Effects of heat and momentum losses on the stability of premixed flames in a narrow channel

Transcription

1 Combustion Theory and Modelling Vol. 10, No. 4, August 2006, Effects of heat and momentum losses on the stability of premixed flames in a narrow channel S. H. KANG,S.W.BAEK and H. G. IM Aeropropulsion Department, Korea Aerospace Research Institute, 45 Eoeun-Dong, Yuseong-Gu, Daejeon , Korea Division of Aerospace Engineering, Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Kusung-Dong, Yusung-Gu, Taejon , Korea Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI , USA (Accepted 14 February 2006) To understand fundamental characteristics of combustion in a small-scale device, the effects of the momentum and heat loss on the stability of laminar premixed flames in a narrow channel are investigated by two-dimensional high-fidelity numerical simulations. A general finding is that momentum loss promotes the Saffman Taylor (S T) instability, which is additive to the Darrieus Landau (D L) instabilities, while the heat loss effects result in an enhancement of the diffusive thermal (D T) instability. It is also found that heat loss amplifies the sensitivity of the growth rate to the Lewis number variation, and the critical Lewis number of neutral stability also increases with heat loss. The two competing effects between the heat and momentum losses play an intricate role in determining the overall instability and cell formation patterns. The simulations of multiple cell interactions revealed that D T instability mechanism is primarily responsible for the splitting into smaller flame cells, while the D L and S T mechanisms favour larger flame cells through merging. The relative sensitivity of the flame instability to the momentum or heat loss effect is also examined by a numerical experiment for different Lewis number cases. Notation c 0 speed of sound D mass diffusivity e internal energy (= 1 2 ρ(u2 + v 2 + w 2 ) + P F 0 initial amplitude of the disturbance F amplitude of flame wrinkle h Hele Shaw cell thickness k wavenumber of the disturbance (= 2π λ ) Le Lewis number (= α/d) L M Markstein length L ref reference acoustic length scale Pe Peclet number (= S Lh α ) Pr Prandtl number γ 1 ) Corresponding author. hgim@umich.edu Combustion Theory and Modelling ISSN: (print), (online) c 2006 Taylor & Francis DOI: /

2 660 S. H. Kang et al. Q R Re a Sc S L S L,a S T ū U c heat release gas constant acoustic Reynolds number (= ρ 0c 0 L ref Schmidt number laminar flame speed speed of a planar adiabatic flame overall flame speed average velocity convective velocity mass fraction of reactant thermal diffusivity (= ν/pr) Zeldovich number (= E a (T RTf 2 f T 0 )) µ 0 ) Y α β γ ratio of specific heats (= C p /C v ) δ n nominal flame thickness (= α S L ) δ th thermal flame thickness [defined in equation (17)] λ wavelength of the disturbance λ c cut-off wavelength at neutral stability λ m wavelength of the disturbance at the maximum growth rate λ T thermal conductivity pre-exponential factor θ nondimensional temperature = T T 0 T f T 0 µ molecular viscosity ν kinematic viscosity (= µ ρ ) σ heat release parameter (= T f T 0 T f ) τ n nominal flame time (= α ) SL 2 τ th the flame time based on the thermal flame thickness (= δ th /S L ) ω growth rate of the flame wrinkle ω R reaction rate Subscripts 0 upstream reactants f flame R reactant Superscripts + normalized by acoustic reference parameters in table 1 ** normalized by the actual thermal flame thickness (δ th ) and flame time (τ th ) 1. Introduction Development of meso- or micro-scale power-generation devices has recently been attracting strong research interests for its wide range of application in the fields of portable electronics, micro-sensors, micro air vehicles (MAV), etc. Many of these systems have been relying on electricity as a major power source. As the system dimension becomes smaller, however, the use of electrical batteries becomes no longer viable owing to their limited energy densities. As an alternative, many researchers have investigated the chemical energy sources, such as fuel cells or micro-combustion systems. Fuel cells have the benefits of rapid recharging time and convenient portability. In moving systems, however, the application of fuel cells may also

