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Response of counterflow premixed flames to oscillating strain rates
Response of Counterflow Premixed Flames to Oscillating Strain Rates H. G. IM,* J. K. BECHTOLD+ and C. K. LAWS Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544
We study the response of premixed counterflow flames to an imposed oscillating strain rate which has potential application to both turbulent combustion and acoustic instabilities. We exploit the limit of large activation energy to resolve the reaction zone structure, and conduct a linear perturbation analysis for small amplitude of oscillation. Reaction-sheet fluctuations and the net heat release rate are investigated for both the symmetric twin-flame configuration and a single flame residing in a counterflow system consisting of a fresh-mixture stream impinging on a cold inert stream. The Lewis number is found to play an important role in the flame response, especially for flames near extinction. For those flames that exhibit turning point behavior, near extinction the effect of finite-rate chemistry overtakes fluid-dynamic effects such that increasing strain rate can lead to a phase reversal of the heat release response with the imposed flow oscillations. Results of the present study suggest the possibility of a wide spectrum of unsteady flame characteristics depending on Lewis number. Our results also demonstrate that extinction can be delayed when the strain rate oscillates about the static extinction point. Thus the laminar flamelet regime of turbulent combustion may be broader than predicted by steady analyses,
INTRODUCTION The laminar flamelet concept has been considered a viable approach to modelling turbulent combustion with a wide range of applicability [l]. It is therefore of fundamental interest to study the effect of nonuniform flow fields, frequently represented by the strain rate, on the characteristics of laminar flames. While many previous studies have been performed based on quasi-steady strained flames [2, 31, it has recently been recognized that the unsteadiness of the flow field can introduce an additional effect that may significantly modify the flame behavior, as reported in several recent studies [4-81. These unsteady effects may indeed be considerable in many practical turbulent reacting flows with a large Reynolds number, in which there exists a spectrum of eddies whose characteristic turnover time becomes comparable to the transport time of the laminar flamelet. To address such an issue, it is appro-
*Present address: Center for Turbulence Research, Stanford University, Stanford, CA 94305. ’ Present address: Department of Mathematics, New Jersey Institute of Technology, Newark, NJ 07102. ’ Corresponding author. OOlO-2180/96/$15.00 SSDI OOlO-2180(95)00217-0
priate to analyze how laminar-flame behavior is modified by unsteadiness arising from a small-amplitude, monochromatic oscillation of the strain rate with respect to its mean value. One must recognize, however, that the inherent randomness involved in turbulence may significantly complicate the direct application of the present results to the laminar flamelet regime of turbulent combustion. An asymptotic analysis of strained diffusion flames subjected to acoustic pressure perturbations has been performed by Kim and Williams [9], who were motivated by acoustic instability phenomena occurring in rocket engines. As a counterpart, and perhaps more relevant to what occurs in turbulent flows, Im et al. [lo] have studied the response of diffusion flames to oscillating strain rates. In both studies it has been shown that the flame response is greatly amplified when the mean flame is close to its extinction state. In this paper we analyze the laminar counterflow premixed flame subject to small-amplitude oscillating strain rates to complement the previous analysis of diffusion flames [lo]. Recognizing that premixed flames usually respond more readily to flow variations, it is expected that, for example, the effect of Lewis numbers may be more pronounced in the study of preCOMBUSTIONAND FLAME 105: 358-372 (1996) Copyright 0 1996 by The Combustion Institute Published by Elsevier Science Inc.
RESPONSE
OF COUNTERFLOW
PREMIXED
mixed flames. We perform an asymptotic analysis for a one-step irreversible Arrhenius reaction with large activation energy. Attention is focused on flames near extinction, so that the time scale of the unsteadiness is chosen to be comparable to that of diffusive transport [lo]. Thus the outer diffusive-convective layer is modified to include the unsteadiness caused by the oscillating strain rate, while the inner reactive-diffusive layer remains quasi-steady. The effect of finite-rate chemistry will then influence the unsteady flame response through instantaneous matching conditions. Both the symmetric twin-flame configuration and a single flame residing in a counterflow system consisting of a fresh mixture stream opposed to a cold inert are studied to examine Lewis number effects. Our primary interest is to predict the response of the reaction sheet and the burning rate to time-varying strain rates in terms of their magnitude and phase. The coupling between flame response and finite-rate chemistry is an important issue, and it will be demonstrated that the flame response can become very sensitive to flow unsteadiness as the mean flame approaches extinction conditions, consistent with the previous studies of diffusion flames. The present results further indicate that the extinction turning point may be shifted from its static value when unsteadiness is considered. In order to verify this observation, we also perform an asymptotic analysis in the low frequency limit with finite amplitude of the strain rate oscillation, and we indeed confirm that unsteadiness can cause extinction delay. This suggests that the range of the laminar flamelet regime may be larger than expected from steady considerations.
FLAMES
359
P." t
P.”
F+O
Twin Flames
t
F+O
single Flame
Fig. 1. Schematics of the counterflow flames: (a) twin premixed flames and (b) single premixed flame with inert downstream.
streamwise coordinate and the nondimensional time are respectively defined as
where k is a geometry factor such that k = 0 or 1 for a two-dimensional or an axisymmetric configuration, respectively. In Eq. 1, a, is the mean velocity gradient at z = ~0, t the time, p the density? Y the kinematic viscosity, and z, the location of the stagnation plane which is identically zero for the twin flame configuration. The transverse velocity component u is proportional to the transverse coordinate x, and is related to a, and to the nondimensional unsteady stream function f(r], 7) by u = a,xf,, where ( ), = d/dq. The strain rate oscillation is imposed at 7 = m as
FORMULATION As shown schematically in Fig. 1, we consider two different unsteady, counterflow configurations: (a) two identical streams of premixed reactants impinging against one another, supporting adiabatic, symmetric twin premixed flames, and (b) a reactant stream opposed by an inert stream, resulting in a single premixed flame subjected to downstream heat loss. The nondimensional similarity variable for the
where (Y is the amplitude of strain rate fluctuation relative to the mean strain rate and w is the oscillation frequency normalized by 2ka,. We consider T_, = T, and assume constant average molecular weight such that the downstream velocity oscillation (at 71= -co), f?, _ m, is also given by Eq. 2. For unity Prandtl number and constant pi, the conservation equations for momentum,
H. G. IM ET AL.
360 species and temperature fVVS+.!& -c
+ ;[Cl - 2f,,]
I&Y&,, + K, I
become [lo]
+ @P(r) (3)
= 0,
- yi, = Da Y,Y,exp( -E/RT),
large activation energy and conduct a linear perturbation analysis for small amplitude of oscillation, a! K 1. Details of the analysis can be found in Ref. 11, and so we shall give only a brief description in the following. For the imposed strain rate oscillation in Eq. 2, the reaction-sheet location nf is expected to respond as qf = 17f+ aeiw7ijf + O(a’),
i = F,O,
(4)
19,,,,+ ff$ - 0, = -Da Y,Yoexp( -E/RT), (5) where
where the overbar denotes mean (steady> values and the quantity $ is a complex number that represents the fluctuation of the reactionsheet. It follows that any dependent variable 9 is expanded in the form [9] fin;
F(r)
= (f,,,)’
+ 2(f,,m4.
(9)
77f>= an;
17f)
(6) + ae’“‘[2+t7j;?jf)
In the above 8 = (T - T,)/(qT,) is the nondimensional temperature, cP the specific heat, q = Q/cpTm the nondimensional heat release, and Q the heat release per unit mass of fuel. YF is the fuel mass fraction, Y. the oxidizer mass fraction scaled by the stoichiometric oxidizer-to-fuel mass ratio (+, E the activation energy, Da = Bo/2ka, the Damkijhler number, and B the frequency factor. Equations 3-5 are to be solved subject to the boundary conditions fV2 + 2& + F(r),
YF + r,,,,
+ ijf537j;Yf)]
+ 0(a2), (10) where 3~; ~~1 = ag/anf. Here & represents the direct inflyence of the oscillatory strain rate, while GfF represents the indirect effect that arises through oscillation of the reaction sheet. Substituting the above expansions into Eqs. 3-5, and collecting terms of like order in (Y,we obtain the appropriate governing equations for the mean and oscillatory states. In the outer regions, where reaction is negligible, the leading-order conservation equations for the mean state are
yo + r,,,, 8 + 0: 77+ M,
(7)
fW = YFll= Yo,, = eq = 0: 77= 0 for twin flames
f,’ + 2f,,T+ F(T), 8+
e_,:
e,,+fe,
Y, = Yo + 0,
7.r+ - w for single flame.
(8) In addition, we choose the origin of the 17 coordinate to lie at the stagnation plane so that f = 0 at 77= 0.
= 0,
(13)
which are to be solved subject to the boundary conditions & = Y,,,,
A)= I,
y, = y,,,,
i=em:q+~,
(14)
fqll = Y& = Y,, = a7 = 0: q = 0 ANALYSIS FOR SMALL-AMPLITUDE OSCILLATION: a -=K1
To describe the reaction sheet response to the oscillatory strain rate, we exploit the limit of
fV = (1 +
ge-Y2,
e= ixm:y+
-co,
for twin flames
FF = Y, = 0, 1 (15)
RESPONSE
OF COUNTERFLOW
and the jump conditions sheet
[B,+ &[:
= 0,
PREMIXED
across the reaction
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361
In the inner reactive layer, the appropriate stretched variables are defined by
for i = F,O,
Cm
Yd,in = Y& + O( p-9, and
e1-,-=elii;= elqf.
(16)
At the next order in (Y,we obtain the governing equations for the unsteady fluctuations as
(23) where the small parameter of expansion, p-l, is the reciprocal of the Zel’dovich number P = @/[ RT,(l + The parameter
(24)
@I~,)*].
m in Eqs. 22 and 23 is (25)
m = Le,( %lV,-)/(Y&+)Y
(18)
f& + fiTv+ f%$-
iw( e’+ $6)
= 0,
(19)
which is the fractional amount of the total heat release lost to the fuel side. Substituting the inner variables into the governing equations, we obtain LiiiLn’s canonical equation and boundary conditions [12] for the inner structure as
with the boundary conditions L=
I,
(26)
I+($=o:n~oo,
(20) where A is the reduced Damkijhler defined as
fq, = cV = (?q= 0: n = 0 for twin flames (1 + iw)(l &=
(1 + q8_,)“*
+ q&,> A = Da
+ iw ’
* = 15= 0:~ + --co for single flame.
