- Email: [email protected]

Effect of Lewis number on flame propagation through vortical flows L. Kagan, G. Sivashinsky ∗ School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel Received 25 October 2004; received in revised form 5 March 2005; accepted 25 March 2005 Available online 22 April 2005

Abstract The premixed gas flame spreading through an array of large-scale vortices is studied numerically. It is found that the flame speed is a nonmonotonic function of the stirring intensity. At sufficiently low Lewis numbers (Le < 1) the system becomes bistable with a hysteretic transition between possible propagation modes. In the presence of volumetric heat losses the stirring invariably promotes extinction (reduces the flammability limits), provided Le > 1. At Le < 1 this holds only for sufficiently strong stirring, whereas moderate stirring actually expands the flammability limits. At Le > 1 the deficient reactant is fully consumed up to the very quenching point. At Le < 1, prior to the total extinction, part of the deficient reactant escapes the reaction zone and remains unconsumed. 2005 The Combustion Institute. Published by Elsevier Inc. All rights reserved. Keywords: Flame–flow interaction; Turbulent flames; Flammability limits

1. Introduction The present work is a continuation of our earlier studies on premixed gas flames spreading through a space-periodic array of large-scale vortices [1,2], and is motivated by the experimentally known phenomenon of flame extinction by turbulence [3,4]. The previous work dealt with the simplest case of unity Lewis number (Le = 1) where the adiabatic flame may be described by a single reaction–diffusion–advection equation. It was found that the flame speed is a nonmonotonic function of the stirring intensity. For moderately strong vortices their intensification results in flame speed enhancement accompanied by shedding of islands of unburned gas (see also [5–7]). Yet, there * Corresponding author.

E-mail address: [email protected] (G. Sivashinsky).

is a certain level of stirring at which the flame speed reaches its maximum. Any further increase in the stirring intensity leads to a drop in the flame speed, followed, for mildly nonadiabatic systems, by flame extinction. The one-scale scheme of the flame–flow interaction thus proved to be capable of capturing at least some basic aspects of turbulent combustion, which, by definition, involves a wide range of spatiotemporal scales. In the present work the discussion is extended over nonunity Lewis numbers. The point is that, other conditions being identical, the Le < 1 turbulent flames are known to be more difficult to quench than the Le > 1 flames [8,9]. It is therefore of interest to ascertain whether this effect also has a counterpart in the one-scale formulation, and what are the basic qualitative changes accompanying the transition from Le > 1 to Le < 1 premixtures, aside from the fact that the Le < 1 flames are prone to cellular instability and self-fragmentation.

0010-2180/$ – see front matter 2005 The Combustion Institute. Published by Elsevier Inc. All rights reserved. doi:10.1016/j.combustflame.2005.03.010

236

L. Kagan, G. Sivashinsky / Combustion and Flame 142 (2005) 235–240

2. Model A conventional one-step, constant-density, nonadiabatic, reaction–diffusion–advection model for a timeindependent periodic vortical flow field is adopted. In appropriately chosen units the corresponding set of equations for the temperature and the deficient reactant concentration reads Tt + u · ∇T = ∇ 2 T + (1 − σ )Ω(C, T ) − Q(T ),

(1)

Ct + u · ∇C = Le−1 ∇ 2 C − Ω(C, T ),

(2)

1 Ω(C, T ) = (1 − σ )2 Le−1 N 2 C 2 × exp N (1 − T −1 ) ,

(3)

Q(T ) = h(T 4 − σ 4 ),

(4)

u = (2A sin kx cos ky, −2A cos kx sin ky).

