Imploding Ignition Waves. I. One-Dimensional Analysis

Abstract

We show that converging spherical and cylindrical shock waves may ignite a detonation wave in a combustible medium, provided the radius at which the shocks become strong exceeds a critical radius, R crit . An approximate analytic expression for R crit is derived for an ideal gas equation of state and a simple (power-law-Arrhenius) reaction law, and shown to reproduce the results of numerical solutions. For typical acetylene-air experiments we find R crit ∼ 100 μm (spherical) and R crit ∼ 1 mm (cylindrical). We suggest that the deflagration to detonation transition (DDT) observed in these systems may be due to converging shocks produced by the turbulent deflagration flow, which reaches sub-(but near) sonic velocities on scales R crit . Our suggested mechanism differs from that proposed by Zel'dovich et al., in which a fine-tuned spatial gradient in the chemical induction time is required to be maintained within the turbulent deflagration flow. Our analysis may be readily extended to more complicated equations of state and reaction laws. An order of magnitude estimate of R crit within a white dwarf at the predetonation conditions believed to lead to Type Ia supernova explosions is 0.1 km, suggesting that our proposed mechanism may be relevant for DDT initiation in these systems. The relevance of our proposed ignition mechanism to DDT initiation may be tested by both experiments and numerical simulations.

A subgrid-scale model for deflagration-to-detonation transitions in Type Ia supernova explosion simulations

“…In all cases, the critical quantity is the strength of turbulence. However, the models of Poludnenko et al (2011) and Kushnir et al (2012) require turbulence speeds close to sonic, which we do not observe in our simulations of deflagrations in white dwarfs. The velocity fluctuations of 10 8 cm s −1 assumed in our DDT-SGS model correspond to Mach numbers of ∼0.3 with respect to the fuel material (∼0.1 with respect to the ashes) in the density range in which DDTs are expected.…”

“…The number of initialized detonations equals the ratio of the potential detonation area to the critical size of a DDT region. Although our model refers to the initiation of the detonation via the Zel'dovich gradient mechanism, we note that other proposed mechanisms for forming a detonation out of a turbulent deflagration burning regime (Poludnenko et al 2011;Kushnir et al 2012) would require a similar parameterization of the DDT-SGS model. In all cases, the critical quantity is the strength of turbulence.…”

“…This would trigger the detonation wave far away from the deflagration front. In our work, however, we assume that the deflagration flame itself produces conditions suitable for a DDT and follow the concept of the Zel'dovich gradient mechanism (Zel'dovich et al 1970), even though it has been suggested that the formation of a preconditioned hot spot may not be a neccessary prerequisite (Poludnenko et al 2011;Kushnir et al 2012). In the gradient mechanism, it is assumed that a spontaneous ignition of the fuel in a region with a shallow spatial gradient of induction times leads to a supersonic reaction wave and the build-up of a shock.…”

The Ignition of Carbon Detonations via Converging Shock Waves in White Dwarfs

“…These types of shocks are referred to as CCW shocks due to the pioneering studies of Chester (1954), Chisnell (1955Chisnell ( , 1957, and Whitham (1957Whitham ( , 1958. Thorough analytic and numeric work on these imploding shock waves has already been performed (e.g., Guderley 1942;Stanyukovich 1960;Zel'dovich & Raizer 1967;Landau & Lifshitz 1987;Ponchaut et al 2006;Kushnir et al 2012), so we only summarize their results.…”

“…4.6. Comparison to other work on C detonation initiation Kushnir et al (2012) performed an earlier study of detonation initiation from imploding shock waves. While they apply a more generalized analysis to the problem and focused on reactions with Arrhenius-type temperature dependences, their order-of-magnitude estimate for C/O fuel with ρ 0 = 10 7 g cm −3 is r(v shock = v CJ ) = 10 3 cm, which is in good agreement with our results.…”