I (21)
The jump conditions for these quantities are identical to Eq. 16 with (-> replaced by (‘1. For large activation energy, the reaction rate term is confined to a very narrow region, the reaction sheet, where a reactive-diffusive balance is maintained. Analysis of this layer using large activation energy asymptotics provides the final condition needed to determine 17f and iif.
%3Y,I,,
number
-P 1
p*(YFllv,+)*exp I 1 + 4%,
’
(27)
which is an eigenvalue of the problem. Equations 26 have been solved numerically [12], and the eigenvalue A has been numerically fitted as a function of m as A = (1 - 1.344m + 0.6307m2)/2.
(28)
Equation 28 provides the final condition needed to determine the relation between strain rate and flame location. In particular, at O((Y) 51f is determined by equating the unsteady parts of Eqs. 27 and 28, which is equivalent to the requirement that the Damkohler
362
H. G. IM ET AL.
number Da does not vary with time [6]. This leads to
interested in two features of the flame response, namely the fluctuations in the reaction-sheet location, ;if, and the net heat release rate. If we define the normalized net heat release rate as the instantaneous fuel consumption rate at the reaction sheet divided by its mean value, i.e., h,,, = YF91~,+/YFVl~~+, then its fluctuating part can be written as
Anet
=
---+
h,
h,
=
-
YF~~fif+/YF&$
(32)
2f&IYj; +_--yF?j [‘ii;
& %II‘iir
4qe'l,
1 + q&/,
- &,. (30)
Here (I) denote the total derivative of a quantity with respect to r.rf at “if, i.e.
A brief summary of the calculation procedure is as follows. First we select a mean reaction-sheet location “ir and solve the steady system Eqs. 11-16. Then using the results of the inner structure analysis, Eqs. 27-28, we find the corresponding Damkijhler number, Da. Next, we proceed to the unsteady system, Eqs. 17-21, together with the additional constraint, Eq. 29, to determine the oscillatory field and ef. By repeating this procedure for different values of ?jf, the flame structure as well as its response to the oscillating strain rate can be calculated as a function of the system Damkohler number. THE REACTION-SHEET
such that if, represents the contribution from fluctuations of the concentration field itself, and A, is due to oscillations of the reaction sheet. In the following we first study the twinflame case, thus eliminating downstream heat loss for simpler interpretation. Then we examine the single-flame with cold downstream, such that the flame is more easily extinguished. To assess the Lewis number effect, we consider Le, = 0.5, 1 and 2 for both configurations, while the Lewis number of the oxidizer is taken to be unity. The Twin-Flame Case: Adiabatic Downstream
Figure 2 shows the steady reaction-sheet response to variations in Da, which is inversely 2.5
.
,
RESPONSE
The calculation procedure outlined in the previous section provides a description of the reaction-sheet response to unsteady strain rates. The parameter values used in the present calculations are E/RT, = 50, q = 100, YF,m = 0.04 and Yo,m = 0.0766. We are mainly
16
Da Fig. 2. Steady reaction-sheet location and temperature as a function of the Damkijhler number for Le., = 1,0.5 and 2; the twin-flame case.
RESPONSE
OF COUNTERFLOW
PREMIXED
proportional to the strain rate, for three different values of Le,. Here the solid and dotted curves denote the reaction sheet location and the normalized temperature, respectively. In agreement with several studies of the steady premixed flame [3], it is seen that for Le, = 0.5 and 1 the reaction sheet monotonically approaches the stagnation plane as the strain rate is increased. In these cases, we may identify extinction as the point at which the two flames merge at the stagnation plane. For the Le, = 2 case, however, due to the temperature decrease with an increase in strain rate, the flame is extinguished before it reaches the stagnation plane, and turning point behavior is observed. Although we refer to this turning point beyond which no steady solution exists as the extinction point, it should be noted that other types of solutions, e.g., solutions describing oscillatory cellular flames, may actually exist beyond this point [13, 141. We shall now analyze the effect of unsteadiness for which these three steady flames represent the mean state. Figure 3 shows the magnitude of the reaction-sheet fluctuation, I’Flfl,as a function of the steady-state Damkijhler number normalized by its extinction value, for various frequencies. We first note that, unlike the diffusion flame response [lo] for which ;if was found to level off as Da -+ 00(i.e., the Burke-Schumann limit), the magnitude of +jf for the premixed flame increases monotonically with Da. This is due to the different stabilization mechanisms of premixed and diffusion flames. That is, the diffusion flame location is constrained by the stoichiometric balance and therefore the mean location “if approaches a constant value as Da increases, whereas the premixed flame always adjusts itself to maintain a dynamic balance in a given flow such that 17f increases monotonically with Da. Consequently, a premixed flame opposed by a smaller mean strain rate (larger Da> will encounter greater relative changes in the normal velocity component when oscillations are introduced, thereby resulting in larger reaction-sheet fluctuations. This also explains the monotonic behavior of I$1 for Le, I 1 in Fig. 3 over the entire range of Da, since a higher strain rate suppresses the reaction-sheet movement as it nears the stagnation plane.
FLAMES
363
Le, =
01 St
“‘8
1
“‘I
1
I
I’.
‘.
2 ..,...,...,...,..~
3
4
Le, = 2
‘:, 1B
011
05
““‘.
1.2
c 1.4
““.
“I,.
1.6
1.8
m
.l
Fig. 3. Magnitude of the reaction-sheet fluctuation as a function of normalized steady Damtihler number for various frequencies for Le, = 1, 0.5, and 2; the twin-flame case.
For Le, = 2 in Fig. 3, however, at low frequencies there is an amplification in the reaction-sheet response as the mean strain rate approaches its extinction value. This behavior reflects the infinite slope of the steady flame response curve at the turning point as shown in Fig. 2. This physically implies that the flame speed variations are more substantial with a given strain rate fluctuation at the near-extinction state. It is further noted from Fig. 3 that, for non-zero frequencies, the value of IGfl remains finite even at the static extinction point. This suggests that the unsteady flame will not be extinguished when the instantaneous Damkijhler number reaches the static extinction value. Rather, the unsteady extinction Damkohler number will be smaller than the corresponding steady value. Furthermore, the fact that the values of I?$ at the static extinction point increases for smaller w indicates
364
that the unsteady extinction Damkijhler number approaches the steady value in the lowfrequency limit. The present linear analysis for small amplitude, however, cannot provide a precise evaluation of this shift of the unsteady extinction point. An analytic evaluation of the shift in extinction point can be made in the low frequency limit by treating w as a small parameer. In the Appendix we perform such an analysis, while allowing for O(1) amplitude of oscillation, thereby giving an explicit description of the extinction shift. This point will be further elaborated in the following section. We next examine the dependence of ]$I on the frequency of oscillation. To demonstrate this clearly, we choose two specific profiles of the plots in Fig. 3 as indicated by the vertical dashed lines. Here case A represents the generic frequency response of the strongly burning flames, and case B the flame near extinction state. The results are plotted in Fig. 4. It is noted that the response of the strongly burning reaction sheet is nonmonotonic, and exhibits a maximum at around w = O(1). This behavior is similar to that observed in previous studies of diffusion flames [lo, 151 and the physical reason is the following. In the quasisteady limit (w .+z l), the reaction sheet responds to the fluctuating flow field instantaneously so that the reaction sheet location can be determined simply by the steady solution for a given value u(r) throughout the entire oscillatory cycle. As w approaches O(1) values, there is a lag bewteen the imposed strain rate and the response of the reaction sheet. That is, with an increase in the strain rate, the reaction sheet experiences an instantaneous rise in the convective flux which leads to an excessive weakening of the reaction relative to the quasi-steady state. Consequently, the reaction-sheet response with some finite time lag is more amplified compared to that with no time lag. As w further increases to very large values, however, the flame eventually stops responding to the oscillating strain rate and it approaches the steady-state limit again. Therefore, the imposed unsteady strain rate is most effective in modifying the flame behavior when its characteristic frequency is comparable to that of transport processes in the flame, which is analogous to resonance phenomena. In the
H. G. IM ET AL. case of the flame near extinction, as shown by case B in Fig. 4, this non-monotonic reactionsheet response is overridden by the finitechemistry effect which is extremely sensitive to the strain-rate fluctuations near the extinction state, thereby exhibiting a monotonic decrease with frequency. The large-w trend observed in Fig. 4 is independent of Lewis number, and the same qualitative behavior is predicted by constant density models [16]. This is not surprising as constant density models have long been known to give correct qualitative descriptions of a large number of flame behaviors. Figure 5 shows the phase lag of ijf with respect to its steady-state response as a function of Da for three different values of Le,. Here we define the phase lag as a negative quantity following conventional phase angle in the complex plane, such that a larger negative value denotes a larger phase lag. Therefore in Fig. 5, the curve-to-curve variation clearly indicates that the phase lag increases with frequency until it reaches -90” in the high frequency limit, as observed in previous studies [17]. The phase lag becomes larger as the mean flame approaches the extinction condition, and this trend is more prominent for the Le, = 2 case. The increase in phase lag for the near-extinction flame is attributed to the longer characteristic chemical time in the reaction zone near extinction, for the same characteristic flow time. We next study the normalized heat release fluctuation in response to the strain rate-oscillation. Figure 6 shows the real parts of h, and
-0.1
1
10
al
Fig. 4. Magnitude of the reaction-sheet fluctuation as a function of normalized frequency for (A) Le, = 1, Da/Da, = 4 and (B) Le, = 2, Da/Da, = 1.05, as indicated in Fig. 3.
RESPONSE
0
I
.
OF COUNTERFLOW
. __----
.
.
