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A flame-bubble analogue and its stability
100
COMBUSTION AND FLAME 7 9 : 1 0 0 - 1 0 9 (1990)
A Flame-Bubble Analogue and Its Stability S. WEERATUNGA,* J. BUCICMASTER, and R. E. J O H N S O N University of Illinois, Urbana, 1L 61801
Lean hydrogen-air mixtures confined to a flammability tube can support "flame-bubbles"--very small, highly curved flames that rise because of buoyancy. An approximate description of the flow field associated with a single unbounded flame is provided by numerical solution of the Navier-Stokes equations for distributed heat sources rising at constant speed in a gravitational field. Zeldovich has proposed a theoretical model valid at 0 g (zero Reynolds number) but it has only unstable solutions. We describe a "flame-cap" model, valid for finite g, and suggest that the flow field is an essential stabilizing mechanism.
NOMENCLATURE a
radius of the flame bubble/cap specific heat D diffusion coefficient E activation energy Fr Froude number k heat conduction coefficient L, Le Lewis number Pe Peclet number Pr Prandtl number Q heat release, (Eq. 1)
Cp
Cl
Q/2rkfa T/
Q r r® R Re T T, u x Y
heat release at the flame spherical radius far-field numerical boundary gas constant Reynolds number temperature flame temperature axial velocity axial distance mass fraction
Greek Symbols ot
e ~ is the time dependence of small disturbances
* Present affiliation: Digital Equipment Corporation.
0010-2180/90/$03.50
6 6*
perturbation of the flame location displacement thickness similarity variable for the wake solution
o
ECp/RQ
# o r ¢ ¢i~
viscosity density cylindrical radius stream function stream function for the irrotational outer flow reaction rate far-field (fresh gas)
fi ( )/
I. I N T R O D U C T I O N If a flammable mixture confined to a vertical flammability tube is ignited at the bottom, the size of the flame is usually comparable to the tube diameter. However, for very lean hydrogen-air mixtures, the flame, although initially large, breaks up as a consequence of thermal-diffusive instabilities to form a number of highly curved flames 4 mm or so in diameter [1]. The precise extent of each flame is not known, but the luminous portion is cap shaped. It rises at what appears to be a constant speed with unchanging shape, suggesting that the local combustion field has a stationary structure. We are concerned with the nature of this structure, and with its stability. Copyright © 1990 by The Combustion Institute Published by Elsevier Science Publishing Co., Inc. 655 Avenue of the Americas. New York, NY 10010
A FLAME-BUBBLE ANALOGUE There is evidence [2], which we shall discuss later, that instability occurs at sufficiently small Reynolds numbers. We will identify stability mechanisms that increase in effectiveness with increasing Reynolds number, and we suggest that these may be responsible for the stable flames observed in a standard gravitational field. The layout of the article is as follows. In Sec. 2 we describe flow field calculations in which the flame is represented by a distribution of heat sources. In this way we can achieve important qualitative insights into the nature of the physical flowfield. In Sec. 3 we describe the Zeldovich limit, in which there is no convection and the combustion field is one dimensional (spherical). We believe that this is the correct steady solution at zero gravity, but it is unstable and so cannot represent a physical solution. In Sec. 4 we examine the neighborhood of the Zeldovich limit, adding convection terms and providing a mathematical description in a one-dimensional context by restricting our description to the axis of symmetry, accounting in an approximate fashion for off-axis diffusion. This shows that uniform convection can suppress the Zeldovich instability at very modest Reynolds numbers. Finally, we note that a mass flux gradient identified by the calculations of Sec. 2 will provide an additional stabilizing influence. 2. AN A P P R O X I M A T E D E S C R I P T I O N OF THE FLOW FIELD As we have noted, a flame bubble is very small. It also tends to be well separated from its peers. It is natural then to seek a solution for the flow field of a single unbounded flame. To carry this out precisely is a difficult task. It requires solution of the Navier-Stokes equations coupled to a temperature equation, accounting in an appropriate fashion for chemically generated heat. The rise speed would be determined by such a calculation. We have carried out calculations that are substantially simpler. By representing the flame as a continuous distribution of heat sources, rising at an assigned speed, solution of the chemical specie equations is not required, nor is it necessary to deal with reaction terms.
101
I U
/
/
f a ! !
! I
v
! I
X Fig. 1. Flamecap (dotted line defines the configuration for the numerical simulations). In principle, proper selection of the rise speed and the source distribution will lead to an exact simulation of the physical flow field. Our results suggest that the essential details do not depend critically on the source distribution, so that the simplest choice will suffice for our essential purpose, that of qualitative insight. We have chosen sources distributed over a spherical surface in such a way that the surface temperature is constant; this surface is transparent to the flow. This configuration, sketched in Fig. 1, has the advantage that the temperature and density within the sphere are constants. Ronney [3] has observed spherical flame bubbles in microgravity experiments, an issue that we will return to later. A full description of the numerical techniques and results is given in Ref. 4, and we shall just summarize the main results here. The equations governing the flow are the Navier-Stokes equations for a variable density (but incompressible) fluid, coupled to the energy (temperature) equation by Charles's law, appropriate for a small Mach number flow. Finite differencing leads to a set of nonlinear equations which are solved using Newton-Raphson iteration. Direct and iterative schemes are used to solve the nonsymmetric linear equations corresponding to the Jacobian matrix. A stream-function-vorticity formulation is adopted, and a nonuniform but
102
S. WEERATUNGA ET AL.
fixed grid is used, chosen according to reasonable preconceptions about the nature of the solution. When the equations are nondimensionalized using 7"i, p/, py, U, and a, three important nondimensional parameters are defined: the Reynolds number Re = p/Ua/#y, the Froude number Fr = U/x/~, and T., the " f l a m e " temperature. These cannot all be arbitrarily assigned, as we shall see. The boundary conditions far upstream correspond to the uniform supply conditions; downstream the situation is more complicated. As noted in Refs. 4 and 5, the heat released into the flow at the sheet is swept into the wake, and the buoyancy forces acting on it generate a velocity defect that persists. In the limit x -* ~ an asymptotic wake solution can be constructed using the following variables: ~="-~x
'
2x ff=~eef(~/), v=
= l + F r 2 • t(~)x ' l 0~ u-o Oo--f"
1 a¢
---
2
oOX R e r
(r/f'-f).
v/--*oo: f ' ~ l ,
~lt'+Prft=O,
(1)
(2)
t-*O, .T-*O: f = 0(7/).
In addition, the heat flux in the wake originates at the " f l a m e " so that
O cl=21rkfaT / = 2 Pr Fr 2
I"0 f ' t d~.
(3)
Some insight into the nature of the wake solution can be achieved by considering the special case when the Prandtl number Pr equals 1. Then it is easily shown that f " ~ t, and the axial velocity on the center line is given by the formula f ' ( 0 ) = [1 - dl/Fr2] ~a.
(~'6")2 =
f
oo
(1 -Ou)27ro do 0
=4xx Re
f"0 (1 - f ' )
dT/.
