Some aspects of the freely propagating premixed flame in a spatially periodic flow field

COMBUSTION A N D F LA ME 97:375-383 (1994)

375

Some Aspects of the Freely Propagating Premixed Flame in a Spatially Periodic Flow Field K. M. YU, C. J. SUNG, and C. K. LAW Department of Mechanical and Aerospace Engineering, Princeton University,Princeton, NJ 08544 The premixed flame situated in a spatially periodic flow field is examined using the passive propagation model with the local flame speed affected by stretch and nonequidiffusion. Numerical solution shows that the average flame speed increases with either increasing fluctuation amplitude or increasing wavelength of the imposed flow field, and that the flame surface can locally extinguish for sufficiently large fluctuation amplitude of the imposed flow. Perturbation solutions in the weakly wrinkled flame and the thin flame limits are presented. The formation of corners on the flame surface in the thin flame limit is illustrated, and the structure of the comer is further found to resemble that of the Bunsen flame. The premixed flame situated in a two-dimensional periodic flow field is also analyzed in the Huygens limit, leading to the observation that flame surface discontinuities exist in the form of cones.

1. INTRODUCTION The response of the laminar premixed flame situated in a unidirectional, spatially periodic and time-independent flow field is of interest to the understanding of the structure and response of turbulent premixed flames. It is well-known that a turbulent flow field is characterized by a wide range of spatial-temporal scales and the associated motion can be visualized as a cascade of eddies. Recently, Sivashinsky [1] has successfully applied the cascade concept to large-scale turbulent premixed flames by modeling the multiple scale turbulent flow as a cascade of one-scale flows with widely separated scales. By successively averaging over increasing scales, a differential expression relating the turbulent flame speed and the turbulent intensity is derived. This expression involves an unknown function that has to be determined by analyzing a flame propagating in a one-scale fluctuating flow. Therefore, there is the need toward understanding the structure and response of a laminar premixed flame in spatial-temporal periodic flows. Berestycki and Sivashinsky [2] analyzed a flame propagating through a one-scale, unidirectional, spatially periodic and time-independent flow field. Under the framework of the diffusional-thermal model with an SVF type scaling [3], a differential equation governing the flame front was derived, which, together Copyright © 1994 by The Combustion Institute Published by Elsevier Science Inc.

with appropriate boundary conditions, constituted a nonlinear eigenvalue problem with the average propagation speed of the wrinkled flame front playing the role of the eigenvalue. It was shown that for a flow field with a given wavelength, there was a level of flow field fluctuation above which no solution for the eigenvalue problem could be found. It was argued that the disappearance of a solution implied local extinction; the crest or the trough of the flame front was extinguished due to stretching and nonequidiffusion (Le ~ 1). The analysis of Berestycki and Sivashinsky [2] was of the diffusional-thermal type, which took into account all relevant dissipative effects in the equation governing the flame front. Ashurst and Sivashinsky [4], on the other hand, analyzed flame propagation in a time-independent system of eddies by utilizing the passive Huygens propagation model, which neglected all dissipative effects. It was found that, for fluctuation amplitudes of the imposed flow less than a critical value, the average propagation speed exhibited a global, integral, dependence on the flow field and the flame front was smooth. On the other hand, for fluctuation amplitudes greater than the critical value, the dependence was local and sharp corners developed o n the flame front due to the nature of the Hamilton-Jacobi-type equation [5-7]. Aldredge [8] numerically studied the flame geometry in a two-dimensional periodic flow 0010-2180/94/$7.00