_.__--_
____._
_
._._
PREMIXED
FLAMES
_______._._._.__
_
,F 0 -30
-__
__________.--------
co=
-
-0.1
-0.5 z~3.Q. -
_~_______.__-._‘-._._.-.-.-._-__2,0
i,:
_x--.
,
-_-._s.o
_________._
_..__..-___
__..--.
-901~ 0
Le.
‘.
=
8,.
. .
__.__
-
1
I
0.
6 ..
_.
.._... -:-_-_JO,o_
_
8..
2 4 -----------________ .. . . ...T . . .
t______
t’
-
_.”
8DalDa,
..
..
.
.
0.01 0.8.
.
___-______j
OA I
0.1 . . . . . ..a
1
10
0
1 .,
10 I
(0 . ,
II,. . . . ..________._.___.........
0.4 :
IC________._._____._._.-.-._._.-.-.-.-.----
%
%a
,___________.___ _..-.__-__.___._...____-.._
I
Le, = 0.5 -90
2
4 8..
8..
I..6
I,,
8D;I/Da,1
_~____________.___.-_..-......-.. --...-..Le, = 2
-90
”
1
“.
1.2
”
”
1.4
3
0.2 :
k
Of -0.2 L............ _ . . . . . .. . . . . . .. . ...__......... h, -0.4 -
‘~.“~~~‘~“~”
1..
g
3 1.6
”
c
”
1.8
Fig. 5. Phase angle of ;Ir as a function of normalized steady Damkiihler number for various frequencies for Le, = 1,0.5, and 2; the twin-flame case.
&, as a function of frequency, for each Lewis number of fuel considered. In each figure two values of the mean Damkohler number are chosen, specifically one is chosen near extinction (Da/Da, = 1.01, the dotted curves) and the other corresponds to a small strain rate (Da/Da, = 2.5, the solid curves). We first observe that for all Lewis numbers, h, > 0 and A, < 0, except at very high frequencies. The change of sign at high-frequencies is due to the phase delay as will be_ discussed later. The physical implication of h, and A, is the following. As the strain rate increases through the oscillatory cycle, the reactant flux to the reaction sheet is increased, yielding an enhanced heat release rate (positive h,) for a fixed reaction-sheet location. However, since the reaction sheet cannot sustain itself in the higher strain rate environment, it is pushed toward the stagnation plane, resulting in a reduction in the burning rate (negative &,I. Results in Fig. 6 then show that, for Le, = 1, i,, and h,
-0.6 0.01 2,
_$I
0.1 I
-
..:‘,,.
-I
[...._.__._... !!y;:::...‘;5 -6 ____._.._..... --...0.01
0.1
Le, = 2 1
1
1
10
m
Fig. 6. Real part of the normalized heat release fluctuations as a function of frequency for the near-extinction and small strain rate cases for Le, = 1, 0.5, and 2; the twinflame case.
exactly cancel out such that there is no net heat release fluctuation for both the near-extinction and small strain rate cases. This is due to the fact that, for unity Lewis number and in the absence of downstream heat loss, the flame structure with quasi-steady reaction zone resembles that of the unrestrained, freely propagating premixed flame with constant, adiabatic flame temperature. Since the reacitivity of such a flame is unaffected by strain rate variations, the reaction sheet simply adjusts to the timevarying strain rate instantaneously such that the net reactant consumption rate remains unchanged. For nonunity Lewis numbers, however, preferential diffusion introduces an additional influence to the flame behavior, and thus the two quantities do not exactly cancel. Specifically, for Le, < l,_it is seen that lh,l is somewhat larger than Ih,l owing to the fact that the flame is not displaced as much by an increase in strain rate because of the corre-
366
H. G. IM ET AL.
sponding increase in the flame temperature and hence the flame speed. Consequently, the net heat release fluctuation is in in phase (positive real) with the strain rate oscillation. The opposite effect is observed for Le, = 2 where the decrease in the flame temperature leads to an amplified reaction sheet response, thereby yielding negative real values of h,,, (out of phase). The net heat release rate fluctuation is seen to be quite small for Le, = 0.5, and in the small strain rate case for Le, = 2. For the flame near the_ extinction turning point (Le, = 21, however, h, overwhelms the net heat release behavior due to its sensitive response to the strain rate oscillation as shown in Fig. 3, t_hereby resulting in a significant amount of h netThe
magnitude and phase lag of &,,t as a function of the normalized Damkijhler number are shown in Figs. 7 and 8 respectively, for both Le, = 0.5 and 2. From Fig. 7 it is seen that significant amplification occurs near the extinction turning point when Le, = 2, while an almost uniform behavior is observed when Le, = 0.5, consistent with the reaction-sheet behaviors shown in Fig. 2. Figure 8 further shows that h,,, for Le, = 0.5 is nearly in phase (0”) while for Le, = 2 it is almost out of phase (180”) in the low-frequency limit. This suggests a qualitatively different response of the pre-
0.3
_ _ _ ~-~-~-~----~-~___._._._._._._~ _._._.-.
.-.-__
‘-._.___
IiiJ 0.1
‘..t_._
0.2 1 0
I
-.
.._.._
~-.-~-.-.-._._._____._.-._._._ _________
. ..-...-..
--...-...
1
Le, = 0.5 .I...&..._...
2
3
1
4
6 Le,
I : -\ : . \‘,..
I 2
.I
6w u)=
.._ 0.1 . - - -03 . --_;; -_-._s:o -..--10.0
0
Fig. 7. Magnitude of the normalized net heat release fluctuation as a function of normalized steady Damkijhler number for various frequencies for Le, = 0.5 and 2, the twin-flame case.
2
4
6
150
Ff #
120
I
90
f
____..-.
-
.__...-.
L+=2 60 1.2
1.4
-
-0.5
I-I ::j _._._5.0 _..._,oo
___...-.
1
.._0.1
__-...-.
,..*
_I
1.6
1.8
Fig. 8. Phase angle of net heat release fluctuation as a function of normalized steady Damkiihler number for various frequencies for Le, = 0.5 and 2; the twin-flame case.
mixed flame for different Lewis numbers from the standpoint of combustion instabilities. The phase of h,,, response is seen to shift downward as the frequency increases, indicating further delay in the response at higher frqeuencies. We further note that, whereas the phase angle remains nearly constant when Le, = 0.5, for Le, = 2 the phase angle becomes smaller as the steady extinction point is approached. That is, the near-extinction flame exhibits a longer phase delay due to the larger fluctuation in the reaction-sheet response, consistent with the result shown in Fig. 5. As a consequence of this phase delay, for Le, = 2 the phase angle of h,,, decreases below 90”, i.e., h,,, takes on positive real values, near extinction at high frequencies (w = 5 and 10 in the bottom part of Fig. 8). This phase reversal with frequency is often observed in acoustic instability problems [9]. Although it is difficult to readily assess the present velocity response in relation to acoustic instability, these results nevertheless suggest the possibility of qualitative changes in the response of near-extinction flames at high frequencies. The Single-Flame Case: Nonadiabatic Downstream
To further substantiate the effect of Lewis number on unsteady premixed flame behavior,
RESPONSE
OF COUNTERFLOW
PREMIXED
we consider the case in which the reactant stream is opposed to an inert stream at the same temperature (T, = T-,1. In this configuration, the flame response exhibits a turning point for all Lewis numbers. Figure 9 shows the basic steady flame response for three different Lewis numbers. It is seen that the flame temperature increases (decreases) with a decrease in Da for Le, < 1 (> 11, while it remains nearly constant for Le, = 1 until the curve starts to turn around. An interesting observation from Fig. 9 is that the values of the extinction Damkohler number do not vary monotonically with the Lewis number. This is due to the additional effect of the heat loss to the cold downstream. That is, for the same boundary conditions, the flame with a larger Lewis number has a higher flame speed and is therefore situated further upstream (larger 17f). Consequently, the downstream heat loss is reduced and the flame can be sustained at higher strain rates. This effect of reduced heat loss for the Le, = 2 case overrides the reduction in the flame temperature, thereby resulting in a smaller extinction Damkohler number as compared to the Le, = 0.5 flame. Based on these steady flame behaviors, in Fig. 10 we plot the magnitude of ijf, i.e., the reaction-sheet fluctuation relative to the mean location, as a function of Da/Da,. Here we first note that while there were significant dif3.5 5.2
2.5
Fig. 9. Steady reaction-sheet location and temperature as a function of the DamkGhler number for Le, = 1, 0.5, and 2; the single-flame case.
367
FLAMES
h.,
8..
8,.
1..
J
0..
Q)= . . . . . . 0.1 -03
l.+=l
r-I-. ;:: -_-._s,o -...10.0
________._.-.---.-~ _____.__..__
1
.
.
.
1.2 . .
. ..-...
.
.
--..
IA .
. Le,
.
.
.
.
I.2 . .
.
.
1.4 .
1.6 . .
.
.
1.8 . ..““B
= 0.5
ot’..‘..‘,“““‘,“.i 1
__..__..
_____.._-...
..,
1.6
1.6 ,
1.6
1.8
%
Le, = 1
ot.“,“‘,“““‘,‘.‘1 1.2
1
1.4
Fig. 10. Magnitude of the reaction-sheet fluctuation as a function of normalized steady DamkChler number for various frequencies for Le, = 1, 0.5, and 2; the single-flame case.
ferences for various Lewis numbers in the twin-flame case, the three plots in Fig. 10 for the single flame behave very similarly in that the reaction-sheet fluctuation is greatly amplified near the static extinction point due to the sensitivity of the flame near extinction. This amplification of the reaction sheet response is again found to be large at low frequencies (e.g., for w = 0.11, while at higher frequencies the value of 1+&lis quite modest, and the magnitude of ;if remains finite even at the extinction point. This allows for the following interpretation. In the present linear perturbation analysis, the time-varying Damkijhler number of the system can be expressed as Da(T)/Da, = u,/u(T)
= (a,/~,>(1
- aeior),
(33)
368
H. G. IM ET AL.
where we have expanded for (Y< 1. The corresponding reaction-sheet response, ~~(71, is given by Eq. 9 such that d+) d[Da(r)/Da,l
Le, = 1
d+)
= d[a,/a(r)l =-
drlf(r) dr
= --
dr d[a,/a(r)l
6f
as/a0
(34) .
