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Flame extinction in a temporally developing mixing layer
Twenty-firstSymposium(International)on Combustion/TheCombustionInstitute, 1986/pp. 1251-1261
FLAME E X T I N C T I O N I N A T E M P O R A L L Y D E V E L O P I N G M I X I N G LAYER P. GIVI, W. -H. JOU ANn R. W. METCALFE Flow Research Company 21414-68th Avenue South Kent, Washington 98032
Nonequilibrium effects leading to the local quenching of a diffusion flame have been investigated by examining the evolution of large-scale structures in a two-dimensional temporally developing mixing layer. Pseudospectral calculations of a temperature-dependent, nonpremixed, constant-density, reacting shear layer indicate that the primary important parameter to be considered for flame extinction is the local instantaneous scalar dissipation rate, conditioned at the scalar stoichiometric value (Xst).At locations where this value is increased beyond a critical value (Xq), the local temperature decreases and the instantaneous reaction rate drops to zero. This is consistent with the results of perturbation methods employing large activation energy asymptotics for the study of flame extinction in nonpremixed flames.
1. Introduction A diffusion flame is characterized by a chemical reaction time that is usually much smaller than a characteristic diffusion time. The chemical reactions occur in a narrow zone between the fuel and the oxidizer, where the concentrations of both reactants are very small. The rate at which the reactants flow into the reaction zone, and therefore the characteristic time of the flame, is d e p e n d e n t on the hydrodynamics of the particular flow. As the characteristic time of the chemical reactions decreases, this reaction region becomes infinitesimally thin. I n this limit the chemical reaction zone approaches a "flame sheet" (local chemical equilibrium1), where the concentrations of both reactants are very low and the rate of combustion is governed by the rate at which fuel and oxidizer flow into the reaction zone. The flame sheet assumption is justified by the very fast chemical reaction rate of the diffusion flame. This assumption significantly reduces the complexity of the problem since it eliminates the analysis associated with the chemical kinetics. For many flows whose characteristic time scale of chemical reaction is much smaller than the hydrodynamic (convective-diffusive) time scale, the assumption of local chemical equilibrium adequately predicts the location and the shape of the flame. 2 One important feature of the calculations based on the fast chemistry model is the introduction of a passive Shvab-Zeldovich2 conserved scalar variable (Z), which is i n d e p e n d e n t of the chemical kinetics. From this quantity, the evolution of the concen-
tration fields of both the reactants and the products can be computed. In turbulent flows, however, the local characteristic flow time scales vary considerably. As a result, many important and interesting problems that cannot be analyzed by local chemical equilibrium assumptions are introduced. Experimental studies of Tsuji 3 show that as the local characteristic diffusion time becomes shorter and approaches the order of magnitude of the chemical time scale, the details of the chemical reactions cannot be neglected. If the flow of reactants into the reaction zone increases further, causing the diffusion time scale to be reduced more, the chemical reaction will not be able to keep pace with the further supply of reactants. The reaction rate will be reduced, and local quenching occurs. As shown by Peters, 4 further reduction of the diffusion time scale leads to lift-off and the blow-off of the entire flame. As noted by both Tsuji and Peters, the consideration of flame extinction cannot be explained by the flame sheet model, which assumes an infinitely fast chemical reaction rate. Therefore, in order to address the quenching p h e n o m e n o n , the structure of a finite reaction rate zone must be studied. Linan 5 has employed a method of matched asymptotic expansions in the limit of large activation energy in an attempt to describe the interaction between the hydrodynamics and chemical reactions in the reaction zone of a counter-flowing laminar diffusion flame. It has been shown that activation energy asymptotics are very useful in predicting flame ignition and extinction characteristics in such flows. This
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TURBULANT COMBUSTION
technique was e x t e n d e d to turbulent flows by Peters 4'6 by considering turbulent diffusion flames as an ensemble of laminar diffusion flamelets. 7 By introducing a local coordinate system that moves with the stoichiometric flame sheet, Peters 4 was able to recognize that the primary "nonequilibrium" p a r a m e t e r for the analysis of the flame extinction is the dissipation rate of the scalar quantity evaluated at stoichiometric conditions (Xst). This quantity is viewed as the inverse o f the diffusion time scale. If this p a r a m e t e r is increased beyond a critical limit (Xq), the heat conducted to both sides of the diffusion flamelets cannot be balanced by the heat production due to chemical reaction. At the critical value o f the dissipation, the finite rate kinetics balance the diffusion. 3 Some numerical calculations, p e r f o r m e d by Liew et al. 8 and c o m p a r e d with experimental data, 9 show also that as the m a x i m u m value of the dissipation (Xmo~)increases, the value of the m a x i m u m temperature decreases, the reaction eventually ceases, and the flame is locally quenched. These results indicate that the dissipation rate of the conserved scalar is a useful characteristic to study in the analysis of nonequilibrium effects leading to flame extinction. In turbulent flows, however, the quantity Xst is a strongly fluctuating quantity that has not yet been numerically computed. Instead, statistical approaches have been chosen by Peters and Williams a~ and Janicka and Peters ll employing different turbulence closures in predicting the lift-off height of a r o u n d diffusion j e t flame. Results obtained using these methods were compared with experi/nental data 12 for both methane and natural gas flames and show some correct o r d e r - o f - m a g n i t u d e predictions. These results are encouraging; however, with the availability of larger computers, it is now possible to employ a m o r e accurate treatment o f the flame extinction a n d local flame quenching that occur in n o n p r e m i x e d flames. It would be very useful to correlate the flame extinction with the nonequilibrium p a r a m e t e r Xst. Also, it would be very interesting to look at the structure o f the diffusion flames at the point of extinction. These are the objectives of this paper. T h e present p a p e r employs a direct numerical simulation a p p r o a c h to investigate the problem of local flame extinction in a time-dependent, two-dimensional mixing layer. In this flow, the governing equations are solved by an accurate numerical m e t h o d without a closure model. The t i m e - d e p e n d e n t flow field can be analyzed statistically to u n d e r s t a n d the underlying physics much as an experimentalist does with laboratory data. T h e direct numerical simulation technique has recently been successfully applied
to chemically reacting flows. Riley at al. 13 considered the three-dimensional temporally growing mixing layer u n d e r the simplest possible assumption o f a constant rate, non-heat-releasing chemical reaction. T h e main contribution of this work is the u n d e r s t a n d i n g of the effects of three-dimensional mixing and the diffusion of the species on the chemical reactions. McMurtry et al. 14 considered the effects of the chemical heat release and the resulting density variation on the fluid motion for a two-dimensional mixing layer. T h e fluid dynamics and the chemical reaction are truly coupled in this work, and the interplay between the two are discussed. However, the assumption of a constant chemical reaction rate is still employed. In the present work, we intend to u n d e r s t a n d the flame extinction problem t h r o u g h a two-dimensional simulation of a mixing layer. In particular, the role of large-scale features of the turbulent flow in flame extinction is studied. Due to limitations o f numerical accuracy, only moderate Reynolds and Damkohler numbers are computed. For flame extinction in a three-dimensional turbulent flow, three-dimensional simulations will be p e r f o r m e d in the future.
y, V
U(yl
x,U L
f CA~ Au FIG. 1. Problem geometry.
I
-I
FLAME EXTINCTION IN A MIXING LAYER 2. Problem
Assuming equal free-stream concentrations, i.e., Ca~ = C~=, the nondimensionalized equations are
Formulation
For simplicity of numerical calculations, we consider a two-dimensional shear layer as shown in Fig. 1. The flow is considered periodic in the horizontal direction (x). Although the splitter plate flow evolves spatially downstream and the numerical simulations evolve temporally, important similarities in the dynamics of these two flows make it useful to study accurate numerical simulations of the temporally growing mixing layers, By simple Galilean transformation, a flow quantity averaged in the x-direction can be related to the time average of the same quantity at a fixed location in a splitter plate configuration. 13'~4 These average quantities are dependent on the transverse coordinate (y) and the time (t). Again, the inverse Galilean transformation relates the time t to the streamwise location in a splitter configuration. T h e presence of periodic boundary conditions allows us to use accurate pseudospectral numerical methods; these methods are discussed in Ref. 13 and will not be described here. T h e chemical reaction between the two species is assumed to be single-step and irreversible and to obey the t e m p e r a t u r e - d e p e n d e n t Arrhenius law. It is assumed that the heat release rate is low so that the effects of the chemical heat release on the flow field can be neglected. This, together with the further assumptions of a low Mach n u m b e r flow, result in a constant-density flow formulation. T o identify the n o n d i m e n s i o n a l groupings, the variables are normalized by a velocity scale (mean velocity difference across the layer A b), a temperature scale (free-stream T~), concentration scales (free-stream Ca, and C~,), and a length scale (1=2.249~r, where ~r is the distance from the plane of symmetry to the transverse plane, where the mean velocity rises to half of its freestream value). T h e value of 2.249 was chosen so that the size of the computational domain would be equal to the wavelength of the most unstable mode and its normalized value would be an integer multiple of 27r. X
_x*= : X
17" = XV
U U*= AU P* =
P
(au) 2 CA
c~= (CA)oo
AU t*=t-X T* -
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T
To~ Ce
c ~ - (cB)~
V.U*=O OU* 9 9 1 ,~ L * ( _ U * , R e ) = - ~ - + _ U - V U - - ~ e V U* = -
VP*
L*(C~, Re Sc) = L ( C ~ , Re Sc) =
(2) -W*
L*(C~, Re Sc) = W* L*(T*, Re Pr) = C e W* where W* = Da e -ze/T'CJCJ Da = A/CA~176 = Damkohler number AU X Ze = Re = Ce=
E
RT= AUX P
Q
CoT=
= Zeldovich number = Reynolds number
(3)
= Heat release parameter
P
Sc = ~ --- Schmidt number P
Pr = ~ = Prandtl number The values of Sc and Pr can be set equal to unity, since this is approximately correct for gaseous diffusion flames. This results in a Lewis n u m b e r of unity. T h e value of other nondimensional parameters is limited by the available resolution of the numerics employed in the computations. Shvab-Zeldovich Variable
For the two-feed diffusion flame considered here, it is possible to consider only three scalar quantities rather than solving for all four scalar variables. The Shvab-Zeldovich variable (Z) and the product concentrations (G), following Givi et al., 15 are defined as
(1) z = cA* + c ~ G = C*
(4)
T h e transport equations governing Z, G, and T* follow:
TURBULANT COMBUSTION
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tion. These b o u n d a r y conditions are n a t u r a l w h e n F o u r i e r series e x p a n s i o n s are used.