376 field for the weakly stretch-affected, nonHuygens flame propagation. Flame wrinkling was found to increase the average flame propagation speed, and the possibility of flame folding was also indicated. The study, however, did not address the possible occurrence of local extinction due to stretch-induced modification of the local flame speed. In the present study, 1 we aim to obtain further understanding of the structure and response of the premixed flame situated in a periodic flow field. The passive propagation approach is utilized with particular emphasis on non-Huygens propagation, which accounts for the effects of both stretch and nonequidiffusion on the local flame speed. Useful insights on several issues of interest have been obtained through the present study. Formulation of the problem is presented in the next section, which is followed in Section 3 by the numerical solution for the average flame propagation speed, and in Section 4 by an analysis of the conditions governing phase reversal. Perturbation solutions, applicable in the limits of either small fluctuations or thin flames, are respectively given in Sections 5 and 6. While the analysis of Sections 3-6 pertains to one-dimensional periodic flows, in Section 7 we present results from a preliminary study of the flame response in two-dimensional periodic flows. The Huygens limit is investigated, and the burning rate and flame geometry are derived. 2. FORMULATION The scalar field formulation has been recently adopted in the study of premixed flame geometry and propagation in non-uniform flow fields [1, 6, 8-10]. If the flame front is represented by an iso-value surface of a scalar field, G, then the scalar field is governed by the following nondimensional evolution equation: OG - - + ulc=c- • V G = su[VGI, (1) Ot where UlG=C- is the flow field velocity evaluated on the unburned side of the flame front

a A preliminary version of the present work was reported in Ref. 10.

K. M. YU ET AL. and s, is the propagation speed of the flame front relative to the unburned mixture. All space and velocity variables are nondimensional in units of a relevant hydrodynamic length l/_/ and the laminar flame speed s, °, respectively. The time variable is in units of

lH/su °. Equation 1 is the equation of motion of the flame front and it is coupled to the external hydrodynamic flow through the term ul~=c-. The coupling represents the interaction between the flame front and the hydrodynamic flow: the flow convects the front while the front affects the flow through thermal expansion. Since this coupled problem is quite complicated, we shall neglect the effect of thermal expansion and consider the flame front as a passive surface convected by the hydrodynamic flow only. When the effect of thermal expansion is neglected, any divergence-free velocity field can be prescribed to Eq. 1. We shall first study a unidirectional, one-dimensional, spatially periodic, and time-independent flow field, specified by: ulc=c -= (0,0,~ + A c o s x ) ,

(2)

where ~ is the average propagation speed of the flame and A is the fluctuation amplitude of the flow field (see Fig. 1). The wavelength of the imposed velocity fluctuation has been chosen to be the hydrodynamic length scale and is used to nondimensionalize all other length variables. Both ~ and A are in units of the laminar flame speed. Moreover, we require (~ A) > 0 so that no reverse flow exists. Having the flow field described by Eq. 2, we need an expression for the propagation speed of the flame front relative to the unburned mixture, s~. From Chung and Law [11], we have: -

S. = 1 + ~ V . n

+ ~-g~

~,

(3)

where n = VG/IVGI is the unit normal on the flame front pointing toward the burned side, 6 the ratio of the laminar flame thickness to the hydrodynamic length scale, e the inverse of the nondimensional activation energy, Le the Lewis number, and K the nondimensional

FREELY

.-

PROPAGATING

PREMIXED

FLAME

I&=~- = (0, 0, t+Acosx) Fig. 1. A flame front, G = C, situated in a unidirectional and spatially periodic flow field.

stretch factor given by [12, 131: I K =

-n-V

X (ul~=~-X

3. NUMERICAL (4)

where U is the nondimensional velocity of the flame front in units of the lamainr flame speed. For the present study, the flame front is stationary, implying that U = 0. Substituting G(x, y, z) = z - f(x) and Eq. 2 into Eq. 4 then into Eq. 3 yields: =

6 & 1 - -03 & +Sp

ij+

A cos x dg

D4

A z--sgsinx

(5) where p = (l/Le - 1)/2~, g = df/a!x, D = (1 + g*)‘/‘, and f is the function describing the flame shape. Substituting Eqs. 2 and 5 into ,Eq. 1 gives an equation governing the geometry of the flame front: u + A cos x D

to insure a 27r-periodic solution. The first-order Eq. 6 together with the two boundary conditions of Eq. 7 constitute an eigenvalue problem for the unknown average flame propagation speed Ij. Due to the nonlinearities associated with the differential equations, a numerical solution of the eigenvalue problem is required. We mention in passing that alternate stretch-affected flame speed expressions exist, such as those of Clavin and Williams [14] and Matalon and Matkowsky [15]. Use of these alternate expressions for s, could lead to quantitative modifications of the predicted flame response, although the methodology of the analysis is the same. SOLUTION

n)

