Accelerating flames in tubes—an analysis

Flame acceleration in tubes is studied. A tube filled with flammable mixture is closed at one end and open to the atmosphere at its second end. When ignition takes place near the closed end, it is well-known from experiments that the flame may accelerate, oscillate and eventually reach considerable speeds. A one-dimensional analysis is presented, based upon the assumption that the flame front propagates at a speed that is small compared to the speed of sound. The analysis leads to a construction of the complete unsteady solution. Results from the analysis and from a numerical simulation are compared. They are similar enough to validate the analysis. The tube acoustics are set in motion by the expansion of the fluid due to ignition at the closed end. Subsequently, both spectrum and amplitude evolve because of the motion of the temperature interface, and because of forcing by the flame front, which the analysis precisely quantifies. Oscillations in the front position are strong enough to result in flow reversal. In addition, the induced periodic acoustic acceleration of the temperature and density interface will periodically make the flame front Rayleigh–Taylor unstable, which should result in the dramatic increase in the propagation speed seen in experiments.