(4)
(5)
Because f(~/) - r/ - Co + o(1)as r/ --' oo, where Co is a numerically determined constant, it follows that 6" = 2(c0/Re)l/2"~x.
This leads to a two-point boundary value problem that describes the flow as x -* 0% o -* oo, o / v ~ fixed: ~f~' + (1 + f ) f " - t = 0,
Thus for a given heat release there is a minimum Froude number for which a solution exists. Presumably at smaller Froude numbers, buoyancy forces are so strong that the heat is carried away in a plume above the source, which makes no physical sense in the present context. For values of Pr different from 1 the Froude number must still be greater than some minimum value determined by tl, but this minimum must be determined numerically. The wake displacement-thickness is defined by
(6)
Thus the flow exterior to the wake is, to leading order, the inviscid flow past a paraboloid of revolution. This can be constructed by superimposing a uniform flow and a line of uniform sources of strength q0 on 0 __< x < o% o = 0. Thus
1 a2_q0 [(x2+a2)~/Z+x]+... ffi"=2 4~"
(7)
valid as x ~ ~ , o/x fixed. This matches with the wake solution provided q0 = 4~rc0/Re. Correct conditions at the downstream computational boundary should properly incorporate this description of the wake and the exterior inviscid flow; this automatically includes the restriction on the Froude number. Related problems (e.g., flow past a heated sphere) have sometimes been solved without accounting for the wake, but this cannot be correct if the goal is to emulate an unbounded flow. It is true that Badr [6] neglected the wake and achieved excellent agreement with experimental results [7], but a fan was located at the downstream experimental boundary.
A FLAME-BUBBLE ANALOGUE
103
|
1 ~
0
I
I
. . . ............ ......
"--'-'--
-2
i
I
i
,
L
-I
0
I
2
3
4
Fig. 2. Streamlines for Re = 5, Pr = 1, T . = 2.5: Fr = 2 . 4 2 , q = 5 . 8 4 (calculated).
We have generated two sets of numerical results. The first set corresponds to a correct prescription of the wake with the Froude number (determined iteratively) having its minimum value, so that the axial velocity on the wake centerline is zero. This case corresponds to a flame bubble rising at its minimum speed for fixed heat release and size, and the flow disturbance generated by the flame will be greatest. However, in all cases the Froude number is relatively high and the streamlines are not grossly perturbed. Fig. 2 is typical of the results obtained for values of T , between 1.5 and 6, Re between 1 and 10. The "flame" displaces the fluid to the side, decreasing the mass flux in the x-direction. At the same time, however, the density drops because of heating, and the net effect is an acceleration of the fluid. Subsequently the flow is retarded downstream by buoyancy forces and cooling, the velocity decay~
!
|
!
!
!
0
5
10
].5
20
ing monotonically to zero along the centerline. This is in marked contrast to the flow field produced by a heated solid sphere, where the pressure field can produce some downstream acceleration. In order to highlight the effect of the Froude number constraint, solutions have also been obtained by ignoring the wake and imposing uniform downstream conditions (at a finite location of course). Figure 3 is typical of results obtained this way, and shows a closed region of recirculating flow. The size and position of this region depends on the parameter values and, for some choices, can completely enclose the " f l a m e . " These solutions depend strongly on the location of the downstream boundary, consistent with results of Shee and Singh [8] for natural convection around a heated sphere. This is marked contrast to the first set of results for which an increase in r~, beyond i
!
I
30
35
i
!
40
45
lO
-5
Fig. 3. Strcam]incs f o r Rc = 5, Pr = (calculated).
25
I, T ,
= 2.0, Fr =
1.5, r ,
50
= 60: q =
3.13
104
S. WEERATUNGA ET AL.
- 25 yields little change. Evidently the unbounded
flow and corresponding wake solution has a character quite different from the bounded flow. The latter is probably not directly relevant to the flame-bubble problem, i.e., physically realistic solutions can only be obtained by properly accounting for the wake at the downstream computational boundary. A final note: The numerical results show that the scale on which the flow is disturbed is significantly larger than the "flame sphere," and gradients are everywhere small. This suggests that we would obtain very similar results for different heat distributions provided the global parameters (q, Re, Fr, etc.) are unchanged. 3. THE Z E L D O V I C H LIMIT A N D THE STABILITY QUESTION
Most observations of flame bubbles have been in a standard gravitational field. The Zeldovich limit [2] provides the most plausible description of the configuration to be expected under microgravity conditions. In the limit there is no motion (in which direction would the flame move?) and the combustion field is spherically symmetric. Then the velocity field is identically zero and if we adopt a simple kinetic model for which the reaction rate f~ depends only on the mass fraction of the deficient component (hydrogen), denoted by Y, together with the temperature T, then the combustion field can be defined by the equations
0 pDd(dY) =7
drr r2 ~
- fl'
(8)
r--,~: T--,T/, Y-,Y/. If we assume that reaction ceases behind the flame because Y vanishes, it is easily deduced that the flame temperature is
T, = 7"/+O Y/ CuL'
(9)
where L is the Lewis number, which is significantly less than unity. This diffusion-controlled flame temperature is also achieved for tubular
flames of vanishingly small radius [9], and is larger than the adiabatic flame temperature Ty + QYI/Cp. Such elevated temperatures are always possible when the deficient component of a mixture is much lighter than the mixture itself. Note that in any region where there is no significant reaction (particularly the far-field), any mass fraction (hydrogen, oxygen, product gas, etc.) has the mathematical representation
Yi= Ai+ Bi/r so that there is a steady diffusive flux (r2a Yi/Or) of reactants from infinity, and a steady flux of products to infinity. The remark of Goldmann [10], that these flames must move to avoid local exhaustion of reactants, is not correct. Unfortunately, the solution of Eq. 8, of which Eq. 9 is an ingredient, appears to be unstable. This is revealed by adding time derivatives and carrying out a linear stability analysis, albeit for a specific and very simple choice of fl, [2, 4]. The flame either collapses or it expands, a fact that has been exploited by Joulin in his ignition studies, [11]. The most reasonable interpretation of this result is that extinction occurs. In the case of collapse the flame radius shrinks to zero; in the case of expansion the decrease in flame curvature lowers the flame temperature to a level at which combustion cannot be sustained. It is in the nature of flame bubbles that they are observed only for mixtures so weak that plane propagation is not possible-the adiabatic flame temperature is too small to sustain the process. These observations raise an interesting question: If at 0 g, when the Reynolds number is zero, there is a fundamental destabilizing mechanism, why are stable flames observed at 1 g, when the Reynolds number is ca. 5-7? What happens to the destabilizing mechanism revealed in the Zeldovich limit as the Reynolds number is increased? The remainder of the article is devoted to this question. 4. A MODEL FOR FLAME BUBBLES
I N C O R P O R A T I N G CONVECTION In the absence of gravity it is natural to seek a spherically symmetric description of the combustion field. In the presence of gravity, buoyancy forces will generate motion of flame, and the
A FLAME-BUBBLE ANALOGUE
105
