Oscillating flames: multiple-scale analysis

A complete multiple-scale solution is constructed for the one-dimensional problem of an oscillating flame in a tube, ignited at a closed end, with the second end open. The flame front moves into the unburnt mixture at a constant burning velocity relative to the mixture ahead, and the heat release is constant. The solution is based upon the assumption that the propagation speed multiplied by the expansion ratio is small compared with the speed of sound. This approximate solution is compared with a numerical solution for the same physical model, assuming a propagation speed of arbitrary magnitude, and the results are close enough to confirm the validity of the approximate solution. Because ignition takes place at the closed end, the effect of thermal expansion is to push the column of fluid in the tube towards the open end. Acoustics set in motion by the impulsive start of the column of fluid play a crucial role in the oscillation. The analytical solution also captures the subsequent interaction between acoustics and the reaction front, the effect of which does not appear to be as significant as that of the impulsive start, however.