+ (U * n)(V * n>,

s,

377

Numerical solution of the nonlinear eigenvalue problem has been performed for various values of 6 and Le with increasing flow field fluctuation. A shooting method using Runge-Kutta integration with adaptive step size is employed. Results with E = 0.1 are shown in Figs. 2 and 3 for Le = 0.7 and 1.2, respectively. It is seen that increasing the flow field fluctuation, A, increases the average flame speed. Furthermore, the flame speed also increases by fixing the fluctuating amplitude and increasing the fluctuation wavelength through reducing 6. These results imply that the turbulent flame speed increases with either increasing turbuE=O. 1, Le=0.7 5.0,

/I

6 dg =l-D3z

‘4

-

.

sgsm x1 *

Equation 6 is supplemented boundary conditions: T(O) = g(7r) = 0

by the following (7)

Fig. 2. Variation of the average propagation speed with the fluctuation amplitude for various values of 6 (E = 0.1, L.e = 0.7): nonexistence of the eigenvalue problem for the fluctuation amplitude beyond some critical values is also exhibited.

378

K . M . YU ET AL. E---0.1, Le=l.2 5.0 0.1 4.0

8=0

3.0

0.2

2.0

1.0

0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

a0

A Fig, 3. Variation of the average propagation speed with the fluctuation amplitude for various values of 8 (~ = 0.1, Le = 1.2).

lence intensity or increasing turbulence length scale because of the increased flame surface area, as expected. Figure 2 (Le = 0.7) also shows that, for a given 3, as the fluctuation amplitude is increased beyond a critical value, solution for the eigenvalue problem ceases to exist. This was also indicated in the similar scalar field analysis of Aldredge [8]. A singular point of the differential Eq. 6 is encountered in the course of integration, and the flame front terminates. This result may be interpreted as local extinction of the flame front. Physically, as the fluctuation amplitude increases, the coupled effect of stretching and nonequidiffusion on the burning intensity amplifies and eventually leads to local extinction. This result also corroborates the finding from the differential analysis of Berestycki and Sivashinsky [2]. In particular, if a nonlinear expression relating the flame stretch and flame response is used in place of the linear expression of Eq. 3, a turning-point behavior similar to that of Berestycki and Sivashinsky [2] is expected to be observed, as was demonstrated for the opening of the Bunsen flame tip in Ref. 10. 4. P H A S E

REVERSAL

It is well known that for a mixture with Lewis number greater than some critical value, concave curvature toward the unburned gas enhances the local burning rate (s u > 1) while

convex curvature toward the unburned gas retards the local burning rate (s, < 1). Therefore, in this Case, the flame front must be in phase with the imposed periodic flow in order to achieve kinematic balance. That is, it must be concave at x = 0 and convex at x = 7r when viewed from the unburned side. On the other hand, if the Lewis number is smaller than this critical value, then it is reasonable to expect the flame front to be out of phase with the imposed flow. Since is the second derivative of the. flame shape, the flame is in phase with the imposed flow if it is negative at x = 0 and positive at x = 7r, and out of phase otherwise. Evaluating at x = 0 and x = 7r using Eqs. 6 and 7 then yields:

dg/dx

dg/dx

(dg) ~

~+A-1 ~=o = 3 [ , f f ( ~ - + A - ) - -

(dg)

1]'

(8)

-A-1 x=~ = 8[/3(~ - A )

- 1]"

(9)

It is apparent that (~ - A ) < 1 < (~ + A ) holds since if the imposed flow speed is everywhere higher or lower than the laminar flame speed, no periodically varying solution can exist. Therefore, if we have Le > 1, which implies / 3 < 0 , then at x = 0 a n d > 0 at x = 7r, and the flame must be in phase with the flow, as anticipated earlier." The situation is more complicated for Le < 1, corresponding to /3 > 0. Here, if [/3(~ + A) 1] < 0 holds, then the flame is in phase with the flow. By the same reasoning, if [/3(~ - A) - 1] > 0 then the flame is out of phase with the flow. In the regime [/3(~ - A) - 1] < 0 <~ [/3(~ + A) - 1], corresponding to the average velocity ~ being in the range (1 + / 3 A ) > /3~ > ( 1 - /3A), there is no solution. Therefore for a given value of /3, and for Le < 1, there is a range of amplitude A over which no periodic flame response is possible. This also corroborates the numerical solutions of Figs. 2 and 3. The result that there may exist situations over which a dynamic balance between the flow and flame velocities cannot be at-, tained over the entire flame surface is not/ unreasonable, considering the asymmetric nature with which stretch and nonequidiffusion,

