A one-dimensional model for deflagration to detonation transition on the tip of elongated flames in tubes.

Abstract

Deflagration-to detonation transition (DDT) of the tip of self-accelerating elongated front of laminar flames in tubes filled with gas mixtures that are very energetic is studied by employing the one-dimensional model sketched in Fig. 1b. The first part of the present work parallels the nonlinear analysis of the self-similar solution of the double discontinuity model performed by Deshaies and Joulin (1989) (termed DJ herein) in the double limit of weak lead shocks and flame speeds that are very sensitive to the flame temperature. The very energetic mixtures addressed herein exhibit only mild flame-speed dependence on flame temperature but large density ratio so that the critical condition concerns a Mach number of the lead-shock exceeding unity by an amount of order unity. A double-feedback mechanism, in which compressional heating by the lead shock is augmented by an effective piston acceleration due to a back-flow of burnt-gas towards the flame-tip, is shown to yield self-similar solutions that exhibit a turning point at a critical propagation velocity as in DJ but for a lead shock which is not weak. Beyond self-similarity, a further analysis of the upstream-running simple waves generated ahead of the self-accelerating flame then predicts the spontaneous formation of a shock wave on the flame front as a consequence of the finite-time singularity of the acceleration of the flame front at the critical velocity (turning point). Such a shock formation is a good candidate to blow up the inner flame structure, producing the abrupt transition of the flame (a subsonic, quasi-isobaric reaction–diffusion wave) into a detonation (a supersonic compressive wave generating rapid chemical heat release), observed in previously reported experiments, the results of which are consistent with the present scaling.