L*(Z) = 0 L*(G) = Da e -zerr* (1 - Z - G) (Z - G) (5)
3. Presentation o f Results
L*(T*) = Ce Da e -zerr* (1 - Z - G) (Z - G) Initial and Boundary Conditions T h e initial a n d b o u n d a r y conditions for the velocity field are given elsewhere 16 a n d are only s u m m a r i z e d here. I n terms o f the stream function, the initial c o n d i t i o n is given by t~ = tOmean -}- qlf Jr" qlsh
(6)
where 0,,~, is the stream function associated with a m e a n hyperbolic t a n g e n t velocity profile, Of is the stream f u n c t i o n for the most unstable m o d e of this m e a n velocity profile, a n d O,h is the stream f u n c t i o n for the first s u b h a r m o n i c of the most unstable mode. T h e properties of these modes have b e e n evaluated from linear stability theory. 17 T h e f u n d a m e n t a l m o d e in this mixing layer p r o d u c e s a single vortex rollup, a n d w h e n the s u b h a r m o n i c is a d d e d in, a second rollup, or pairing, can o c c u r ] 6 I n the computations r e p o r t e d in this paper, we have employed two initial conditions for the hydrodynamics: (1) the m e a n flow a n d the f u n d a m e n t a l m o d e alone (Case I), a n d (2) the m e a n flow, the f u n d a m e n t a l mode, a n d the first s u b h a r m o n i c m o d e a d d e d together in phase (Case II). In this m a n n e r , the effects that the vortex rollup has o n the chemical reaction can be examined. T h e initial conditions for the reactant concentration are given by
C~(x*,y*,O)=y--~ j_
exp~-~-;
C~(x*,y *, O) = 1 - C,T
A pseudospectral n u m e r i c a l code was develo p e d for the calculations o f the incompressible reacting flow c o n s i d e r e d here. C o m p u t a t i o n s were p e r f o r m e d in a square d o m a i n with the size 0 < x < 2rrA, -~rA < y < ~'A for the single vortex rollup a n d the d o m a i n 0 < x < 47rA, -27rA < y < 2~rA for the double rollup computations. T h e spatial resolution was 64 x 64 F o u r i e r modes. T h e values of the Reynolds a n d D a m k o h l e r n u m b e r s were set equal to 200 a n d 10, respectively, so that the simulation would be accurately resolved o n the 64 x 64-point grid e m p l o y e d here. T h e value of the Zeldovich n u m b e r was set equal to 20. T h e value of the heat release p a r a m e t e r was selected to be small e n o u g h so that, w h e n multiplied by the D a m k o h l e r n u m b e r , it would be resolvable by the n u m e r i c s a n d so that it would also be reasonable to neglect the effects of heat release o n the flow field. However, this value should be large e n o u g h to allow the effects of t e m p e r a -
d( (7)
C~(x*,y*, O) = 0 Note that there are no initial fluctuations in the scalar profiles. T h e t e m p e r a t u r e in the free stream is equal to T=. Near y = 0, this value is increased to a n ignition t e m p e r a t u r e to allow the chemical reactions to occur. T h e initial profile for T* is given by an e x p o n e n t i a l function.
T*(x*,y*, 0 ) = 1 + 4 exp ( - 4y'2 )
(8)
A periodic b o u n d a r y c o n d i t i o n is e m p l o y e d in the streamwise direction, a n d a free-slip b o u n d a r y c o n d i t i o n in the cross-stream direc-
FIG. 2. Plots of Shvab-Zeldovich variable contours (Case I), (a) t* = 9, contour minimum is 0, contour maximum is 1, contour interval is 0.1. (b) t* = 15, contour minimum is 0, contour maximum is 1, contour interval is 0.1.
FLAME EXTINCTION IN A MIXING LAYER ture d e p e n d e n c e on the chemical reaction term. A value o f Ce8 is satisfactory, giving a m a x i m u m (adiabatic) t e m p e r a t u r e o f T'adiabatic = 7. T h e profiles o f the c o n s e r v e d Shvab-Zeldovich scalar variable are chosen f o r the p u r p o s e o f flow visualization. I n Figs. 2 and 3, we p r e s e n t the time s e q u e n c e d e v e l o p m e n t o f this profile for Cases I a n d II, respectively. Initially, the p e r t u r b a t i o n associated with the f u n d a m e n t a l m o d e grows until a time of t* = 9, w h e r e the first v o r t e x r o l l u p occurs. Proc e e d i n g f u r t h e r in time results in diffusion of the c o r e o f the v o r t e x with no additional rollup. A d d i n g the s u b h a r m o n i c associated with this unstable m o d e results in a second rollup (Fig. 3) that initiates at a time o f about t* = 12 a n d is c o m p l e t e d by t* = 24. As shown in these figures, the effect o f the vortex dynamics is to increase the strain rate at the braids b e t w e e n the v o r t e x cores. I f a fast chemistry m o d e l was assumed to describe the chemical reactions, the surface o f Z = Z~t = 0.5 would c o r r e s p o n d to the flame location. T h e vortex dynamics plays an i m p o r tant role in increasing the a r e a o f the flame
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surface (Z,t surface) and in increasing the total a m o u n t o f reaction p r o d u c t c o m p a r e d to the case w h e r e there is no r o l l u p a n d the chemical reaction is only limited to the i n t e r d i f f u s i o n o f the scalars at the reaction surface. In the p r e s e n t calculations, the increase o f the equilibr i u m flame sheet surface d u e to r o l l u p (Case I, Fig. 2b) is 153 p e r c e n t in c o m p a r i s o n with the u n f o r c e d case, M e r g i n g o f the vortices (Case II, Fig. 3b) increases this surface by 340 p e r c e n t w h e n c o m p a r e d with the u n f o r c e d case. T h e time sequence o f the t e m p e r a t u r e is shown o n Figs. 4 and 5 for Cases I and II, respectively. T h e m e c h a n i s m of v o r t e x rollup is to d r a w the reactants f r o m the two streams to the reaction surface. T h e m a x i m u m t e m p e r a ture a l o n g the Z = Z,t surface occurs at the c o r e o f the v o r t e x (the hottest location), w h e r e vorticity a n d the p r o d u c t c o n c e n t r a t i o n are also highest, a n d decreases in the braids, where the g r a d i e n t s o f the scalar are at their h i g h e r values. At the time w h e n the c o m p u t a t i o n is stopped, the m a x i m u m p r o d u c t c o n c e n t r a t i o n has not yet r e a c h e d its e q u i l i b r i u m value. T h e r e still r e m a i n reactants, e v e n at the core o f
(a)
(b)
/ FIG. 3. Plots of Shvab-Zeldovich variable contours (Case II). (a) t* = 12, contour minimum is 0, contour maximum is 1, contour interval is 0.1. (b) t* = 24, contour minimum is 0, contour maximum is 1, contour interval is 0.1.
FIG. 4. Plots of normalized temperature contours (Case I). (a) t* = 9, contour minimum is 0, contour maximum is 6, contour interval is 0.6. (b) t* = 15, contour minimum is 0, contour maximum is 6, contour interval is 0.6.
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T U R B U L A N T COMBUSTION
the mixing layer, a n d the flame s h o r t e n i n g p h e n o m e n o n , which is usually caused by the d e p l e t i o n of the reactants, has not b e e n yet observed. T h e reason for this b e h a v i o r of the chemical p r o d u c t and t e m p e r a t u r e can be e x p l a i n e d by the c o n t o u r plots o f the instantaneous reaction rate W, p r e s e n t e d in Fig. 6 for Case I. As shown in this figure, the r e a c t i o n rate initially is very u n i f o r m along the species i n t e r d i f f u s i o n zone and is m a x i m i z e d at the reaction surface, w h e r e the reactants are in contact. At later times, the reaction rate decreases, a n d at points w h e r e the strain rate is sufficiently large, the reaction rate goes to zero a n d the flame locally quenches, This m e c h a n i s m o f extinction o f the diffusion flame is consistent with the e x p e r i m e n t a l observations o f Tsuji. 3 I n the braids, as the inverse o f the diffusion time scale (1/X~t) decreases (as the result of the v o r t e x rollup), the supply o f the reactants is g r e a t e r t h a n the chemical reaction can utilize. Notice f r o m Figs. 2 t h r o u g h 5 that the reactants are well m i x e d on the braids, b u t the reaction rates at these places are zero. T h i s n o n e q u i l i b r i u m p h e n o m e n o n can be e x p l a i n e d by e x a m i n i n g the r e a c t i o n rate e q u a t i o n [Eq. (3)].
(e)
(0)
FtG. 5. Plots of normalized temperature contours (Case II). (a) t* = 12, contour minimum is 0, contour maximum is 6, contour interval is 0.6. (b) t* = 24, contour minimum is 0, contour maximum is 6.6, contour interval is 0.6.