adverse convective flux of mixture at the rear will reduce the vigor of the reaction there. At large enough rise speeds this will be sufficient to extinguish the rear portion of the flame, so that reaction is confined to an open cap (Fig. 1), rather than a closed surface. We initially proposed this model in [12], but did not seriously address the stability question. The mathematical ingredients of the flame-cap model reside in the equations
/ d2T 2 d T )
c.p. d T = k ~--~x2 +x-~x +Qf~,
(
pu -~xx=PD \ ~ x 2+-x-d-xx - f l ,
(10a)
(10b)
which we adopt as an approximate description of the combustion field on the axis of symmetry (o = 0) for x < 0. The mass flux on the centerline, pu, is an assigned function o f x that we will take as a constant in the initial part of our discussion; later a more realistic choice will be described. The derivatives on the right side are approximations to the Laplacian on the centerline, with the first derivatives accounting for off-axis diffusion generated by the curvature of the combustion field. The rationale for this has been discussed previously [12], and can be briefly summarized: (i) the equations need only be valid over the range o f x that defines the flame thickness (5; (ii) at 1 g, 6 is significantly less than a, the cap radius, since Re a/6 where Re, the Reynolds number, is approximately 5-7 for the 1 g flames (the estimate comes from noting that the mass flux ( M ) through the flame is - # R e ~ a , the flame thickness is - k / MCp, and the Prandtl number is - 1); (iii) when (5 is significantly less than a, surfaces of constant Y and T within the flame are, to a good approximation, locally spherical with a common center, the situation for which the model is exact; items (i)(iii) combined imply that the model is a reasonable approximation for 1 g flames; (iv) the model is also correct in the limit Re ---, 0, for it then reduces to the 0 g Zeldovich limit, Eq. 1. For these reasons, we adopt the model as a plausible interpolation, valid for 0 g to 1 g flames. To complete the description, a choice must be
made for ft. We shall adopt a flame-sheet model for which fl is a Dirac 6 function whose strength is such that
1 [,Y]
= Be- E/2Rr,,
[ ' ] ~ (')hot--(')cold,
( 11a) (llb)
where T . is the flame temperature. Arrhenius kinetics for large activation energy leads to such a model, but it has also proven valuable in the discussion of finite activation energy effects [ 13], so that there is no assumption here that E/R T, is large. In the steady state the 6 function is located at x = - a, where a, in general, is related to the rise speed. Steady Solution When pu -- constant The steady solution is easily found for any choice of the function pu, but we shall write down results only for pu = constant. The key characteristics are the flame temperature, and the flame radius, given by the formulas
T,-T/=-QY/ G(Pe) G -1 (LPe)
CpL
(12a)
a = pD Yf ee/2Rr*G- i (LPe), B G(P)
--
I~ ' d x e P(x+ 1)
(12b)
where Pe = PrRe is the Peclet number and Re = pua/#. In the limit Pe ~ 0 we recover Zeldovich's result (Eq. 9) with
a =oD ~ e e/2Rr*,
(13)
B
its minimum value. The radius a can be treated as a free parameter of the model and can be assigned any value not less than this, with Eq. 12b determining the rise speed. In the limit Pe ~ ~ , T, is equal to the adiabatic flame temperature, and
a = pD Yy LPeeE/2Rr. ' B
(14)
106
S. WEERATUNGA ET AL.
AT
0
2
4
6
~,4~" !0
8
Pe
Fig. 4. A T , (T, - T/)Cp/Y/vsPeforLe = 0.2(0.1)0.7. which, since Pe is proportional to a, is actually a formula for the adiabatic flame speed. Values of T . for intermediate Pe are shown in Fig. 4. At the values relevant to 1 g flames (Pc 3-5) there is significant enhancement over the adiabatic flame temperature, but much less than that achieved in the Zeldovich limit.
Stability Analysis As we noted in Sec. 3, the steady solution for Pe = 0 is unstable. We want to examine the effect of finite Pe on this result, and some progress can be made on this problem when the mass flow pu is constant. The additional terms pCpOT/Ot and oOY/Ot must be added to Eq. 10a and 10b, and a linear stability analysis can be carded out in the context of the constant density model. This uncouples the fluid mechanics from the thermal processes (the flow affects the flame but not viceversa) and has proven valuable for problems where the coupling plays no important qualitative role [141. Because, even now, the constant density approximation is sometimes misunderstood, it is worth adding some additional remarks. Clearly the density is not constant, and is not so approximated, in the base flow whose stability is of concern; it is, after all, the buoyancy forces that are responsible
for the flame motion. The constant density approximation is only applied to the linear equations that describe infinitesimal perturbations to this base flow. The essential idea is that these equations contain a number of physical ingredients, but only a subset of these are responsible for generating a specific mode of instability. For example, hydrodynamic effects are responsible for the DarrieusLandau instability of plane flames; the differential diffusion of heat and mass (of the deficient species) is responsible for the cellular instability. We can learn a great deal about the first by ignoring diffusive effects, and about the second by ignoring the hydrodynamics. This idea plays a key role in intuitive discussions. The usual "physical" explanation of the cellular instability invokes only the curvature of the combustion field and the diffusion of heat and reactants--no mention is made of the hydrodynamics. A mathematical treatment that ignores the hydrodynamics is, therefore, in the same spirit, but provides a much richer description of the consequences of nonhydrodynamic interactions. Many examples that support this claim are discussed in Ref. [13], and [141. In the present context, when we adopt the constant density approximation, we recognize that we may be discarding a hydrodynamic instability (after all, perturbed buoyancy effects have been discarded along with the momentum equation), so that on that basis alone our discussion is incomplete. But we will find that those physical ingredients retained can lead to an instability (thermaldiffusive in origin) that is affected by the Reynolds number of the base flow. It is useful to nondimensionalize the equations, and to do this we use the characteristic length a, time a2oCp/)~, and temperature Q/C.. Then infinitesimal one-dimensional perturbations with a time dependence e '~t satisfy the following equations on each side of the reaction zone:
dx2+
-Pe
- dx 2 +
--a~r-- et Tl = 0
- LPe
~-
Let YI.
(15)
A FLAME-BUBBLE A N A L O G U E
107
The equation for 7"1 has independent solutions
simplifies to
Yt=eX~l Jo dse~X \ ~ - ~ ]
2
'
(16a)
Y2 = f"s2 J dseSX - kS-S2~
(16b)
where sL2 = Pe/2 +_ ¶/2, ¶ - (Pe 2 + 4a) I/2, and, for complex or, the integral defining Y2 is in the complex s-plane. Yt (unlike Y2) vanishes as x - o o and so describes the cold-gas perturbations; Y2 (unlike Yt) is bounded as x --, 0 and is appropriate for perturbations in the hot gas. The corresponding' solutions for Y~ are obtained by making the substitutions ct --, otL, Pe ~ PeL. The perturbed flame-sheet is located at x = - 1 + tS, and the linearized connection conditions there are T,t = {ST o + 7"1}-I±0, (17)
0 = {tSYo + Y1}- I-0
k [tSTo' + T~ ] = Cp
BaO T , i
e -°/2r*o
2 T , o2 p
=PD[SYo + Y1 l, where 0 = E C p / R Q and ( )o denotes steady-state values. When the general solutions for Tt and Yl are substituted into Eq. 17 a homogeneous set of equations is obtained that have nontrivial solution if the following dispersion relation is satisfied:
(
- Yo + Yo YIL +(-To
-I-0
\Ky2
+ ToY(ly,/_,_o=O,
Kyl
,)
L
(18)
where YlL is yl with the replacements et --, a L , Pe ---, PeL, and
K ~-
Ba 0 -e-°/2r*o. pD 2 T , o 2
(19)
When the Reynolds number is zero, Eq. 18
7",02 0
(1-x/-~L)(l+cothx/~)
= (T,0 - Tf)(I - x/L).
