Self-accelerating flames in long narrow open channels

In this work we extend our earlier asymptotic one-dimensional analysis of flame propagation in long narrow channels open at both ends to two-dimensional flames. The analysis follows two tracks; a multi-scale asymptotic study and a full numerical study of the unsteady propagation. We show that during the early stages of propagation the flame accelerates at a nearly constant rate, independent of the channel height. In sufficiently narrow channels, the flame retains a constant acceleration until it reaches the end of the channel, consistent with our earlier work. In wider channels, however, the flame beyond a certain distance begins to accelerate at a nearly-exponential rate, reaching exceedingly large speeds at the end of the channel. The flame self-acceleration arises from the combined effects of gas expansion and lateral confinement. The gas expansion that results from the heat released by the chemical reactions produces a continuous flow of burned gas directed towards the ignition end of the channel. Due to the frictional forces at the walls and, since the pressure at both ends is maintained constant, the gas motion that develops in the burned gas sets a pressure gradient that drives the fresh unburned gas towards the other end of the channel. Stretching out to reach additional fuel, the flame extends towards the fresh mixture propagating faster. And because of lateral confinement, the gas expansion induces large straining on the elongated flame surface that further increases its propagation speed. The asymptotic approximation properly predicts the initial propagation stage, the location within the channel where the sudden acceleration begins and the early stages of the self-accelerating process. The full numerical study confirms and extends the asymptotic results, showing that in long but finite channels premixed flames could self-accelerate reaching velocities that are ten-to-twenty times larger than the laminar flame speed.