(dg/dx)

(dg/dx)
FREELY PROPAGATING PREMIXED FLAME affect the concave and convex segments of the flame. • 5. THE WEAKLY WRINKLED FLAME

379

The boundary conditions are gi = 0 at x = 0 and x = 7r for i = 0,1,2,.-- . The leading-order solution is simply

LIMIT (A --* 0)

go = +[(v0 + A c ° s x ) 2 - 1] '/2

In the limit, A --, 0, the solution is a perturbation to the planar flame solution. Expansion is sought in ascending integral power of A:

It is apparent that only one of the two boundary conditions, g0(0) = 0 and g0(~-) = 0, can be satisfied. The solution therefore develops a cusp and hence a boundary layer at one of the boundaries. This is to be expected since in the limit fi --* 0, the highest order derivative of Eq. 6 is dropped. Thus, in order to satisfy g0(Tr) = 0, we require T0 -- 1 + A. On the other hand, if the boundary condition g0(0) = 0 is satisfied, then we have To = 1 - A . However, if T0 = 1 - A , then Eq. 15 implies that go is imaginary which is physically unreasonable. Therefore, we require T0 = 1 + A and the boundary condition at x = rr is to be satisfied. It is easily seen that the positive and negative branches of the solution go correspond to the in-phase and the out-of-phase flames, respectively. Subsequently, by analyzing the outer region up to the first order, we obtain the following expression for the average propagation speed of the flame front:

g = go + Agl + A2g2 + o(A2)

(10)

= T0 + A ~ 1 + A e u 2 + o ( A 2 ) ,

(11)

with the boundary conditions gi = 0 at x = 0 and x = 7r for i = 0, 1 , 2 , . . . . Substituting these expansions into Eq. 6 • readily yields the planar flame solution, T0 = 1, to the leading order in A. By utilizing the leading-order solution, the first-order correction, ~1, is found to be identically zero. The second-order correction is then found to be nontrivial. The final solution is given by

[ A ]2 = 1+

2 ( / 3 - 1)6

+ °(A2)"

(12)

Equation 12 shows that the first nontrivial correction is quadratic in A, which agrees with predictions from previous theoretical studies of weakly turbulent flames [1, 16].

(15)

T0 = 1 + A ___ 6(1 - fl)v~'- + 0 ( 6 ) forLe><(1 +2e) -l.

(16)

The boundary layer that forms at x = 0 is of thickness 6 and can be investigated by introducing the stretched coordinate

6. THE THIN FLAME LIMIT (8 ~ 0) Since 6 is the ratio between the flame thickness and the wavelength of the imposed velocity fluctuation, in the limit of vanishing 6, the effect of flow nonuniformity and flame wrinkling on the flame structure is negligible, and the local flame speed at every point of the flame front, s,, approaches unity. Hence, this limit corresponds to Huygens propagation, which has been studied by Sethian [5], Kerstein et al. [6], Osher and Sethian [7], and Ashurst and Sivashinsky [4]. An expansion in ascending integral power of 6 can be used in the limit 6 ~ 0, namely g = go + 6gl + o ( 6 ) ,

(13)

l; = t' 0 + (~U1 + 0 ( 6 ) .

(14)

x

sc = -~,

(17)

and the inner expansion g = 4' + o(1).

