Finite-time singularity associated with the deflagration-to-detonation transition on the tip of an elongated flame-front in a tube

Abstract

The critical condition for the deflagration to detonation transition (DDT) on the tip of elongated flames in tubes is revisited. Outstanding experiments and numerics, performed in 2010, have shown that a train of successive shock waves is produced by the self-accelerating flame-front. A pre-conditioned state is thus generated just prior to the DDT in a slice of unreacted gas adjacent to the flame front with a temperature too low for self-ignition, leaving unexplained the abrupt transition. Using a simplified one-dimensional model, the objective of the theoretical analysis of this short note is to identify the pre-conditioned state just prior to the spontaneous formation of a flow singularity on the flame front. This finite-time singularity responsible for the DDT is produced by the least further increase in propagation velocity beyond a critical value. The attention is focused on the unsteady compression waves emitted in the unreacted gas by the self-accelerating flame front. The unsteady effects were overlooked by the previous theoretical analyses.