Therefore, if we plot the unsteady reactionsheet ~~(7) as a function of Da(r) similar to I will indicate the absoFig. 9, the quantity 1535 lute value of the slope of the unsteady response curve. For example, the local slope variation along the quasi-steady response curves in Fig. 9 can be approximately represented by the dotted curves in Fig. 10, whose blowup at the extinction point (Da/Da, = 1) indicates the infinite slope at the turning point in Fig. 9. Therefore, as in Fig. 3, the finite value of 151 at the static extinction point for nonzero frequencies implies that the unsteady flame will not be extinguished when the instantaneous Damkohler number reaches the static extinction value. Rather, since the slope of the curve is smaller than that of the steady situation, the unsteady extinction Damkohler number will be shifted to a smaller value. This physically implies that flames become more resistant to extinction as the strain rate fluctuation is applied within a shorter duration. Figure 11 shows the real part of the heat release fluctuation as a function of frequency for the single flame case. Both near-extinction (Da/Da, = 1.01, the dotted curves) and small strain rate (Da/Da, = 2.5, the solid curves) cases are shown. Similar to the twin flame case, it is found that the real parts of h,,, for the flame with small strain rate is relatively small due to the cancelling of h, and 6, regardless of the Lewis number, w_hileh, is larger (smaller) in magnitude than h, for Le, > 1 (Le, < 1). The behavior of the Le, = 1 case is qualitatively similar to that of the Le, = 2 case because the presence of downstream heat loss causes the flame temperature to decrease with an increase in strain rate. For the near-extinc-
~~~
Y:::::,,.“.
..._______....__ L _______........ -3 4
__..L ___________...............~~~~~~~ ;;I'
0.01
0.1
1
10
co
Fig. 11. Real part of the normalized heat release fluctuations as a function of frequency for the near-extinction and small strain rate cases for Le, = 1, 0.5, and 2; the singleflame case.
tion flames, &, dominates the net heat release response due to the sensitive reaction sheet response. Figure 12 shows the phase of &,,, for the three Lewis numbers considered. Although the behavior of the three plots in Figs. 10 and 11 appear to be similar, the phase responses are seen to be quite different for each Lewis number. Consider first the Le, 2 1 case. The behavior is similar to that of the twin flame situation for Le, = 2 (Fig. 81, in that the heat release is mostly out of phase with the imposed strain rate, and undergoes a phase reversal at high frequency, although it is more confined near the extinction state for higher Lewis number. For Le, < 1, however, the phase response is significantly different between the flame near extinction and the flame opposed by small strain rate, especially at low frequencies. As shown in Fig. 11, in the Le, < 1 case h, over-
RESPONSE
OF COUNTERFLOW
PREMIXED
s
369
of stability characteristics depending on Lewis number, especially as extinction conditions are approached.
150
+
FLAMES
ml
I,
CONCLUSIONS
30 180
1
1.2 . . . . .., ,___...--- .. . ..___._..
‘.
...
-60 1
a
; go
1
.
1.4 , .
..---.-...-
.
1.6 . .
Le,
= 0.5
.._...
-..
’
8
1.2
1.4
/’ :’
Le, 1.2
.
1.4
.
.
1.8 .
DalDrg a=
‘-,---c:‘---i----r
1.6
1.8
1.6
1.8
DrJDis
= 2 DalDa,
Fig. 12. Phase angle of net heat release fluctuation as a function of normalized steady Damkiihler number for various frequencies for Le, = 1, 0.5, and 2; the single-flame case.
rides A, for small strain rates, and therefore h,,, is positive. Near the static extinction state, however, &, becomes dominant due to the sensitive reaction-sheet response, resulting in large negative values of h,,,. Consequently, h,,, for the flame with Le, = 0.5 experiences a reversal of phase response from in phase (< 90”) to out of phase (> 90”) as the steady strain rate increases. At higher frequencies, this jump in the phase lag becomes less prominent as the reaction-sheet response is reduced, and the near-extinction phase reversal does not occur. In summary of Fig. 12, in the presence of downstream heat loss, the heat release fluctuation for the Le, 2 1 (I 1) flame is mostly out of phase (in phase) with the imposed strain rate fluctuation, while near-extinction reversal of phase occurs at high (lowj frequencies. From the standpoint of acoustic instabilities, this result suggests the possibility of a wide spectrum
In this paper, we have employed large activation energy asymptotics to study the response of counterflow premixed flames to imposed oscillating strain rates. In particular, we have investigated the reaction-sheet fluctuations and the net heat release rate for two different flow configurations, namely symmetric twin flames and a single flame residing in a mixture stream impinging on a cold inert stream. As expected, the reaction-sheet response was found to be more sensitive to unsteadiness when the flames were near the extinction (turning) point. The Lewis number was found to play a major role in the response characteristics, particularly for the twin-flame configuration. In the twin-flame case, for Le, 5 1 both the mean location and the reaction-sheet fluctuation decrease monotonically with strain rate, while for Le, sufficiently greater than unity the mean location displays turning point behavior and the response of the fluctuation is amplified as the mean-state extinction condition is approached, most notably at low frequencies. For all Lewis numbers, the phase lag of the reaction-sheet fluctuation becomes larger as the static extinction condition is approached, and again this trend was found to be most prominent for Le, > 1. Furthermore, the near-extinction phase lag of the net heat release rate was found to be in phase with the imposed oscillation for Le, < 1, while it is out of phase for Le, > 1. This suggests that acoustic instability characteristics may be influenced significantly by Lewis number. The cold-downstream configuration supports a single flame which always displays turning point behavior due to downstream heat loss. Consequently, whereas flames with Lewis numbers greater than unity exhibit a similar response as their twin-flame counterparts, for Lewis numbers less than unity, the flame in this configuration is much more sensitive to imposed unsteadiness than the twin flames. In particular, its heat-release rate may undergo a phase reversal from in-phase with the imposed
H. G. IM ET AL.
370 oscillation to out-of-phase as the strain rate is increased toward the extinction value. Consequently, these flames may exhibit a reversal of acoustic characteristics as the flame nears extinction. We have also performed an asymptotic analysis in the limit of low frequency and have demonstrated that extinction can be delayed when the strain rate oscillates about the static extinction point. From the standpoint of the laminar flamelet concept of turbulent combustion, this result suggests that the laminar flamelet regime may be broader than the range predicted from steady considerations. This work was supported by the United States Air Force OfJice of Scientific Research. The authors would like to thank Dr. J. S. Kim of University of California, San Diego for helpful discussions.
REFERENCES 1. Peters, N., Twenty-First Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1987, pp. 1231-1250. 2. Libby, P. A., and Williams, F. A., Combust. Flame 44:287-303 (1982). 3. Buckmaster, J. D., Seventeenth Symposium (Zntemational) on Combustion, The Combustion Institute, Pittsburgh, 1979, pp. 835-842. 4. Haworth, D. C., Drake, M. C., Pope, S. B., and Blint, R. J., Twenty-Second Symposium (International1 on Combustion, The Combustion Institute, Pittsburgh, 1988, pp. 589-597. Stahl, G., and Wamatz, J., Combust. Flame 85: 285-299 (1991). Darabiha, N., Combust. Sci. Technol. 86:163-181 (1992). Ghoniem, A. F., Soteriou, M. C., and Knio, 0. M.: Twenty-Fourth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1992, pp. 223-230. 8. Egolfopolous, F. N., and Campbell, C. S., Twenty-Fifth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1994, in press. 9. Kim, J. S., and Williams, F. A., Cornbust. Flame 98~279-299 (1994). 10. Im, H. G., Law, C. K., Kim, J. S., and Williams, F. A., Co.mbust. Flame, 100:21-30 (1995). 11. Im, H. G., Ph.D. thesis, Princeton University, Princeton, New Jersey, November 1994. 12. Liii&n, A., Acta Astronaut. 1:1007-1039 (1974). 13. Joulin, G., and Sivashinsky, G. I., Cornbust. Sci. Technol. 31:75-90 (1983). 14. Kaper, H. G., Leaf, G. K., Matkowsky, B. J., and
15.
16. 17. 18. 19.
Olmstead, W. E., S&‘t4J. Appl. Math. 48:1054-1063 (1988). Strahle, W. C., Tenth Symposium Untemational) on Combustion, The Combustion Institute, Pittsburgh, 1965, pp. 1315-1325. Joulin, G., Cornbust. Sci. Technol. 97:219-229 (1994). Saitoh, T. and Otsuka, Y., Cornbust. Sci. Technol. 12:135-146 (1976). Kapila, A. K., S&VJ. Appl. Math. 41:29-42 (1981). Matkowsky, B. J., and Sivashinsb, G. I., SZAM J. Appl. Math. 37:686-699 (1979).
