The Premixed Flame in a Counterflow

COMBUSTION AND FLAME 47: 191-204 (1982) 191

The Premixed Flame in a Counterflow

J. BUCKMASTER and D. MIKOLAITIS

Department of Theoretical and Applied Mechanics. University of Illinois, Urbana, IL, 61801

We examine a premixed flame located in a counterflow of fresh cold mixture and hot burnt gas, the latter at a temperature close to that of adiabatic deflagration. A constant density, high activation energy model is adopted, and the steady state response depends essentially on two parameters, the Lewis number, and the temperature of the remote hot gas. If the second is sufficiently small, the response of flame position to changes in straining rate is a variation on the familiar S-shaped response for systems that display ignition and extinction. The possible implications for a flammability-limit model are discussed. The stability of the steady state under the influence of one~limensional disturbances is examined in detail, using the numerical method of weighted residuals.

1. INTRODUCTION

The behavior of a flame in a stagnation point flow is of fundamental interest to anyone who wishes to understand how flames respond to nonuniform flows. Thus Buckmaster [1] using a constant density model, examined a flame located in the stagnation point flow generated by the im- pingement of fresh mixture against an adiabatic wall, a problem equivalent to the counter-flow of two streams of flesh mixture supporting a double flame. More generally, the analysis is restricted only to those situations in which conditions behind the flame are adiabatic. (This problem was also treated in [2], but for values of Lewis number bounded away from 1 that are probably not directly relevant to real flames in the context of large activation energy asymptotics.) Among other things it was shown in [1] that sufficiently large straining rates can reduce the flame speed to zero, one of the key ingredients of a model for flammability limits proposed by Buckmaster and Mikolaitis [3]. In addition, the results revealed profound Lewis number (L) effects, which make it clear that for problems of this type setting L = 1 has little to commend it, except for simplicity of analysis. In some circumstances this can be of

Copyright © 1982 by The Combustion Institute Published by Elsevier Science Publishing Co., Inc. 52 Vanderbilt Avenue, New York, NY 10017

overriding importance of course, and Libby and Williams [4], in their study of counterflows of fresh mixture and burnt gas (which incorporates the full fluid-mechanical coupling) make this choice. They note that there is a solution for all straining rates, the flame speed being negative at sufficiently large values, so that the gas flow at the reaction zone is then directed from the hot side to the cold side, the fresh mixture reaching the zone by diffusion alone. Their analysis is mathematically quite special since when L = 1 and the remote temperature of the hot counterflow is equal to the adiabatic flame temperature (another choice they make), the temperature on the hot side of the flame is always uniform, ir- respective of the straining rate. Thus in the flame- work of a constant density model the results of [1 ] still apply (the response in the negative flame speed regime is shown in [3] and complements the earlier results). When the Lewis number is different from 1 the flame temperature will change with the straining rate so that conditions behind the flame will only be adiabatic if commensurate ad- justment of the remote temperature of the hot products is made. This is unlikely to be a physically interesting situation. Thus a flesh analysis is necessary which will enable us to predict the re-

oo 10-2180/82/08191 + 14502.75

192 J. BUCKMASTER and D. MIKOLAITIS

sponse of the flame to different straining rates for arbitrary, but fixed, remote temperatures.

The major motivation that we have for ex- amining this problem stems from a conjecture made by us [3] in the discussion of a model for flammability limits in upward propagation through lean CHa/air mixtures in a standard flammability tube. There it was argued that the flame will be quenched if the buoyancy-induced straining rate is sufficient to reduce the flame speed to zero. There is experimental evidence favoring this conjecture, so that its consequences may legitimately be discussed even if a causal mechanism is not identified. Nevertheless, it was noted that for straining rates larger than the critical the flame speed will be negative, so that stale burnt gas, gas that has had an opportunity to cool significantly, will be drawn back through the flame; and it was thus conjectured that the reduced temperature of this gas provides the essential quenching mechanism. Here we provide supporting evidence by demonstrating that for a sufficiently cool counterflow of burnt gas, the steady response displays the same kind of extinction phenomenon as is found in fuel-drop burning as the Damkohler number is decreased.

Libby and Williams [4] were motivated by their interest in turbulent flames; such justification extends to the situation analyzed here.