(5)

Here T is the nondimensional temperature in units of Tb , the adiabatic temperature of combustion products; C is the nondimensional concentration of the deficient reactant in units of C0 , its value in the fresh premixture; x and y are the nondimensional spatial coordinates in units of lth = Dth /Ub , the thermal width of the flame; Dth is the thermal diffusivity of the mixture; Ub is the burning velocity of a planar adiabatic flame; t is the nondimensional time in units of lth /Ub , σ = T0 /Tb , T0 being the fresh mixture temperature; N = Ta /Tb is the nondimensional activation energy, Ta being the activation temperature; Le = Dth /Dmol is the Lewis number, Dmol being the molecular diffusivity of the deficient reactant; u is the √ prescribed flow field and A = (1/2)u · u its intensity in units of Ub ; k is the periodic flow wavenumber −1 ; Q(T ) is the term responsible for the in units of lth radiative heat losses with h being the scaled Stefan– Boltzmann constant in units of ρb cp lp Ub /4Tb3 lth , where lp is the Planck mean absorption length, cp is the specific heat, and ρb is the burned gas density; Ω(C, T ) is the appropriately normalized reaction rate to ensure that at large N the nondimensional speed of a well-settled planar adiabatic flame is close to unity. The streamlines of the adopted flow field (5) are depicted in Fig. 1. Equations (1) and (2) are considered in the strip 0 < x < π/k, −∞ < y < ∞ and subjected to the insulating boundary conditions, Tx = Cx = 0 at x = 0, π/k.

(6)

The flame is assumed to propagate upward. Hence the boundary conditions at y = ±∞, C = 1,

T =σ

at y → +∞,

Cy = Ty = 0 at y → −∞.

(7)

(a)

(b)

(c)

Fig. 1. Deficient reactant concentration (C) distributions for Le = 0.5, A = 0.2 (a), A = 5 (b), and Le = 1.25, A = 5 (c). Darker shading corresponds to higher level of C. Combustion wave propagates upward. Closed lines represent streamlines. Table 1 Speed of a planar adiabatic flame V0 versus Lewis number Le Le 0.5

0.7

0.75

0.8

0.9

1

1.1

1.25

V0 0.94 0.960 0.957 0.955 0.949 0.944 0.939 0.931

The problem (1)–(7) is solved for N = 20, 0.5 Le 1.25, σ = 0.2, k = 0.125. The computational strategy follows that described in [1].

3. Numerical simulations 3.1. Adiabatic case The normalizing factor (1/2)(1 − σ )2 Le−1 N 2 in expression (3) for the reaction rate is introduced to ensure that in the limit of infinitely high activation energy (N → ∞) the propagation speed of a planar adiabatic flame V0 is held at unity. For high but finite activation energy (N = 20) employed in the numerical simulations there is naturally a slight variation of V0 with the Lewis number. Some representative values of V0 (Le) are given in Table 1. These numbers are obtained by numerical simulations of the onedimensional version of the model (1)–(7) at u = 0 and h = 0.

L. Kagan, G. Sivashinsky / Combustion and Flame 142 (2005) 235–240

237

Fig. 2. Flame speed V versus stirring intensity A for different Lewis numbers Le. Adiabatic limit.

In order to single out the impact of Lewis number in its “purest” form, in the following discussion we deal with the renormalized effective flame speed V referred to as V0 (Le). The evaluation of V follows the procedure described in [1]. Fig. 2 depicts V versus A curves calculated for several Lewis numbers in the absence of heat losses (h = 0). As expected [1], the emerging V (A) dependencies are nonmonotonic. At relatively weak stirring its intensification results in the flame speed enhancement. Yet, there is a crossover intensity Acr (Le) where the flame speed reaches its maximum. As is readily seen Acr is an increasing function of the inverse Lewis number Le−1 , which agrees with the experimentally observed tendencies [8,9]. At A Acr the curves V (A) nearly coincide with each other. Moreover, they closely follow the Huygens picture [1,7] valid in the limit of large-scale vortices (k 1), and where the flame structure is considered as locally planar and unaffected by the underlying flow field. At Le = 0.7 the flame speed maximum reaches the level of about 50Ub (Fig. 2). For larger vortices, other conditions being identical, the maximum is expected to be even higher [1], and in principle should not have an upper bound [7]. Note that in experiments with turbulent flames the flame speed is known to rise up to 20Ub [8,9]. At Le = 1.25 the maximum flame speed is only slightly above 1.5Ub (Fig. 2). In this situation the shedding of islands does not occur (Fig. 1c). It is interesting, that at sufficiently low Lewis numbers the system becomes bistable with a hysteretic transition between possible propagation modes.1 The broken lines (Fig. 2) pertain to presumably existing 1 A similar effect was identified in Le > 1 flames spreading through periodic shear flows [10].

