Asymptotic Solutions of Two Fundamental Problems in Gaseous Detonations

ABSTRACT

Two recent analytical results concerning i) the direct initiation of gaseous detonations in free space and ii) the deflagration to detonation transition in tubes are revisited in this article. The first problem is treated by an asymptotic analysis in the limit of small heat release, enlightening the mechanisms controlling the direct initiation process of real detonations. The second one is limited to a one-dimensional mechanism of transition at work on the tip of elongated flames in tubes. Particular attention is paid to the physical mechanisms more than to the technical details of the analyses. However, the mathematical formulation as well as the different steps of the analysis are clearly presented in a synthetic manner.*

KEYWORDS:

• Direct initiation
• integral equation
• finite-time singularity in DDT

Acknowledgments

The authors are grateful to A. Liñán and F.A. Williams for enlightening discussions and to B. Denet, R. Hernández-Sánchez, G. Lodato, H. Tofaili and L. Vervisch for fruitful collaboration.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Notes

¹ By definition, a mode is said downstream-running (upstream-running) when it propagates in the same (opposite) direction as (of) the flow relative to the lead shock $u-\mathcal{D}$. The propagation of the detonation is here from left to right (in the direction of increasing radius), the shocked gas corresponding to $x<0$.

² According to the Russian literature, the marginal character of the regime called the Chapman-Jouguet was first pointed out in 1893 by Mikhel’son; the analysis of Chapman was published independently in 1899 and that of Jouguet in 1904, see the text book of Shchelkin and Troshin (1965).

Additional information

Funding

The work was partially supported by the French agency Agence Nationale de la Recherche ANR through grant #18-CE05-0030.