Received 20 March 1995; revised 18 September 1995
APPENDIX: ANALYSIS FOR LOW FREQUENCY (o -c 11, O(1) AMPLITUDE OSCILLATION The results of the main body of the text suggest that imposed unsteady oscillations will result in smaller Damkohler numbers (larger strain rates) at extinction, as compared to the steady values, thus resulting in an extinction delay. Furthermore, Figs. 3 and 10 indicate that this delay is shortened at low frequencies. However, the linearized analysis of the section on the reaction-sheet response for small amplitude of oscillation is unable to evaluate this shift in extinction conditions. Thus we consider here the low-frequency limit (w +Z l>, while allowing for O(1) amplitude, and we explicitly determine the extinction delay. Our approach is similar to the ignition delay problem of Kapila [18], which can be consulted for additional details. We consider only the twin-flame configuration sketched in Fig. 1. Furthermore, our aim is to give a qualitative description of this delay and for this purpose it is appropriate to assume weak thermal expansion and employ the diffusional-thermal model [19] of flame propagation. In this model, the equations for the transport properties decouple from those of fluid dynamics and thus the stream function is given by Eq. 2 for all n. The remaining equations and jump conditions can be written as 1191 t$, + y7& - we, = 0,
77> 7fP
(Al)
H,,,, + yr/H,, - OH, = -le,,,,
(A3
ti= 1,
W)
@Q
We study the response of premixed counterflow flames to an imposed oscillating strain rate which has potential application to both turbulent combustion and acoustic instabilities. We exploit the limit of large activation energy to resolve the reaction zone structure, and conduct a linear perturbation analysis for small amplitude of oscillation. Reaction-sheet fluctuations and the net heat release rate are investigated for both the symmetric twin-flame configuration and a single flame residing in a counterflow system consisting of a fresh-mixture stream impinging on a cold inert stream. The Lewis number is found to play an important role in the flame response, especially for flames near extinction. For those flames that exhibit turning point behavior, near extinction the effect of finite-rate chemistry overtakes fluid-dynamic effects such that increasing strain rate can lead to a phase reversal of the heat release response with the imposed flow oscillations. Results of the present study suggest the possibility of a wide spectrum of unsteady flame characteristics depending on Lewis number. Our results also demonstrate that extinction can be delayed when the strain rate oscillates about the static extinction point. Thus the laminar flamelet regime of turbulent combustion may be broader than predicted by steady analyses,
INTRODUCTION The laminar flamelet concept has been considered a viable approach to modelling turbulent combustion with a wide range of applicability [l]. It is therefore of fundamental interest to study the effect of nonuniform flow fields, frequently represented by the strain rate, on the characteristics of laminar flames. While many previous studies have been performed based on quasi-steady strained flames [2, 31, it has recently been recognized that the unsteadiness of the flow field can introduce an additional effect that may significantly modify the flame behavior, as reported in several recent studies [4-81. These unsteady effects may indeed be considerable in many practical turbulent reacting flows with a large Reynolds number, in which there exists a spectrum of eddies whose characteristic turnover time becomes comparable to the transport time of the laminar flamelet. To address such an issue, it is appro-
*Present address: Center for Turbulence Research, Stanford University, Stanford, CA 94305. ’ Present address: Department of Mathematics, New Jersey Institute of Technology, Newark, NJ 07102. ’ Corresponding author. OOlO-2180/96/$15.00 SSDI OOlO-2180(95)00217-0
priate to analyze how laminar-flame behavior is modified by unsteadiness arising from a small-amplitude, monochromatic oscillation of the strain rate with respect to its mean value. One must recognize, however, that the inherent randomness involved in turbulence may significantly complicate the direct application of the present results to the laminar flamelet regime of turbulent combustion. An asymptotic analysis of strained diffusion flames subjected to acoustic pressure perturbations has been performed by Kim and Williams [9], who were motivated by acoustic instability phenomena occurring in rocket engines. As a counterpart, and perhaps more relevant to what occurs in turbulent flows, Im et al. [lo] have studied the response of diffusion flames to oscillating strain rates. In both studies it has been shown that the flame response is greatly amplified when the mean flame is close to its extinction state. In this paper we analyze the laminar counterflow premixed flame subject to small-amplitude oscillating strain rates to complement the previous analysis of diffusion flames [lo]. Recognizing that premixed flames usually respond more readily to flow variations, it is expected that, for example, the effect of Lewis numbers may be more pronounced in the study of preCOMBUSTIONAND FLAME 105: 358-372 (1996) Copyright 0 1996 by The Combustion Institute Published by Elsevier Science Inc.
RESPONSE
OF COUNTERFLOW
PREMIXED
mixed flames. We perform an asymptotic analysis for a one-step irreversible Arrhenius reaction with large activation energy. Attention is focused on flames near extinction, so that the time scale of the unsteadiness is chosen to be comparable to that of diffusive transport [lo]. Thus the outer diffusive-convective layer is modified to include the unsteadiness caused by the oscillating strain rate, while the inner reactive-diffusive layer remains quasi-steady. The effect of finite-rate chemistry will then influence the unsteady flame response through instantaneous matching conditions. Both the symmetric twin-flame configuration and a single flame residing in a counterflow system consisting of a fresh mixture stream opposed to a cold inert are studied to examine Lewis number effects. Our primary interest is to predict the response of the reaction sheet and the burning rate to time-varying strain rates in terms of their magnitude and phase. The coupling between flame response and finite-rate chemistry is an important issue, and it will be demonstrated that the flame response can become very sensitive to flow unsteadiness as the mean flame approaches extinction conditions, consistent with the previous studies of diffusion flames. The present results further indicate that the extinction turning point may be shifted from its static value when unsteadiness is considered. In order to verify this observation, we also perform an asymptotic analysis in the low frequency limit with finite amplitude of the strain rate oscillation, and we indeed confirm that unsteadiness can cause extinction delay. This suggests that the range of the laminar flamelet regime may be larger than expected from steady considerations.
FLAMES
359
P." t
P.”
F+O
Twin Flames
t
F+O
single Flame
Fig. 1. Schematics of the counterflow flames: (a) twin premixed flames and (b) single premixed flame with inert downstream.
streamwise coordinate and the nondimensional time are respectively defined as
where k is a geometry factor such that k = 0 or 1 for a two-dimensional or an axisymmetric configuration, respectively. In Eq. 1, a, is the mean velocity gradient at z = ~0, t the time, p the density? Y the kinematic viscosity, and z, the location of the stagnation plane which is identically zero for the twin flame configuration. The transverse velocity component u is proportional to the transverse coordinate x, and is related to a, and to the nondimensional unsteady stream function f(r], 7) by u = a,xf,, where ( ), = d/dq. The strain rate oscillation is imposed at 7 = m as
FORMULATION As shown schematically in Fig. 1, we consider two different unsteady, counterflow configurations: (a) two identical streams of premixed reactants impinging against one another, supporting adiabatic, symmetric twin premixed flames, and (b) a reactant stream opposed by an inert stream, resulting in a single premixed flame subjected to downstream heat loss. The nondimensional similarity variable for the
where (Y is the amplitude of strain rate fluctuation relative to the mean strain rate and w is the oscillation frequency normalized by 2ka,. We consider T_, = T, and assume constant average molecular weight such that the downstream velocity oscillation (at 71= -co), f?, _ m, is also given by Eq. 2. For unity Prandtl number and constant pi, the conservation equations for momentum,
H. G. IM ET AL.
360 species and temperature fVVS+.!& -c
+ ;[Cl - 2f,,]
I&Y&,, + K, I
become [lo]
+ @P(r) (3)
= 0,
- yi, = Da Y,Y,exp( -E/RT),
large activation energy and conduct a linear perturbation analysis for small amplitude of oscillation, a! K 1. Details of the analysis can be found in Ref. 11, and so we shall give only a brief description in the following. For the imposed strain rate oscillation in Eq. 2, the reaction-sheet location nf is expected to respond as qf = 17f+ aeiw7ijf + O(a’),
i = F,O,
(4)
19,,,,+ ff$ - 0, = -Da Y,Yoexp( -E/RT), (5) where
where the overbar denotes mean (steady> values and the quantity $ is a complex number that represents the fluctuation of the reactionsheet. It follows that any dependent variable 9 is expanded in the form [9] fin;
F(r)
= (f,,,)’
+ 2(f,,m4.
(9)
77f>= an;
17f)
(6) + ae’“‘[2+t7j;?jf)
In the above 8 = (T - T,)/(qT,) is the nondimensional temperature, cP the specific heat, q = Q/cpTm the nondimensional heat release, and Q the heat release per unit mass of fuel. YF is the fuel mass fraction, Y. the oxidizer mass fraction scaled by the stoichiometric oxidizer-to-fuel mass ratio (+, E the activation energy, Da = Bo/2ka, the Damkijhler number, and B the frequency factor. Equations 3-5 are to be solved subject to the boundary conditions fV2 + 2& + F(r),
YF + r,,,,
+ ijf537j;Yf)]
+ 0(a2), (10) where 3~; ~~1 = ag/anf. Here & represents the direct inflyence of the oscillatory strain rate, while GfF represents the indirect effect that arises through oscillation of the reaction sheet. Substituting the above expansions into Eqs. 3-5, and collecting terms of like order in (Y,we obtain the appropriate governing equations for the mean and oscillatory states. In the outer regions, where reaction is negligible, the leading-order conservation equations for the mean state are
yo + r,,,, 8 + 0: 77+ M,
(7)
fW = YFll= Yo,, = eq = 0: 77= 0 for twin flames
f,’ + 2f,,T+ F(T), 8+
e_,:
e,,+fe,
Y, = Yo + 0,
7.r+ - w for single flame.
(8) In addition, we choose the origin of the 17 coordinate to lie at the stagnation plane so that f = 0 at 77= 0.
= 0,
(13)
which are to be solved subject to the boundary conditions & = Y,,,,
A)= I,
y, = y,,,,
i=em:q+~,
(14)
fqll = Y& = Y,, = a7 = 0: q = 0 ANALYSIS FOR SMALL-AMPLITUDE OSCILLATION: a -=K1
To describe the reaction sheet response to the oscillatory strain rate, we exploit the limit of
fV = (1 +
ge-Y2,
e= ixm:y+
-co,
for twin flames
FF = Y, = 0, 1 (15)
RESPONSE
OF COUNTERFLOW
and the jump conditions sheet
[B,+ &[:
= 0,
PREMIXED
across the reaction
FLAMES
361
In the inner reactive layer, the appropriate stretched variables are defined by
for i = F,O,
Cm
Yd,in = Y& + O( p-9, and
e1-,-=elii;= elqf.
(16)
At the next order in (Y,we obtain the governing equations for the unsteady fluctuations as
(23) where the small parameter of expansion, p-l, is the reciprocal of the Zel’dovich number P = @/[ RT,(l + The parameter
(24)
@I~,)*].
m in Eqs. 22 and 23 is (25)
m = Le,( %lV,-)/(Y&+)Y
(18)
f& + fiTv+ f%$-
iw( e’+ $6)
= 0,
(19)
which is the fractional amount of the total heat release lost to the fuel side. Substituting the inner variables into the governing equations, we obtain LiiiLn’s canonical equation and boundary conditions [12] for the inner structure as
with the boundary conditions L=
I,
(26)
I+($=o:n~oo,
(20) where A is the reduced Damkijhler defined as
fq, = cV = (?q= 0: n = 0 for twin flames (1 + iw)(l &=
(1 + q8_,)“*
+ q&,> A = Da
+ iw ’
* = 15= 0:~ + --co for single flame.
