Flame Propagation in a Nonuniform Mixture: The Structure of Anchored Triple-Flames

Abstract

A model, which includes upstream conduction and diffusion, is used to describe the structure of flames in a nonuniform medium. Such flames typically consist of fuel-rich and fuel-lean premixed flames, followed by a diffusion flame which starts where the two premixed flames meet. Such formations have been observed experimentally and probably occur as laminar flamelets in turbulent nonpremixed combustion. The equations are solved in a limit of low heat release (so that hydrodynamic effects need not be considered), while the Zel'dovich number is considered to be large. In this context, it is found that freely propagating triple-flames have propagation speeds that are bounded above by the maximum adiabatic flame speed of the system. In the context of a diffusion flame behind a splitter-plate, the flames are considered for blowing velocities slightly less than the maximum flame speed. For greater blowing velocities, the structure can only be maintained if combustion is initiated by some means other than upstream propagation. Solutions are obtained for such anchored flames. As the blowing velocity is reduced towards the triple-flame propagation speed, the freely propagating triple-flame solution is recovered. Introduction Given a region of nonuniformly premixed fuel and oxidant, the adiabatic laminar flame speed, as determined by the local mixture properties of the gas, varies from point to point. In particular, for similar initial temperatures of fuel and oxidant, the path of greatest adiabatic laminar flame speed tends to lie close to any stoichiometric boundary in the system. Thus, one should expect that the propagation of a premixed flame would tend to surge ahead primarily around this boundary. After the flame has passed, hot excess fuel remains unburnt on one side of the stoichiometric line, while excess oxidant remains on the other side. As a result, a diffusion flame forms at the boundary where these excess react ants meet. Away from the stoichiometric boundary, the propagation of the premixed flame becomes slower and eventually stops as the flame enters regions of weakening mixture strength. In fact, three distinct flames can be identified: a fuel-rich premixed flame, leaving unburnt fuel behind it; a fuel-lean flame, leaving oxidant; and a diffusion Copyright © 1988 by the American Institute of Aeronautics and Astronautics, inc. All rights reserved. *Mathematics Department. 240 ANCHORED TRIPLE-FLAMES 241 flame along the stoichiometric boundary, beginning where all three flames meet. As seen in Fig. 1, such triple-flames have been observed experimentally [Phillips (1965), Ishizuka (1986)] and probably play a role in the combustion of turbulent diffusion flames, where a possible extinction of a locally laminar flamelet may lead to diffusive mixing and subsequent reignition of a region of nonuniformly premixed gases [Peters (1986)]. Until recently [Bold (1988)], modelling of flames in nonuniform media has relied on neglecting upstream conduction and diffusion [Linan and Crespo (1976), Dold and Clarke (1986)]. Although simplifying the mathematics, this effectively rules out the possibility of upstream flame propagation and cannot be justified for flows comparable with adiabatic flame speeds. Combustion must therefore be initiated in some other way, such as an ignition through thermal runaway or perhaps a hot wire. Subsequent combustion then takes the form of transversely propagating premixed flames (which slow down and eventually extinguish through propagating into regions of weakening mixture ratio), and a diffusion flame which is created as one of these flames crosses a stoichiometric boundary. Thus, some aspects of triple-flame structure are retained, without the property of upstream propagation. In this paper, we use a small heat-release model (developed elsewhere [Dold (1988)]) to examine the structure of the triple-flame in a slowly varying regime of solution. The results are valid for large Zel'dovich number /?, and for transverse mixture-fraction gradients (on the length scale of a typical preheat-zone thickness) which are small compared with jfl. Results are considered in the context of a diffusion flame behind a splitter-plate, with a blowing velocity not much less than the maximum adiabatic flame speed (at which stage the assumption of slow variation is valid). For blowing velocities greater than this maximum speed, solutions are found only if combustion is initiated in some way besides upstream propagation (thermal runaway or a hot wire as before). These solutions are examined as the blowing velocity is increased and are compared with solutions obtained through neglecting upstream conduction and diffusion. As the blowing velocity is progressively reduced, the flame structure abruptly recovers its freely propagating form. Fig. 1 Triple-flame propagating in a nonuniform medium. Note the presence of threedimensional effects. (British Crown Copyright: Reproduced from Phillips (1965) by kind permission of H. Phillips, Health and Safety Executive, Buxton, U.K.)