Nonequidiffusive premixed-flame propagation in obstructed channels with open, nonreflecting ends

Equidiffusive premixed combustion in obstructed channels with both open, nonreflecting ends exhibits various forms of flame propagation: oscillations, acceleration or a combination of both regimes. Given the limited practicality of equidiffusive premixed combustion, it is important to understand how these modes of combustion are altered at nonequidiffusive conditions, characterized by a nonunity Lewis number (thermal to mass diffusivity ratio) $\mathrm{Le}\ne 1$. To achieve this, the impacts of $\mathrm{Le}$ on the flame dynamics and morphology are analyzed by means of the computational simulations of the reacting flow equations, with Arrhenius chemical kinetics, fully compressible hydrodynamics, and transport properties. In addition to varying $\mathrm{Le}$, the parametric study includes various blockage ratios, channel widths, obstacle spacing and thermal expansion ratios. It is identified how these parameters influence the burning velocities as well as the scaled oscillation amplitude and frequency. Specifically, in the narrow channels with small blockage ratios, the amplitude and frequency of the oscillations vary with $\mathrm{Le}$, with the frequency decreasing and the amplitude increasing as $\mathrm{Le}$ grows from 0.3 to 2. In other conditions, a transition from the flame oscillations to sudden flame acceleration or its propagation at a constant velocity is singularly influenced by $\mathrm{Le}$, or by the interplay of $\mathrm{Le}$ with the geometric parameters of a channel. The delay time before the onset of flame acceleration, especially at $\mathrm{Le}<1$, also varies as the channel width and the blockage ratio change. In all cases, $\mathrm{Le}$ has both quantitative and qualitative effects on flame propagation in obstructed channels with both open, nonreflecting ends.

Figure 1

An illustration of flame propagation in an obstructed channel with both extremes open (only an upper half is shown).

Figure 2

The typical temperature snapshots for the evolution of a flame with $\mathrm{\Theta }=8$ during one oscillation in a channel with $R=12L_f$, $\alpha =1/3$ for $\mathrm{Le}=0.3$ (a) and $\mathrm{Le}=2$ (b). The white lines with the arrows represent the streamlines of the flow.

Figure 3

The scaled burning rate $U_w/S_L$ vs the scaled time $t{S}_L/R$ for the thermal expansion ratio $\mathrm{\Theta }=8$, the blockage ratio $\alpha =1/3$, the obstacle spacing $\mathrm{\Delta }\mathrm{Z}=R/4$, the channel width $R=12L_f$, and various Lewis numbers $\mathrm{Le}=$ 0.3 (a), 0.6 (b), 1 (c), and 2 (d).

Figure 4

The scaled oscillation amplitude, $\mathrm{\Delta }{U}_w/S_L$, and frequency, $f_p$ (scaled by $S_L/R$), vs the Lewis number, $\mathrm{Le}$, for the thermal expansion ratio $\mathrm{\Theta }=8$, the channel half-width $R=12L_f$, the blockage ratio $\alpha =1/3$, and the obstacle spacing $\mathrm{\Delta }\mathrm{Z}=R/4$. The amplitudes were calculated using the maximal and minimal values of the scaled burning rates in Fig. 3, $\mathrm{\Delta }U=\left(U_{w,\max }-U_{w,\min }\right)/2$, while the frequencies were determined as the number of complete oscillations per period from the same plot.

Figure 5

The color temperature snapshots for the evolution of a flame with $\mathrm{Le}=0.3,$ $\mathrm{\Theta }=8$, in a channel with $\alpha =2/3$ and $R=12L_f$ (a) and $R=24L_f$ (b). The white lines with the arrows represent the streamlines of the flow.

Figure 6

The scaled burning rate $U_w/S_L$ vs the scaled time $\tau =t{S}_L/R$ for the $\mathrm{\Theta }=8$ flames at various Lewis numbers $\mathrm{Le}=0.3,0.6,1,2$, propagating in the channels with the blockage ratio $\alpha =2/3$, the obstacle spacing $\mathrm{\Delta }Z=R/4$, at various channel half-widths: $R=12L_f$ (a) and $R=24L_f$ (b).