(20)
The left side ranges from +oo to - o o as a increases through real values from 0 to + oo, so there is always a real positive root. When L = 1, for example, c~ = 1. As noted earlier, the Zeldovich solution is unstable. Numerical methods can be used to follow the movement of this root in the complex c~ plane as Pe changes. This is easily done when the root remains real, which is true for small Pe. We have found that, as Pe increases, the root moves to the left along the real axis into the left half-plane, so that at some critical value of Pe, ~ becomes negative, and the mode is stable. For example, with L = 0.4, Yf = 2.9 x 10 -3, Tf = 1.3 x 10 -3, 0 = 2.53 x 10 -2 (the adiabatic flame temperature is 3.23 times the temperature of the remote gas and 0.17 times E / R ) the critical Peclet number is approximately 1.05. That is, f l o w
effects, even f o r constant pu, eliminate the Zeldovich-limit instability at quite modest Reynolds numbers, and this alone could be responsible f o r the stability o f the 1 g flames. At the same time it is possible that flow effects could introduce their own instabilities, corresponding to additional roots of Eq. l 1. A complete search for these roots in the complex plane is not a simple task and we have not attempted it, since there are other aspects of the flow, not accounted for here, that will have a strong stabilizing effect.
The Centerline Mass Flux So far in this section we have taken pu on the centerline to be constant. In fact, it will vary with x, and these variations can be determined from the numerical computations of Sec. 2. In Fig. 5, the centedine mass flux is plotted against x for a representative set of parameter values (again, with the Froude number equal to its minimum). There is a significant gradient in the neighborhood of the
108
S. WEERATUNGA ET AL.
X 0.1m
I 0.400.
0.200
• 41
i ,4
-
| .~
-
i -2
-
i -I
-
i O
-
i I
•
i 2
•
i 3
•
i 4
• 5
x
Fig. 5. Mass flux at the centerline vs x for Re = 5, Pr = 0.72, T, = 4: Fr = 3.35, q = 9.66 (calculated).
flame at x = - 1 , especially upstream in the preheat zone, and this will have a strong stabilizing effect. Expansion of the flame corresponds to a leftward displacement into a region of increased mass flux, which will drive it back. Contraction and displacement to the right will similarly be resisted. This is so clear on intuitive grounds that we provide no mathematical evidence, but the stability analysis can easily be modified to include a perturbation gradient, and this shifts the root towards stability, as expected. 1. CONCLUDING REMARKS In this article we have provided an approximate description of the flow field associated with a single unbounded flame bubble. Although, in general, buoyancy forces can generate closed regions of recirculating flow, the naturally arising restriction on the Froude number prevents this. In the Zeldovich limit, which we believe corresponds to the stationary solution under microgravity conditions, the flame is unstable. We have explored, albeit tentatively, possible stabilizing mechanisms that could account for the stable flames observed at 1 g. These stabilizing mechanisms are only relevant in the context of the flamecap model, where it is assumed that the Reynolds number is large enough to ensure extinction of the rear portion of the flame. Our results suggest that a sufficiently large uniform convective flux through the flame can overcome the instability that prevails
at zero Reynolds number (zero convective flux). An additional stabilizing mechanism is revealed by the numerical simulations which show that the centerline mass flux decreases significantly in the neighborhood of the flame. Put another way, we might expect, on intuitive grounds, that a flame bubble could act as an ignition source, giving rise to an expanding unsteady combustion-field. Experiment shows that at 1 g this does not occur, implying the existence of a preventive mechanism. We have argued that buoyancy-induced convection is such a mechanism. Experiments under low-gravity conditions that could confirm or refute our discussion appear, at the present time, to be inconclusive. Ronney [3] has carried out experiments in the 2.2-s NASA drop tower. Lean hydrogen-air mixtures are ignited by a spark. As the flame propagates radially outwards, it breaks up into cells, which subsequently form what appear to be closed spherical flame bubbles. Throughout most of the 2.2 s there is unambiguous radial motion from the ignition source, but at the end of this period the bubbles are moving very slowly. Only a longer test period can reveal whether or not the bubbles survive as stationary objects. If they do, there must be an additional stabilizing mechanism not identified here; if they do not, that is consistent with our analysis. It is, of course, possible that there is an unidentified stabilizing mechanism responsible for stability at zero Reynolds numbers that is inadequate under 1 g conditions.
This work was supported by the Air Force Office o f Scientific Research. J. Neves provided valuable assistance with the stability computations of Sec. 2. REFERENCES I.
2.
Lewis, B., and yon Elbe, G., Combustion Flames and Explosions of Gases, Academic Press, New York, 1961, p. 314. Zeldovich, Ya. B., Barenblatt, G. I., Librovich, V. B., and Makhviladze, G. M., The Mathematical Theory of Combustion and Explosions, Consultants Bureau, New York, 1985, p. 327.
S. W E E R A T U N G A 3. 4. 5. 6. 7.
8.
ET AL.
Ronney, P., Private communication. Weeratunga, S., Ph.D. thesis, University of Illinois at Urbana-Champaign, Urbana, IL, 1986. Buckmaster, J. D., and Weeratunga, S., Combust. Sci. Technol. 35:287-296 (1984). Badr, H. M., Int. J. Heat Mass Trans. 27:15-27 (1984). Krause, J. R., and Tarasuk, J. D., in Fundamentals o f Forced and Mixed Convection (F. A. Kulacki and R. D. Boyd, Ed.), The 23rd National Heat Transfer Conference, The Heat Transfer Division, ASME, Denver, CO, 1985, pp. 171-179. Shee, Y.-T., and Singh, S. N., in Natural Convection (I. Cotten and R. N. Smith, Ed.), ASME, New York, 1981, pp. 67-74.
109 9.
Takeno, T., Ishizuka, S., and Nishioka, M., Combust. Flame ( in press). 10. Goldmann, F., Z. Physik. Chem. B5:307 (1929). 1 I. Champion, M., Deshaies, B., Joulin, G., and Kinochita, K., Combust. Flame 65:319-337 (1986). 12. Buckmaster, J.. Johnson, R. E., and Weeratunga, S., in Mathematical Modeling in Combustion Science, Springer-Verlag, New York, 1988, p. 112. 13.
14.
Buckmaster, J. D., and Luford, G. S. S., Lectures on Mathematical Combustion, SIAM. Philadelphia. 1983, p. 73. Buckmaster, J. D., Ed., The Mathematics o f Combustion, SIAM. Philadelphia, 1985, p. 26.