unstable solutions unattainable in numerical simulations of the dynamical model such as (1)–(7). The emerging multiplicity is evidence of two mechanisms governing the wave spread: Huygenslike, advection dominated, and non-Huygens, dominated by eddy diffusivity. In order to isolate the second mechanism one may consider the limit A 1 where the adiabatic flame again speeds up with the velocity V ∼ A1/4 [2,11]. 3.2. Nonadiabatic case For all the intricacies of the flame–vortical flow interaction, the adiabatic flame seems to withstand any level of stirring however high. Incorporation of volumetric heat losses changes the picture essentially. Figs. 3 and 4 show the V (A) dependency for nonzero heat losses calculated for Le = 1.25 and Le = 0.75. At Le = 1.25, as expected, the heat losses exert an inhibiting influence on combustion, resulting in reduction of the flame speed, and complete extinction at sufficiently strong stirring. Nevertheless, the deficient reactant is fully consumed up to the very extinction point. The situation therefore is qualitatively similar to that observed in equidiffusive premixtures [1]. At Le = 0.75 the dynamical picture acquires a new ingredient. At relatively high heat losses h > 0.013, where the quiescent premixture becomes effectively nonreactive, combustion may be revived by stirring, provided its intensity falls within a suitable range. All the curves with nonzero heat losses (Figs. 4 and 5) have the right end points marked by the open circle. Some of them also have the left end points. This implies that in order to sustain combustion one needs a certain level of stirring. An overly weak (or strong) stirring leads to the flame extinction. In

238

L. Kagan, G. Sivashinsky / Combustion and Flame 142 (2005) 235–240

Fig. 3. Flame speed V versus stirring intensity A at different levels of heat losses h; Le = 1.25. Open circles correspond to the extinction points.

Fig. 4. Flame speed V versus stirring intensity A at different levels of heat losses h; Le = 0.75. Open circles correspond to the extinction points.

other words, under appropriate conditions stirring extends the flammability limit. A similar effect has recently been identified in the theory of thermal explosion [12]. Fig. 6 plots heat loss intensity at the quenching point hq (A) evaluated for several Lewis numbers. At Le = 0.5 the intensity h0q = hq (0) appears to be considerably higher than its counterparts corresponding to Le = 0.75, 1, and 1.25, which are nearly identical. Such a disparity is a direct consequence of the cellular instability, rather prominent at Le = 0.5 (Fig. 1a). The cellular flame spreads faster than the planar one and is therefore more difficult to quench [13]. As one may anticipate, sufficiently strong stirring stretches the flame and suppresses the cells (Fig. 1b). At Le = 1

and 1.25 the flame is unconditionally stable, while at Le = 0.75 it is unstable but, by virtue of the boundary condition (6) and relatively large k, only just slightly. With regard to the Le = 0.5 curve, Fig. 6 shows only its initial (growing) section. The decaying section, emerging at A > 100, falls beyond the figure’s frame. At Le < 1, at any nonzero heat loss, part of the deficient reactant escapes the reaction zone and remains unconsumed. The effect is especially prominent at high heat losses, where the flame breaks up into separate fragments [13]. Fig. 5 plots the residual concentration C∞ versus the stirring intensity A. There are two different types of responses. At h < 0.013 the concentration C∞ in-

L. Kagan, G. Sivashinsky / Combustion and Flame 142 (2005) 235–240

239

Fig. 5. Residual concentration C∞ versus stirring intensity A at different levels of heat losses h; Le = 0.75. Open circles correspond to the extinction points.

