Fingering instability in forward smolder combustion

Abstract

We use numerical strategies to examine the linear and nonlinear stability of forward smolder waves in the framework of a simplified thermal–diffusive model, with the hydrodynamic effects completely filtered out. The configuration consists of a horizontal thin solid fuel, over which air blows in the same direction as the smolder front propagation. It is found that, in the absence of convective heat losses, the whole one-dimensional adiabatic solution branch is linearly stable; in contrast, when the convective heat loss effect is taken into account, fingering instability emerges provided the incoming air flow rate is within a narrow range near the one-dimensional extinction limit, a manifestation that is reminiscent of the familiar cellular instability occurring in the context of low-Lewis-number diffusion flames. Accordingly, the fingering instability herein identified in forward smolder combustion is purely thermal–diffusive in nature. Furthermore, a heuristic analysis by drawing an analogy with premixed flame suggests that the occurrence of such fingering instability is the joint consequence of the Lewis number effects and convective heat losses. It is proposed that a Hele–Shaw-type combustion channel may be adopted to experimentally reveal the fingering patterns predicted by current numerical simulations.

Keywords:

• forward smolder
• thermal–diffusive
• fingering instability
• Lewis number
• heat loss

Acknowledgments

This work was supported by the National Natural Science Foundation of China, under grant No.50706024, and the Shanghai Rising-Star Program, under grant No.09QA1402300. We are grateful to Aiming Yang for suggestions and helpful discussions on the WENO scheme. We also wish to thank Sandra L. Olson for bringing to our attention the concurrent flame spread experiment discussed in [19].

Notes

¹Henceforth the term upper (lower) solution branch will be associated with the non-adiabatic temperature response curve in Figure 1.

²The current Lewis number of O₂ is less than the normal value (≈1). Practically this low Lewis number may be achieved in an atmosphere diluted with certain heavier gases, e.g. CO₂ [18]. However, according to the heuristic analysis proposed in the text, fingering instability may equally be expected for higher Lewis number cases, as long as the heat loss effect becomes dominant.