I (21)
The jump conditions for these quantities are identical to Eq. 16 with (-> replaced by (‘1. For large activation energy, the reaction rate term is confined to a very narrow region, the reaction sheet, where a reactive-diffusive balance is maintained. Analysis of this layer using large activation energy asymptotics provides the final condition needed to determine 17f and iif.
%3Y,I,,
number
-P 1
p*(YFllv,+)*exp I 1 + 4%,
’
(27)
which is an eigenvalue of the problem. Equations 26 have been solved numerically [12], and the eigenvalue A has been numerically fitted as a function of m as A = (1 - 1.344m + 0.6307m2)/2.
(28)
Equation 28 provides the final condition needed to determine the relation between strain rate and flame location. In particular, at O((Y) 51f is determined by equating the unsteady parts of Eqs. 27 and 28, which is equivalent to the requirement that the Damkohler
362
H. G. IM ET AL.
number Da does not vary with time [6]. This leads to
interested in two features of the flame response, namely the fluctuations in the reaction-sheet location, ;if, and the net heat release rate. If we define the normalized net heat release rate as the instantaneous fuel consumption rate at the reaction sheet divided by its mean value, i.e., h,,, = YF91~,+/YFVl~~+, then its fluctuating part can be written as
Anet
=
---+
h,
h,
=
-
YF~~fif+/YF&$
(32)
2f&IYj; +_--yF?j [‘ii;
& %II‘iir
4qe'l,
1 + q&/,
- &,. (30)
Here (I) denote the total derivative of a quantity with respect to r.rf at “if, i.e.
A brief summary of the calculation procedure is as follows. First we select a mean reaction-sheet location “ir and solve the steady system Eqs. 11-16. Then using the results of the inner structure analysis, Eqs. 27-28, we find the corresponding Damkijhler number, Da. Next, we proceed to the unsteady system, Eqs. 17-21, together with the additional constraint, Eq. 29, to determine the oscillatory field and ef. By repeating this procedure for different values of ?jf, the flame structure as well as its response to the oscillating strain rate can be calculated as a function of the system Damkohler number. THE REACTION-SHEET
such that if, represents the contribution from fluctuations of the concentration field itself, and A, is due to oscillations of the reaction sheet. In the following we first study the twinflame case, thus eliminating downstream heat loss for simpler interpretation. Then we examine the single-flame with cold downstream, such that the flame is more easily extinguished. To assess the Lewis number effect, we consider Le, = 0.5, 1 and 2 for both configurations, while the Lewis number of the oxidizer is taken to be unity. The Twin-Flame Case: Adiabatic Downstream
Figure 2 shows the steady reaction-sheet response to variations in Da, which is inversely 2.5
.
,
RESPONSE
The calculation procedure outlined in the previous section provides a description of the reaction-sheet response to unsteady strain rates. The parameter values used in the present calculations are E/RT, = 50, q = 100, YF,m = 0.04 and Yo,m = 0.0766. We are mainly
16
Da Fig. 2. Steady reaction-sheet location and temperature as a function of the Damkijhler number for Le., = 1,0.5 and 2; the twin-flame case.
RESPONSE
OF COUNTERFLOW
PREMIXED
proportional to the strain rate, for three different values of Le,. Here the solid and dotted curves denote the reaction sheet location and the normalized temperature, respectively. In agreement with several studies of the steady premixed flame [3], it is seen that for Le, = 0.5 and 1 the reaction sheet monotonically approaches the stagnation plane as the strain rate is increased. In these cases, we may identify extinction as the point at which the two flames merge at the stagnation plane. For the Le, = 2 case, however, due to the temperature decrease with an increase in strain rate, the flame is extinguished before it reaches the stagnation plane, and turning point behavior is observed. Although we refer to this turning point beyond which no steady solution exists as the extinction point, it should be noted that other types of solutions, e.g., solutions describing oscillatory cellular flames, may actually exist beyond this point [13, 141. We shall now analyze the effect of unsteadiness for which these three steady flames represent the mean state. Figure 3 shows the magnitude of the reaction-sheet fluctuation, I’Flfl,as a function of the steady-state Damkijhler number normalized by its extinction value, for various frequencies. We first note that, unlike the diffusion flame response [lo] for which ;if was found to level off as Da -+ 00(i.e., the Burke-Schumann limit), the magnitude of +jf for the premixed flame increases monotonically with Da. This is due to the different stabilization mechanisms of premixed and diffusion flames. That is, the diffusion flame location is constrained by the stoichiometric balance and therefore the mean location “if approaches a constant value as Da increases, whereas the premixed flame always adjusts itself to maintain a dynamic balance in a given flow such that 17f increases monotonically with Da. Consequently, a premixed flame opposed by a smaller mean strain rate (larger Da> will encounter greater relative changes in the normal velocity component when oscillations are introduced, thereby resulting in larger reaction-sheet fluctuations. This also explains the monotonic behavior of I$1 for Le, I 1 in Fig. 3 over the entire range of Da, since a higher strain rate suppresses the reaction-sheet movement as it nears the stagnation plane.
FLAMES
363
Le, =
01 St
“‘8
1
“‘I
1
I
I’.
‘.
2 ..,...,...,...,..~
3
4
Le, = 2
‘:, 1B
011
05
““‘.
1.2
c 1.4
““.
“I,.
1.6
1.8
m
.l
Fig. 3. Magnitude of the reaction-sheet fluctuation as a function of normalized steady Damtihler number for various frequencies for Le, = 1, 0.5, and 2; the twin-flame case.
For Le, = 2 in Fig. 3, however, at low frequencies there is an amplification in the reaction-sheet response as the mean strain rate approaches its extinction value. This behavior reflects the infinite slope of the steady flame response curve at the turning point as shown in Fig. 2. This physically implies that the flame speed variations are more substantial with a given strain rate fluctuation at the near-extinction state. It is further noted from Fig. 3 that, for non-zero frequencies, the value of IGfl remains finite even at the static extinction point. This suggests that the unsteady flame will not be extinguished when the instantaneous Damkijhler number reaches the static extinction value. Rather, the unsteady extinction Damkohler number will be smaller than the corresponding steady value. Furthermore, the fact that the values of I?$ at the static extinction point increases for smaller w indicates
364
that the unsteady extinction Damkijhler number approaches the steady value in the lowfrequency limit. The present linear analysis for small amplitude, however, cannot provide a precise evaluation of this shift of the unsteady extinction point. An analytic evaluation of the shift in extinction point can be made in the low frequency limit by treating w as a small parameer. In the Appendix we perform such an analysis, while allowing for O(1) amplitude of oscillation, thereby giving an explicit description of the extinction shift. This point will be further elaborated in the following section. We next examine the dependence of ]$I on the frequency of oscillation. To demonstrate this clearly, we choose two specific profiles of the plots in Fig. 3 as indicated by the vertical dashed lines. Here case A represents the generic frequency response of the strongly burning flames, and case B the flame near extinction state. The results are plotted in Fig. 4. It is noted that the response of the strongly burning reaction sheet is nonmonotonic, and exhibits a maximum at around w = O(1). This behavior is similar to that observed in previous studies of diffusion flames [lo, 151 and the physical reason is the following. In the quasisteady limit (w .+z l), the reaction sheet responds to the fluctuating flow field instantaneously so that the reaction sheet location can be determined simply by the steady solution for a given value u(r) throughout the entire oscillatory cycle. As w approaches O(1) values, there is a lag bewteen the imposed strain rate and the response of the reaction sheet. That is, with an increase in the strain rate, the reaction sheet experiences an instantaneous rise in the convective flux which leads to an excessive weakening of the reaction relative to the quasi-steady state. Consequently, the reaction-sheet response with some finite time lag is more amplified compared to that with no time lag. As w further increases to very large values, however, the flame eventually stops responding to the oscillating strain rate and it approaches the steady-state limit again. Therefore, the imposed unsteady strain rate is most effective in modifying the flame behavior when its characteristic frequency is comparable to that of transport processes in the flame, which is analogous to resonance phenomena. In the
H. G. IM ET AL. case of the flame near extinction, as shown by case B in Fig. 4, this non-monotonic reactionsheet response is overridden by the finitechemistry effect which is extremely sensitive to the strain-rate fluctuations near the extinction state, thereby exhibiting a monotonic decrease with frequency. The large-w trend observed in Fig. 4 is independent of Lewis number, and the same qualitative behavior is predicted by constant density models [16]. This is not surprising as constant density models have long been known to give correct qualitative descriptions of a large number of flame behaviors. Figure 5 shows the phase lag of ijf with respect to its steady-state response as a function of Da for three different values of Le,. Here we define the phase lag as a negative quantity following conventional phase angle in the complex plane, such that a larger negative value denotes a larger phase lag. Therefore in Fig. 5, the curve-to-curve variation clearly indicates that the phase lag increases with frequency until it reaches -90” in the high frequency limit, as observed in previous studies [17]. The phase lag becomes larger as the mean flame approaches the extinction condition, and this trend is more prominent for the Le, = 2 case. The increase in phase lag for the near-extinction flame is attributed to the longer characteristic chemical time in the reaction zone near extinction, for the same characteristic flow time. We next study the normalized heat release fluctuation in response to the strain rate-oscillation. Figure 6 shows the real parts of h, and
-0.1
1
10
al
Fig. 4. Magnitude of the reaction-sheet fluctuation as a function of normalized frequency for (A) Le, = 1, Da/Da, = 4 and (B) Le, = 2, Da/Da, = 1.05, as indicated in Fig. 3.
RESPONSE
0
I
.
OF COUNTERFLOW
. __----
.
.
_.__--_
____._
_
._._
PREMIXED
FLAMES
_______._._._.__
_
,F 0 -30
-__
__________.--------
co=
-
-0.1
-0.5 z~3.Q. -
_~_______.__-._‘-._._.-.-.-._-__2,0
i,:
_x--.