2. STEADY STATE SOLUTION

The mathematical model that we shall discuss is identical to that considered by Buckmaster [ 1 ], so that our discussion of it will be brief. The basic equations are

d T d 2 T 0 = 2fin - - + - - + BTexp (--O/T), (1)

dn dn 2

d Y 1 d 2 y 0 = 2fin - - + B Y exp (--0/T), (2)

dn L dn 2

where

B - 02 y 2

2(Y~ + T~,) 4 I°t exp Y ~ + T ~

which must be solved subject to the boundary conditions

n ~ + ~ : Y-,'. Y ~ , T ~ T ~ , (3)

n ~ ---~: Y ~ 0, T ~ T r. (4)

Here fl is the nondimensional straining rate, corresponding to a gas speed in the n direction of -2f ln (Fig. 1); L is the Lewis number; 0 is the activation energy; and T, Y are, respectively, the temperature and mass fraction of mixture. The specification of the parameter B simply reflects the manner in which the equations have been nondimensional- ized. The reference velocity is taken to be the adiabatic flame speed when L = 1, 0 ~ ~ [5], and the reference length is chosen to be the thickness of such a flame.

Analysis in the limit 0 -+oo is particularly simple if L is close to 1, and the enthalpy (T + Y) is close to (T~. + Y~). Then if we write

x = 1 - 0, x = o 0 ) , ( s )

1 T+ Y= T~. + Y~. + - ~ , (6)

0

the equations on each side of a thin reaction zone, or flame-sheet, become

2t inT '+ T" = 0, (7)

2finch' + ~b" + XT" = 0, (8)

where T, q~ here stand for the leading terms in their asymptotic expansions for large 0. Note that (4b) and (6) are only consistent if T r is close to (T~ + Y~.), in recognition of which we write

1 Tr= T= + Y= + o q~r. (9)

For these near-equidiffusion flames the boundary conditions (3), (4) become

n ~ +oo: T ~ T~, ~b ~ 0, (10)

n~--- ,~: T - - , ' T * * + Y ~ , ¢ ~ b r. (11)

PREMIXED FLAME IN A COUNTERFLOW 193

n

FRESH ]

n--h

FLAME SHEET

BURNT PRODUCTS

Fig. 1. Coordinate system (the flame location as drawn corresponds to a negative flame speed).

Analysis o f the flame sheet shows that Y must vanish identically behind it, whence the appropriate solution of (7) in this region is

n < h : T = T~ + Yo~, (12)

and the corresponding equation for ~ is homogeneous. Here h defines the location of the flame sheet. In addition the flame sheet analysis pre- scribes jump conditions across the sheet, and these are

5 ( r ) = o, 6(q~) =o, (13)

E oj 6(4') =-)t~i(T') = XY~ exp 2(Yoo + T~) 2 (14)

where ~ refers to values on the cold side (n = h + 0) minus values on the hot side (n ; h - 0).

Equations (7)-(14) provide the complete formu- lation of the problem. It differs from the problem considered in [1] in two respects: Here we are concerned with the entire region - ~ < n < oo rather than n > 0 alone; and 4~r appears as an ad- ditional parameter of the problem.

Use of the variables ~ - ¢p/(Y~ + T~) 2 and (T - T ~ ) / Y ~ shows that this system contains but three parameters:

13, X=- X Y ~ / ( Y ~ + T~) 2,

and ~r = o J ( r ~ + T~)2. (15)


In particular the value of Y~ is not, by itself, sig- nificant, and we could, without loss of generality, assign it the value I as in the earlier analysis [1 ].

Our major concern will be with variations of h with /3 for different values of ~., and we shall be particularly concerned with the limit ~. ~ corresponding to large deviations (on the 0 - 1 scale) of the remote temperature below the adiabatic flame temperature. In this connection we might have chosen to analyze the problem for values of T~ less than, and differing by order O(1) values from, Too + Y~. Such an analysis would, for parts of the response, lead to flame-sheet structure quite different from that considered here, one for which an analytical description is not possible, so that among other things the condition (14b) must be discarded. (The different kinds of structure that can arise in problems of this type are discussed in a different context by Linan [6]). The difficulties and complexities of such an approach are avoided in the present analysis by re- stricting T r to values close to T~ + Y~. Andffet , by considering the nature of the solution as ~ ---~, important qualitative information about the solution for T r - Too - Yoo = O(1) can be un- covered. This approach has its counterpart in the analysis of fuel-drop burning. The general treat- ment of the latter is complicated and requires ex- tensive numerical computations [7]. But if the ambient temperature is restricted to values within O(0 - 1 ) of the adiabatic flame temperature, and then this difference is allowed to go to infinity on the 0 -1 scale, essential features of the general response, including ignition and extinction phenomena, are revealed [8].