3 Stability of premixed flames in a narrow channel 661 be restricted considering the efficiency associated with the conversion of electrical energy into mechanical energy. On the other hand, the micro-combustion system can directly convert chemical energy into mechanical work as in internal combustion engines. Even for the purpose of generating electricity, the micro-combustion system shows a greater promise in favour of its higher energy density. Therefore, the concept of micro-combustion devices has a tremendous potential in today s advanced technology. For a successful development of micro-combustors, it is essential to achieve stable burning in a small-size chamber. For a premixed system, it is well known that various kinds of intrinsic instabilities influence the flame propagation and dynamics. As the primary mechanisms among them, the Darrieus Landau (D L) [1, 2] and diffusive thermal (D T) instabilities have been extensively studied for decades [3 11]. In small-scale systems, however, additional effects such as viscous friction and heat losses become important and can substantially modify the flame instability behaviour. Therefore, further work is needed to understand fundamental characteristics of premixed flame instability in the context of micro-combustor application. The instability of a propagating front by the friction force was first recognized by Saffman and Taylor [12] in their study on the viscous fingering of the interface between two immiscible fluids within a narrow channel called the Hele Shaw cell. Joulin and Sivashinsky [13] later performed an asymptotic analysis of the Saffman Taylor (S T) instability in premixed flame propagation, demonstrating that the momentum loss can contribute to an additional amount of flame instability. However, their analysis was limited to a linear regime with a neglect of the D T instability mode. This work has been followed by an experimental investigation by Abid et al. [14], although the difficulties in achieving flame propagation in a sufficiently narrow channel prevented a thorough characterization of the S T instability mechanism. Motivated by these earlier efforts, we have recently performed an extensive parametric study using high-fidelity numerical simulations [15] which accounts for realistic finite-rate chemistry and temperature-dependent transport properties. Quantitative assessment of the individual effects from the D L, D T and S T instability mechanisms was undertaken and the effects of the Lewis and Peclet numbers were extensively examined. It was demonstrated that the S T instability mode is inherently hydrodynamic in terms of the physical mechanism and the wavenumber selection behaviour. While this study provided important baseline results regarding the role of momentum loss on flame instability, the analysis was limited to adiabatic walls, hence neglecting another important issue of heat loss. Joulin and Clavin [16] incorporated volumetric heat loss effects in their linear stability analysis of the D T mechanism, where it was predicted that heat loss changes the critical Lewis number at which the flame becomes neutrally stable. Joulin and Sivashinsky [13] also addressed the heat loss effects on the S T instability in a crude way, but the results lack practical significance because the D T mode, which is expected to be most affected by heat loss among all instability modes, was not considered in the analysis. Kagan and Sivashinsky [17] explained the radiative heat loss effects on the nonlinear behaviour of cellular flames. They showed local quenching phenomena owing to the heat loss, but mechanisms of the cell splitting and their dependence on heat loss were not clearly explained. A recent study by Bechtold et al. [18] provided a systematic asymptotic analysis to show that radiative heat loss can directly affect the Markstein number and thus the self-wrinkling of expanding flames via the D T mechanism. It would be of interest to investigate similar aspects in the microcombustor application where the effect of viscous friction is significant. Therefore, this paper extends our previous work to study in particular how the coupled D L, D T and S T instability modes are affected by the heat loss through the large surface area. Numerical simulations of two-dimensional (2D) premixed flames in a narrow channel subjected to momentum and heat losses are performed under various parametric conditions to assess the qualitative and quantitative impact of various loss mechanisms.

4 662 S. H. Kang et al. Figure 1. Schematic of the flame propagation through the Hele Shaw cell. 2. Numerical methods Figure 1 shows the schematic of the premixed flame propagation between the two parallel plates. Neglecting the z-directional velocity and assuming symmetry in this direction, that is w = 0, P/ z = 0 and Y i / z = 0, the conservation equations for mass, x- and y-momentum, energy, and species are written as t + (ρ+ u + ) + t + (ρ+ v + ) + e + t + + ρ + t + + x + ( ρ + u +2 ) + x + (ρ+ u + v + ) + x + (ρ+ u + ) + y + (ρ+ v + ) = 0 (1) [ ] y + (ρ+ u + v + ) = P+ x + 1 τ xx + + Re a x + τ+ xy + y + τ+ zx + z + [ ] ( ρ + ) P + v +2 = y + y + 1 τ + xy + Re a x + τ+ yy + y + τ+ yz + z + x + [(e+ + P + )u + ] + y + [(e+ + P + )v + ] = 1 [ ( u + τ + ) ( Re a x + xx + u + τ + ) ( y + xy + u + τ + ) ] z + zx + 1 [ ( v + τ + ) ( Re a x + xy + v + τ + ) ( y + yy + v + τ + ) ] z + yz + 1 [ (µ + T + ) + (µ + T + ) + (µ + T + Re a Pr x + x + y + y + z + z + ( ρ + Y + ) ( t + i + u + ρ + Y + x + i = 1 Re a Sc [ x + ) + (µ + Ȳ + i x + ( v + ρ + Y + y + i ) + y + ) (2) (3) )] + Q + R ω+ R (4) (µ + Ȳ + )] i ω + y + R (5)