Figure 7

The scaled total burning rate $U_w/S_L$ vs the scaled time $t{S}_L/R$ for the thermal expansion ratio $\mathrm{\Theta }=8$, the obstacle spacing $\mathrm{\Delta }Z=R/4$, the channel half-width $R=24L_f$, various blockage ratios $\alpha =1/3,1/2,2/3$, at various Lewis numbers $\mathrm{Le}=0.3$ (a) and $\mathrm{Le}=2$ (b).

Figure 8

The scaled total burning rate $U_w/S_L$ vs the scaled time $t{S}_L/R$ for the thermal expansion ratio $\mathrm{\Theta }=8$, the obstacle spacing $\mathrm{\Delta }\mathrm{Z}=R/4$, the blockage ratio $\alpha =2/3$, for the channels of half-widths $R=12L_f,24L_f$, $36L_f$, $48L_f$, at various Lewis numbers $\mathrm{Le}=0.3$ (a) and $\mathrm{Le}=2$ (b).

Figure 9

The scaled total burning rate $U_w/S_L$ vs the scaled time $t{S}_L/R$ for the thermal expansion ratio $\mathrm{\Theta }=8$, the channel half-width $R=24L_f$, the Lewis number $\mathrm{Le}=0.3$, for various obstacle spacing $\mathrm{\Delta }\mathrm{Z}=R/4,R/2,R$, and various blockage ratios $\alpha =1/2$ (a) and $2/3$ (b).

Figure 10

Oscillating and accelerating regimes of flame propagation: $R\text{−}\mathrm{Le}$ diagram at $\alpha =1/2$, $\mathrm{\Delta }Z=R/4$, and $\mathrm{\Theta }=8$ (a); $\alpha \text{−}\mathrm{Le}$ diagram at $R=24L_f$, $\mathrm{\Delta }Z=R/4,$ and $\mathrm{\Theta }=8$ (b); and $\mathrm{\Theta }\text{−}\mathrm{Le}$ diagram at $R=24L_f$, $\mathrm{\Delta }Z=R/4,$ and $\alpha =1/3$ (c). Colors represent the scaled burning rate $U_w/S_L.$

Figure 11

The color temperature snapshots for the evolution of a flame with $\mathrm{\Theta }=8$, $\mathrm{Le}=0.3$ propagating in a channel with $\alpha =2/3$, $R=12L_f$ (a partial section view). The white lines with the arrows represent the streamlines of the flow.

Figure 12

The scaled burning rate $U_w/S_L$ (a), the fuel temperature at the flame tip $T_{\mathrm{tip}}$ (b), and the flame tip Mach number ${\mathrm{Ma}}_{\mathrm{tip}}$ (c) vs the scaled time $t{S}_L/R$ for the $\mathrm{Le}=0.3$, $\mathrm{\Theta }=8$ flames propagating in the obstructed channels with $\alpha =2/3$, $\mathrm{\Delta }\mathrm{Z}=R/4$, and various $R/L_f=12,24,36$.

Figure 13

The resolution test: the scaled burning rate $U_w/S_L$ vs the scaled time $t{S}_L/R$ for the thermal expansion ratio $\mathrm{\Theta }=8$, the half-width of the channel $R=24L_f$, the blockage ratio $\alpha =1/2$, the obstacle spacing $\mathrm{\Delta }\mathrm{Z}=R/4$, for various Lewis numbers $\mathrm{Le}=0.3$ (a), 1 (b), and 2 (c), and various mesh sizes $\mathrm{\Delta }{Z}_f=0.1L_f$ (dashed-dotted), $\mathrm{\Delta }{Z}_f=0.2L_f$ (dashed), and $\mathrm{\Delta }{Z}_f=0.4L_f$ (solid) in each panel.