Pulsating Instabilities of Combustion Waves in a Chain-Branching Reaction Model

Abstract

In this paper we investigate the properties and linear stability of traveling premixed combustion waves and the formation of pulsating combustion waves in a model with two-step chain-branching reaction mechanism. These calculations are undertaken in the adiabatic limit, in one spatial dimension and for the case of arbitrary Lewis numbers for fuel and radicals. It is shown that the Lewis number for fuel has a significant effect on the properties and stability of premixed flames, whereas varying the Lewis number for the radicals has only qualitative (but not qualitative) effect on the combustion waves. We demonstrate that when the Lewis number for fuel is less than unity, the flame speed is unique and is a monotonically decreasing function of the dimensionless activation energy. Moreover, in this case, the combustion wave is stable and exhibits extinction for finite values of activation energy as the flame speed decreases to zero. However, for the fuel Lewis number greater than unity, the flame speed is a C-shaped and double valued function. The linear stability of the traveling wave solution was determined using the Evans function method. The slow solution branch is shown to be unstable whereas the fast solution branch is stable or exhibits the onset of pulsating instabilities via a Hopf bifurcation. The critical parameter values for the Hopf bifurcation and extinction are found and the detailed map for the onset of pulsating instabilities is determined. We show that a Bogdanov—Takens bifurcation is responsible for both the change in the behavior of the traveling wave solution near the point of extinction from unique to double valued type as well as for the onset of pulsating instabilities. We investigate the properties of the Hopf bifurcation and the emerging pulsating combustion wave solutions. It is demonstrated that the Hopf bifurcation observed in our present study is of supercritical type. We show that the pulsating combustion wave propagates with the average speed smaller than the speed of the traveling combustion wave and at certain parameter values the pulsating wave exhibits a period doubling bifurcation.

The differential diffusion effect of the intermediate species on the stability of premixed flames propagating in microchannels

“…The time-dependent calculation agrees well with the solution given by the linear stability analysis. Period-two and period-four solutions have been reported earlier in the chain-branching kinetics of planar flame fronts [42].…”

“…The case m = 0 also exhibits extinction. It seems to be a characteristic of the chain-branching kinetics, not seen in the one-step model, that the flame becomes unstable to extinction before reaching the theoretical flammability limit for Le F > 1 cases [42]. The role of the differential diffusion of the radical is completely reversed for Le F > 1 in comparison to Le F < 1.…”

“…However, the dependence of the solutions with the activation energy of the branching reaction is known to be weak, a surprising characteristic that differs from the behaviour of the one-step model. For Le F > 1, the chain-branching model predicts oscillatory instability, but only below a threshold heat of reaction, found to be very close to the flammability limit and for too high Le F [32,33,42]. The radical concentration at this limit is very small, indicating the significant role of the intermediate species on flame stabilisation.…”

Period doubling and chaotic transient in a model of chain-branching combustion wave propagation

“…The linear stability of the travelling combustion waves is investigated in Gubernov et al (2009) in detail for the case u a = 0. Here, we extend the previously obtained results to include the effect of non-zero ambient temperature on the flame stability.…”

“…Since the variation of u a does not change the qualitative behaviour of the combustion waves, in the rest of the paper we fix the value of the ambient temperature to zero. Gubernov et al (2009) …”