5 Stability of premixed flames in a narrow channel 663 Table 1. Reference parameters for nondimensionalization. Variables Symbol Reference scale Velocity u i c 0 Length x i L ref Time t L ref /c 0 Energy E c0 2 Density ρ ρ 0 Pressure P ρ 0 c0 2 Mass gas constant R c p,0 Mass fraction Y i Y i,0 Viscosity µ µ 0 Mass diffusivity D i c 0 L Thermal conductivity λ µ 0 c p,0 Frequency factor c 0 /L ref Temperature T T ref = (γ 1)T 0 P + = ρ + R + T + (6) where superscript + indicates a dimensionless variable based on the acoustic length and time scales. The reference parameters for nondimensionalization are listed in table 1. We further assume that the solution profiles in the z-direction are parabolic. Equations (1) to (6) are then considered to represent the solutions on the mid-plane, z = h/2, with all the z-directional derivative terms being simplified into explicit forms as [15] τ zx + z + = [µ + 2 u + + u+ µ + T + ] ( ) = 8µ+ u + U c z=h/2 z + T + z + (7) z=h/2 z +2 τ yz + z + = [µ + 2 v + z=h/2 (u + τ + zx ) z + z +2 z=h/2 + v+ µ + T + ] z + T + z + z=h/2 = 8µ+ v + h +2 h +2 (8) z=h/2 [ = µ + u + 2 u + ( u + µ + + ) ] 2 + u + u+ µ + T + z=h/2 z +2 z + z + T + z + z=h/2 = 8µ+ u ( ) + u + U c (9) z=h/2 h +2 (v + τ yz + ) [ z + = µ + v + 2 v + ( v + µ + + ) ] 2 + v + v+ µ + T + z +2 z + z + T + z + z=h/2 z=h/2 = 8µ+ v + 2 (10) z=h/2 h + 2 (µ + T + ) [ = µ z + z z=h/2 + 2 T + + z µ+ T + ( T + ) ] 2 z + z=h/2 = 8µ+ (T + T w ) h + 2 z=h/2 In the above, a Galilean transformation is used such that the entire plates and the flow within the channel are moving at a convective velocity, U c,inorder to retain the flame within the computational domain for a long period of time. T w is the wall temperature and h is the Hele Shaw cell thickness. (11)

6 664 S. H. Kang et al. For the consideration of heat loss effect, T w is adjusted by the heat loss parameter H, such that T w + = T (1 H) (T + T + 0 ) (12) where T + 0 is the fresh mixture temperature and T + is the local gas temperature. Therefore, an adiabatic wall condition can be achieved by setting H = 0, and an isothermal wall of T w + = T + 0 can be obtained by setting H = 1. As for the reaction term, a one-step global chemical reaction model with a single species is employed, written as Q + R = 1 c 2 ( 0 T + f T + ) 1 0 = c0 2 Y R,0 Y R,0 ( σ )( 1 ) 1 σ γ 1 ( ) ω + R = + ρ + Y + R exp β(1 θ) (14) 1 σ (1 θ) where β = E a RTf 2 (T f T 0 ) is the Zeldovich number and σ = T f T 0 is the heat release parameter. T f Furthermore, we use a temperature-dependent viscosity model µ + = µ ( ) T 0.76 = (15) µ 0 The 2D system of governing equations is discretized by a sixth-order explicit finite difference scheme [19] and is integrated by a third-order explicit Runge Kutta method [20]. For a nonreflecting inlet with fixed velocity, a soft-inflow boundary condition is used at the inlet [21]. At the outflow boundary, the nonreflecting characteristic outflow boundary condition is employed [22]. All the transverse boundaries are set to be periodic. To handle the large amount of computation efficiently, the numerical code has been parallelized using the message passing interface (MPI) protocol with excellent linear scalability up to over ten processors, which was sufficient for the present study. For the generation of the initial condition, one-dimensional (1D) premixed flame propagation is calculated and mapped into the 2D domain. A small sinusoidal disturbance is then imposed on the initial flame front in the following form T 0 (13) x f = F 0 sin (ky) (16) where x f is the perturbation of the flame location, F 0 is the initial amplitude of the disturbance, and k is the wavenumber of the disturbance. Unless otherwise noted, in all the calculations we use Re a = 1700, Pr = 0.7, σ = 0.8, β = 10, + = 15 and the horizontal domain length is set to be about 50 times the flame thickness. A sufficient level of spatial resolution is used with approximately 15 grid points across one thermal flame thickness. The length and time scales are normalized by the thermal flame thickness (δ th ) and flame time (τ th ) which are defined as T f T 0 δ th = (dt/dx) T =(Tf +T 0 )/2 (17) τ th = δ th /S L (18)