FIG. 6. Plots of normalized reaction rate contours (Case I). (a) t* = 6, contour minimum is 0, contour maximum is 0.03, contour interval is 0.003. (b) t* = 9, contour minimum is 0, contour maximum is 0.024, contour interval is 0,002. (c) t* = 12, contour minimum is 0, contour maximum is 0.02, contour interval is 0.002. (d) t* = 24, contour minimum is 0, contour maximum is 0.017, contour interval is 0.001. Unlike the t e m p e r a t u r e - i n d e p e n d e n t reacting m i x i n g layer calculation r e p o r t e d by Givi et al., 18 the m a x i m u m value o f the reaction rate
FLAME EXTINCTION IN A MIXING LAYER does not necessarily c o r r e s p o n d to the maxim u m value o f the p r o d u c t o f the reactants concentration, since the effects o f the temperature variations influence the local conversion rate o f the chemistry. I f the flame temperature goes below a critical characteristic temperature, the flame becomes very rich with both reactants. However, since the value of the dissipation rate is greater than the critical value, this p r e m i x e d region of the reactants cannot be reignited from the high t e m p e r a t u r e zone of the core. A higher dissipation rate results in a further decrease in t e m p e r a t u r e until it becomes equal to the b a c k g r o u n d temperature. At this point, the vortex has reached the b o u n d a r y of the computational domain and, therefore, further computation is not realistic. A direct quantitative comparison of Xq, obtained in our numerical simulations (Xq is a p p r o x i m a t e d here as the value o f the critical instantaneous dissipation rate evaluated at the stoichiometric surface Z,t) with that obtained u n d e r the large activation energy asymptotics is not expected to agree exactly for several reasons. First, in asymptotic analysis, the flame thickness is very small and is only b r o a d e n e d near the reaction zone by a small parameter. In the direct numerical simulations reported here, the reactants have a fairly "thick" overlap region in the reaction zone d u e to the numerical accuracy limitations. F u r t h e r m o r e , the initial conditions for the t e m p e r a t u r e profile employed here near the reaction zone are not the same as the u n i f o r m initial temperature distributions o f Peters. 4 However, making the assumption that the initial concentration of the reactant and the t e m p e r a t u r e o f the reactants are at some average values in the range between the free stream and the reaction zone, correct order-of-magnitude asymptotic results are verified by the results o f the simulations. T h e t e m p e r a t u r e profiles presented in Fig. 4 show that by a time o f t* = 15, the value o f the t e m p e r a t u r e at the braids falls to about onefifth the flame t e m p e r a t u r e (approximately equal to the free-stream temperature). This corresponds to local quenching. The corresponding c o m p u t e d normalized dissipation rate X~ (X* = X,tA/AU) at the point of quenching, for the given kinetics data, is about 7 for both Cases I and II. T h e estimated corresponding value for the dissipation rate, c o m p u t e d with the asymptotic methods described above, is about 8, which is in good a g r e e m e n t with the results of the numerical simulation. 4, C o n c l u s i o n s
A pseudospectral algorithm has been used for the numerical calculations of a constant-
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density, chemically reacting, temperature-dep e n d e n t mixing layer. T h e nonequilibrium effects leading to the local flame quenching have been simulated in a case with fairly large Zeldovich n u m b e r a n d m o d e r a t e Reynolds and Damkohler numbers. It has been found that the rollup of an unstable shear layer creates regions with high dissipation rates at the braids, where local flame extinction occurs according to the theory of Peters 4'1~ T h e temperature contour shows that the t e m p e r a t u r e drops to a value close to ambient at the braids and the reaction rate reduces to zero, even though the reactants are well mixed there. Within the limitations of numerical resolution, comparison o f the critical scalar dissipation rate with that obtained by large activation energy asymptotics shows reasonable agreement. We are presently e x p a n d i n g this work to investigate a spatially developing mixing layer and three-dimensional turbulent flows. This should further illuminate the basic mechanisms o f such p h e n o m e n a as the diffusion j e t flame lift-off. Nomenclature
Af C Cv Ce D Da E G K L P Pr Q R Re Sc T t U V W X x y Z Ze
Preexponential factor Concentration Specific heat Heat release p a r a m e t e r Molecular diffusivity Damkohler n u m b e r Activation energy Normalized p r o d u c t concentration T h e r m a l conductivity Convective-diffusive o p e r a t o r Pressure Prandtl n u m b e r " Heat of reaction Universal gas constant Reynolds n u m b e r Schmidt n u m b e r Temperature Time Streamwise velocity c o m p o n e n t Cross-stream velocity c o m p o n e n t Reaction rate Dissipation Streamwise coordinate Cross-stream coordinate Shvab-Zeldovich variable Zeldovich n u m b e r
Greek A v tO
Length scale Molecular viscosity Stream function
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T U R B U L A N T COMBUSTION
Superscript *
Non-dimensionalized
3. 4. 5. 6.
Subscript A,B f P q
sh st
7.
Reactants Fundamental Product Quenching Subharmonic Stoichiometric Free stream
8.
Acknowledgments The calculations reported here were performed under research sponsored by the Air Force Office of Scientific Research u n d e r Contract F 4 9 6 2 0 - 8 5 - C 00067. The United States Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation hereon. The authors have also appreciated the support of NASA Lewis Research Center in providing computer time.
REFERENCES: 1. BILGER, R. W.: Turbulent Reacting Flows (P. A. Libby and F. A. Williams, Eds.), p. 65, SpringerVerlag, 1980. 2. WILLIAMS, F. A.: Combustion Theory, Second edi-
Lion, The Benjamin/Cummings Publishing Company, Inc., 1985. Tsvj1, H.: Prog. Energy Comb. Sci. 8, 93 (1982). PETERS, N.: Comb. Sci. Tech. 30, 1 (1983). LINAN, A.: Acta Astronautica 1, 1007 (1974). PETERS, N.: Prog. Energy Comb. Sci. 10, 319 (1984). WILLIAMS,F. A. : Turbulent Mixing in Non-Reactive and Reactive Flows (S. N. B. Murthy, Ed.), p. 189, Plenum Press, t975. LIEW, S. K., Moss, J. B. AYD BRAY, K. N. C.: Dynamics of Flames and Reactive Systems (J. R. Bowen, N. Manson, A. K. Oppenheime and R. I. Sdoukhin, Eds.), Progress in Astronautics and Aeronautics, Vol. 95, p. 305 (1984).
9. MITCHELL, R., SAROFIN, A. AND CLOMBURG, L.: Comb. and Flame 37, 337 (1980). 10. PETERS, N. AND WILLIAMS, F. A.: AIAAJ. 21,423 (1983). 11. JANICKA, J. AND PETERS, N.: Nineteenth Symposium (International) on Combustion, p. 367, T h e Combustion Institute, 1982~ 12. DONNERHACK, S. AND PETERS, N.: Comb. Sci. Tech. 41, 101 (1984). 13. RZLrY,J. j., METCALVE, R. W. AZ~'OORSZaG, S. A.: Phys. Fluids 29, 406 (1986). 14. McMURTRV, P. A., JOLT, W. -H., RILEY, J. J. ANn METCALVE, R. W.: AIAA Paper 85-0143, 1985. 15. Gtvt, P., SIRIGNANO, W. A. AND POPE, S. B.: Comb. Sci. Tech. 37, 59 (1984). 16. RILEe, J. J. AND METCALFE, R. W.: AIAA Paper 80-0274, 1980. 17. MICHALKE, A.: J. Fluid Mech. 19, 543 (1964). 18. GIvI, P., RAMOS, J. I. ANn SIRIGNANO, W. A.: J. Non-Equilibrium Thermodynamics 10, 75 (1985).
COMMENTS j. P. Boris, Naval Research Laboratory, USA. Fluid compressibitlity (expansion with heat release), acoustic waves, and spatial variation with a realistic difference between inflow and outflow are known to have profound effects on flame propagation and extinction. Your calculations have none of these important effects. Expansion, for example, changes roll up and merging by suppressing exactly the modes you simulated. How are your results to be related to reality in view of these computational short comings? Author's Reply. In addition to the influences of fluid compressibility, acoustic waves and spatial variation, there are also other p h e n o m e n a that can have very important effects on the mechanism of flame stabilization, structure, and extinction. Namely, the influences of multi-step chemical kinetics, the non-equilib-
rium effects of radiation, 1 gravity ~'3 and Lewis number, 4 the role of three-dimensional turbulence motion, etc. Incorporating all these effects in the accurate simulations of reacting flow problems is not practical computationally. Our approach to the investigation of the physically important, complex p h e n o m e n o n of flame extinction is to first study a simpler system. After developing insight into some of the basic phenomena, the degree of complexity can then be increased. Our present calculations have focused on the influence of scalar strain rate on the structure of the flame, and the results of our simulation are consistent with experimental observations. 5 Work is in progress at Flow Research and elsewhere to address the effects of some more complex physical phenomena on the
FLAME EXTINCTION IN A MIXING LAYER evolution of reacting flows. For example, Givi and Jou 6 reported some calculations for spatially evolving flows with different inflow/ourflow conditions and addressed the asymmetric nature of the mixing mechanism. Riley et al 7 studied the effects of the three-dimensional turbulence motion on the structure of the flame in a constant finite rate chemical reaction, and McMurtry et al 8 examined the role of expansion due to heat release in a reacting mixing layer. Thus, different aspects of the physics of the problem are isolated and studied separately for better understanding. As computational capabilities continue to improve, more realistic simulations combining several of these p h e n o m e n a will become feasible.