Fig. 6. Flammability limit hq versus stirring intensity A at different Lewis numbers Le.

creases monotonously with A, and becomes quite appreciable at A > 5. At h > 0.013 the C∞ (A) dependency turns nonmonotonic. A relatively weak stirring reduces C∞ , while, as in the case of low heat losses, strong stirring enhances C∞ . In the quiescent premixture (A = 0) the flame extinguishes. Apart from the nonmonotonicity, one ends up with the upper and lower flammability limits, marked by open circles in Figs. 4 and 5. Note that for Le > 1 the T –C fields are largely similar to those of Le = 1, discussed in [1,2]. Fig. 1 shows the T –C fields for Le = 0.75. Here h = 0.0125 is the case of relatively low heat losses, allowing, in the absence of stirring (A = 0), for planar flames. At h = 0.075 the planar flames are ruled out, and in order to sustain combustion one needs some level of steering.

4. Concluding remarks The above findings show that the dual influence of the large-scale turbulence on premixed combustion (flame speed enhancement followed by its reduction and extinction) and higher resilience of turbulent flames for lower Lewis numbers are actually not related to the multiple-scale nature of the flow field, and the effects may be captured within the framework of a one-scale flame–flow interaction scheme. Comparison of the quenching curves for Le = 0.75, 1, and 1.25 (Fig. 6) shows that at mild heat losses (h < h0q ) the stirring intensity at the quenching point increases with the inverse Lewis number (Le−1 ), which is quite in line with the experimental observations noted earlier [8,9]. One should realize, of course, that in the real-life situation any change

240

L. Kagan, G. Sivashinsky / Combustion and Flame 142 (2005) 235–240

in the Lewis number (normally attained by changing the system’s equivalence ratio) affects the adiabatic temperature Tb , the burning velocity Ub , as well as the radiative properties of the mixture. In the present discussion these ingredients are treated as prescribed and effectively unrelated, which, judging by the results obtained, is not too detrimental for understanding the overall physical picture. Yet, it would certainly be of interest to extend the model employed over a physically more realistic formulation, based on a nonstoichiometric bimolecular reaction, and to utilize the mixture’s equivalence ratio as a control parameter.

Acknowledgments The authors gratefully acknowledge the support of the U.S.–Israel Binational Science Foundation (Grant 2002008), the German-Israel Foundation (Grant G-695-15.10/2001), the Israel Science Foundation (Grants 67-01 and 278-03), and the European Community Program RTN-NPRN-CT-2002-00274. The numerical simulations were performed at the Israel Inter University Computer Center.

References [1] L. Kagan, G. Sivashinsky, Combust. Flame 120 (2000) 222. [2] L. Kagan, P.D. Ronney, G. Sivashinsky, Combust. Theory Modelling 6 (2002) 479. [3] D. Bradley, Proc. Combust. Inst. 24 (1992) 247. [4] D. Bradley, A.K. Lau, M. Lawes, Philos. Trans. R. Soc. London Ser. A 338 (1992) 359. [5] W.T. Ashurst, G. Sivashinsky, Combust. Sci. Technol. 80 (1991) 159. [6] J. Zhu, P.D. Ronney, Combust. Sci. Technol. 100 (1994) 183. [7] R.C. Aldredge, Combust. Flame 106 (1996) 29. [8] V.P. Karpov, E.S. Severin, Combust. Explos. Shock Waves 16 (1980) 41. [9] R.G. Abdel-Gayed, D. Bradley, M.N. Hamid, M. Lawes, Proc. Combust. Inst. 20 (1984) 505. [10] L. Kagan, G. Sivashinsky, G. Makhviladze, Combust. Theory Modelling 2 (1998) 399. [11] B. Audoly, H. Berestycki, Y. Pomeau, C. R. Acad. Sci. Ser. II B 328 (2000) 255. [12] L. Kagan, H. Berestycki, G. Joulin, G. Sivashinsky, Combust. Theory Modelling 1 (1997) 97. [13] L. Kagan, G. Sivashinsky, Combust Flame 108 (1997) 220.