7 Stability of premixed flames in a narrow channel 665 Figure 2. Growth rate versus the wavenumber for different Lewis number and heat loss conditions: h = 10, F 0 /λ = These normalized quantities are denoted by superscript **, and will be primarily used in subsequent presentations of the results. As shown in our previous study [15], the accuracy of the numerical code was validated by comparing the baseline dispersion relation with another numerical study [23]. 3. Instability behaviour: linear regime 3.1 Effects of heat loss The physical mechanism of the diffusive-thermal instability is the imbalance between mass and heat diffusion; as such, heat loss is expected to play an important role in the instability behaviour. In this section, the effect of heat loss on the D T instability is considered as a baseline case study by varying the Lewis number of the reactant. Numerical experiments were performed for various conditions to generate the dispersion relation curve (the growth rate versus the wavenumber of perturbation) in the linear regime. Figure 2 shows the dispersion relation curves for various Lewis numbers, with and without heat loss. The overall behaviour appears to be consistent throughout the range of parameters. However, the effect of heat loss is seen such that the growth rate is increased for Le = 0.7 and 1.0, and is decreased for Le = 1.3, respectively, compared to the corresponding cases without heat loss. To elaborate on this observation, the linear growth rate for a fixed wavenumber of disturbance (k = or λ = 20) was collected and plotted in figure 3 as a function of the Lewis number for two different levels of heat loss. As expected, it is first noted that the growth rate increases for smaller values of the Lewis number due to the enhanced diffusivethermal instability. Consideration of heat loss shows that the growth rate becomes larger for

8 666 S. H. Kang et al. Figure 3. The growth rate as a function of the Lewis number and heat loss parameter: h = 10,λ = 20, F 0 /λ = diffusive-thermally unstable cases (Le < 1) while the opposite is true for D T stable cases (Le > 1). In other words, heat loss amplifies the sensitivity of the growth rate to the Lewis number variation; namely, heat loss shows a more stabilizing effect for higher Lewis numbers, but amore destabilizing effect for lower Lewis numbers. While no simple phenomenological explanation has been found for this behaviour, a recent asymptotic analysis [18] demonstrated in an explicit formula that the presence of heat loss effectively modifies the Markstein length in a manner consistent with the present observation. Joulin and Clavin [16] predicted that heat loss changes the critical Lewis number, which is defined as the crossover Lewis number at which the flame is neutrally stable. Such a trend is not clearly seen in figure 3, because the present study takes into account both D L and D T instability modes, unlike Joulin and Clavin s analysis which considered the D T mode only. Therefore, the additional instability caused by the D L mode, which is insensitive to the Lewis number variation, masks the distinction that is unique to the D T instability. While it is difficult to separate the effects of the two instability modes, the following assumptions are made as an attempt to systematically eliminate the D L instability effect: (a) for the adiabatic case, l 1,c 2 (i.e. Le c 0.8) and (b) the D L instability mode is independent of the Lewis number variation and heat loss. With this approximation, the magnitude of the growth rate resulting from the pure D L effect for all Lewis numbers and heat loss is found to be , denoted by a horizontal line in figure 3. It is then clearly seen that the critical Lewis number for neutral D T instability indeed increases with additional heat loss, thereby supporting the theoretical prediction. The above discussion is a heuristic extrapolation to identify the pure effect of the D L instability. In particular, assumption (b) is not correct because the D L instability also depends on heat loss. Nevertheless, the increased amount of heat loss will only reduce the growth rate associated with the D L instability mode, hence resulting in a slightly negative slope of the

9 Stability of premixed flames in a narrow channel 667 Figure 4. Effects of heat and momentum loss on the growth rate for different Lewis numbers: h = 10, H = 1,λ = 20, F 0 /λ = crossing line rather than the horizontal line as shown in figure 3. This will result in even greater separations between the critical Lewis numbers at various levels of heat loss. Therefore, the conclusion that heat loss increases the critical Lewis number remains valid. 3.2 Effects of heat and momentum loss We now study the combined effects of the heat and momentum losses on the premixed flames propagating in a Hele Shaw cell. The results discussed in the previous section will be compared. Figure 4 shows the linear growth rate for the same conditions shown in figure 3, with various combinations of heat and momentum losses. In what follows, the baseline case represents the reference condition without heat or momentum loss. For case (3), the heat loss parameter of H = 1.0 and h = 8 were used to represent a significant effect of the S T mode contribution. As discussed in the previous section, heat loss amplifies the sensitivity of the flame instability to the Lewis number. The addition of the momentum loss further contributes to the overall growth rate, making the flame even more unstable. Curves (5) and (6) indicate, respectively, the difference between curves (2) and (1), and (4) and (3), thereby representing the growth rate increase due to the momentum loss only. Consistent with the results in [15] the effect of momentum loss is found to be insensitive to the Lewis number variation, even in the presence of heat loss. Furthermore, comparing curves (5) and (6) in figure 4, it is also found that the additional momentum loss effect on the growth rate becomes smaller in the presence of heat loss. This is because heat loss lowers the temperature of burnt gas, resulting in a reduced viscosity variation across the flame. Since the viscosity variation is the main driving force for the S T instability, the amount of flame instability is attenuated with heat loss. Figures 5(a) (c) show this behaviour more clearly by plotting the growth rate versus the heat loss level for the three Lewis number cases. While the combined effect of heat release on the growth rate depends on

