Growth of a Diffusion Flame in the Field of a Vortex

Abstract

A simple diffusion flame with fast chemical kinetics is initiated along the horizontal axis between a fuel occupying the upper half-plane and an oxidizer below. Simultaneously a vortex of circulation Γ is established at the origin. As time progresses the flame is extended and “wound up” by the vortex flow field and the viscous core of the vortex spreads, converting the motion in the core to a solid-body rotation.

The kinematics of the flame extension and distortion is described and the effect of the local-flow field upon local-flame structure is analyzed in detail. It is shown that the combustion field consists of a totally reacted core region, whose radius is time dependent, and an external flame region consisting of a pair of spiral arms extending off at large radii toward their original positions on the horizontal axis.

The growth of the reacted core, and the reactant-consumption rate augmentation by the vortex field in both core and outer-flame regions were determined for values of the Reynolds number (Γ/2πν) between 1 and 10³ and for a wide range of Schmidt numbers (ν/D) covering both gas and liquid reactions.

For large values of Reynolds number the radius r _* of the reactant grows much more rapidly than the viscous core so that only the nearly inviscid portion of the flow is involved. The more accurate condition for this behavior is that R(Sc)^1/2 > 50 and, under this restriction, the similarity rule for the core radius growth is shown to be

$$\frac{{{r_*}}}{{{{({\Gamma ^{{\raise0.7ex\hbox{$2$} \!\mathord{\left/ {\vphantom {2 3}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$3$}}}}{D^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$3$}}}}t)}^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}}}} = 0.5092 + O{(D/\Gamma )^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}}$$

In this case also the reactant consumption rate becomes independent of time and, for the complete diffusion flame in the vortex field, the augmentation of reactant-consumption rate due to the vortex field satisfies

$$\frac{{{\rm{Augmented consumption rate}}}}{{{\Gamma ^{{\raise0.7ex\hbox{$2$} \!\mathord{\left/ {\vphantom {2 3}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$3$}}}}{D^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$3$}}}}}} = 1.2327 - 1.4527{(D/\Gamma )^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 6}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$6$}}}} + O{(D/\Gamma )^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}}$$

Both of these similarity rules are, as is appropriate for high Reynolds number, independent of kinematic viscosity.