(18)

The leading-order governing equation in the inner region is given by de - -

d~:

D - (1 + 2A) =

D

3

D - /3(1 + 2 A ) '

(19)

where D = (1 + th2) 1/2. Equation 19 is similar to the equation governing the geometry of the Bunsen flame in a uniform flow [10]. This is physically reasonable since in the thin boundary layer region, the

380

K. M. YU ET AL.

flow velocity is quasi-uniform. The behavior of the solution of Eq. 19 can be examined by using the critical point analysis, which is briefly discussed in the Appendix. The analysis yields the following critical Lewis numbers:

where a = 1 + 2A. Then, by recognizing that opening of the Bunsen flame tip is mathematically related to the singularities of Eq. 19, we can determine the width of the opening w as 2(/32a2

1 Lel = 1 + 2~'

(20)

1 LeE = 1 + 2E/(1 + 2 A ) "

(21)

W -----

2(1 - /3)

- - 1 ) 1/2

a

a

× tan- l( /3 2aZ - 1) 1/2 For Le < Lel, the solution th ~ [(1 + 2A) 2 1]1/2 as ~ ~ ~, hence, the positive branch of the solution for go given by Eq. 15 is appropriate. By similar reasoning, for Le > Le2, the negative branch should be chosen. For Lea < Le < Le2, the solution of Eq. 19 takes the shape of an open-tip Bunsen flame [10]. Since the effect of flame stretch and nonequidiffusion on the flame structure is retained in the inner boundary layer region, this result, in the present context, can be interpreted as the local quenching of the flame front due to stretching and nonequidiffusion (Le =~ 1). Furthermore, we notice that Le 2 is bounded between Le 1 and 1, hence, local quenching occurs only for a bounded range of the Lewis number between Lel and 1. Therefore, for a given Lewis number satisfying Le 1 < Le < 1, local quenching occurs for fluctuation amplitudes above a critical value A*, which can be obtained by substituting Le 2 = Le in Eq. 21 and then solving for A:

1(: )

A* = - ~

- 1 .

(22)

Equation 19 can be further integrated exactly by substituting 4~ = tan(0 + ~r). The integration yields:

~:=/3sin0-

(1-/3)

/° --a

(a z - 1) 1/2

a+a'lJ21}

I/3a + 1 1/2

[a-1] 1) /3a + 1 I 1/2 + [ a + 1 ) 1 / 2

/3a -

× In

/3a -

1

J

~a-1]

(24) Figure 4 shows the variation of w with A for = 0.1 and various values of the Lewis number. It is apparent that for A = A*, the flame surface is at the incipient state of opening and w -- 0. Further increasing A causes w to increase. This is physically reasonable since increasing the fluctuation amplitude of the imposed flow inevitably increases the extent of stretching of the flame front. As A ~ ~, Eq. 24 shows that w approaches the value 2/3. It is significant to note that tip opening, and hence local extinction, is described at the thin flame limit of 8 ~ 0. It is therefore conceivable that local extinction of a wrinkled flame can take place even for a moderate extent of wrinkling. 7. T W O - D I M E N S I O N A L PERIODIC

HUYGENS SOLUTION A flow field that is periodic in both x and y directions can be specified as:

a ( a 2 _ 1 ) 1/2

ulc= ¢- = (0, 0, ~ + A cos x cos ky), tan----~- + \ a - 1] ×In

0 + ~r tan~

"

a + 1 1/2 ( a----2~ )

,

(23)

(25)

where k is the ratio between the wavelengths of the imposed flow fluctuation in the x and y directions. We shall restrict this analysis to the Huygens limit, for which every element of the flame surface propagates relative to the un-

FREELY PROPAGATING PREMIXED FLAME ~=0.1

2.0

1.5

f

05/ °'°0o

'

o95 0.5

,.o

~.5

21o

2.5

A Fig. 4. Variation of the width of the opening with the fluctuation amplitude in the thin flame limit (6 ---, 0) for various values of the Lewis number (e = 0.1).

burned mixture with speed unity (s u = 1). Further writing G(x, y, z) = z - f(x, y) and substituting Eq. 25 into Eq. 1 yields an equation governing the geometry of the flame surface fx~ + k2f, 2 = (~ + A cos x cos "q)2 - 1,

(26)

where '0 = ky and the subscripts of x and '0 represent partial derivatives. Before boundary conditions are imposed on Eq. 26 to specify the solution, we first notice that Eq. 26 is invariant to the following two transformations: (i) Reflection in the x or the "0 direction, which can be represented as X--~ --x

or

'0-:, --'0.