10 668 S. H. Kang et al. Figure 5. Growth rate versus the heat loss level and S T mode contribution for (a) Le = 0.7, (b) Le = 1.0, and (c) Le = 1.3: h = 10, H = 1,λ = 20, F 0 /λ = the Lewis number (as discussed in the previous section), the contribution from the momentum loss [difference between curves (1) and (2)] consistently decreases with heat loss for all Lewis numbers. This result implies that the S T instability mode in a micro-combustion device may not be as significant as one might expect from large scale combustors, as the small dimension is often subjected to a large amount of heat loss. 4. Instability behaviour: nonlinear regime 4.1 Cell splitting by D T instability In the previous section, heat loss was found to amplify the D T instability. This result, however, was within the linear instability regime with a small amplitude of perturbation, and it is unclear if this result can be extended to a general nonlinear behaviour. In this section, the nonlinear behaviour of flame instabilities with heat loss is investigated by simulating the evolution of flame wrinkles for an extended period of time. As in previous sections, the effects of the D L and D T instability modes without momentum loss are investigated first as a baseline case. The local flame speed is affected by flame curvature due to the diffusive-thermal effect. In a simplest form, Markstein s analysis [24] predicted that the actual flame speed (u n )ismodified from the laminar flame speed (u L )bythe relation u n = u L [1 (L M /R)] (19)

11 Stability of premixed flames in a narrow channel 669 Figure 6. Time evolution of the normalized reaction rate contours in nonlinear behaviour at (c) t + = 200, (c) t + = 400, (c) t + = 600, and (d) t + = 800, without heat and momentum loss, for Le = 1.0,λ = 20, F 0 /λ = where L M is the Markstein length and R is the radius of curvature, which is defined positive when the flame is convex towards the upstream. More detailed asymptotic analysis [25] shows a formal derivation for the flame stretch, which includes a consistent dependence on curvature. While the effect of curvature is not significant in the linear instability regime due to the small flame wrinkles, consideration of the diffusive-thermal effect associated with curvature is essential in describing the nonlinear behaviour of the flame. Yuan et al. [26] reported that diffusive-thermal effect can lead to a cell splitting in the flame front. Figures 6 and 7 show the evolution of the reaction rate contours along the premixed flame front for Lewis number of 1 and 0.7, respectively. In these cases, neither momentum nor heat loss was included. In figure 6, only one cell develops and the local reaction rate variation is found to be negligible for Le = 1. For Le = 0.7 (figure 7), however, development of two cells by the splitting mechanism is clearly seen. The mechanism of cell splitting can be explained as follows. Owing to the D T instability, a small disturbance becomes large and the flame develops troughs and crests. As the trough

12 670 S. H. Kang et al. Figure 7. Time evolution of the normalized reaction rate contours in nonlinear behaviour at (a) t + = 200, (b) t + = 300, (c) t + = 700, and (d) t + = 900, without heat and momentum loss, for Le = 0.7,λ = 20, F 0 /λ = region retreats, the curvature of the cell edge is increased, resulting in an increase in the reaction rate around the flame segments with a positive curvature [see figure 7(b)]. This results in a transverse acceleration of the flame segments with enhanced reaction rates, creating a stretching in the flatter frontal section of the flame front. Eventually, the crest part weakens and decelerates due to the reduced reaction rate, thereby forming a secondary trough. Comparing figures 6 and 7, it is evident that only the diffusive-thermally unstable mixture can lead to a cell splitting via the D T instability mechanism. 4.2 Effects of heat and momentum loss As we examine the cell splitting processes, the D T effect is found to be an important mechanism. Recognizing that the heat loss effect amplifies the D T instability as discussed in

13 Stability of premixed flames in a narrow channel 671 Figure 8. Time evolution of the normalized reaction rate at the flame surface for (a) the baseline case without heat or momentum loss, and (b) with heat loss only, for Le = 1.0,λ = 20, F 0 /λ = section 3.1., it is anticipated that this cell splitting behaviour will be promoted for a flame with a lower Lewis number in the presence of heat loss. This is investigated by repeating the same calculations as in figures 6 and 7 with an addition of heat loss. In the following results, the Hele Shaw cell thickness of h = 5 and heat loss parameter of H = 0.4 are used. Note that we choose parameter values that are different from those used in section 3, such that the S T instability mode becomes more pronounced as a result of the reduced cell thickness. Considering equations (11) and (12), the heat loss parameter scales as H/h 2 and thus the net heat loss effect is maintained at a similar level as before with the new choice of parameters. First, as an overall observation, figures 8(a) and (b) show the variation in the maximum and minimum values of the normalized reaction rate along the flame surface. These values can be regarded as the local flame strength. In these results, the Lewis number is fixed as unity and momentum loss was not considered. It is found that, if there is no heat loss [figure 8(a)], the local flame strength variation due to the D T effect is small. In the presence of heat loss [figure 8(b)], however, the local maximum and minimum flame strength changes in time even for Le = 1. This result further provides an alternative explanation as to how the D T instability is promoted by heat loss. Figure 9 shows the temporal evolution of a curved flame, and cell splitting is observed even in the flame with Le = 1. Since the role of heat loss is found to magnify the D T instability mode, it is expected that such an effect would be more prominent with smaller Lewis numbers. Figure 10 shows the nonlinear behaviour of the flame in the presence of heat loss for Le = 0.7, in which the enhancement of D T instability by heat loss is remarkable. Owing to the added effect of heat loss, the splitting processes are more active and complex than those shown in figure 9. The variation in the reaction intensity along the flame surface is so large that a local quenching