REFERENCES 1. BONNY, U.: Combustion and Flame 16, 147 (1971). 2. COCHRAN, T.H. AND MASICA, W.J.: Thirteenth Symposium (International) on Combustion, p. 821, The Combustion Institute, 1971. 3. ALTENKIRCH, R.A.: "The Use of a Low-Gravity Environment in Combustion Research," presented at the Spring Technical Meeting of the Combustion Institute, Central States Section, Cleveland, Ohio, May, 1986. 4. CHUNG, S.H. AND LAW, C.K.: Combustion and Flame 52, 59 (1983). 5. TsuJi, H.: Prog. Energy Comb. Sci. 8, 93 (1982) 6. GIvI, P. AND JOU, W.-H.: "Mixing and Chemical Reactions in a Spatially Developing Mixing Layer", presented at the Spring Technical Meeting of the Combustion institute, Central States Section, Cleveland, Ohio, May, 1986. 7. RILEY, J.J.; METCALFE, R.W. AND ORSZAG, S.A.: Phys. Fluids 29, 406 (1986). 8. MCMURTRY, P.A.; W.-H.; RILEY, J.J. AND METCALFE, R.W.: AIAA Journal 24, 962 (1986).
E. F. Brown, Office of Naval Research, U.K. You indicated in your presentation that the flow and chemistry parameters (Re, Da, etc.) you used in your calculations were chosen for computational ease and not for physical reality. What values of these parameters would have made your calculations physically realistic, what computational difficulties (specifically) would this have caused you, and how do you expect your results would have differed? Author's Reply. T h e physically realistic values of hydrochemical parameters are dependent on the choice of reactants and the laboratory conditions u n d e r which experiments are performed. For a typical laboratory hydrocarbon flame, the approximate values of the Damkohler number, Zeldovich n u m b e r and heat release parameter are about 1000,
1259
50, and 200 respectively. The typical Reynolds number (in the experiments of Hermanson ~) is about 2000. For gaseous flows, the values of the Schmidt and Prandtl numbers are of order unity. Direct numerical simulations of flows with higher values of hydrochemical parameters than those employed in this paper require higher resolution (than 64 • 64 grids) and smaller computational time increments in order to resolve all the scales accurately. T h e values of the sample parameters selected here, however, are realistic in exhibiting some of the important physical features of the hydrodynamic and the chemical reactions. The value of the Reynolds n u m b e r is high enough for the simulation of vortexrollup and pairing (Ho and Huerre2), The value of the Damkohler n u m b e r was selected to be low enough so that the flame is extinguished at moderate values of the dissipation rate. Finally, the magnitude of the Zeldovich n u m b e r is sufficiently high to address the extinction p h e n o m e n a and to compare the numerical results with that obtained by asymptotic analysis 3 in the limit of high activation energy. REFERENCES 1. HERMANSON,J.C.: Ph.D. Thesis, California Institute of Technology, Pasadena, CA, 1985. 2. Ho, C.-H. ANn HUERRE, P.: AnnualReview of Fluid Mechanics 16, 363 (1984). 3. BUCKMASTER,J.D. AND LUDFORD, G.S.S.: Theory of Laminar Flames, Cambridge University Press (1982).
A. F. Ghoniem, Massachusetts Institute of Technology, USA. Since your main concern here is the effect of stretch on the propagation and quenching of the flame, the Reynolds n u m b e r which determines the highest value of strain within the braids should be of prime importance in comparing your results with either experimental or analytical results. What is the effect of the Reynolds number on the evolution of the mixing layer in your simulation and what is the maximum value of the Reynolds number that you can accomodate with your scheme?
Author's Reply. The maximum value of the Reynolds n u m b e r that can be simulated by pseudospectral methods using Fourier transforms depends on the n u m b e r of Fourier modes used to solve the transport equations. In the present 64 • 64 grid calculation, a Reynolds n u m b e r of about 200 can be accurately simulated. For the present two-dimensional simulations, this value is large enough to simulate vortex roll-up and pairing since the vortex dynamics is fairly independent of the Reynolds n u m b e r and only the shape of the vortices depends on the magnitude of the
1260
TURBULANT COMBUSTION
Reynolds number employed. ~ In simulating flows with higher Reynolds numbers, one cannot assume two-dimensionality since the essential features of turbulence are three-dimensional. With regard to the chemical aspects of the problem, the value of the reduced Damkohler number 2 is proportional to the inverse of the strain rate. Extinction occurs if the reduced Damkohler number is below the critical quenching value. In other words, laminar diffusion flamelets exist if the dissipation rate is below the critical value (Xq). Therefore, for given kinetics parameters, the flame can exist for higher values of the scalar gradient as the value of the Reynolds number is increased. The Reynolds number of 200, used in the present simulation, is comparable with the Reynolds number based on the thickness of laminar diffusion flamelets in laboratory turbulent diffusion flames.
Layer," presented at the Spring Technical Meeting of the Combustion Institute, Central States Section, Cleveland, Ohio, May, 1986.
R. W. Bilger, University of Sydney, Australia. Measurements on a turbulent jet diffusion flame of methane near extinction using laser Raman/Raleigh techniques show joint pdfs (e.g., temperature versus mixture fraction) that are very interesting: they are bimodal-burnt and unburnt---on the lean side of (instantaneous) stoichiometric and centrally distributed-partly burnt---on the rich side of stoichiometric (instantaneous)) What do your joint pdfs look like, and what is the inlfuence of three-dimensionality on the joint pdfs?
REFERENCES 1. Ho, C.-M., ANt) HUERRE, P.: Annual Review of Fluid Mechanics, 16, 365 (1984). 2. PETERS, N., ANt) WILLInMS., F.A.: A1AA Journal, 21,423 (1983).
S. R. GollahaUi, University of Oklahoma, USA. The analysis assumes that the reaction zone (flame) exists at the shear layer. In the case of hydrocarbon fuel/air shear layer, the flame zone is shifted towards the air side away from the shear layer. How can we treat the extinction in such a case? Author's Reply. Treatment of extinction in diffusion flames requires the analysis of the strain rate near the stoichiometric surface (Z~t). In the present calculations, the whole range of Z (0 ~< Z ~ 1) is considered and the stoichiometric surface is at Zst = 0.5. In the case of hydrocarbon fuel/air shear layer, the stoichiometric surface is at lower Z values (for example Zst = 0.055 for methane flame) and can be simulated. In the present calculations using a temporally evolving code, however, the asymmetric aspect of mixing as discussed by Givi and Jou 1 are missing because of the implicit symmetry of the computational domain. Generally, numerical simulations of reacting flow problems require a large enough computational domain to include the flow field and the scalar fields. The location of the flame is not preassigned and the flame exists where the conditions are favorable for combustion. If the flame is located on the "air side", the numerical simulations would show that. REFERENCE 1. GIvI, P. ANt)Jou, W.-H.: "Mixing and Chemical Reactions in a Spatially Developing Mixing
REFERENCE 1. MASRI, A.R., BILGER R.W. ANt) DIBBLE, R.W.: Turbulent Nonpremixed Flames of Methean Near Extinction, Part I: Instantaneous Structure (submitted to Combustion and Flame).
Author's Reply. Statistical analysis of the variables, such as the joint pdfs of the scalar quantities, are usually appropriate for flows under the influence of random, three-dimensional turbulence. In the present two-dimensional simulation, there are no random motions, and only the influences of large coherent structures are considered. Presently, we are performing some three-dimensional calculations in order to simulate turbulent reacting jets and mixing layers. It would be very interesting to compare the calculated statistical quantities with those obtained experimentally by Dr. Assad Masri when our threedimensional simulations are available.
J. w. Daily, University of California at Berkeley, USA. 1. Since one expects spacial development, threedimensional effects, etc. to have a substantial influence on the integral parameters of the flow, it would be extremely interesting to have a systematic comparison of integral parameters as the various effects are incorporated into numerical calculations. 2. We do not see any place in a spacially evolving layer which is fully two-dimensional, even prior to the mixing transition.
Author's Reply. 1. Such studies have been and are being performed extensively in our previous and present efforts. Riley
FLAME EXTINCTION IN A MIXING LAYER and Metcalfe I examined the effects of three-dimensionality on incompressible non-reacting flows. Riley et al 2 studied such effects in reacting temporally evolving mixing layers, and Givi and Jou 3 addressed some of the aspects of the spatial development of a mixing layer with different inflow/outflow boundary conditions. 2. Your observations are in disagreement with those reported by Ho and Huerre 4 and Browand and Troutt. 5 These experiments clearly indicate that, at least in the beginning stages of the flow evolution, the mixing layer can exhibit some features that are strongly two-dimensional. REFERENCES 1. RILEY, J J . A~'D METCALFE, R.W.: A1AA Paper 80-0274 (1980). 2. RILEY, J.J, METCALFE, R.W., AND ORSZAC,, S.A.: Physics of Fluids 29, 406 (1986). 3. GIVI, P, ANn Jou, W.-H.: "Mixing and Chemical Reactions in a Spatially Developing Mixing Layer", presented at the Spring Technical Meeting of the Combustion Institute, Central States Sections, Cleveland, Ohio, May, 1986.
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4. Ho, C.-M., AND HUERRE, P.: Annual Review of Fluid Mechanics 16, 365 (1984). 5. BROWAND, F.K, AND TROUTT, T,R.:J. FluidMech. 97, 771 (1980).
N. Peters, RWTH Aachen, West Germany. In the two-variable formulation of diffusion flamelets generally only the two steady states, the burning and the non-burning state, are considered. Since you are calculating the entire instationary process, can you estimate the amount of time where unsteady transitions take place? Author's Reply. In order to give the exact value of this time, the unsteady evolution of the flamelet during transition must be considered. With the available data records that were stored, a rough estimate of nondimensional transition time is about 3. This value is about 25% of the time that is required for the first roll-up to occur.