(ii) Translation by a distance ~r in the x and the '0 directions simultaneously. This transformation can be represented as

x~x+rr

or

381

are particularly interested in the geometry of the discontinuities, whether they are in the form of ridges or isolated sharp comers on the flame surface. Since Eq. 26 is first order in both x and '0, one boundary condition is needed in each direction to specify the solution. Owing to the nature of the imposed periodic flow, it is natural to impose the following boundary conditions: x=0:

fx=0,

(27)

n = 0:

f~ = 0.

(28)

It may be noted that the two boundary conditions together with the transformational invariance (i) and (ii) imply the following: fx = 0 on x= ~-andfn=0on'0= rr. Imposition of the two boundary conditions of Eqs. 27 and 28, and the particular condition: fx = fn --- 0 at x = "0 = 0 immediately yields the average propagation speed as 99 = 1 - A, which then implies that fx 2 + kZf, ~ < 0 from Eq. 26. This solution is thus physically unreasonable. In order to obtain a physically reasonable solution to Eq. 26, we try a different set of boundary conditions: x=0:fx=0 '0=~:

(29)

fn=O.

(30)

Imposition of the boundary conditions Eqs. 29 and 30, and the particular condition: fx = f , = 0 at x = 0, '0 = zr gives the average propagation speed as ~0= 1 +A,

(31)

"0~'0+7r.

Moreover, Eq. 26 is invariant with a translation by a distance 2¢r in either the x or the '0 direction. This result is implied by the transformational invariance (i) and (ii). The main goal of the subsequent analysis is the determination of the average propagation speed of the flame surface ~0, but not the complete solution of the nonlinear first-order partial differential Eq. 26. Moreover, we expect the solution of Eq. 26 to exhibit discontinuities akin to those observed in Section 6. We

which immediately yields the following upon substituting into Eq. 26:

L 2 + k 2 f È 2 = (1 + A + A c o s x c o s ' 0 ) 2 -

1. (32)

The boundary conditions of Eqs. 29 and 30 together with the transformational invariance (i) and (ii) imply that the following conditions on the boundaries are satisfied: fx = 0 on x = 7r and fn = 0 on 77 = 0. Hence, the solution exhibits no discontinuous behavior at the point:

382

K. M. YU ET AL.

x = 7r, 7/= 0. Moreover, since the following: fx = fn = 0 at x = 0, 7/= 7r is satisfied by the solution, the solution exhibits no discontinuous behavior here either. The solution is however discontinuous at x = 0, 77 = 0 since at this singular point, Eq. 32 implies that: fx 2 + k 2 f n 2 = (1 +

2A) 2 - 1.

(33)

Therefore, if we approach the singular point along x = 0 on which fx = 0, we find that f~ approaches a finite value. On the other hand, if we approach the singular point along r~ = 0 on which fn = 0, then we find that fx approaches in general a different nonvanishing value. Hence, the geometry of the flame surface in the neighborhood of the singular point resembles a cone elongated in one direction. CONCLUDING REMARKS By using the passive flame propagation formulation with the local flame speed affected by stretch and nonequidiffusion, we have theoretically investigated several issues regarding the structure and response of a premixed flame propagating in a spatially periodic flow field. Specifically, we have shown that the average flame propagation speed increases with increasing fluctuation amplitude and wavelength, that local extinction occurs when the fluctuating amplitude exceeds a critical value, that in the weakly wrinkled flame limit the increase in the average flame speed varies quadratically with the fluctuation amplitude, that the structure of the hydrodynamic sharp corners resembles that of the Bunsen flame tip, and that these sharp corners take the form of cones for flames in two-dimensional periodic flow fields. In the present investigation we have considered only weak stretch effects in that the flame speed varies only linearly with the stretch rate, as given by Eq. 3. Consequently, the analysis breaks down for large stretch rates, especially those responsible for local extinction. It is therefore of interest to extend the present analysis to include the physics of nonlinear stretch in order to properly describe local extinction. Of particular interest is the behavior of the average flame speed with increasing fluctuation amplitude subsequent to the onset

of local quenching, and its relation to the experimental observation of the plateauing of the turbulent flame speed with increasing turbulence intensity [17-19].