Solution of the system (7)-(14) is straightfor- ward and leads to the following results. Defining

~" = nx/ff, ~'o =h X/if, (16)

the value of ~ at the flame-sheet, denoted by ¢~, is given by

~f =.~1 (~r erfc (~'o) -- gx ~X erfc (--~'o)

1 × [~'o 2 +-ff -- foe-ro2/rrll z erfc (~'o)], (17)

and the straining rate by

Vr~-= ~V~ e~o 2 erfc (~'o) exp [ ~ ~ f ] . 2

(18)

, ~'o varies monotonically along the h-/3 response curve and the location of points on the latter is most easily determined by assigning ~o, calculating/3 from (17) and (18), and thence calculating h from (16b).

The combustion field is described by the formulas

× erfc (-~')/erfc (-S'o), (19)

n > h: T = T~ + Yoo erfc (f)/erfc (fo), (20)

= ~f erfc (f)/erfc (~'o) + x/~erfc2 (~'o)

× [~'e -~'2 erfc (~ 'o ) - foe-~'o2erfc(f)] . (21)

If ~. is not fixed, but is set equal to ~ , the results of [ 1 ] are recovered.

All of the response curves defined by these formulas originate at h = 0%/3 = 0, cross the 13 axis at some finite straining rate, and approach h = 0, /3 --- oo from below. Typical curves for~ = 0 and different values of ~r are shown in Fig. 2; for ~ = 0 and different values of ~ in Fig. 3. The flame speed may be defined as the value of the normal gas velocity at the flame sheet, and so equals 2/3h. Corresponding variations of this quantity are shown in Figs. 4 and 5.

Solut ion in the Limit ~r ~ --oo

The solution described here is valid in the limit 0 --> oo for arbitrarily large (but fixed) values of L~r L. Examination now of the limit t~ r ~ --oo provides an approximate picture of the response for large negative values of ~r (e.g., the value - 2 4 used to construct one of the curves in Fig. 2).

We shall confine attention to the case ~ = 0, since this contains those ingredients of the general case that we wish to uncover. Different parts of the response are determined by different dis-


h

2 0

I0

0

[ - 2 4

- 4 q

~r = 0

- I 0

- 8

- 2 0 - 2 0

16 -12

12 - 2 0

- 5 - 4 - 3 - 2 -I

In B Fig. 2. Var ia t ion o f f l ame loca t ion wi th s t ra ining rate, "~ = O.

0 I

tinguished limits in which ~'o is taken to be a function of ~r-

(i) {'o = O(~r-1)

When ~'o vanishes,/3 has the value/3o, where

130 = exp [~gSrl. (22)

In the neighborhood of this point

/3 ~/30 exp [-S'o~r/x/-~] • (23)

In this range /3 increases with ~'o and so does h for /3 < /3o e2. When this inequality is first vio- lated, h achieves a local maximum of value -2x/~-/ eX/fO~r (Fig. 6).

(ii) ~o = O(1)

Here we note the exact expressions

It =-- e2~'°2 erfc2 (~'o) exp [½ ~r erfc(~'o)], (24)

/3 4

h = [2~'oe-~'o2/x/-~ erfc (~'o)]

X exp [--¼ ~r erfc (~'o)1, (25)

so that /3 is exponentially small, h exponentially large./3 is an increasing function of ~*o, but h'(fo) is negative when ~'o is positive, and vice versa (Fig. 6).


I0

8

I 6 - -

I i

4 -

2

, , ,

-I.5 - I .0 - O 5 0

In/~ Fig. 3, Variation of flame location with straining rate, ~t = O.