COMBUSTION AND FLAME 7 9 : 1 0 0 - 1 0 9 (1990)
A Flame-Bubble Analogue and Its Stability S. WEERATUNGA,* J. BUCICMASTER, and R. E. J O H N S O N University of Illinois, Urbana, 1L 61801
Lean hydrogen-air mixtures confined to a flammability tube can support "flame-bubbles"--very small, highly curved flames that rise because of buoyancy. An approximate description of the flow field associated with a single unbounded flame is provided by numerical solution of the Navier-Stokes equations for distributed heat sources rising at constant speed in a gravitational field. Zeldovich has proposed a theoretical model valid at 0 g (zero Reynolds number) but it has only unstable solutions. We describe a "flame-cap" model, valid for finite g, and suggest that the flow field is an essential stabilizing mechanism.
NOMENCLATURE a
radius of the flame bubble/cap specific heat D diffusion coefficient E activation energy Fr Froude number k heat conduction coefficient L, Le Lewis number Pe Peclet number Pr Prandtl number Q heat release, (Eq. 1)
Cp
Cl
Q/2rkfa T/
Q r r® R Re T T, u x Y
heat release at the flame spherical radius far-field numerical boundary gas constant Reynolds number temperature flame temperature axial velocity axial distance mass fraction
Greek Symbols ot
e ~ is the time dependence of small disturbances
* Present affiliation: Digital Equipment Corporation.
0010-2180/90/$03.50
6 6*
perturbation of the flame location displacement thickness similarity variable for the wake solution
o
ECp/RQ
# o r ¢ ¢i~
viscosity density cylindrical radius stream function stream function for the irrotational outer flow reaction rate far-field (fresh gas)
fi ( )/
I. I N T R O D U C T I O N If a flammable mixture confined to a vertical flammability tube is ignited at the bottom, the size of the flame is usually comparable to the tube diameter. However, for very lean hydrogen-air mixtures, the flame, although initially large, breaks up as a consequence of thermal-diffusive instabilities to form a number of highly curved flames 4 mm or so in diameter [1]. The precise extent of each flame is not known, but the luminous portion is cap shaped. It rises at what appears to be a constant speed with unchanging shape, suggesting that the local combustion field has a stationary structure. We are concerned with the nature of this structure, and with its stability. Copyright © 1990 by The Combustion Institute Published by Elsevier Science Publishing Co., Inc. 655 Avenue of the Americas. New York, NY 10010
A FLAME-BUBBLE ANALOGUE There is evidence [2], which we shall discuss later, that instability occurs at sufficiently small Reynolds numbers. We will identify stability mechanisms that increase in effectiveness with increasing Reynolds number, and we suggest that these may be responsible for the stable flames observed in a standard gravitational field. The layout of the article is as follows. In Sec. 2 we describe flow field calculations in which the flame is represented by a distribution of heat sources. In this way we can achieve important qualitative insights into the nature of the physical flowfield. In Sec. 3 we describe the Zeldovich limit, in which there is no convection and the combustion field is one dimensional (spherical). We believe that this is the correct steady solution at zero gravity, but it is unstable and so cannot represent a physical solution. In Sec. 4 we examine the neighborhood of the Zeldovich limit, adding convection terms and providing a mathematical description in a one-dimensional context by restricting our description to the axis of symmetry, accounting in an approximate fashion for off-axis diffusion. This shows that uniform convection can suppress the Zeldovich instability at very modest Reynolds numbers. Finally, we note that a mass flux gradient identified by the calculations of Sec. 2 will provide an additional stabilizing influence. 2. AN A P P R O X I M A T E D E S C R I P T I O N OF THE FLOW FIELD As we have noted, a flame bubble is very small. It also tends to be well separated from its peers. It is natural then to seek a solution for the flow field of a single unbounded flame. To carry this out precisely is a difficult task. It requires solution of the Navier-Stokes equations coupled to a temperature equation, accounting in an appropriate fashion for chemically generated heat. The rise speed would be determined by such a calculation. We have carried out calculations that are substantially simpler. By representing the flame as a continuous distribution of heat sources, rising at an assigned speed, solution of the chemical specie equations is not required, nor is it necessary to deal with reaction terms.
101
I U
/
/
f a ! !
! I
v
! I
X Fig. 1. Flamecap (dotted line defines the configuration for the numerical simulations). In principle, proper selection of the rise speed and the source distribution will lead to an exact simulation of the physical flow field. Our results suggest that the essential details do not depend critically on the source distribution, so that the simplest choice will suffice for our essential purpose, that of qualitative insight. We have chosen sources distributed over a spherical surface in such a way that the surface temperature is constant; this surface is transparent to the flow. This configuration, sketched in Fig. 1, has the advantage that the temperature and density within the sphere are constants. Ronney [3] has observed spherical flame bubbles in microgravity experiments, an issue that we will return to later. A full description of the numerical techniques and results is given in Ref. 4, and we shall just summarize the main results here. The equations governing the flow are the Navier-Stokes equations for a variable density (but incompressible) fluid, coupled to the energy (temperature) equation by Charles's law, appropriate for a small Mach number flow. Finite differencing leads to a set of nonlinear equations which are solved using Newton-Raphson iteration. Direct and iterative schemes are used to solve the nonsymmetric linear equations corresponding to the Jacobian matrix. A stream-function-vorticity formulation is adopted, and a nonuniform but
102
S. WEERATUNGA ET AL.
fixed grid is used, chosen according to reasonable preconceptions about the nature of the solution. When the equations are nondimensionalized using 7"i, p/, py, U, and a, three important nondimensional parameters are defined: the Reynolds number Re = p/Ua/#y, the Froude number Fr = U/x/~, and T., the " f l a m e " temperature. These cannot all be arbitrarily assigned, as we shall see. The boundary conditions far upstream correspond to the uniform supply conditions; downstream the situation is more complicated. As noted in Refs. 4 and 5, the heat released into the flow at the sheet is swept into the wake, and the buoyancy forces acting on it generate a velocity defect that persists. In the limit x -* ~ an asymptotic wake solution can be constructed using the following variables: ~="-~x
'
2x ff=~eef(~/), v=
= l + F r 2 • t(~)x ' l 0~ u-o Oo--f"
1 a¢
---
2
oOX R e r
(r/f'-f).
v/--*oo: f ' ~ l ,
~lt'+Prft=O,
(1)
(2)
t-*O, .T-*O: f = 0(7/).
In addition, the heat flux in the wake originates at the " f l a m e " so that
O cl=21rkfaT / = 2 Pr Fr 2
I"0 f ' t d~.
(3)
Some insight into the nature of the wake solution can be achieved by considering the special case when the Prandtl number Pr equals 1. Then it is easily shown that f " ~ t, and the axial velocity on the center line is given by the formula f ' ( 0 ) = [1 - dl/Fr2] ~a.
(~'6")2 =
f
oo
(1 -Ou)27ro do 0
=4xx Re
f"0 (1 - f ' )
dT/.
(4)
(5)
Because f(~/) - r/ - Co + o(1)as r/ --' oo, where Co is a numerically determined constant, it follows that 6" = 2(c0/Re)l/2"~x.