,
-_-._s.o
_________._
_..__..-___
__..--.
-901~ 0
Le.
‘.
=
8,.
. .
__.__
-
1
I
0.
6 ..
_.
.._... -:-_-_JO,o_
_
8..
2 4 -----------________ .. . . ...T . . .
t______
t’
-
_.”
8DalDa,
..
..
.
.
0.01 0.8.
.
___-______j
OA I
0.1 . . . . . ..a
1
10
0
1 .,
10 I
(0 . ,
II,. . . . ..________._.___.........
0.4 :
IC________._._____._._.-.-._._.-.-.-.-.----
%
%a
,___________.___ _..-.__-__.___._...____-.._
I
Le, = 0.5 -90
2
4 8..
8..
I..6
I,,
8D;I/Da,1
_~____________.___.-_..-......-.. --...-..Le, = 2
-90
”
1
“.
1.2
”
”
1.4
3
0.2 :
k
Of -0.2 L............ _ . . . . . .. . . . . . .. . ...__......... h, -0.4 -
‘~.“~~~‘~“~”
1..
g
3 1.6
”
c
”
1.8
Fig. 5. Phase angle of ;Ir as a function of normalized steady Damkiihler number for various frequencies for Le, = 1,0.5, and 2; the twin-flame case.
&, as a function of frequency, for each Lewis number of fuel considered. In each figure two values of the mean Damkohler number are chosen, specifically one is chosen near extinction (Da/Da, = 1.01, the dotted curves) and the other corresponds to a small strain rate (Da/Da, = 2.5, the solid curves). We first observe that for all Lewis numbers, h, > 0 and A, < 0, except at very high frequencies. The change of sign at high-frequencies is due to the phase delay as will be_ discussed later. The physical implication of h, and A, is the following. As the strain rate increases through the oscillatory cycle, the reactant flux to the reaction sheet is increased, yielding an enhanced heat release rate (positive h,) for a fixed reaction-sheet location. However, since the reaction sheet cannot sustain itself in the higher strain rate environment, it is pushed toward the stagnation plane, resulting in a reduction in the burning rate (negative &,I. Results in Fig. 6 then show that, for Le, = 1, i,, and h,
-0.6 0.01 2,
_$I
0.1 I
-
..:‘,,.
-I
[...._.__._... !!y;:::...‘;5 -6 ____._.._..... --...0.01
0.1
Le, = 2 1
1
1
10
m
Fig. 6. Real part of the normalized heat release fluctuations as a function of frequency for the near-extinction and small strain rate cases for Le, = 1, 0.5, and 2; the twinflame case.
exactly cancel out such that there is no net heat release fluctuation for both the near-extinction and small strain rate cases. This is due to the fact that, for unity Lewis number and in the absence of downstream heat loss, the flame structure with quasi-steady reaction zone resembles that of the unrestrained, freely propagating premixed flame with constant, adiabatic flame temperature. Since the reacitivity of such a flame is unaffected by strain rate variations, the reaction sheet simply adjusts to the timevarying strain rate instantaneously such that the net reactant consumption rate remains unchanged. For nonunity Lewis numbers, however, preferential diffusion introduces an additional influence to the flame behavior, and thus the two quantities do not exactly cancel. Specifically, for Le, < l,_it is seen that lh,l is somewhat larger than Ih,l owing to the fact that the flame is not displaced as much by an increase in strain rate because of the corre-
366
H. G. IM ET AL.
sponding increase in the flame temperature and hence the flame speed. Consequently, the net heat release fluctuation is in in phase (positive real) with the strain rate oscillation. The opposite effect is observed for Le, = 2 where the decrease in the flame temperature leads to an amplified reaction sheet response, thereby yielding negative real values of h,,, (out of phase). The net heat release rate fluctuation is seen to be quite small for Le, = 0.5, and in the small strain rate case for Le, = 2. For the flame near the_ extinction turning point (Le, = 21, however, h, overwhelms the net heat release behavior due to its sensitive response to the strain rate oscillation as shown in Fig. 3, t_hereby resulting in a significant amount of h netThe
magnitude and phase lag of &,,t as a function of the normalized Damkijhler number are shown in Figs. 7 and 8 respectively, for both Le, = 0.5 and 2. From Fig. 7 it is seen that significant amplification occurs near the extinction turning point when Le, = 2, while an almost uniform behavior is observed when Le, = 0.5, consistent with the reaction-sheet behaviors shown in Fig. 2. Figure 8 further shows that h,,, for Le, = 0.5 is nearly in phase (0”) while for Le, = 2 it is almost out of phase (180”) in the low-frequency limit. This suggests a qualitatively different response of the pre-
0.3
_ _ _ ~-~-~-~----~-~___._._._._._._~ _._._.-.
.-.-__
‘-._.___
IiiJ 0.1
‘..t_._
0.2 1 0
I
-.
.._.._
~-.-~-.-.-._._._____._.-._._._ _________
. ..-...-..
--...-...
1
Le, = 0.5 .I...&..._...
2
3
1
4
6 Le,
I : -\ : . \‘,..
I 2
.I
6w u)=
.._ 0.1 . - - -03 . --_;; -_-._s:o -..--10.0
0
Fig. 7. Magnitude of the normalized net heat release fluctuation as a function of normalized steady Damkijhler number for various frequencies for Le, = 0.5 and 2, the twin-flame case.
2
4
6
150
Ff #
120
I
90
f
____..-.
-
.__...-.
L+=2 60 1.2
1.4
-
-0.5
I-I ::j _._._5.0 _..._,oo
___...-.
1
.._0.1
__-...-.
,..*
_I
1.6
1.8
Fig. 8. Phase angle of net heat release fluctuation as a function of normalized steady Damkiihler number for various frequencies for Le, = 0.5 and 2; the twin-flame case.
mixed flame for different Lewis numbers from the standpoint of combustion instabilities. The phase of h,,, response is seen to shift downward as the frequency increases, indicating further delay in the response at higher frqeuencies. We further note that, whereas the phase angle remains nearly constant when Le, = 0.5, for Le, = 2 the phase angle becomes smaller as the steady extinction point is approached. That is, the near-extinction flame exhibits a longer phase delay due to the larger fluctuation in the reaction-sheet response, consistent with the result shown in Fig. 5. As a consequence of this phase delay, for Le, = 2 the phase angle of h,,, decreases below 90”, i.e., h,,, takes on positive real values, near extinction at high frequencies (w = 5 and 10 in the bottom part of Fig. 8). This phase reversal with frequency is often observed in acoustic instability problems [9]. Although it is difficult to readily assess the present velocity response in relation to acoustic instability, these results nevertheless suggest the possibility of qualitative changes in the response of near-extinction flames at high frequencies. The Single-Flame Case: Nonadiabatic Downstream
To further substantiate the effect of Lewis number on unsteady premixed flame behavior,
RESPONSE
OF COUNTERFLOW
PREMIXED
we consider the case in which the reactant stream is opposed to an inert stream at the same temperature (T, = T-,1. In this configuration, the flame response exhibits a turning point for all Lewis numbers. Figure 9 shows the basic steady flame response for three different Lewis numbers. It is seen that the flame temperature increases (decreases) with a decrease in Da for Le, < 1 (> 11, while it remains nearly constant for Le, = 1 until the curve starts to turn around. An interesting observation from Fig. 9 is that the values of the extinction Damkohler number do not vary monotonically with the Lewis number. This is due to the additional effect of the heat loss to the cold downstream. That is, for the same boundary conditions, the flame with a larger Lewis number has a higher flame speed and is therefore situated further upstream (larger 17f). Consequently, the downstream heat loss is reduced and the flame can be sustained at higher strain rates. This effect of reduced heat loss for the Le, = 2 case overrides the reduction in the flame temperature, thereby resulting in a smaller extinction Damkohler number as compared to the Le, = 0.5 flame. Based on these steady flame behaviors, in Fig. 10 we plot the magnitude of ijf, i.e., the reaction-sheet fluctuation relative to the mean location, as a function of Da/Da,. Here we first note that while there were significant dif3.5 5.2
2.5
Fig. 9. Steady reaction-sheet location and temperature as a function of the DamkGhler number for Le, = 1, 0.5, and 2; the single-flame case.
367
FLAMES
h.,
8..
8,.
1..
J
0..
Q)= . . . . . . 0.1 -03
l.+=l
r-I-. ;:: -_-._s,o -...10.0
________._.-.---.-~ _____.__..__
1
.
.
.
1.2 . .
. ..-...
.
.
--..
IA .
. Le,
.
.
.
.
I.2 . .
.
.
1.4 .
1.6 . .
.
.
1.8 . ..““B
= 0.5
ot’..‘..‘,“““‘,“.i 1
__..__..
_____.._-...
..,
1.6
1.6 ,
1.6
1.8
%
Le, = 1
ot.“,“‘,“““‘,‘.‘1 1.2
1
1.4
Fig. 10. Magnitude of the reaction-sheet fluctuation as a function of normalized steady DamkChler number for various frequencies for Le, = 1, 0.5, and 2; the single-flame case.
ferences for various Lewis numbers in the twin-flame case, the three plots in Fig. 10 for the single flame behave very similarly in that the reaction-sheet fluctuation is greatly amplified near the static extinction point due to the sensitivity of the flame near extinction. This amplification of the reaction sheet response is again found to be large at low frequencies (e.g., for w = 0.11, while at higher frequencies the value of 1+&lis quite modest, and the magnitude of ;if remains finite even at the extinction point. This allows for the following interpretation. In the present linear perturbation analysis, the time-varying Damkijhler number of the system can be expressed as Da(T)/Da, = u,/u(T)
= (a,/~,>(1
- aeior),
(33)
368
H. G. IM ET AL.
where we have expanded for (Y< 1. The corresponding reaction-sheet response, ~~(71, is given by Eq. 9 such that d+) d[Da(r)/Da,l
Le, = 1
d+)
= d[a,/a(r)l =-
drlf(r) dr
= --
dr d[a,/a(r)l
6f
as/a0
(34) .