0.5 1.0

(iii) ~'o > 0, ~'o = O x / ~ ( -¢r) When ~'o is large and positive, (24) and (25) may be approximated by

1 ~ exp [½ ~r erfc (~'o)1, (26)

4~'o 2

h ~ 2~'02 exp [ - ¼ ~ erfc (~'o)] • (27)

As ~'0 ~ ~ we recover the adiabatic flame, corresponding to h -~ ~, ~3 -~ 0, 2/3h - 1. Equation (26) shows that ~'(~'o) vanishes when

~'oe--s'o 2~r ~ --2X/-~, (28)

corresponding to

~'o ~ ~ * / ~ r ) , /3 ~ 1/4 In (--~r),

h ~ 2 in ( - -~0 . (29)

Equation (27) shows that h'(~'o) vanishes when

~'oe-~'o 2 ~ ~ --4X/~ (30)

with, again, the estimates (29). Thus in this range of ~'o the response has both a horizontal and vertical tangent (Fig. 6).

(iv) ~'o < o, ~'o = o l n ~ - ( ~ )

When ~'o is large and negative, (24) and (25) may be approximated by

/3 ~ rre2~'o 2 exp [½ ~ erfc (S'o)], (31)

~-oe-~'o 2 h ~ exp 1--¼ ~r erfc(~'o)], (32)

Then /3'(fo) and h'(fo) vanish at separate points whose location is defined to leading order by

fo - le -~ 'o2~ r ~41r, (33)


1.0

f I I I I

0.5

0

Qf -0.5

-I.0

-I.5

- 2 . 0 t ~ ~ ~ ~ - -3.0 -2.5 -2.0

-4

-20

-24

-1.5 -I.0 -0.5

In,B

~r = O

Fig. 4. Variation of flame speed with straining rate,'h = 0.

0 05 1.0

corresponding to

~o - - ~ ,

13 ~qS~ 2 exp(2 + ~) /16 in ( - ~ ) ,

_1 ~). h ~ 4 in ( - -~) exp (--1

Thus both/3 and h have local minima for this range of S'o (Fig. 6). This minimum value of/3 is much smaller than Go, and the minimum of h is much larger in magnitude than the local maximum un- covered for ~'o = O(Or-1) •

(v) ~'o < o, ~'o = o ( x / 2 ~

For large negative values of ~'o, (31) and (32) may be simplified further to

I3 ~ 7r exp (2~'o 2 + ~ ) ,

~'o h ~ - exp (--~'o 2 - ½~r), (36)

so that for O(1) values of (~'0 2 + ½~r ) [i.e., O ( x / ~ r ) values of~'o ] ,/3 is O(1) and h is O(x/~-~.).

These limiting conclusions are in accord with (34) the numerical calculations of Fig. 2 and reveal the

classical phenomena of ignition and extinction. On the upper branch the flame speed is equal to 1, to leading order, and the flame temperature 47f is O(1). An increase in ~ drives the response toward A (Fig. 6), where these estimates are still valid, and any further increase in/3 will cause the flame to jump to A', where ~f is O(~). On this lower branch the flame temperature is everywhere much less than the adiabatic flame temperature and the reaction rate is correspondingly much lower. We may legitimately infer that when Tr is less than

(35) T~ + Y~ by an O(1) amount, the reaction rate on


Qf

3

0

I I I I I

- 12

- 4

- 8

= 12-

- I . 5 I I I \ \ \

- 1.0 -0.5 0 0.5

In,8 Fig. 5. Variation of flame speed with straining rate, ~r = 0.

1.0 1.5

the lower branch is several orders of magnitude less than that on the upper branch.

If at A ' the straining rate is decreased, ignition will occur at the point B with transition to the point B' on the upper branch.

It is the extinction phenomenon that is o f interest insofar as our flammability limit model is concerned [3]. True, the present analysis does not predict complete extinction, in the sense of no steady burning, but the latter is mathematically an elusive phenomenon. For example, fuel or mono- propellant drops undoubtedly cease to burn if the Damkohler number (D) is reduced sufficiently, and yet the usual mathematical model of this situation, the counterpart to the one examined here, does not predict elimination of all reaction at the so-called frozen limit, but rather predicts

weak burning in the far-field at essentially the ambient temperature, with a positive heat flux to the environment if the adiabatic flame temperature exceeds the ambient. Of course, weak burning would be turned into true extinction if a cut-off temperature was introduced into the reaction term.

