Modelling the Fluctuations of Reactive Shock Waves in Heterogeneous Solid Explosives as Stochastic Processes

Abstract

While current phenomenological burn models are useful for describing the average or bulk reactive flow behaviour of heterogeneous explosives, one fundamental weakness inherent to these models is the loss of detailed microstructural information at the scale of the calculation. In order to include the effects of the microstructure, and in particular the underlying material heterogeneities that influence the build‐up to detonation, a new paradigm is put forth for modelling sub‐grid, reaction‐induced fluctuations (i. e. “hot spots”) at the continuum level. This modelling approach assumes that the reaction rate is stochastic, rather than deterministic, and it uses Langevin‐type equations with a mathematical framework built upon Itô calculus and Lambourn's CIM model for shocked heterogeneous explosives. This approach follows directly from our previous letter, Ref. [1], and is inspired by the probability density function (pdf) methods used in turbulent reactive flows. Here, the stochastic burn model is derived, implemented, and exercised far beyond what has been shown in previous work. New hydrocode simulation results demonstrate the role of stochastic fluctuations during shock initiation; these fluctuations are approximated by collections of discrete particles, that evolve with drift (i. e. deterministic) and diffusion (i. e. stochastic) coefficients. Additionally, the particle values are propagated and averaged to calculate the heat release, yield strength, and material impedance in each computational cell. Hydrocode simulation results further show how the fluctuating hot spot energy may or may not be transmitted to the wave front, and result in a detonation wave‐like structure. The fundamental stochastic nature of this model permits simulations to have varying outcomes with the same initial conditions; this allows for go/no‐go estimation (e. g., marginal or failed detonations), which might possibly be calibrated using the statistical distributions from real materials. Mesoscale calculations of shocked heterogeneous explosives also show that these fluctuations are physically justified (i. e., Ref. [2]), and it is hypothesized that the pdf functions provide a link between the meso(grain) and continuum scales for practical engineering calculations. Thus, our approach is a paradigm shift for building efficient, reduced order continuum burn models, that represent the sub‐grid stochastic behaviour of shocked heterogeneous solid explosives.

Effect of spatial distribution of mesoscale heterogeneities on the shock-to-detonation transition in liquid nitromethane

“…This study thus offers a caveat that, considering only one or a small number of cavities, a mesoscale calculation might feed limited or biased information into a continuum-level simulation. Some recent modelling efforts have been made by Kittel et al 60,61 and Bakarji and Tartakovsky 62 to link the statistical properties of mesoscale heterogeneities to continuum-level simulations of an explosive via a stochastic burn model. As suggested by the current study, the stochastic information fed into such a model can be obtained from meso-resolved simulations wherein a statistically significant distribution of heterogeneities is explicitly considered.…”