This leads to a two-point boundary value problem that describes the flow as x -* 0% o -* oo, o / v ~ fixed: ~f~' + (1 + f ) f " - t = 0,
Thus for a given heat release there is a minimum Froude number for which a solution exists. Presumably at smaller Froude numbers, buoyancy forces are so strong that the heat is carried away in a plume above the source, which makes no physical sense in the present context. For values of Pr different from 1 the Froude number must still be greater than some minimum value determined by tl, but this minimum must be determined numerically. The wake displacement-thickness is defined by
(6)
Thus the flow exterior to the wake is, to leading order, the inviscid flow past a paraboloid of revolution. This can be constructed by superimposing a uniform flow and a line of uniform sources of strength q0 on 0 __< x < o% o = 0. Thus
1 a2_q0 [(x2+a2)~/Z+x]+... ffi"=2 4~"
(7)
valid as x ~ ~ , o/x fixed. This matches with the wake solution provided q0 = 4~rc0/Re. Correct conditions at the downstream computational boundary should properly incorporate this description of the wake and the exterior inviscid flow; this automatically includes the restriction on the Froude number. Related problems (e.g., flow past a heated sphere) have sometimes been solved without accounting for the wake, but this cannot be correct if the goal is to emulate an unbounded flow. It is true that Badr [6] neglected the wake and achieved excellent agreement with experimental results [7], but a fan was located at the downstream experimental boundary.
A FLAME-BUBBLE ANALOGUE
103
|
1 ~
0
I
I
. . . ............ ......
"--'-'--
-2
i
I
i
,
L
-I
0
I
2
3
4
Fig. 2. Streamlines for Re = 5, Pr = 1, T . = 2.5: Fr = 2 . 4 2 , q = 5 . 8 4 (calculated).
We have generated two sets of numerical results. The first set corresponds to a correct prescription of the wake with the Froude number (determined iteratively) having its minimum value, so that the axial velocity on the wake centerline is zero. This case corresponds to a flame bubble rising at its minimum speed for fixed heat release and size, and the flow disturbance generated by the flame will be greatest. However, in all cases the Froude number is relatively high and the streamlines are not grossly perturbed. Fig. 2 is typical of the results obtained for values of T , between 1.5 and 6, Re between 1 and 10. The "flame" displaces the fluid to the side, decreasing the mass flux in the x-direction. At the same time, however, the density drops because of heating, and the net effect is an acceleration of the fluid. Subsequently the flow is retarded downstream by buoyancy forces and cooling, the velocity decay~
!
|
!
!
!
0
5
10
].5
20
ing monotonically to zero along the centerline. This is in marked contrast to the flow field produced by a heated solid sphere, where the pressure field can produce some downstream acceleration. In order to highlight the effect of the Froude number constraint, solutions have also been obtained by ignoring the wake and imposing uniform downstream conditions (at a finite location of course). Figure 3 is typical of results obtained this way, and shows a closed region of recirculating flow. The size and position of this region depends on the parameter values and, for some choices, can completely enclose the " f l a m e . " These solutions depend strongly on the location of the downstream boundary, consistent with results of Shee and Singh [8] for natural convection around a heated sphere. This is marked contrast to the first set of results for which an increase in r~, beyond i
!
I
30
35
i
!
40
45
lO
-5
Fig. 3. Strcam]incs f o r Rc = 5, Pr = (calculated).
25
I, T ,
= 2.0, Fr =
1.5, r ,
50
= 60: q =
3.13
104
S. WEERATUNGA ET AL.
- 25 yields little change. Evidently the unbounded
flow and corresponding wake solution has a character quite different from the bounded flow. The latter is probably not directly relevant to the flame-bubble problem, i.e., physically realistic solutions can only be obtained by properly accounting for the wake at the downstream computational boundary. A final note: The numerical results show that the scale on which the flow is disturbed is significantly larger than the "flame sphere," and gradients are everywhere small. This suggests that we would obtain very similar results for different heat distributions provided the global parameters (q, Re, Fr, etc.) are unchanged. 3. THE Z E L D O V I C H LIMIT A N D THE STABILITY QUESTION
Most observations of flame bubbles have been in a standard gravitational field. The Zeldovich limit [2] provides the most plausible description of the configuration to be expected under microgravity conditions. In the limit there is no motion (in which direction would the flame move?) and the combustion field is spherically symmetric. Then the velocity field is identically zero and if we adopt a simple kinetic model for which the reaction rate f~ depends only on the mass fraction of the deficient component (hydrogen), denoted by Y, together with the temperature T, then the combustion field can be defined by the equations
0 pDd(dY) =7
drr r2 ~
- fl'
(8)
r--,~: T--,T/, Y-,Y/. If we assume that reaction ceases behind the flame because Y vanishes, it is easily deduced that the flame temperature is
T, = 7"/+O Y/ CuL'
(9)
where L is the Lewis number, which is significantly less than unity. This diffusion-controlled flame temperature is also achieved for tubular
flames of vanishingly small radius [9], and is larger than the adiabatic flame temperature Ty + QYI/Cp. Such elevated temperatures are always possible when the deficient component of a mixture is much lighter than the mixture itself. Note that in any region where there is no significant reaction (particularly the far-field), any mass fraction (hydrogen, oxygen, product gas, etc.) has the mathematical representation
Yi= Ai+ Bi/r so that there is a steady diffusive flux (r2a Yi/Or) of reactants from infinity, and a steady flux of products to infinity. The remark of Goldmann [10], that these flames must move to avoid local exhaustion of reactants, is not correct. Unfortunately, the solution of Eq. 8, of which Eq. 9 is an ingredient, appears to be unstable. This is revealed by adding time derivatives and carrying out a linear stability analysis, albeit for a specific and very simple choice of fl, [2, 4]. The flame either collapses or it expands, a fact that has been exploited by Joulin in his ignition studies, [11]. The most reasonable interpretation of this result is that extinction occurs. In the case of collapse the flame radius shrinks to zero; in the case of expansion the decrease in flame curvature lowers the flame temperature to a level at which combustion cannot be sustained. It is in the nature of flame bubbles that they are observed only for mixtures so weak that plane propagation is not possible-the adiabatic flame temperature is too small to sustain the process. These observations raise an interesting question: If at 0 g, when the Reynolds number is zero, there is a fundamental destabilizing mechanism, why are stable flames observed at 1 g, when the Reynolds number is ca. 5-7? What happens to the destabilizing mechanism revealed in the Zeldovich limit as the Reynolds number is increased? The remainder of the article is devoted to this question. 4. A MODEL FOR FLAME BUBBLES
I N C O R P O R A T I N G CONVECTION In the absence of gravity it is natural to seek a spherically symmetric description of the combustion field. In the presence of gravity, buoyancy forces will generate motion of flame, and the
A FLAME-BUBBLE ANALOGUE
105
