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Rayleigh–Bénard instability generated by a diffusion flame

Published online by Cambridge University Press:  08 November 2013

P. Pearce  and
J. Daou
P. Pearce
Affiliation:
School of Mathematics, University of Manchester, Manchester M13 9PL, UK
J. Daou*
Affiliation:
School of Mathematics, University of Manchester, Manchester M13 9PL, UK
*
†Email address for correspondence: [email protected]
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Abstract

We investigate the Rayleigh–Bénard convection problem within the context of a diffusion flame formed in a horizontal channel where the fuel and oxidizer concentrations are prescribed at the porous walls. This problem seems to have received no attention in the literature. When formulated in the low-Mach-number approximation the model depends on two main non-dimensional parameters, the Rayleigh number and the Damköhler number, which govern gravitational strength and reaction speed respectively. In the steady state the system admits a planar diffusion flame solution; the aim is to find the critical Rayleigh number at which this solution becomes unstable to infinitesimal perturbations. In the Boussinesq approximation, a linear stability analysis reduces the system to a matrix equation with a solution comparable to that of the well-studied non-reactive case of Rayleigh–Bénard convection with a hot lower boundary. The planar Burke–Schumann diffusion flame, which has been previously considered unconditionally stable in studies disregarding gravity, is shown to become unstable when the Rayleigh number exceeds a critical value. A numerical treatment is performed to test the effects of compressibility and finite chemistry on the stability of the system. For weak values of the thermal expansion coefficient $\alpha $, the numerical results show strong agreement with those of the linear stability analysis. It is found that as $\alpha $ increases to a more realistic value the system becomes considerably more stable, and also exhibits hysteresis at the onset of instability. Finally, a reduction in the Damköhler number is found to decrease the stability of the system.

JFM classification

Convection: Buoyancy-driven instability Reacting Flows: Combustion Reacting Flows: Flames
Type
Papers
Information
Journal of Fluid Mechanics , Volume 736 , 10 December 2013 , pp. 464 - 494
Copyright
©2013 Cambridge University Press 

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