Triple-flame propagation in a parallel flow: an analytical study

Abstract

We present an analytical study of triple-flame propagation in a two-dimensional mixing layer against a parallel flow. The problem is formulated within a constant density thermo-diffusive model, and solved analytically in the asymptotic limit of large activation energy of the chemical reaction for flames thin compared with their typical radius of curvature. Explicit expressions are obtained in this limit, describing the influence of the flow on the triple-flame. The results are expected to be applicable when the ratio between the flow-scale and the flame-front radius of curvature (which is mainly dictated by concentration gradients) is of order unity, or larger. When this ratio is large, as in the illustrative case of a Poiseuille flow in a porous channel considered here, the flow is found to negligibly affect the flame structure except for a change in its speed by an amount which depends on the stoichiometric conditions of the mixture. On the other hand, when this ratio is of order unity, the flow is able to significantly wrinkle the flame-front, modify its propagation speed, and shift its leading edge away from the stoichiometric line. The latter situation is investigated in the illustrative case of spatially harmonic flows. The results presented describe, in particular, how the leading-edge of the flame-front can be determined in terms of the flow amplitude A which is critical in determining the flame speed. The latter is found to depend linearly on A in the first approximation with a correction proportional to the flame thickness multiplied by , for |A| sufficiently large. The effect of varying the flow-scale on flame propagation in this context is also described, with explicit formulae provided, and interesting behaviours, such as non-monotonic dependence on the scale, identified.

Keywords:

• triple-flames
• flame-flow interaction
• partially premixed flames

Acknowledgments

The authors would like to thank Dr. Guy Joulin and Prof. John Dold for helpful discussions and suggestions concerning this work.

Notes

See e.g. Equation 45.

Strictly speaking, y* should be denoted by y*₀ since it stands for the leading-order location of the leading edge. It is to be determined precisely as the location of a global maximum of S _L0−u. Since we only need a leading-order approximation to y*, we will not use any additional subscript.

For this ε-expansion to make sense, we further assume that ε|A|^1/2/ℓ≪1.

It may be useful to point out that 72 and 73 imply that y*→y*_min as A→−∞, y*→y*_max as A→+∞ if is even, and y*→y*_max as A→−∞, y*→y*_min as A→+∞ if n is odd. This explains why y* is a decreasing function of A for ℓ=1/3, but not for the other values of ℓ considered.