Two-phase modeling of deflagration-to-detonation transition in granular materials: Reduced equations

Abstract

Of the two-phase mixture models used to study deflagration-to-detonation transition in granular explosives, the Baer–Nunziato model is the most highly developed. It allows for unequal phase velocities and phase pressures, and includes source terms for drag and compaction that strive to erase velocity and pressure disequilibria. Since typical time scales associated with the equilibrating processes are small, source terms are stiff. This stiffness motivates the present work where we derive two reduced models in sequence, one with a single velocity and the other with both a single velocity and a single pressure. These reductions constitute outer solutions in the sense of matched asymptotic expansions, with the corresponding inner layers being just the partly dispersed shocks of the full model. The reduced models are hyperbolic and are mechanically as well as thermodynamically consistent with the parent model. However, they cannot be expressed in conservation form and hence require a regularization in order to fully specify the jump conditions across shock waves. Analysis of the inner layers of the full model provides one such regularization [Kapila et al., Phys. Fluids 9, 3885 (1997)], although other choices are also possible. Dissipation associated with degrees of freedom that have been eliminated is restricted to the thin layers and is accounted for by the jump conditions.

A new formulation of Kapila’s five-equation model for compressible two-fluid flow, and its numerical treatment

“…In the seven-equation model of Baer and Nunziato [4], two different speeds of sound may occur in a single finite volume. The wave speeds (51) are the same as in Kapila's model [10], of course, and the same as in the reduced model of [14].…”

“…In this paper, a new formulation has been derived for Kapila's known five-equation model [10] two-fluid flow formulation does not explicitly consider the two-fluid interface. It assumes that the flow is a mixture of the two fluids.…”

“…Assumption (1) prescribes that the pressures and velocity vectors on both sides of a two-fluid interface are equal. It is the known step to reduce a seven-equation model to a five-equation model, see, e.g., [1,20,10,3,14,6]. Further, viscosity and heat conduction are neglected.…”

“…doi:10.1016/j.jcp.2010.04.025 contact discontinuity, a practically relevant case. The archetype five-equation model is that of Kapila et al [10]. It has already found many applications, a non-exhaustive list of excellent references is [12,3,14,2,21,17].…”

Extended Noble–Abel Stiffened-Gas Equation of State for Sub-and-Supercritical Liquid-Gas Systems Far from the Critical Point

“…Another approach relies on continuous models with extra internal variables, such as volume and mass fractions and extended equation of state. Examples of such models are the Kapila et al [7] one and its extension with phase transition (Saurel et al [8]) to cite a few. With these formulations, the same equations are solved everywhere routinely, in pure liquid, pure gas and interface which becomes a diffuse zone.…”