14 672 S. H. Kang et al. Figure 9. Time evolution of the normalized reaction rate contours in nonlinear behaviour at (a) t + = 480, (b) t + = 960, (c) t + = 1560, and (d) t + = 2160, with heat loss only, for Le = 1.0,λ = 20, F 0 /λ = of the trough region is observed [figure 10(a)]. In practical application, this may lead to a leakage of unburned fuel, resulting in a reduced overall burning rate and increased emission. This behaviour is in line with the results reported in [17]. We next consider the effect of the momentum loss on the nonlinear flame instability response. Figure 11 shows the evolution of the flame with the momentum loss only, without heat loss for Le = 0.7. Even for the same Lewis number as in figure 10, drastically different behaviour is observed. In this case, the strong D L and S T instability cause a large amount of convective flow concentrated near the trough region, hence relatively a small amount of convective flow is applied near the crest region. Consequently, larger-scale wrinkles are formed and the cell splitting event is greatly suppressed. It is of practical interest to assess the net impact of various parameters on the overall flame speed, which is defined as the volume-averaged consumption speed [15]. Figure 12 shows the time evolution of the overall flame speed for various test cases considered. The flame speed was normalized by that of the planar premixed flame for Le = 1.0 without heat loss, such

15 Stability of premixed flames in a narrow channel 673 Figure 10. Time evolution of the normalized reaction rate contours in nonlinear behaviour at (a) t + = 560 and (b) t + = 900, with heat loss only, for Le = 0.7, λ = 20, F 0 /λ = that the relative magnitudes for various conditions can be directly compared. In early periods, heat loss in general results in a reduction in the flame speed, as found in previous studies [17]. After the initial transients, however, the flames with heat loss are subjected to a larger level of the D T instability, increasing the overall flame speed. Line (iv) denoting the case for Le = 1.0 without heat loss shows the least amount of fluctuation caused by flame instability. On the other hand, the case of Le = 0.7 with heat loss, denoted by line (i), has the most active cell-splitting characteristics, thereby compensating for the reduced flame strength by heat loss and resulting in the overall flame speed enhanced up to the level of case (iv). Therefore, it is evident that the magnitude of the overall flame speed and the level of fluctuation are strongly correlated with the cell-splitting tendency, which in turn depends strongly on the strength of the D T instability mode. An alternative interpretation of the cell-splitting tendency arising from the D T instability mode can be made based on the dispersion relation curve as shown in figure 2. Bychkov et al. [27] found that the D L instability behaviour depends strongly on the cut-off wavelength, λ c,g defined as the wavenumber at neutral stability (as seen in figure 2). Travnikov et al. [28] also reported that flame cell split occurs if λ/λ c is larger than 4 5. From figure 2 and many other simulation cases not reported here, we have consistently found that the maximum growth rate occurs approximately at λ m 2λ c. Therefore, the observation by Travnikov et al. may interpreted such that the disturbances with λ 4λ c tend to split into two cells more easily. While it is difficult to determine the sensitivity of the cut-off wavenumber to various parameters when multiple instability modes are combined, figure 12 suggests that the level of flame speed fluctuation and the ultimate overall flame speed increases monotonically with an increase in the resultant cut-off wavenumber, consistent with earlier studies. In summary, the D T instability mode is found to induce cell splitting when the mixture is diffusive-thermally unstable (Le < Le c ). The heat loss effect tends to amplify this behaviour and thus promotes cell splitting into smaller sizes. The momentum loss, on the other hand, is hydrodynamic in nature, and tends to form a larger-scale bulges rather than splitting the cell. The combination of these two competing effects can lead to an interesting flame dynamics, as will be discussed in the next section.