3. ONE-DIMENSIONAL STABILITY ANALYSIS

The discussion of Section 2 is predicated on the as- sumption that the portion of the steady state response between the points A and B (Fig. 6) is not physically attainable, in contrast to the remainder of the response. There is good reason to believe that this is essentially true, but only a stability analysis can confirm it. Moreover, it is known that


for some combustion systems stability considerations can move the extinction point away from the vertical tangent point if the Lewis number differs from 1 [9]. Thus, in this section, we shall examine the stability of the steady state when subject to in- finitesimal one-dimensional disturbances. Some preliminary calculations of this type were made in [1 ] when Or = ~,.

To the steady state equations (1), (2) are now added terms a T / S t , a Y / 3 t , and small disturbances are introduced by writing

h = h s + ee w t fi, T = T s + ee w t] '(n),

ga = ga s + ee ~ t~ (n) (37)

and linearizing with respect to the small parameter e. Here the subscript s denotes the steady state solution. We are concerned with identifying those portions of the response for which the real part of co is positive, for then instability is assured.

It is convenient to introduce new variables and parameters by the formulas

- - exp - , T hYo~

~e--~'o 2 A = - , K 1 =

/3 X/~- erfc (~'o)

e - - ~ - o 2

K~- 2 ~

X/~ erfc (~'o) /

(38)

Then it is necessary to solve the equations

r " + 2 ~ r ' - - a r = O o } r ~ ' ~ o ,

• " + 2 ~ ' - " - - A c b - - - X r ,

(39)

subject to the boundary conditions

r - + 0 , ¢ ' - + 0 as~--+oo;

• (~o + o ) - ~ ( ~ o - o) = - x ,

• ' (~o + o) - ,V(~o - o )

= 4~'oX + K 2 ~ + Kx q~(~'o - 0).

a b e 0 a s ~ ' - + - - -~ ,

r(~'o + 0) = 1,

r'(~ o + 0) =--2~ o - -K 2 - - ( K 1 / ~ , ) q b ( ~ ' o - - 0). (40)

When k does not vanish its explicit appearance can be eliminated by use of the variable D/X, leaving the three parameters Ks, K2, and ~'o ; this is equivalent to writing k = 1 in (39) and (40). Since there are seven boundary conditions for the sixth-order system, solutions can only be expected for certain choices of A; instability corresponds to Re(A) > 0.

It is convenient to write the boundary conditions in homogeneous form by using (40e), so that

r - 0 , ~ 0 as~ ' -+~; ~ 0 as~-+---~,

(I'6"o + 0) - ~(~'o - 0) = - ~ , r 6 " o + 0),

* ' (~ 'o + 0) - 'I"(~'o - 0)

= X(4~" o + K2)r(~" o + 0) + KI ~(~" o -- 0),

r'(~" o + 0) = -- (2~" o + K2)r(~" o + 0)

- - ( K ~ / X ) ~ ( ~ o - - 0 ) . ( 4 1 )

Solutions for which r(~'o + 0) vanish are then spurious, but it is easy to show that they neces- sarily correspond to stable solutions.

When k = 0 the eigenvalues are defined by the system

r" + 2~'r' - Ar = 0,

r - + 0 as ~'-+ oo, r '(~'o + 0)

= - (2~'o +K2)r(~'o + 0).

(42)

Numerical determination of the eigenvalues defined by the system (42) or (39) and (41) is easily carried out using, respectively, the Galer- kin method and the method of weighted residuals;

200

h


\ \

.

/ '-I o : O(Oo!_ I / I " :

[Oo

i= o( O(l/In (:~r)) I l [ O(In (-~r)) I

B

!

I,:o, \ I I

~ : o(~;,2exp(2 ÷ ~;,I/,n (-~,11 I l h : O('n(-(~r)exp (-'---'-~ ~r))l

Fig. 6. Response in the limit ~'r -'~ --=~.

#

these approaches worked very well in the similar problem of a flame stabilized in a rear stagnation point flow [9]. Details are given in the Appendix.

