Analytical studies of the dynamics of gaseous detonations

The dynamics of gaseous detonation is revisited on the basis of analytical studies. Problems of initiation, quenching, pulsation and cellular structures are addressed. The objective is to improve our physical understanding of the development, stability and structure of gaseous detonations. New insights that have been gained from analytical investigations are emphasized. Specific problems discussed are the direct initiation of detonations in spherical geometry, the spontaneous soft initiation and quenching of detonations in a temperature gradient, the stability threshold and dynamics of galloping detonations, and the multi-dimensional instability threshold and cellular structures of both overdriven and near-Chapman-Jouguet detonations. It will be seen that, although there have been many accomplishments, some outstanding questions remain.

Asymptotic behavior of the Rayleigh-Taylor instability

We investigate long time numerical simulations of the inviscid Rayleigh-Taylor instability at Atwood number one using a boundary integral method. We are able to attain the asymptotic behavior for the spikes predicted by Clavin and Williams for which we give a simplified demonstration. In particular, we observe that the spike's curvature evolves as t(3), while the overshoot in acceleration shows good agreement with the suggested 1/t(5) law. Moreover, we obtain consistent results for the prefactor coefficients of the asymptotic laws. Eventually we exhibit the self-similar behavior of the interface profile near the spike.

Theory of gaseous detonations

The objective of the present paper is to review some developments that have occurred in detonation theory over the last ten years. They concern nonlinear dynamics of detonation fronts, namely patterns of pulsating and/or cellular fronts, selection of the cell size, dynamical self-quenching, direct (blast) or spontaneous initiation, and transition from deflagration to detonation. These phenomena are all well documented by experiments since the sixties but remained unexplained until recently. In the first part of the paper, the patterns of cellular detonations are described by an asymptotic solution to nonlinear hyperbolic equations (reactive Euler equations) in the form of unsteady (sometime chaotic) and multidimensional traveling-waves. In the second part, turning points of quasi-steady solutions are shown to correspond to critical conditions of fully unsteady problems, either for (direct or spontaneous) initiation or for spontaneous failure (self-quenching). Physical insights are tentatively presented rather than technical aspects. The challenge is to identify the physical mechanisms with their relevant parameters, and more specifically to explain how the length-scales involved in detonation dynamics are larger by two order of magnitude (at least) than the length-scale involved in the steady planar traveling-wave solution (detonation thickness).

Diamond patterns in the cellular front of an overdriven detonation

A nonlinear integral-differential equation describing the cellular front of an overdriven detonation is obtained by an analysis carried out in the neighborhood of the instability threshold. The analysis reveals both an unusual mean streaming motion, resulting from the rotational part of the oscillatory flow, and pressure bursts generated by the crossover of cusps representative of Mach stems propagating on the detonation front. A numerical study of the nonlinear equation exhibits the "diamond" patterns observed in experiments. An overall physical understanding is provided.