adverse convective flux of mixture at the rear will reduce the vigor of the reaction there. At large enough rise speeds this will be sufficient to extinguish the rear portion of the flame, so that reaction is confined to an open cap (Fig. 1), rather than a closed surface. We initially proposed this model in [12], but did not seriously address the stability question. The mathematical ingredients of the flame-cap model reside in the equations
/ d2T 2 d T )
c.p. d T = k ~--~x2 +x-~x +Qf~,
(
pu -~xx=PD \ ~ x 2+-x-d-xx - f l ,
(10a)
(10b)
which we adopt as an approximate description of the combustion field on the axis of symmetry (o = 0) for x < 0. The mass flux on the centerline, pu, is an assigned function o f x that we will take as a constant in the initial part of our discussion; later a more realistic choice will be described. The derivatives on the right side are approximations to the Laplacian on the centerline, with the first derivatives accounting for off-axis diffusion generated by the curvature of the combustion field. The rationale for this has been discussed previously [12], and can be briefly summarized: (i) the equations need only be valid over the range o f x that defines the flame thickness (5; (ii) at 1 g, 6 is significantly less than a, the cap radius, since Re a/6 where Re, the Reynolds number, is approximately 5-7 for the 1 g flames (the estimate comes from noting that the mass flux ( M ) through the flame is - # R e ~ a , the flame thickness is - k / MCp, and the Prandtl number is - 1); (iii) when (5 is significantly less than a, surfaces of constant Y and T within the flame are, to a good approximation, locally spherical with a common center, the situation for which the model is exact; items (i)(iii) combined imply that the model is a reasonable approximation for 1 g flames; (iv) the model is also correct in the limit Re ---, 0, for it then reduces to the 0 g Zeldovich limit, Eq. 1. For these reasons, we adopt the model as a plausible interpolation, valid for 0 g to 1 g flames. To complete the description, a choice must be
made for ft. We shall adopt a flame-sheet model for which fl is a Dirac 6 function whose strength is such that
1 [,Y]
= Be- E/2Rr,,
[ ' ] ~ (')hot--(')cold,
( 11a) (llb)
where T . is the flame temperature. Arrhenius kinetics for large activation energy leads to such a model, but it has also proven valuable in the discussion of finite activation energy effects [ 13], so that there is no assumption here that E/R T, is large. In the steady state the 6 function is located at x = - a, where a, in general, is related to the rise speed. Steady Solution When pu -- constant The steady solution is easily found for any choice of the function pu, but we shall write down results only for pu = constant. The key characteristics are the flame temperature, and the flame radius, given by the formulas
T,-T/=-QY/ G(Pe) G -1 (LPe)
CpL
(12a)
a = pD Yf ee/2Rr*G- i (LPe), B G(P)
--
I~ ' d x e P(x+ 1)
(12b)
where Pe = PrRe is the Peclet number and Re = pua/#. In the limit Pe ~ 0 we recover Zeldovich's result (Eq. 9) with
a =oD ~ e e/2Rr*,
(13)
B
its minimum value. The radius a can be treated as a free parameter of the model and can be assigned any value not less than this, with Eq. 12b determining the rise speed. In the limit Pe ~ ~ , T, is equal to the adiabatic flame temperature, and
a = pD Yy LPeeE/2Rr. ' B
(14)
106
S. WEERATUNGA ET AL.
AT
0
2
4
6
~,4~" !0
8
Pe
Fig. 4. A T , (T, - T/)Cp/Y/vsPeforLe = 0.2(0.1)0.7. which, since Pe is proportional to a, is actually a formula for the adiabatic flame speed. Values of T . for intermediate Pe are shown in Fig. 4. At the values relevant to 1 g flames (Pc 3-5) there is significant enhancement over the adiabatic flame temperature, but much less than that achieved in the Zeldovich limit.
Stability Analysis As we noted in Sec. 3, the steady solution for Pe = 0 is unstable. We want to examine the effect of finite Pe on this result, and some progress can be made on this problem when the mass flow pu is constant. The additional terms pCpOT/Ot and oOY/Ot must be added to Eq. 10a and 10b, and a linear stability analysis can be carded out in the context of the constant density model. This uncouples the fluid mechanics from the thermal processes (the flow affects the flame but not viceversa) and has proven valuable for problems where the coupling plays no important qualitative role [141. Because, even now, the constant density approximation is sometimes misunderstood, it is worth adding some additional remarks. Clearly the density is not constant, and is not so approximated, in the base flow whose stability is of concern; it is, after all, the buoyancy forces that are responsible
for the flame motion. The constant density approximation is only applied to the linear equations that describe infinitesimal perturbations to this base flow. The essential idea is that these equations contain a number of physical ingredients, but only a subset of these are responsible for generating a specific mode of instability. For example, hydrodynamic effects are responsible for the DarrieusLandau instability of plane flames; the differential diffusion of heat and mass (of the deficient species) is responsible for the cellular instability. We can learn a great deal about the first by ignoring diffusive effects, and about the second by ignoring the hydrodynamics. This idea plays a key role in intuitive discussions. The usual "physical" explanation of the cellular instability invokes only the curvature of the combustion field and the diffusion of heat and reactants--no mention is made of the hydrodynamics. A mathematical treatment that ignores the hydrodynamics is, therefore, in the same spirit, but provides a much richer description of the consequences of nonhydrodynamic interactions. Many examples that support this claim are discussed in Ref. [13], and [141. In the present context, when we adopt the constant density approximation, we recognize that we may be discarding a hydrodynamic instability (after all, perturbed buoyancy effects have been discarded along with the momentum equation), so that on that basis alone our discussion is incomplete. But we will find that those physical ingredients retained can lead to an instability (thermaldiffusive in origin) that is affected by the Reynolds number of the base flow. It is useful to nondimensionalize the equations, and to do this we use the characteristic length a, time a2oCp/)~, and temperature Q/C.. Then infinitesimal one-dimensional perturbations with a time dependence e '~t satisfy the following equations on each side of the reaction zone:
dx2+
-Pe
- dx 2 +
--a~r-- et Tl = 0
- LPe
~-
Let YI.
(15)
A FLAME-BUBBLE A N A L O G U E
107
The equation for 7"1 has independent solutions
simplifies to
Yt=eX~l Jo dse~X \ ~ - ~ ]
2
'
(16a)
Y2 = f"s2 J dseSX - kS-S2~
(16b)
where sL2 = Pe/2 +_ ¶/2, ¶ - (Pe 2 + 4a) I/2, and, for complex or, the integral defining Y2 is in the complex s-plane. Yt (unlike Y2) vanishes as x - o o and so describes the cold-gas perturbations; Y2 (unlike Yt) is bounded as x --, 0 and is appropriate for perturbations in the hot gas. The corresponding' solutions for Y~ are obtained by making the substitutions ct --, otL, Pe ~ PeL. The perturbed flame-sheet is located at x = - 1 + tS, and the linearized connection conditions there are T,t = {ST o + 7"1}-I±0, (17)
0 = {tSYo + Y1}- I-0
k [tSTo' + T~ ] = Cp
BaO T , i
e -°/2r*o
2 T , o2 p
=PD[SYo + Y1 l, where 0 = E C p / R Q and ( )o denotes steady-state values. When the general solutions for Tt and Yl are substituted into Eq. 17 a homogeneous set of equations is obtained that have nontrivial solution if the following dispersion relation is satisfied:
(
- Yo + Yo YIL +(-To
-I-0
\Ky2
+ ToY(ly,/_,_o=O,
Kyl
,)
L
(18)
where YlL is yl with the replacements et --, a L , Pe ---, PeL, and
K ~-
Ba 0 -e-°/2r*o. pD 2 T , o 2
(19)
When the Reynolds number is zero, Eq. 18
7",02 0
(1-x/-~L)(l+cothx/~)
= (T,0 - Tf)(I - x/L).
