A scaling analysis of the critical conditions for a deflagration-to-detonations transition

In this paper an intuitive criterion for the deflagration-to-detonations transition (DDT) by turbulent mixing was deduced. An analysis was first carried out to determine the critical conditions for detonation initiation within a reactive mixture, which is non-uniformly heated. It was demonstrated that the critical conditions depend on both the size and the characteristic heating time of the energy source. If the characteristic heating time is larger than a critical value, no detonation initiation can be expected, no matter how large the total energy of the heat source is. The critical parameters obtained were then applied to determine the critical conditions for detonation initiation via turbulent mixing. It was found that the DDT depends on both the mixing time and length scales. According to the Damkohler number, there are two regimes. (a) When the Damkohler number is smaller than unity, the critical integral length scale for DDT decreases as the inverse of the turbulent velocity. (b) When the Damkohler number is larger than unity, the critical integral length scale increases according to the cube of the turbulent velocity. These critical conditions were explained in a turbulent phase diagram of Borghi.(Some figures in this article appear in colour in the electronic version; see www.iop.org)