  
t u r b u l e n t  eddy s t r u c t u r e s  on bo th  s ides  o f  t h e  j e t  w i t h  a  p o t e n t i a l  
core i ns i de .  Few j e t  w id ths  downstream , the  two l a y e r s  s t a r t  t o  
i n t e r a c t  and t he  f l o w  becomes dominated by t h e  p a i r i n g  o f  eddies f rom 
bo th  s ides.  The t u r b u l e n t  eddies grow by  e n t r a i n i n g  t h e  non- tu rbu len t  
f l u i d ,  w h i l e  t h e i r  t r a j e c t o r i e s  become more and more convoluted as a  
r e s u l t  o f  t h e  i n t e r a c t i o n  between pos i  ti ve and negat i ve  v o r t i  c i  ty. 
These p i c t u r e s  d i s p l a y  a  remarkable resemblence t o  exper imenta l  photo- 
graphs ob ta ined  by Dimotakis,  e t  a l .  (5) across t h e  p lane  o f  symmetry 
of an ex isymmetr ic  j e t .  
The concept o f  f lame aerodynamics has been developed on t he  bas i s  
o f  our  numerical  model ing s tud ies  , supported by exper imenta l  observa- 
t i o n s  o f  t u r b u l e n t  f lames. The u l t i m a t e  conc lus ion  o f  these s tud ies  
i s  the  dominance o f  l a r g e  sca le  eddies over  t h e  f l o w  f i e l d ,  w h i l e  t he  
f lame i t s e l f  becomes es tab l i shed  a t  t h e  o u t e r  edges o f  these eddies 
and ac t s  as a  semi-permeable membrane encompassing tHe b u r n t  gases. 
I t  i s  the  i n t e r a c t i o n  between-these eddies and t h e  expansion assoc ia ted  
w i t h  t h e  exo thermic i  t y  o f  combustion process t h a t  produces t h e  char- 
(6) a c t e r i s t i c  aerodynamic p a t t e r n  t h a t  we a re  now s tudy ing  . 
REFERENCES 
1. Ghoniem, A. F . ,  Chorin, A. J . ,  and Oppenheim, A. K . ,  "Numerical 
Modeling of Turbulent Flow in  a Combustion Tunnel ," Phi l .  Trans. 
Roy. Soc. Lond., A304, pp. 303-325, 1982. 
2. Ghoniem, A . F . ,  Oppenheim, A. K . ,  "Random Element Method f o r  
Numerical Modeling of Diffusional Processes ," 8th International  
Conference on Numerical Methods in Fluid Dynamics, Aachen, W .  
Germany, June 1982. 
3. Ghoniem, A. F . ,  Sherman, F . ,  "Random Walk Simulation of Diffusion 
Processes ," J .  Comp. Phys. ( in  preparation).  
4. Ghoniem, A. F . ,  Oppenheim A. K . ,  "Solution of the Problem of 
Flame Propagation by the Use of the  Random Element Method," 
AIAA 21st Aerospace Sciences Meeting, AIAA Paper No. 83-0600, 
Reno, Nevada, January 1983. 
5. Dimotakis, P.  E . ,  Miake-Lye, R. C . ,  Papantonic, D. A s ,  "Structure 
and Dynamics of Round Turbulent J e t s , "  CALCIT Report FM82-01. 
6. Oppenheim, A.  K . ,  and Ghoniern, A.  F . ,  "Aerodynamic Features of 
Turbulent Flames," AIAA 21st Aerospace Sciences Meeting, AIAA 
Paper No. 83-0470, Reno, Nevada, January 1983. 
TABLE 1. FUNDAMENTAL IDEA 
Nomenclature 
H - Heavyside s tep  f u n c t i o n  
6 - D i rac  d e l t a  f u n c t i o n  
E - Expecte'd va lue 
- Gaussian random v a r i a b l e  
a - Standard d e v i a t i o n  
D i f f e r e n t i a l  Equat ion 
Boundary Condi t ions 
Cons t ra i n t  
Formal Sol u t i o n  
S tochas t i c  So lu t i on  
Local  Sampl i ng 
Global  Sampl i ng 
2 &La u 
a t  
ay 
$(Y,O) = 6 (0 )  ; @(krn, t )  = 0 
Q -1 "  $ d y = l  
-00 
- 1 $ = -   exp[- ( y / a )  2 I ; a = 42at 
d2na 
C 6Qi = 1 
yi ( t  + A t )  = Yi ( t )  + ni 
2 E(ni) = 0 ; E(ni ) = 2a A t  
1 @ = - E 6Q. 6(y-yi) 
&Y 1 
@ = B 6 Qi H(y-yi ) 
TABLE 2. A HYBRID SCHEME FOR TWO-DIFIENSIONS 
6 = 20 ; thickness of l-D diffusion layer 2 
6Ai = area element 
DOMA IN 
Differential  Equation 
Boundary Condi t i  on 
Diffusing Element 
Coup1 i n g  
Stochastic Solution 
Sampl i ng 
BOUNDARY 
Y < 6s 
aT 
-= 8 ' ~  
a t  ag 
T = l  
&Ti 
6ei = 6Ti * (yi-GS) 
r . ( t + ~ t )  = r . ( t )  + oi 
-1 -1 
C 6ei 
T = -  + C &Ti H (y-yi ) 6s * d x  
INTERIOR 
Y ' 6, 
a T s a V T  2 
a t  
T = 6(y-6,) 
6ei 
6T. 1 = ? 6ei/6, 
-1 -1 -1 r .  ( t + ~ t )  = r .  ( t )  + TI. 