(20)
The left side ranges from +oo to - o o as a increases through real values from 0 to + oo, so there is always a real positive root. When L = 1, for example, c~ = 1. As noted earlier, the Zeldovich solution is unstable. Numerical methods can be used to follow the movement of this root in the complex c~ plane as Pe changes. This is easily done when the root remains real, which is true for small Pe. We have found that, as Pe increases, the root moves to the left along the real axis into the left half-plane, so that at some critical value of Pe, ~ becomes negative, and the mode is stable. For example, with L = 0.4, Yf = 2.9 x 10 -3, Tf = 1.3 x 10 -3, 0 = 2.53 x 10 -2 (the adiabatic flame temperature is 3.23 times the temperature of the remote gas and 0.17 times E / R ) the critical Peclet number is approximately 1.05. That is, f l o w
effects, even f o r constant pu, eliminate the Zeldovich-limit instability at quite modest Reynolds numbers, and this alone could be responsible f o r the stability o f the 1 g flames. At the same time it is possible that flow effects could introduce their own instabilities, corresponding to additional roots of Eq. l 1. A complete search for these roots in the complex plane is not a simple task and we have not attempted it, since there are other aspects of the flow, not accounted for here, that will have a strong stabilizing effect.
The Centerline Mass Flux So far in this section we have taken pu on the centerline to be constant. In fact, it will vary with x, and these variations can be determined from the numerical computations of Sec. 2. In Fig. 5, the centedine mass flux is plotted against x for a representative set of parameter values (again, with the Froude number equal to its minimum). There is a significant gradient in the neighborhood of the
108
S. WEERATUNGA ET AL.
X 0.1m
I 0.400.
0.200
• 41
i ,4
-
| .~
-
i -2
-
i -I
-
i O
-
i I
•
i 2
•
i 3
•
i 4
• 5
x
Fig. 5. Mass flux at the centerline vs x for Re = 5, Pr = 0.72, T, = 4: Fr = 3.35, q = 9.66 (calculated).
flame at x = - 1 , especially upstream in the preheat zone, and this will have a strong stabilizing effect. Expansion of the flame corresponds to a leftward displacement into a region of increased mass flux, which will drive it back. Contraction and displacement to the right will similarly be resisted. This is so clear on intuitive grounds that we provide no mathematical evidence, but the stability analysis can easily be modified to include a perturbation gradient, and this shifts the root towards stability, as expected. 1. CONCLUDING REMARKS In this article we have provided an approximate description of the flow field associated with a single unbounded flame bubble. Although, in general, buoyancy forces can generate closed regions of recirculating flow, the naturally arising restriction on the Froude number prevents this. In the Zeldovich limit, which we believe corresponds to the stationary solution under microgravity conditions, the flame is unstable. We have explored, albeit tentatively, possible stabilizing mechanisms that could account for the stable flames observed at 1 g. These stabilizing mechanisms are only relevant in the context of the flamecap model, where it is assumed that the Reynolds number is large enough to ensure extinction of the rear portion of the flame. Our results suggest that a sufficiently large uniform convective flux through the flame can overcome the instability that prevails
at zero Reynolds number (zero convective flux). An additional stabilizing mechanism is revealed by the numerical simulations which show that the centerline mass flux decreases significantly in the neighborhood of the flame. Put another way, we might expect, on intuitive grounds, that a flame bubble could act as an ignition source, giving rise to an expanding unsteady combustion-field. Experiment shows that at 1 g this does not occur, implying the existence of a preventive mechanism. We have argued that buoyancy-induced convection is such a mechanism. Experiments under low-gravity conditions that could confirm or refute our discussion appear, at the present time, to be inconclusive. Ronney [3] has carried out experiments in the 2.2-s NASA drop tower. Lean hydrogen-air mixtures are ignited by a spark. As the flame propagates radially outwards, it breaks up into cells, which subsequently form what appear to be closed spherical flame bubbles. Throughout most of the 2.2 s there is unambiguous radial motion from the ignition source, but at the end of this period the bubbles are moving very slowly. Only a longer test period can reveal whether or not the bubbles survive as stationary objects. If they do, there must be an additional stabilizing mechanism not identified here; if they do not, that is consistent with our analysis. It is, of course, possible that there is an unidentified stabilizing mechanism responsible for stability at zero Reynolds numbers that is inadequate under 1 g conditions.
This work was supported by the Air Force Office o f Scientific Research. J. Neves provided valuable assistance with the stability computations of Sec. 2. REFERENCES I.
2.
Lewis, B., and yon Elbe, G., Combustion Flames and Explosions of Gases, Academic Press, New York, 1961, p. 314. Zeldovich, Ya. B., Barenblatt, G. I., Librovich, V. B., and Makhviladze, G. M., The Mathematical Theory of Combustion and Explosions, Consultants Bureau, New York, 1985, p. 327.
S. W E E R A T U N G A 3. 4. 5. 6. 7.
8.
ET AL.
Ronney, P., Private communication. Weeratunga, S., Ph.D. thesis, University of Illinois at Urbana-Champaign, Urbana, IL, 1986. Buckmaster, J. D., and Weeratunga, S., Combust. Sci. Technol. 35:287-296 (1984). Badr, H. M., Int. J. Heat Mass Trans. 27:15-27 (1984). Krause, J. R., and Tarasuk, J. D., in Fundamentals o f Forced and Mixed Convection (F. A. Kulacki and R. D. Boyd, Ed.), The 23rd National Heat Transfer Conference, The Heat Transfer Division, ASME, Denver, CO, 1985, pp. 171-179. Shee, Y.-T., and Singh, S. N., in Natural Convection (I. Cotten and R. N. Smith, Ed.), ASME, New York, 1981, pp. 67-74.
109 9.
Takeno, T., Ishizuka, S., and Nishioka, M., Combust. Flame ( in press). 10. Goldmann, F., Z. Physik. Chem. B5:307 (1929). 1 I. Champion, M., Deshaies, B., Joulin, G., and Kinochita, K., Combust. Flame 65:319-337 (1986). 12. Buckmaster, J.. Johnson, R. E., and Weeratunga, S., in Mathematical Modeling in Combustion Science, Springer-Verlag, New York, 1988, p. 112. 13.
14.
Buckmaster, J. D., and Luford, G. S. S., Lectures on Mathematical Combustion, SIAM. Philadelphia. 1983, p. 73. Buckmaster, J. D., Ed., The Mathematics o f Combustion, SIAM. Philadelphia, 1985, p. 26.