FLAME E X T I N C T I O N I N A T E M P O R A L L Y D E V E L O P I N G M I X I N G LAYER P. GIVI, W. -H. JOU ANn R. W. METCALFE Flow Research Company 21414-68th Avenue South Kent, Washington 98032
Nonequilibrium effects leading to the local quenching of a diffusion flame have been investigated by examining the evolution of large-scale structures in a two-dimensional temporally developing mixing layer. Pseudospectral calculations of a temperature-dependent, nonpremixed, constant-density, reacting shear layer indicate that the primary important parameter to be considered for flame extinction is the local instantaneous scalar dissipation rate, conditioned at the scalar stoichiometric value (Xst).At locations where this value is increased beyond a critical value (Xq), the local temperature decreases and the instantaneous reaction rate drops to zero. This is consistent with the results of perturbation methods employing large activation energy asymptotics for the study of flame extinction in nonpremixed flames.
1. Introduction A diffusion flame is characterized by a chemical reaction time that is usually much smaller than a characteristic diffusion time. The chemical reactions occur in a narrow zone between the fuel and the oxidizer, where the concentrations of both reactants are very small. The rate at which the reactants flow into the reaction zone, and therefore the characteristic time of the flame, is d e p e n d e n t on the hydrodynamics of the particular flow. As the characteristic time of the chemical reactions decreases, this reaction region becomes infinitesimally thin. I n this limit the chemical reaction zone approaches a "flame sheet" (local chemical equilibrium1), where the concentrations of both reactants are very low and the rate of combustion is governed by the rate at which fuel and oxidizer flow into the reaction zone. The flame sheet assumption is justified by the very fast chemical reaction rate of the diffusion flame. This assumption significantly reduces the complexity of the problem since it eliminates the analysis associated with the chemical kinetics. For many flows whose characteristic time scale of chemical reaction is much smaller than the hydrodynamic (convective-diffusive) time scale, the assumption of local chemical equilibrium adequately predicts the location and the shape of the flame. 2 One important feature of the calculations based on the fast chemistry model is the introduction of a passive Shvab-Zeldovich2 conserved scalar variable (Z), which is i n d e p e n d e n t of the chemical kinetics. From this quantity, the evolution of the concen-
tration fields of both the reactants and the products can be computed. In turbulent flows, however, the local characteristic flow time scales vary considerably. As a result, many important and interesting problems that cannot be analyzed by local chemical equilibrium assumptions are introduced. Experimental studies of Tsuji 3 show that as the local characteristic diffusion time becomes shorter and approaches the order of magnitude of the chemical time scale, the details of the chemical reactions cannot be neglected. If the flow of reactants into the reaction zone increases further, causing the diffusion time scale to be reduced more, the chemical reaction will not be able to keep pace with the further supply of reactants. The reaction rate will be reduced, and local quenching occurs. As shown by Peters, 4 further reduction of the diffusion time scale leads to lift-off and the blow-off of the entire flame. As noted by both Tsuji and Peters, the consideration of flame extinction cannot be explained by the flame sheet model, which assumes an infinitely fast chemical reaction rate. Therefore, in order to address the quenching p h e n o m e n o n , the structure of a finite reaction rate zone must be studied. Linan 5 has employed a method of matched asymptotic expansions in the limit of large activation energy in an attempt to describe the interaction between the hydrodynamics and chemical reactions in the reaction zone of a counter-flowing laminar diffusion flame. It has been shown that activation energy asymptotics are very useful in predicting flame ignition and extinction characteristics in such flows. This
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1252
TURBULANT COMBUSTION
technique was e x t e n d e d to turbulent flows by Peters 4'6 by considering turbulent diffusion flames as an ensemble of laminar diffusion flamelets. 7 By introducing a local coordinate system that moves with the stoichiometric flame sheet, Peters 4 was able to recognize that the primary "nonequilibrium" p a r a m e t e r for the analysis of the flame extinction is the dissipation rate of the scalar quantity evaluated at stoichiometric conditions (Xst). This quantity is viewed as the inverse o f the diffusion time scale. If this p a r a m e t e r is increased beyond a critical limit (Xq), the heat conducted to both sides of the diffusion flamelets cannot be balanced by the heat production due to chemical reaction. At the critical value o f the dissipation, the finite rate kinetics balance the diffusion. 3 Some numerical calculations, p e r f o r m e d by Liew et al. 8 and c o m p a r e d with experimental data, 9 show also that as the m a x i m u m value of the dissipation (Xmo~)increases, the value of the m a x i m u m temperature decreases, the reaction eventually ceases, and the flame is locally quenched. These results indicate that the dissipation rate of the conserved scalar is a useful characteristic to study in the analysis of nonequilibrium effects leading to flame extinction. In turbulent flows, however, the quantity Xst is a strongly fluctuating quantity that has not yet been numerically computed. Instead, statistical approaches have been chosen by Peters and Williams a~ and Janicka and Peters ll employing different turbulence closures in predicting the lift-off height of a r o u n d diffusion j e t flame. Results obtained using these methods were compared with experi/nental data 12 for both methane and natural gas flames and show some correct o r d e r - o f - m a g n i t u d e predictions. These results are encouraging; however, with the availability of larger computers, it is now possible to employ a m o r e accurate treatment o f the flame extinction a n d local flame quenching that occur in n o n p r e m i x e d flames. It would be very useful to correlate the flame extinction with the nonequilibrium p a r a m e t e r Xst. Also, it would be very interesting to look at the structure o f the diffusion flames at the point of extinction. These are the objectives of this paper. T h e present p a p e r employs a direct numerical simulation a p p r o a c h to investigate the problem of local flame extinction in a time-dependent, two-dimensional mixing layer. In this flow, the governing equations are solved by an accurate numerical m e t h o d without a closure model. The t i m e - d e p e n d e n t flow field can be analyzed statistically to u n d e r s t a n d the underlying physics much as an experimentalist does with laboratory data. T h e direct numerical simulation technique has recently been successfully applied
to chemically reacting flows. Riley at al. 13 considered the three-dimensional temporally growing mixing layer u n d e r the simplest possible assumption o f a constant rate, non-heat-releasing chemical reaction. T h e main contribution of this work is the u n d e r s t a n d i n g of the effects of three-dimensional mixing and the diffusion of the species on the chemical reactions. McMurtry et al. 14 considered the effects of the chemical heat release and the resulting density variation on the fluid motion for a two-dimensional mixing layer. T h e fluid dynamics and the chemical reaction are truly coupled in this work, and the interplay between the two are discussed. However, the assumption of a constant chemical reaction rate is still employed. In the present work, we intend to u n d e r s t a n d the flame extinction problem t h r o u g h a two-dimensional simulation of a mixing layer. In particular, the role of large-scale features of the turbulent flow in flame extinction is studied. Due to limitations o f numerical accuracy, only moderate Reynolds and Damkohler numbers are computed. For flame extinction in a three-dimensional turbulent flow, three-dimensional simulations will be p e r f o r m e d in the future.
y, V
U(yl
x,U L
f CA~ Au FIG. 1. Problem geometry.
I
-I
FLAME EXTINCTION IN A MIXING LAYER 2. Problem
Assuming equal free-stream concentrations, i.e., Ca~ = C~=, the nondimensionalized equations are
Formulation
For simplicity of numerical calculations, we consider a two-dimensional shear layer as shown in Fig. 1. The flow is considered periodic in the horizontal direction (x). Although the splitter plate flow evolves spatially downstream and the numerical simulations evolve temporally, important similarities in the dynamics of these two flows make it useful to study accurate numerical simulations of the temporally growing mixing layers, By simple Galilean transformation, a flow quantity averaged in the x-direction can be related to the time average of the same quantity at a fixed location in a splitter plate configuration. 13'~4 These average quantities are dependent on the transverse coordinate (y) and the time (t). Again, the inverse Galilean transformation relates the time t to the streamwise location in a splitter configuration. T h e presence of periodic boundary conditions allows us to use accurate pseudospectral numerical methods; these methods are discussed in Ref. 13 and will not be described here. T h e chemical reaction between the two species is assumed to be single-step and irreversible and to obey the t e m p e r a t u r e - d e p e n d e n t Arrhenius law. It is assumed that the heat release rate is low so that the effects of the chemical heat release on the flow field can be neglected. This, together with the further assumptions of a low Mach n u m b e r flow, result in a constant-density flow formulation. T o identify the n o n d i m e n s i o n a l groupings, the variables are normalized by a velocity scale (mean velocity difference across the layer A b), a temperature scale (free-stream T~), concentration scales (free-stream Ca, and C~,), and a length scale (1=2.249~r, where ~r is the distance from the plane of symmetry to the transverse plane, where the mean velocity rises to half of its freestream value). T h e value of 2.249 was chosen so that the size of the computational domain would be equal to the wavelength of the most unstable mode and its normalized value would be an integer multiple of 27r. X
_x*= : X
17" = XV
U U*= AU P* =
P
(au) 2 CA
c~= (CA)oo
AU t*=t-X T* -
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T
To~ Ce
c ~ - (cB)~
V.U*=O OU* 9 9 1 ,~ L * ( _ U * , R e ) = - ~ - + _ U - V U - - ~ e V U* = -
VP*
L*(C~, Re Sc) = L ( C ~ , Re Sc) =
(2) -W*
L*(C~, Re Sc) = W* L*(T*, Re Pr) = C e W* where W* = Da e -ze/T'CJCJ Da = A/CA~176 = Damkohler number AU X Ze = Re = Ce=
E
RT= AUX P
Q
CoT=
= Zeldovich number = Reynolds number
(3)
= Heat release parameter
P
Sc = ~ --- Schmidt number P
Pr = ~ = Prandtl number The values of Sc and Pr can be set equal to unity, since this is approximately correct for gaseous diffusion flames. This results in a Lewis n u m b e r of unity. T h e value of other nondimensional parameters is limited by the available resolution of the numerics employed in the computations. Shvab-Zeldovich Variable
For the two-feed diffusion flame considered here, it is possible to consider only three scalar quantities rather than solving for all four scalar variables. The Shvab-Zeldovich variable (Z) and the product concentrations (G), following Givi et al., 15 are defined as
(1) z = cA* + c ~ G = C*
(4)
T h e transport equations governing Z, G, and T* follow:
TURBULANT COMBUSTION
1254
tion. These b o u n d a r y conditions are n a t u r a l w h e n F o u r i e r series e x p a n s i o n s are used.