16 674 S. H. Kang et al. Figure 11. Time evolution of the normalized reaction rate contours in nonlinear behaviour at (a) t + = 200, (b) t + = 300, (c) t + = 700, (d) t + = 900, with momentum loss only, for Le = 0.7, λ = 20, F 0 /λ = Multiple-cell interaction In the previous section, a basic cell development pattern was studied for a flame with a single sinusoidal wave across the channel. To better understand a large-scale dynamics and interactions between the cells, numerical calculations were performed with the transverse (y-direction) domain size increased to 54 times the flame thickness. As before, a single sine wave of disturbance was imposed as the initial condition. Four cases were computed: (i) the baseline adiabatic case without momentum loss; (ii) with momentum loss only; (iii) heat loss only; and (iv) with both heat and momentum losses. For all cases when applicable, the parameter values of h = 5.7 and H = 0.5 were chosen in order to yield adequate levels of visual demonstration of the instability behaviour. Temporal evolutions of these four cases are

17 Stability of premixed flames in a narrow channel 675 Figure 12. Time evolution of the overall flame speed for various conditions for λ = 20, F 0 /λ = shown in the animation files 1 through 4, and their instantaneous snapshots are summarizes in figures 13 through 16. Animation 1 and figure 13 show the multiple cell interactions for the baseline case. At an early stage, the initial sinusoidal disturbance grows and flame surface is split into five cells [figure 13(a) and (b)]. Once the cells are formed, the flame speed at the regions of a large curvature increases and the cell grows as it propagates toward the transverse direction. This subsequently leads to the stretching of the flame at the crest part, forming a secondary cell [figure 13(c)]. This cell-splitting mechanism is consistent with the discussion in section 4. During the splitting process, cell merging also occurs at another region. In figure 13(c) and (d), it is seen that two cells merge together at a region near y/δ th = 40. The lateral movement of the cells that leads to the cell merging observed in animation 1 can arise from two effects. First, each cell crest has a natural tendency to grow in size owing to propagation (Huygens principle). According to the dispersion relations, the rate of the flame cell growth depends on the cell size. If one flame cell grows faster than the other, it merges with the other flame cell. Second, the local flow field can also advect the flame front around it, as the tangential component of the convective flow increases with the flame front being more curved. This results in a travelling of smaller waves riding on a larger scale wave. In most cases under study, the latter effect of the convection-driven cell movement appears to be the dominant mode because the growth rate of the flame cell is usually comparable, while the tangential convective flow component is always generated due to the growth of a flame wrinkle as a result of D L instability. List of Animations All animation files are in AVI format and were created by Tecplot v8.0 in the Windows 2000 system. Animation 1. Time evolution of the normalized reaction rate contours with multiple cell interactions for the baseline case without heat or momentum loss, for Le = 0.7,λ = 54, F 0 /λ = 10 2.Aduration of t + = is shown. Animation 2. Time evolution of the normalized reaction rate contours with multiple cell interactions with momentum loss only, for Le = 0.7,λ = 54, F 0 /λ = 10 2.Aduration of t + = is shown. Animation 3. Time evolution of the normalized reaction rate contours with multiple cell interactions with heat loss only, for Le = 0.7,λ = 54, F 0 /λ = 10 2.Aduration of t + = is shown. Animation 4. Time evolution of the normalized reaction rate contours with multiple cell interactions with both heat and momentum losses, for Le = 0.7,λ = 54, F 0 /λ = 10 2.Aduration of t + = is shown.

18 676 S. H. Kang et al. Figure 13. Time evolution of the normalized reaction rate contours with multiple cell interactions for the baseline case without heat or momentum loss, for Le = 0.7, λ = 54, F 0 /λ = The results suggest that cell splitting is mainly induced by the D T instability, while cell merging results from the increased tangential convective flow owing to the D L instability. Therefore, it is expected that the relative dominance between the two events will be affected by the presence of heat or momentum loss, which will be discussed next. Animation 2 and figure 14 show the multiple cell interactions with momentum loss only. Figure 14(a) corresponds to an enhanced initial disturbance that is split into several cells. These smaller cells, however, soon travel along the flame front by the lateral convective flow, and disappear at the trough [figure 14(b)]. A similar pattern repeats in the subsequent event [figures 14(c) and (d)]. All of these results look very similar to those in figure 13, but the flame wrinkle development and the subsequent merging occur at a faster rate. Moreover, cell merging is found to be more prominent than cell splitting. This can be attributed to an increased level of S T instability which is additive to the D L mode, thereby enhancing flame cell merging processes. Consequently, the ultimate flame front shape consists of even larger scale cells [compare figure 13(d) and figure 14(d)].