These calculations reveal a very simple stability picture. That part of the response that lies between the points of infinite slope A and B (Fig. 6) is always unstable, a real eigenvalue changing sign as the points are traversed. For moderate values of

the remainder of the response is stable, all of it if its slope is finite for all nonvanishing/3. However, for sufficiently large ~, an interval with left limit /3 ~ 0, h ~ oo is unstable, the extent of this interval increasing with increasing~. Indeed, for sufficiently

large ~ the entire upper branch of an S-shaped response can be unstable (Fig. 7). This instability is associated with complex eigenvalues and is easily understood. For ~ > 4(1 + x/~-) (~10.93) one- dimensional adiabatic deflagration (the essential character of the present combustion field in the limit/3 ~ 0, h -~ oo) is unstable, and finite but weak straining will have a stabilizing effect since a small displacement of the flame toward the fresh mixture essentially leaves the flame speed unchanged so that the increased gas velocity at the sheet drives it back. Thus as /3 is increased an everincreasing Lewis number is needed for the instability with


h

15

14- -

13 - -

12--

II -I .800

I -I.775

i i i / / / I / / / / ~ t I

I

I / I / / I / / It t I - I. ' /50 - 1.725 - 1.7OO - 1.675 -I .650

I n B Fig. 7. Stability near the extinction point for large X, Cr = -8.

~ S T A B L E

. . . . UNSTABLE

I -I.625 -I.600

which it is associated to overcome the stabilizing effect of the strain. These Lewis numbers are sufficiently large that the instability may seldom, if ever, be manifest physically. This is in contrast to the rear stagnation point problem [9], for which the strain decreases the Lewis number for which instability is first achieved.

CONCLUDING REMARKS

The steady state analysis of a premixed flame in a counterflow of fresh mixture and hot products [the latter at a temperature close, on an O(1) scale, to the adiabatic flame temperature] reveals classical ignition and extinction phenomena if the product stream is sufficiently cool. The quantita- tive details of these conclusions are not altered by

one-dimensional stability considerations for all but unphysically large Lewis numbers. This then provides a mechanism for extinction both for turbulent flames (at least those that can be modeled using laminar flamelets) and for upward propagation in flammability tubes [3]. If burnt gas that is generated at earlier times has an opportunity to cool, it can then effectively quench the flame. In the case of flammability limits, the question whether cooling mechanisms of sufficient magnitude exist is an important one, unresolved at the present time.

APPENDIX-STABILITY CALCULATIONS

)~=0

The spectrum defined by (42) can be found using

202

a Galerkin method. Defining

~n = (an~ n + bn~n-1) e-~2 ,

an = (1 -- n)~'o n - 2 -- K 2 ~'0 n -1 ,

bn = n~o n -1 + K2~'o n, (A.1)

r can be approximated by

M

T ~ TM = Z An t~n. (A.2) n = l

The residual is defined by

RM = TM't -I- 2~T M' -- ATM, (A.3)

and the coefficients A , are determined by the orthogonality condition

f ~ R M ~ i d ~ = O , ] = l , 2 , ' " , M . (A.4)

Nontrivial solutions only exist for certain discrete values of A.

~,-¢o To calculate the eigenvalues defined by (39) and (41) we set ~ equal to 1 whenever it appears ex- plicitly and write for ~" > ~'o

r ~ TM = e--(~'--to )2 AI(~" -- ~'o)

1 (Ax +BI C1)+ E An(~--~o) 2~'o n = 2

(a.5)


q~ ~ ¢I~M+ = e--(~'--~'o) 2

E A1 X BI(~ ' -~ 'o) 2~oK1

× (4~" o + K 2 + K 1)+ (C1 - B 1 )

2~'oK 1

× (2~0 +K2 +K1) + ~ Bn(~-- ~o) n=2

(A.6)

and for ~" < ~'o

I A1 O ~ ¢ M - - = e - - G ' - - i ' o ) 2 CI(~'--~'0) 2~.oK x

(C 1 -- B 1 ) × (4~'o + K2) + (2S'o + K2)

2~oKa

+ ~ c . ( f - ~ o ) " • n=2

(A.7)

These satisfy all of the boundary conditions. The residuals are

R 1 = T M ' ' + 2~7" M' -- ArM, (A.8)

r¢ R2 = q~M +' ' + 2~'~M +' -- AqbM+ + r M , (A.9)

R3 = qbM--" + 2~'~M--' -- A ~ M - . (A.10)

TABLE 1

Confirmation That A = 0 Is an Eigenvalue at Points of Vertical Tangency for ~¢ = -6,'~ = 0

~0 -0.37 -0.38 -0.39 -0.40 -0.41 3 0.030392 0.030379 0.030378 0.030390 0.030415 A 0.07728 0.02767 -0.02169 --0.07078 -0.11955 ~0 0.96 0.97 0.98 0.99 1.00 3 0.089557 0.089587 0.089600 0.089596 0.089576 A 0.07845 0.04367 0.00913 --0.02518 -0.05924