L*(Z) = 0 L*(G) = Da e -zerr* (1 - Z - G) (Z - G) (5)
3. Presentation o f Results
L*(T*) = Ce Da e -zerr* (1 - Z - G) (Z - G) Initial and Boundary Conditions T h e initial a n d b o u n d a r y conditions for the velocity field are given elsewhere 16 a n d are only s u m m a r i z e d here. I n terms o f the stream function, the initial c o n d i t i o n is given by t~ = tOmean -}- qlf Jr" qlsh
(6)
where 0,,~, is the stream function associated with a m e a n hyperbolic t a n g e n t velocity profile, Of is the stream f u n c t i o n for the most unstable m o d e of this m e a n velocity profile, a n d O,h is the stream f u n c t i o n for the first s u b h a r m o n i c of the most unstable mode. T h e properties of these modes have b e e n evaluated from linear stability theory. 17 T h e f u n d a m e n t a l m o d e in this mixing layer p r o d u c e s a single vortex rollup, a n d w h e n the s u b h a r m o n i c is a d d e d in, a second rollup, or pairing, can o c c u r ] 6 I n the computations r e p o r t e d in this paper, we have employed two initial conditions for the hydrodynamics: (1) the m e a n flow a n d the f u n d a m e n t a l m o d e alone (Case I), a n d (2) the m e a n flow, the f u n d a m e n t a l mode, a n d the first s u b h a r m o n i c m o d e a d d e d together in phase (Case II). In this m a n n e r , the effects that the vortex rollup has o n the chemical reaction can be examined. T h e initial conditions for the reactant concentration are given by
C~(x*,y*,O)=y--~ j_
exp~-~-;
C~(x*,y *, O) = 1 - C,T
A pseudospectral n u m e r i c a l code was develo p e d for the calculations o f the incompressible reacting flow c o n s i d e r e d here. C o m p u t a t i o n s were p e r f o r m e d in a square d o m a i n with the size 0 < x < 2rrA, -~rA < y < ~'A for the single vortex rollup a n d the d o m a i n 0 < x < 47rA, -27rA < y < 2~rA for the double rollup computations. T h e spatial resolution was 64 x 64 F o u r i e r modes. T h e values of the Reynolds a n d D a m k o h l e r n u m b e r s were set equal to 200 a n d 10, respectively, so that the simulation would be accurately resolved o n the 64 x 64-point grid e m p l o y e d here. T h e value of the Zeldovich n u m b e r was set equal to 20. T h e value of the heat release p a r a m e t e r was selected to be small e n o u g h so that, w h e n multiplied by the D a m k o h l e r n u m b e r , it would be resolvable by the n u m e r i c s a n d so that it would also be reasonable to neglect the effects of heat release o n the flow field. However, this value should be large e n o u g h to allow the effects of t e m p e r a -
d( (7)
C~(x*,y*, O) = 0 Note that there are no initial fluctuations in the scalar profiles. T h e t e m p e r a t u r e in the free stream is equal to T=. Near y = 0, this value is increased to a n ignition t e m p e r a t u r e to allow the chemical reactions to occur. T h e initial profile for T* is given by an e x p o n e n t i a l function.
T*(x*,y*, 0 ) = 1 + 4 exp ( - 4y'2 )
(8)
A periodic b o u n d a r y c o n d i t i o n is e m p l o y e d in the streamwise direction, a n d a free-slip b o u n d a r y c o n d i t i o n in the cross-stream direc-
FIG. 2. Plots of Shvab-Zeldovich variable contours (Case I), (a) t* = 9, contour minimum is 0, contour maximum is 1, contour interval is 0.1. (b) t* = 15, contour minimum is 0, contour maximum is 1, contour interval is 0.1.
FLAME EXTINCTION IN A MIXING LAYER ture d e p e n d e n c e on the chemical reaction term. A value o f Ce8 is satisfactory, giving a m a x i m u m (adiabatic) t e m p e r a t u r e o f T'adiabatic = 7. T h e profiles o f the c o n s e r v e d Shvab-Zeldovich scalar variable are chosen f o r the p u r p o s e o f flow visualization. I n Figs. 2 and 3, we p r e s e n t the time s e q u e n c e d e v e l o p m e n t o f this profile for Cases I a n d II, respectively. Initially, the p e r t u r b a t i o n associated with the f u n d a m e n t a l m o d e grows until a time of t* = 9, w h e r e the first v o r t e x r o l l u p occurs. Proc e e d i n g f u r t h e r in time results in diffusion of the c o r e o f the v o r t e x with no additional rollup. A d d i n g the s u b h a r m o n i c associated with this unstable m o d e results in a second rollup (Fig. 3) that initiates at a time o f about t* = 12 a n d is c o m p l e t e d by t* = 24. As shown in these figures, the effect o f the vortex dynamics is to increase the strain rate at the braids b e t w e e n the v o r t e x cores. I f a fast chemistry m o d e l was assumed to describe the chemical reactions, the surface o f Z = Z~t = 0.5 would c o r r e s p o n d to the flame location. T h e vortex dynamics plays an i m p o r tant role in increasing the a r e a o f the flame
1255
surface (Z,t surface) and in increasing the total a m o u n t o f reaction p r o d u c t c o m p a r e d to the case w h e r e there is no r o l l u p a n d the chemical reaction is only limited to the i n t e r d i f f u s i o n o f the scalars at the reaction surface. In the p r e s e n t calculations, the increase o f the equilibr i u m flame sheet surface d u e to r o l l u p (Case I, Fig. 2b) is 153 p e r c e n t in c o m p a r i s o n with the u n f o r c e d case, M e r g i n g o f the vortices (Case II, Fig. 3b) increases this surface by 340 p e r c e n t w h e n c o m p a r e d with the u n f o r c e d case. T h e time sequence o f the t e m p e r a t u r e is shown o n Figs. 4 and 5 for Cases I and II, respectively. T h e m e c h a n i s m of v o r t e x rollup is to d r a w the reactants f r o m the two streams to the reaction surface. T h e m a x i m u m t e m p e r a ture a l o n g the Z = Z,t surface occurs at the c o r e o f the v o r t e x (the hottest location), w h e r e vorticity a n d the p r o d u c t c o n c e n t r a t i o n are also highest, a n d decreases in the braids, where the g r a d i e n t s o f the scalar are at their h i g h e r values. At the time w h e n the c o m p u t a t i o n is stopped, the m a x i m u m p r o d u c t c o n c e n t r a t i o n has not yet r e a c h e d its e q u i l i b r i u m value. T h e r e still r e m a i n reactants, e v e n at the core o f
(a)
(b)
/ FIG. 3. Plots of Shvab-Zeldovich variable contours (Case II). (a) t* = 12, contour minimum is 0, contour maximum is 1, contour interval is 0.1. (b) t* = 24, contour minimum is 0, contour maximum is 1, contour interval is 0.1.
FIG. 4. Plots of normalized temperature contours (Case I). (a) t* = 9, contour minimum is 0, contour maximum is 6, contour interval is 0.6. (b) t* = 15, contour minimum is 0, contour maximum is 6, contour interval is 0.6.
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T U R B U L A N T COMBUSTION
the mixing layer, a n d the flame s h o r t e n i n g p h e n o m e n o n , which is usually caused by the d e p l e t i o n of the reactants, has not b e e n yet observed. T h e reason for this b e h a v i o r of the chemical p r o d u c t and t e m p e r a t u r e can be e x p l a i n e d by the c o n t o u r plots o f the instantaneous reaction rate W, p r e s e n t e d in Fig. 6 for Case I. As shown in this figure, the r e a c t i o n rate initially is very u n i f o r m along the species i n t e r d i f f u s i o n zone and is m a x i m i z e d at the reaction surface, w h e r e the reactants are in contact. At later times, the reaction rate decreases, a n d at points w h e r e the strain rate is sufficiently large, the reaction rate goes to zero a n d the flame locally quenches, This m e c h a n i s m o f extinction o f the diffusion flame is consistent with the e x p e r i m e n t a l observations o f Tsuji. 3 I n the braids, as the inverse o f the diffusion time scale (1/X~t) decreases (as the result of the v o r t e x rollup), the supply o f the reactants is g r e a t e r t h a n the chemical reaction can utilize. Notice f r o m Figs. 2 t h r o u g h 5 that the reactants are well m i x e d on the braids, b u t the reaction rates at these places are zero. T h i s n o n e q u i l i b r i u m p h e n o m e n o n can be e x p l a i n e d by e x a m i n i n g the r e a c t i o n rate e q u a t i o n [Eq. (3)].
(e)
(0)
FtG. 5. Plots of normalized temperature contours (Case II). (a) t* = 12, contour minimum is 0, contour maximum is 6, contour interval is 0.6. (b) t* = 24, contour minimum is 0, contour maximum is 6.6, contour interval is 0.6.