T = -  e i - r . )  . &Ai - -1 
TABLE 3.  PRESSURE-DENSITY INTERACTION 
n = number of temperature jump elements 
N = Number of vortex elements 
DIFFERENTIAL EQUATIONS 
INITIAL CONDITIONS 
BOUNDARY CONDITIONS 
FRACTIONAL STEPS 
VORTIC ITY PRODUCTION 
- -  a T - a  - a 2 ~  
at ax 2 
2 
- - v au + ge (T-T_ ) 
at ax 2 
T = O  ; u = O  
x = 0 T = l  ; u = O  
x = c o  T = O  ; u = O  
aT - a2r . 2~- - v 7  a25 
a t  
at ax 
2 6 
-
a T 
at=@% 
6y = gB 6T 
T = 1 6T H(x-xi) 
n 
y = 1 6y H(x-xi ) 
N 
N(t+~t) = N(t) + 2n 
TABLE 4. REACTION-DIFFUSION EQUATION 
DIFFERENTIAL EQUATION 
INITIAL CONDITION 
BOUNDARY CONDITIONS 
SOLUTION 
FRACTIONAL STEPS 
REACTION 
DIFFUSION 
DETERMINISTIC 
STOCHASTIC 
2 
3T R - 2  - a + f(T) 
T(xYo) = To(x) 
T(+m,O) = 0 ; T(-a,O) = 1 
~ ( n ~ t )  = [R(A~) o(at)ln T~ 
- dT = f (T) 
d t 
a2T ?I=- 
at ax 2 
6Ti(t + ~ t )  = 6Ti(t) + f(Ti) At 
T = 1 6Ti H(x-xi ) 
xi(t + at) = xi(t) + T-I~ 
E[nl = 0 ; E[:] = 2 at 
F ig .  1 
F i g .  2 
Fig.  3 
F i g .  4 
258 
Fig. 6 
259 
Fig.  7 
Fig.  8 
260 
F i g .  9 
Near Field 
1 
BOUNDARY 
Vorticity Generation, 
F ig .  10 
261 
TIME .- 14 .OO TIVt: 2 I 00 TI \It: ) I  00 TIME 4 1  00 TIME 51 00 TlUE 6.1 00 
TIME i 3  00 TIME 5; 00 TIME 5 7  00 TIME - S R  00 TIME 61  00 TIUE 6 7  00 
F i g .  11 
COMBUSTOR FLAME FLASHBACK 
James S. T'ien and Margaret Proctor 
Case Western Reserve University 
Cleveland, Ohio 44135 
Objective: Analytically model and conduct basic experimental 'tests to study 
the fundamentals of flame flashbacks in combustors. 
Background: Flashback has been a recurrent problem with the present Chrysler 
automotive gas turbine combustor. It can be a potential problem for 
- 
other types and applications of advanced combustors as well. An improved 
understanding of the phenomenon would lead to improved design techniques 
to avoid its occurrence. 
Approach: It is proposed to model, test, and develop the fundamental 
conditions and process by which flashback occurs on combustor wall 
surfaces. An atmospheric rig consisting of a small dump combustor with a 
premixing channel would be required. Operation would be on a gaseous 
fuel. The primary variables would be inlet air and wall temperatures (up 
to 1500' F), boundary layer thickness, gas stream velocities, and 
controlled pressure disturbance level in the combustor. Both 
steady-state and transient tests will be conducted. 
Status: A steady-state flame flashback model is in existence. A transient 
model is being considered. The-experimental rig is being designed. The 
experiment will be performed at NASA Lewis Research Center by a Case 
Western Reserve University student. 
W . S .  GOVERNMENT PRINTING OFFICE: 1983/65g-0g4/328 


Numerical Modeling of Turbulent Combustion

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