Therefore, if we plot the unsteady reactionsheet ~~(7) as a function of Da(r) similar to I will indicate the absoFig. 9, the quantity 1535 lute value of the slope of the unsteady response curve. For example, the local slope variation along the quasi-steady response curves in Fig. 9 can be approximately represented by the dotted curves in Fig. 10, whose blowup at the extinction point (Da/Da, = 1) indicates the infinite slope at the turning point in Fig. 9. Therefore, as in Fig. 3, the finite value of 151 at the static extinction point for nonzero frequencies implies that the unsteady flame will not be extinguished when the instantaneous Damkohler number reaches the static extinction value. Rather, since the slope of the curve is smaller than that of the steady situation, the unsteady extinction Damkohler number will be shifted to a smaller value. This physically implies that flames become more resistant to extinction as the strain rate fluctuation is applied within a shorter duration. Figure 11 shows the real part of the heat release fluctuation as a function of frequency for the single flame case. Both near-extinction (Da/Da, = 1.01, the dotted curves) and small strain rate (Da/Da, = 2.5, the solid curves) cases are shown. Similar to the twin flame case, it is found that the real parts of h,,, for the flame with small strain rate is relatively small due to the cancelling of h, and 6, regardless of the Lewis number, w_hileh, is larger (smaller) in magnitude than h, for Le, > 1 (Le, < 1). The behavior of the Le, = 1 case is qualitatively similar to that of the Le, = 2 case because the presence of downstream heat loss causes the flame temperature to decrease with an increase in strain rate. For the near-extinc-
~~~
Y:::::,,.“.
..._______....__ L _______........ -3 4
__..L ___________...............~~~~~~~ ;;I'
0.01
0.1
1
10
co
Fig. 11. Real part of the normalized heat release fluctuations as a function of frequency for the near-extinction and small strain rate cases for Le, = 1, 0.5, and 2; the singleflame case.
tion flames, &, dominates the net heat release response due to the sensitive reaction sheet response. Figure 12 shows the phase of &,,, for the three Lewis numbers considered. Although the behavior of the three plots in Figs. 10 and 11 appear to be similar, the phase responses are seen to be quite different for each Lewis number. Consider first the Le, 2 1 case. The behavior is similar to that of the twin flame situation for Le, = 2 (Fig. 81, in that the heat release is mostly out of phase with the imposed strain rate, and undergoes a phase reversal at high frequency, although it is more confined near the extinction state for higher Lewis number. For Le, < 1, however, the phase response is significantly different between the flame near extinction and the flame opposed by small strain rate, especially at low frequencies. As shown in Fig. 11, in the Le, < 1 case h, over-
RESPONSE
OF COUNTERFLOW
PREMIXED
s
369
of stability characteristics depending on Lewis number, especially as extinction conditions are approached.
150
+
FLAMES
ml
I,
CONCLUSIONS
30 180
1
1.2 . . . . .., ,___...--- .. . ..___._..
‘.
...
-60 1
a
; go
1
.
1.4 , .
..---.-...-
.
1.6 . .
Le,
= 0.5
.._...
-..
’
8
1.2
1.4
/’ :’
Le, 1.2
.
1.4
.
.
1.8 .
DalDrg a=
‘-,---c:‘---i----r
1.6
1.8
1.6
1.8
DrJDis
= 2 DalDa,
Fig. 12. Phase angle of net heat release fluctuation as a function of normalized steady Damkiihler number for various frequencies for Le, = 1, 0.5, and 2; the single-flame case.
rides A, for small strain rates, and therefore h,,, is positive. Near the static extinction state, however, &, becomes dominant due to the sensitive reaction-sheet response, resulting in large negative values of h,,,. Consequently, h,,, for the flame with Le, = 0.5 experiences a reversal of phase response from in phase (< 90”) to out of phase (> 90”) as the steady strain rate increases. At higher frequencies, this jump in the phase lag becomes less prominent as the reaction-sheet response is reduced, and the near-extinction phase reversal does not occur. In summary of Fig. 12, in the presence of downstream heat loss, the heat release fluctuation for the Le, 2 1 (I 1) flame is mostly out of phase (in phase) with the imposed strain rate fluctuation, while near-extinction reversal of phase occurs at high (lowj frequencies. From the standpoint of acoustic instabilities, this result suggests the possibility of a wide spectrum
In this paper, we have employed large activation energy asymptotics to study the response of counterflow premixed flames to imposed oscillating strain rates. In particular, we have investigated the reaction-sheet fluctuations and the net heat release rate for two different flow configurations, namely symmetric twin flames and a single flame residing in a mixture stream impinging on a cold inert stream. As expected, the reaction-sheet response was found to be more sensitive to unsteadiness when the flames were near the extinction (turning) point. The Lewis number was found to play a major role in the response characteristics, particularly for the twin-flame configuration. In the twin-flame case, for Le, 5 1 both the mean location and the reaction-sheet fluctuation decrease monotonically with strain rate, while for Le, sufficiently greater than unity the mean location displays turning point behavior and the response of the fluctuation is amplified as the mean-state extinction condition is approached, most notably at low frequencies. For all Lewis numbers, the phase lag of the reaction-sheet fluctuation becomes larger as the static extinction condition is approached, and again this trend was found to be most prominent for Le, > 1. Furthermore, the near-extinction phase lag of the net heat release rate was found to be in phase with the imposed oscillation for Le, < 1, while it is out of phase for Le, > 1. This suggests that acoustic instability characteristics may be influenced significantly by Lewis number. The cold-downstream configuration supports a single flame which always displays turning point behavior due to downstream heat loss. Consequently, whereas flames with Lewis numbers greater than unity exhibit a similar response as their twin-flame counterparts, for Lewis numbers less than unity, the flame in this configuration is much more sensitive to imposed unsteadiness than the twin flames. In particular, its heat-release rate may undergo a phase reversal from in-phase with the imposed
H. G. IM ET AL.
370 oscillation to out-of-phase as the strain rate is increased toward the extinction value. Consequently, these flames may exhibit a reversal of acoustic characteristics as the flame nears extinction. We have also performed an asymptotic analysis in the limit of low frequency and have demonstrated that extinction can be delayed when the strain rate oscillates about the static extinction point. From the standpoint of the laminar flamelet concept of turbulent combustion, this result suggests that the laminar flamelet regime may be broader than the range predicted from steady considerations. This work was supported by the United States Air Force OfJice of Scientific Research. The authors would like to thank Dr. J. S. Kim of University of California, San Diego for helpful discussions.
REFERENCES 1. Peters, N., Twenty-First Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1987, pp. 1231-1250. 2. Libby, P. A., and Williams, F. A., Combust. Flame 44:287-303 (1982). 3. Buckmaster, J. D., Seventeenth Symposium (Zntemational) on Combustion, The Combustion Institute, Pittsburgh, 1979, pp. 835-842. 4. Haworth, D. C., Drake, M. C., Pope, S. B., and Blint, R. J., Twenty-Second Symposium (International1 on Combustion, The Combustion Institute, Pittsburgh, 1988, pp. 589-597. Stahl, G., and Wamatz, J., Combust. Flame 85: 285-299 (1991). Darabiha, N., Combust. Sci. Technol. 86:163-181 (1992). Ghoniem, A. F., Soteriou, M. C., and Knio, 0. M.: Twenty-Fourth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1992, pp. 223-230. 8. Egolfopolous, F. N., and Campbell, C. S., Twenty-Fifth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1994, in press. 9. Kim, J. S., and Williams, F. A., Cornbust. Flame 98~279-299 (1994). 10. Im, H. G., Law, C. K., Kim, J. S., and Williams, F. A., Co.mbust. Flame, 100:21-30 (1995). 11. Im, H. G., Ph.D. thesis, Princeton University, Princeton, New Jersey, November 1994. 12. Liii&n, A., Acta Astronaut. 1:1007-1039 (1974). 13. Joulin, G., and Sivashinsky, G. I., Cornbust. Sci. Technol. 31:75-90 (1983). 14. Kaper, H. G., Leaf, G. K., Matkowsky, B. J., and
15.
16. 17. 18. 19.
Olmstead, W. E., S&‘t4J. Appl. Math. 48:1054-1063 (1988). Strahle, W. C., Tenth Symposium Untemational) on Combustion, The Combustion Institute, Pittsburgh, 1965, pp. 1315-1325. Joulin, G., Cornbust. Sci. Technol. 97:219-229 (1994). Saitoh, T. and Otsuka, Y., Cornbust. Sci. Technol. 12:135-146 (1976). Kapila, A. K., S&VJ. Appl. Math. 41:29-42 (1981). Matkowsky, B. J., and Sivashinsb, G. I., SZAM J. Appl. Math. 37:686-699 (1979).
Received 20 March 1995; revised 18 September 1995
APPENDIX: ANALYSIS FOR LOW FREQUENCY (o -c 11, O(1) AMPLITUDE OSCILLATION The results of the main body of the text suggest that imposed unsteady oscillations will result in smaller Damkohler numbers (larger strain rates) at extinction, as compared to the steady values, thus resulting in an extinction delay. Furthermore, Figs. 3 and 10 indicate that this delay is shortened at low frequencies. However, the linearized analysis of the section on the reaction-sheet response for small amplitude of oscillation is unable to evaluate this shift in extinction conditions. Thus we consider here the low-frequency limit (w +Z l>, while allowing for O(1) amplitude, and we explicitly determine the extinction delay. Our approach is similar to the ignition delay problem of Kapila [18], which can be consulted for additional details. We consider only the twin-flame configuration sketched in Fig. 1. Furthermore, our aim is to give a qualitative description of this delay and for this purpose it is appropriate to assume weak thermal expansion and employ the diffusional-thermal model [19] of flame propagation. In this model, the equations for the transport properties decouple from those of fluid dynamics and thus the stream function is given by Eq. 2 for all n. The remaining equations and jump conditions can be written as 1191 t$, + y7& - we, = 0,
77> 7fP
(Al)
H,,,, + yr/H,, - OH, = -le,,,,
(A3
ti= 1,
W)
@Q