FIG. 6. Plots of normalized reaction rate contours (Case I). (a) t* = 6, contour minimum is 0, contour maximum is 0.03, contour interval is 0.003. (b) t* = 9, contour minimum is 0, contour maximum is 0.024, contour interval is 0,002. (c) t* = 12, contour minimum is 0, contour maximum is 0.02, contour interval is 0.002. (d) t* = 24, contour minimum is 0, contour maximum is 0.017, contour interval is 0.001. Unlike the t e m p e r a t u r e - i n d e p e n d e n t reacting m i x i n g layer calculation r e p o r t e d by Givi et al., 18 the m a x i m u m value o f the reaction rate
FLAME EXTINCTION IN A MIXING LAYER does not necessarily c o r r e s p o n d to the maxim u m value o f the p r o d u c t o f the reactants concentration, since the effects o f the temperature variations influence the local conversion rate o f the chemistry. I f the flame temperature goes below a critical characteristic temperature, the flame becomes very rich with both reactants. However, since the value of the dissipation rate is greater than the critical value, this p r e m i x e d region of the reactants cannot be reignited from the high t e m p e r a t u r e zone of the core. A higher dissipation rate results in a further decrease in t e m p e r a t u r e until it becomes equal to the b a c k g r o u n d temperature. At this point, the vortex has reached the b o u n d a r y of the computational domain and, therefore, further computation is not realistic. A direct quantitative comparison of Xq, obtained in our numerical simulations (Xq is a p p r o x i m a t e d here as the value o f the critical instantaneous dissipation rate evaluated at the stoichiometric surface Z,t) with that obtained u n d e r the large activation energy asymptotics is not expected to agree exactly for several reasons. First, in asymptotic analysis, the flame thickness is very small and is only b r o a d e n e d near the reaction zone by a small parameter. In the direct numerical simulations reported here, the reactants have a fairly "thick" overlap region in the reaction zone d u e to the numerical accuracy limitations. F u r t h e r m o r e , the initial conditions for the t e m p e r a t u r e profile employed here near the reaction zone are not the same as the u n i f o r m initial temperature distributions o f Peters. 4 However, making the assumption that the initial concentration of the reactant and the t e m p e r a t u r e o f the reactants are at some average values in the range between the free stream and the reaction zone, correct order-of-magnitude asymptotic results are verified by the results o f the simulations. T h e t e m p e r a t u r e profiles presented in Fig. 4 show that by a time o f t* = 15, the value o f the t e m p e r a t u r e at the braids falls to about onefifth the flame t e m p e r a t u r e (approximately equal to the free-stream temperature). This corresponds to local quenching. The corresponding c o m p u t e d normalized dissipation rate X~ (X* = X,tA/AU) at the point of quenching, for the given kinetics data, is about 7 for both Cases I and II. T h e estimated corresponding value for the dissipation rate, c o m p u t e d with the asymptotic methods described above, is about 8, which is in good a g r e e m e n t with the results of the numerical simulation. 4, C o n c l u s i o n s
A pseudospectral algorithm has been used for the numerical calculations of a constant-
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density, chemically reacting, temperature-dep e n d e n t mixing layer. T h e nonequilibrium effects leading to the local flame quenching have been simulated in a case with fairly large Zeldovich n u m b e r a n d m o d e r a t e Reynolds and Damkohler numbers. It has been found that the rollup of an unstable shear layer creates regions with high dissipation rates at the braids, where local flame extinction occurs according to the theory of Peters 4'1~ T h e temperature contour shows that the t e m p e r a t u r e drops to a value close to ambient at the braids and the reaction rate reduces to zero, even though the reactants are well mixed there. Within the limitations of numerical resolution, comparison o f the critical scalar dissipation rate with that obtained by large activation energy asymptotics shows reasonable agreement. We are presently e x p a n d i n g this work to investigate a spatially developing mixing layer and three-dimensional turbulent flows. This should further illuminate the basic mechanisms o f such p h e n o m e n a as the diffusion j e t flame lift-off. Nomenclature
Af C Cv Ce D Da E G K L P Pr Q R Re Sc T t U V W X x y Z Ze
Preexponential factor Concentration Specific heat Heat release p a r a m e t e r Molecular diffusivity Damkohler n u m b e r Activation energy Normalized p r o d u c t concentration T h e r m a l conductivity Convective-diffusive o p e r a t o r Pressure Prandtl n u m b e r " Heat of reaction Universal gas constant Reynolds n u m b e r Schmidt n u m b e r Temperature Time Streamwise velocity c o m p o n e n t Cross-stream velocity c o m p o n e n t Reaction rate Dissipation Streamwise coordinate Cross-stream coordinate Shvab-Zeldovich variable Zeldovich n u m b e r
Greek A v tO
Length scale Molecular viscosity Stream function
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T U R B U L A N T COMBUSTION
Superscript *
Non-dimensionalized
3. 4. 5. 6.
Subscript A,B f P q
sh st
7.
Reactants Fundamental Product Quenching Subharmonic Stoichiometric Free stream
8.
Acknowledgments The calculations reported here were performed under research sponsored by the Air Force Office of Scientific Research u n d e r Contract F 4 9 6 2 0 - 8 5 - C 00067. The United States Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation hereon. The authors have also appreciated the support of NASA Lewis Research Center in providing computer time.
REFERENCES: 1. BILGER, R. W.: Turbulent Reacting Flows (P. A. Libby and F. A. Williams, Eds.), p. 65, SpringerVerlag, 1980. 2. WILLIAMS, F. A.: Combustion Theory, Second edi-
Lion, The Benjamin/Cummings Publishing Company, Inc., 1985. Tsvj1, H.: Prog. Energy Comb. Sci. 8, 93 (1982). PETERS, N.: Comb. Sci. Tech. 30, 1 (1983). LINAN, A.: Acta Astronautica 1, 1007 (1974). PETERS, N.: Prog. Energy Comb. Sci. 10, 319 (1984). WILLIAMS,F. A. : Turbulent Mixing in Non-Reactive and Reactive Flows (S. N. B. Murthy, Ed.), p. 189, Plenum Press, t975. LIEW, S. K., Moss, J. B. AYD BRAY, K. N. C.: Dynamics of Flames and Reactive Systems (J. R. Bowen, N. Manson, A. K. Oppenheime and R. I. Sdoukhin, Eds.), Progress in Astronautics and Aeronautics, Vol. 95, p. 305 (1984).
9. MITCHELL, R., SAROFIN, A. AND CLOMBURG, L.: Comb. and Flame 37, 337 (1980). 10. PETERS, N. AND WILLIAMS, F. A.: AIAAJ. 21,423 (1983). 11. JANICKA, J. AND PETERS, N.: Nineteenth Symposium (International) on Combustion, p. 367, T h e Combustion Institute, 1982~ 12. DONNERHACK, S. AND PETERS, N.: Comb. Sci. Tech. 41, 101 (1984). 13. RZLrY,J. j., METCALVE, R. W. AZ~'OORSZaG, S. A.: Phys. Fluids 29, 406 (1986). 14. McMURTRV, P. A., JOLT, W. -H., RILEY, J. J. ANn METCALVE, R. W.: AIAA Paper 85-0143, 1985. 15. Gtvt, P., SIRIGNANO, W. A. AND POPE, S. B.: Comb. Sci. Tech. 37, 59 (1984). 16. RILEe, J. J. AND METCALFE, R. W.: AIAA Paper 80-0274, 1980. 17. MICHALKE, A.: J. Fluid Mech. 19, 543 (1964). 18. GIvI, P., RAMOS, J. I. ANn SIRIGNANO, W. A.: J. Non-Equilibrium Thermodynamics 10, 75 (1985).
COMMENTS j. P. Boris, Naval Research Laboratory, USA. Fluid compressibitlity (expansion with heat release), acoustic waves, and spatial variation with a realistic difference between inflow and outflow are known to have profound effects on flame propagation and extinction. Your calculations have none of these important effects. Expansion, for example, changes roll up and merging by suppressing exactly the modes you simulated. How are your results to be related to reality in view of these computational short comings? Author's Reply. In addition to the influences of fluid compressibility, acoustic waves and spatial variation, there are also other p h e n o m e n a that can have very important effects on the mechanism of flame stabilization, structure, and extinction. Namely, the influences of multi-step chemical kinetics, the non-equilib-
rium effects of radiation, 1 gravity ~'3 and Lewis number, 4 the role of three-dimensional turbulence motion, etc. Incorporating all these effects in the accurate simulations of reacting flow problems is not practical computationally. Our approach to the investigation of the physically important, complex p h e n o m e n o n of flame extinction is to first study a simpler system. After developing insight into some of the basic phenomena, the degree of complexity can then be increased. Our present calculations have focused on the influence of scalar strain rate on the structure of the flame, and the results of our simulation are consistent with experimental observations. 5 Work is in progress at Flow Research and elsewhere to address the effects of some more complex physical phenomena on the
FLAME EXTINCTION IN A MIXING LAYER evolution of reacting flows. For example, Givi and Jou 6 reported some calculations for spatially evolving flows with different inflow/ourflow conditions and addressed the asymmetric nature of the mixing mechanism. Riley et al 7 studied the effects of the three-dimensional turbulence motion on the structure of the flame in a constant finite rate chemical reaction, and McMurtry et al 8 examined the role of expansion due to heat release in a reacting mixing layer. Thus, different aspects of the physics of the problem are isolated and studied separately for better understanding. As computational capabilities continue to improve, more realistic simulations combining several of these p h e n o m e n a will become feasible.