This work was supported by the Office of Basic Energy Sciences of the Department of Energy under the technical management of Dr. Oscar P. Manley.

REFERENCES 1. Sivashinsky, G. I., Combust. Sci. Technol. 62:77-96 (1988). 2. Berestycki, H., and Sivashinsky, G. I., SIAM J. Appl. Math. 51(2):344-350 (1991). 3. Buckmaster, J. D., and Ludford, G. S. S., Lectures on Mathematical Combustion, Society of Industrial and Applied Mathematics, 1983, pp. 34-38. 4. Ashurst, W. T., and Sivashinsky, G. I., Combust. Sci. Technol. 80:159-164 (1991). 5. Sethian, J. A., Commun. Math. Phys. 101:487-499 (1985). 6. Kerstein, A. R., Ashurst, W. T., and Williams, F. A., Phys. Rev. A 37:2728-2731 (1988). 7. Osher, S., and Sethian, J. A., J. Comput. Phys. 79:12-49 (1988). 8. Aldredge, R. C., Combust. Flame 90:121-133 (1992). 9. Ashurst, W. T., Sivashinsky, G. I., and Yakhot, V., Combust. Sci. Technol. 62:273-284 (1988). 10. Sung, C. J., Yu, K. M., and Law, C. K. Paper No._ 92-0238, Thirtieth Aerospace Sciences Meeting, The American Institute of Aeronautics and Astronautics, Reno, NV, 1992. 11. Chung, S. H., and Law, C. K., Combust. Flame 72:325-336 (1988). 12. Matalon. M., Combust. Sci. Technol. 31:169-181 (1983). 13. Chung, S. H., and Law, C. K., Combust. Flame 55:123-125 (1984). 14. Clavin, P., and Williams, F. A., J. Fluid Mech. 116:251-282 (1982). 15. Matalon, M., and Matkowsky, B. J., J. Fluid Mech. 124:239-259 (1982). 16. Clavin, P., and Williams, F. A., J. Fluid Mech. 90:589-604 (1979). 17. Adbel-Gayed, R. G., and Bradley, D., Combust. Flame 62:61-68 (1985). 18. Peters, N., Twenty-First Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1986, pp. 1231-1250. 19. Adbel-Gayed, R. G., Bradley, D., and Lawes, M.~ Proc. Roy. Soc. Lond. A 414:389-413 (1987). Received 27 May 1993; revised 19 January 1994

FREELY PROPAGATING PREMIXED FLAME APPENDIX: CRITICAL POINT ANALYSIS

Equating the numerator and the denominator of Eq. 19 to zero yields, respectively, the following two critical points: ~ + = +[(1 + 2A) 2 - 1] 1/2

(34)

~b*= [/32(1 + 2A) 2 - 1] 1/2

(35)

Further setting ~ ± = ~b* and ~b~ = 0 yields the two critical Lewis numbers, Le 1 and Le 2, respectively. We note that for Le > Le2, dqb/d£ is negative in the range q~_ < ~b < 0. Hence, an integration starting from qb = 0 eventually reaches ~b = ~b_ as sc ~ ~, and the solution of Eq. 19 resembles half of a normal Bunsen flame with the slope q~_ on the shoulder. Furthermore, ~_ = g0(0) with the negative branch of Eq. 15 chosen. Hence, the inner solution matches with the outer solution in accordance with the concept of asymptotic matching. We therefore conclude that for Le > Le2, the negative branch of Eq. 15 is to be chosen to achieve proper matching. Similar argument applies for Le < Lel, in which case dc~/d£ is positive in the range 0 < ~+ < ~b, and the positive branch of Eq. 15 has to be chosen to achieve matching. For the intermediate range of the Lewis number, Le I < Le < Le 2, the singular points

383

of Eq. 19, ~b*, situate inside the range ~_ < 4) < ~b+. The singular points represent termination points of the flame surface. Hence, in this range of the Lewis number, the flame surface is discontinuous and resembles an open tip Bunsen flame. Figure A1 shows the qualitative dependence of the flame shape on the Lewis number. Le > Le2

Lel < Le < Le2

L//

Le

< Ee 1

Fig. A1. A schematic showing the configuration of the flame front and its structure for various values of the Lewis number.