19 Stability of premixed flames in a narrow channel 677 Figure 14. Time evolution of the normalized reaction rate contours with multiple cell interactions with momentum loss only, for Le = 0.7, λ = 54, F 0 /λ = Animation 3 and figure 15 shows the multiple cell interactions with heat loss only, revealing contrasting behaviour. From the early stage, the flame shape is dominated by relatively smaller size cells, which spawn into a larger number of cells by the splitting mechanism [figure 15(b)] as described earlier. This is owing to the fact that the added heat loss reduces the D L instability, such that the D T instability mode becomes more prominent to generate an increased number of cell-splitting events. In this case, cell merging is still observed [figure 15(c) near y/δ th = 30)], but it is mainly owing to the competition between the two flame cells with unequal strength rather than the tangential convective flow. This further confirms that the heat loss effect promotes the D T instability mode that enhances multiple cell-splitting events. Finally, animation 4 and figure 16 show the results with both heat and momentum loss effects included. As the two competing effects are combined, the results are expected to be somewhere between those in figures 14 and 15. Figure 16(d) demonstrates that the results are as expected, exhibiting the number of smaller cells as many as those shown in figure 15, yet revealing a larger scale wavy structure arising from the additional S T mechanism.

20 678 S. H. Kang et al. Figure 15. Time evolution of the normalized reaction rate contours with multiple cell interactions with heat loss only, for Le = 0.7, λ = 54, F 0 /λ = The above results lead to the conclusion that the net flame dynamics behaviour with both heat and momentum loss mechanisms depend strongly on the relative dominance between the two effects. As an attempt to identify the relative sensitivity of the flame instability to the two important parameters, the Peclet number and the heat loss parameter (H), the following numerical experiments were performed. For two representative Lewis number conditions (Le > 1 and Le < 1), the dispersion relation curve was generated for the baseline condition (no momentum or heat loss) as shown in figure 2. The wavenumber at the neutral stability is then selected, which are found to be k = 1.74 for Le = 0.7 and k = 0.41 for Le = 1.3. Once the wavenumber of the perturbation is fixed, then by trial and error the combination of Pe and H values is sought at which the dispersion relation finds the neutral stability point at the same wavenumber. The procedure is repeated and the results are plotted in figure 17 in terms of 1/Pe and H for the two Lewis number cases. For each case of the Lewis numbers, the region above the curve represents that the flames are more unstable (or the neutral stability occurs at a higher wavenumber) compared to the baseline case. As discussed before, for Le < 1 the heat loss amplifies the D T instability mode such that a negative slope of the curve results, and vice

21 Stability of premixed flames in a narrow channel 679 Figure 16. Time evolution of the normalized reaction rate contours with multiple cell interactions with both heat and momentum losses, for Le = 0.7, λ = 54, F 0 /λ = versa for Le > 1. The overall slope of the curves therefore represents the relative sensitivity of the flame instability behaviour to the effect of momentum loss (1/Pe) and the heat loss (H). 6. Conclusions As a potential application to the development of micro-scale combustion devices, the fundamental mechanisms and characteristics of the momentum and heat loss effects on the stability of laminar premixed flames in a Hele Shaw cell were investigated using two-dimensional high-fidelity numerical simulations. The effects of heat loss to the wall have been considered by a modelling approximation of quadratic temperature profiles. In general, the momentum loss promotes the S T instability which is additive to the hydrodynamic instabilities, while the heat loss effects result in an enhancement of the D T instability. In the linear regime, heat loss increases the sensitivity of the growth rate to the Lewis number variation. In other words, heat loss shows a stabilizing effect for larger Lewis numbers and a destabilizing effect for smaller Lewis numbers. Furthermore,

22 680 S. H. Kang et al. Figure 17. Iso-stability lines for Le = 0.7 and Le = 1.3 at fixed wavenumber of perturbation (k = 1.74 for Le = 0.7 and k = 0.41 for Le = 1.3), showing the relative sensitivity of the flame instability to the Peclet number and heat loss parameter. the critical Lewis number, at which the flame becomes neutrally stable, was found to increase with an additional heat loss. In the nonlinear regime analysis, it was found that an enhancement of the D T instability results in a complex flame cell dynamics. In general, heat loss promotes cell splitting by the amplified D T instability mechanism. On the other hand, an increase in the D L or S T instability favours a growth to larger cells by the induced convective flow tangent to the flame surface. The two competing effects between the heat and momentum losses play an intricate role in determining the overall instability and cell formation patterns. It was also found that the overall flame speed and its transient fluctuation level depend strongly on the resultant cut-off wavenumber determined by the combined effects of various instability modes. These characteristics were further confirmed in the simulations of multiple cell interactions. The D T instability is found to be enhanced by the heat loss effects and the D L instability is increased by the momentum loss effects. An increase in the D L or S T instability mechanisms results in larger flame cells by the merging process, while a strong D T instability effect produces smaller flame cells by splitting. The relative sensitivity of the flame instability to the momentum or heat loss effect was also examined by a numerical experiment for the Lewis number greater or less than unity. Acknowledgements SHK and SWB were supported by the Combustion Engineering Research Center at the Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology,

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