P R E M I X E D F L A M E IN A C O U N T E R F L O W

TABLE 2

Confirmation That A = 0 is an Eigenvalue at Points of Vertical Tangency for ~'r = -6, '~ = 8

203

~o 1.50 1.51 1.52 -0.23 -0.24 -0.25 fl 0.02891555 0.02891883 0.02891882 0.00457865 0.00457736 0.00457866 A 0.1140 0.0375 -0.0375 0.05982 0.00007 -0.058484

TABLE 3

Confirmation That the Results for Small~ Determined from (A.11) Are Consistent with Those for'~ = 0 from (A.4), ~'0 = 0.9, ~'r = - 6

X -0.200 -0.020 -0.002 0 0.002 0.020 0.200 A 0.2460 0.2868 0.2909 0.2914 0.2918 0.2960 0.3383

A is then a generalized eigenvalue o f the system

f~Rl (~- -~o)J - ld~ =0,

f oo R2(~" ~'0) i - 1 d~" O, o

f ~'0 R3(~ _ ~ ' 0 ) y - 1 d~" = 0, (A.11)

where ] = 1, "", M.

TABLE4

ConvergenceoftheLargestEigenvalueinFourCases a

M A{a) A(b) A(c) A(d)

1 - - 1.366335 2.428083 -0.15032 2 3.171481 -0.3268024 -1.703066 0.31144 3 4.093459 -l .113251 -1.599506 0.28437 4 4.366677 -1.135761 -1.609482 0.29202 5 4.435154 -1.142369 -1.608782 0.29117 6 4.449189 -1.142007 -1.608820 0.29139 7 4.451711 -1.142187 -1.608817 0.29136 8 4.452127 -1.142162 --1.608817 - - 9 4.452191 -1.142166 - - - -

10 4.452199 - - - - - -

a (a) ~ = -8 , '~ = 4, ~'0 = 0.7; (b) ~'r = - 4 , ~ = - 2 , rO = 1.1; (c) ~r = -2 , '~ = 6, g'O = -0.3; (d) ~r = -6 , 'k = O, g'O 0.9.

R E S U L T S

Certain e lementary expecta t ions were verified as a

check on the numerical computa t ions . Thus there

must be a zero eigenvalue at points where the

steady state response has a vertical tangent; and

the eigenvalues defined by (A.11) must recapture

those defined by (A.4) a s~ ~ 0. Typical results are

shown in Tables 1-3. Typical convergence checks

are shown in Table 4.

This work was supported by the National Sci- ence Foundation, NSF ENG 78-28086, and by the Army Research Office.

N O M E N C L A T U R E

B see (2)

h flame sheet locat ion

Ka,2 see (38) L Lewis number

n distance f rom stagnation point (Fig. 1)

Qf flame speed

T tempera ture

Y mass fract ion o f combust ib le mix ture

fl straining rate

e small parameter ; see (27)

¢ see (6)

~. see (15) qb see (38)

X see (5)


see (15) A see (38)

~', ~'o see (16) z see (38) 0 activation energy

see (27) ( )r remote conditions for the hot counterflow

( )s steady state ( )~ fresh mixture ( - ) see (27)

2. Sivashinsky, G. I.,ActaAstronautica 3:889 (1976). 3. Buckmaster, J. D., and Mikolaitis, D., Combust.

Flame (in press). 4. Libby, P., and Williams, F. A., Combust. Flame (in

press). 5. Bush, W. B., and Fendell, F. E., Combust. ScL Tech.

1:421 (1970). 6. Linan, A., Acta Astronautica 2:1009 (1975). 7. Law, C. K., Combust. Flame 24:89 (1975). 8. Buckmaster, J. D., Letters in Applied & Engineering

Sciences 3:365 (1975). 9. Mikolaitis, D., and Buckmaster, J. D., Combust. ScL

Tech. (in press).

REFERENCES

1. Buckmaster, J. D., 1 7th International Symposium on Combustion, The Combustion Institute, Pittsburgh, 1979, p. 835. Received 2 June 1981; revised 15 October 1981

Documents

The Premixed Flame in a Counterflow