REFERENCES 1. BONNY, U.: Combustion and Flame 16, 147 (1971). 2. COCHRAN, T.H. AND MASICA, W.J.: Thirteenth Symposium (International) on Combustion, p. 821, The Combustion Institute, 1971. 3. ALTENKIRCH, R.A.: "The Use of a Low-Gravity Environment in Combustion Research," presented at the Spring Technical Meeting of the Combustion Institute, Central States Section, Cleveland, Ohio, May, 1986. 4. CHUNG, S.H. AND LAW, C.K.: Combustion and Flame 52, 59 (1983). 5. TsuJi, H.: Prog. Energy Comb. Sci. 8, 93 (1982) 6. GIvI, P. AND JOU, W.-H.: "Mixing and Chemical Reactions in a Spatially Developing Mixing Layer", presented at the Spring Technical Meeting of the Combustion institute, Central States Section, Cleveland, Ohio, May, 1986. 7. RILEY, J.J.; METCALFE, R.W. AND ORSZAG, S.A.: Phys. Fluids 29, 406 (1986). 8. MCMURTRY, P.A.; W.-H.; RILEY, J.J. AND METCALFE, R.W.: AIAA Journal 24, 962 (1986).
E. F. Brown, Office of Naval Research, U.K. You indicated in your presentation that the flow and chemistry parameters (Re, Da, etc.) you used in your calculations were chosen for computational ease and not for physical reality. What values of these parameters would have made your calculations physically realistic, what computational difficulties (specifically) would this have caused you, and how do you expect your results would have differed? Author's Reply. T h e physically realistic values of hydrochemical parameters are dependent on the choice of reactants and the laboratory conditions u n d e r which experiments are performed. For a typical laboratory hydrocarbon flame, the approximate values of the Damkohler number, Zeldovich n u m b e r and heat release parameter are about 1000,
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50, and 200 respectively. The typical Reynolds number (in the experiments of Hermanson ~) is about 2000. For gaseous flows, the values of the Schmidt and Prandtl numbers are of order unity. Direct numerical simulations of flows with higher values of hydrochemical parameters than those employed in this paper require higher resolution (than 64 • 64 grids) and smaller computational time increments in order to resolve all the scales accurately. T h e values of the sample parameters selected here, however, are realistic in exhibiting some of the important physical features of the hydrodynamic and the chemical reactions. The value of the Reynolds n u m b e r is high enough for the simulation of vortexrollup and pairing (Ho and Huerre2), The value of the Damkohler n u m b e r was selected to be low enough so that the flame is extinguished at moderate values of the dissipation rate. Finally, the magnitude of the Zeldovich n u m b e r is sufficiently high to address the extinction p h e n o m e n a and to compare the numerical results with that obtained by asymptotic analysis 3 in the limit of high activation energy. REFERENCES 1. HERMANSON,J.C.: Ph.D. Thesis, California Institute of Technology, Pasadena, CA, 1985. 2. Ho, C.-H. ANn HUERRE, P.: AnnualReview of Fluid Mechanics 16, 363 (1984). 3. BUCKMASTER,J.D. AND LUDFORD, G.S.S.: Theory of Laminar Flames, Cambridge University Press (1982).
A. F. Ghoniem, Massachusetts Institute of Technology, USA. Since your main concern here is the effect of stretch on the propagation and quenching of the flame, the Reynolds n u m b e r which determines the highest value of strain within the braids should be of prime importance in comparing your results with either experimental or analytical results. What is the effect of the Reynolds number on the evolution of the mixing layer in your simulation and what is the maximum value of the Reynolds number that you can accomodate with your scheme?
Author's Reply. The maximum value of the Reynolds n u m b e r that can be simulated by pseudospectral methods using Fourier transforms depends on the n u m b e r of Fourier modes used to solve the transport equations. In the present 64 • 64 grid calculation, a Reynolds n u m b e r of about 200 can be accurately simulated. For the present two-dimensional simulations, this value is large enough to simulate vortex roll-up and pairing since the vortex dynamics is fairly independent of the Reynolds n u m b e r and only the shape of the vortices depends on the magnitude of the
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TURBULANT COMBUSTION
Reynolds number employed. ~ In simulating flows with higher Reynolds numbers, one cannot assume two-dimensionality since the essential features of turbulence are three-dimensional. With regard to the chemical aspects of the problem, the value of the reduced Damkohler number 2 is proportional to the inverse of the strain rate. Extinction occurs if the reduced Damkohler number is below the critical quenching value. In other words, laminar diffusion flamelets exist if the dissipation rate is below the critical value (Xq). Therefore, for given kinetics parameters, the flame can exist for higher values of the scalar gradient as the value of the Reynolds number is increased. The Reynolds number of 200, used in the present simulation, is comparable with the Reynolds number based on the thickness of laminar diffusion flamelets in laboratory turbulent diffusion flames.
Layer," presented at the Spring Technical Meeting of the Combustion Institute, Central States Section, Cleveland, Ohio, May, 1986.
R. W. Bilger, University of Sydney, Australia. Measurements on a turbulent jet diffusion flame of methane near extinction using laser Raman/Raleigh techniques show joint pdfs (e.g., temperature versus mixture fraction) that are very interesting: they are bimodal-burnt and unburnt---on the lean side of (instantaneous) stoichiometric and centrally distributed-partly burnt---on the rich side of stoichiometric (instantaneous)) What do your joint pdfs look like, and what is the inlfuence of three-dimensionality on the joint pdfs?
REFERENCES 1. Ho, C.-M., ANt) HUERRE, P.: Annual Review of Fluid Mechanics, 16, 365 (1984). 2. PETERS, N., ANt) WILLInMS., F.A.: A1AA Journal, 21,423 (1983).
S. R. GollahaUi, University of Oklahoma, USA. The analysis assumes that the reaction zone (flame) exists at the shear layer. In the case of hydrocarbon fuel/air shear layer, the flame zone is shifted towards the air side away from the shear layer. How can we treat the extinction in such a case? Author's Reply. Treatment of extinction in diffusion flames requires the analysis of the strain rate near the stoichiometric surface (Z~t). In the present calculations, the whole range of Z (0 ~< Z ~ 1) is considered and the stoichiometric surface is at Zst = 0.5. In the case of hydrocarbon fuel/air shear layer, the stoichiometric surface is at lower Z values (for example Zst = 0.055 for methane flame) and can be simulated. In the present calculations using a temporally evolving code, however, the asymmetric aspect of mixing as discussed by Givi and Jou 1 are missing because of the implicit symmetry of the computational domain. Generally, numerical simulations of reacting flow problems require a large enough computational domain to include the flow field and the scalar fields. The location of the flame is not preassigned and the flame exists where the conditions are favorable for combustion. If the flame is located on the "air side", the numerical simulations would show that. REFERENCE 1. GIvI, P. ANt)Jou, W.-H.: "Mixing and Chemical Reactions in a Spatially Developing Mixing
REFERENCE 1. MASRI, A.R., BILGER R.W. ANt) DIBBLE, R.W.: Turbulent Nonpremixed Flames of Methean Near Extinction, Part I: Instantaneous Structure (submitted to Combustion and Flame).
Author's Reply. Statistical analysis of the variables, such as the joint pdfs of the scalar quantities, are usually appropriate for flows under the influence of random, three-dimensional turbulence. In the present two-dimensional simulation, there are no random motions, and only the influences of large coherent structures are considered. Presently, we are performing some three-dimensional calculations in order to simulate turbulent reacting jets and mixing layers. It would be very interesting to compare the calculated statistical quantities with those obtained experimentally by Dr. Assad Masri when our threedimensional simulations are available.
J. w. Daily, University of California at Berkeley, USA. 1. Since one expects spacial development, threedimensional effects, etc. to have a substantial influence on the integral parameters of the flow, it would be extremely interesting to have a systematic comparison of integral parameters as the various effects are incorporated into numerical calculations. 2. We do not see any place in a spacially evolving layer which is fully two-dimensional, even prior to the mixing transition.
Author's Reply. 1. Such studies have been and are being performed extensively in our previous and present efforts. Riley
FLAME EXTINCTION IN A MIXING LAYER and Metcalfe I examined the effects of three-dimensionality on incompressible non-reacting flows. Riley et al 2 studied such effects in reacting temporally evolving mixing layers, and Givi and Jou 3 addressed some of the aspects of the spatial development of a mixing layer with different inflow/outflow boundary conditions. 2. Your observations are in disagreement with those reported by Ho and Huerre 4 and Browand and Troutt. 5 These experiments clearly indicate that, at least in the beginning stages of the flow evolution, the mixing layer can exhibit some features that are strongly two-dimensional. REFERENCES 1. RILEY, J J . A~'D METCALFE, R.W.: A1AA Paper 80-0274 (1980). 2. RILEY, J.J, METCALFE, R.W., AND ORSZAC,, S.A.: Physics of Fluids 29, 406 (1986). 3. GIVI, P, ANn Jou, W.-H.: "Mixing and Chemical Reactions in a Spatially Developing Mixing Layer", presented at the Spring Technical Meeting of the Combustion Institute, Central States Sections, Cleveland, Ohio, May, 1986.
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4. Ho, C.-M., AND HUERRE, P.: Annual Review of Fluid Mechanics 16, 365 (1984). 5. BROWAND, F.K, AND TROUTT, T,R.:J. FluidMech. 97, 771 (1980).
N. Peters, RWTH Aachen, West Germany. In the two-variable formulation of diffusion flamelets generally only the two steady states, the burning and the non-burning state, are considered. Since you are calculating the entire instationary process, can you estimate the amount of time where unsteady transitions take place? Author's Reply. In order to give the exact value of this time, the unsteady evolution of the flamelet during transition must be considered. With the available data records that were stored, a rough estimate of nondimensional transition time is about 3. This value is about 25% of the time that is required for the first roll-up to occur.