John B. Bdzil

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

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.

Two-phase modeling of deflagration-to-detonation transition in granular materials: A critical examination of modeling issues

The two-phase mixture model developed by Baer and Nunziato (BN) to study the deflagration-to-detonation transition (DDT) in granular explosives is critically reviewed. The continuum-mixture theory foundation of the model is examined, with particular attention paid to the manner in which its constitutive functions are formulated. Connections between the mechanical and energetic phenomena occurring at the scales of the grains, and their manifestations on the continuum averaged scale, are explored. The nature and extent of approximations inherent in formulating the constitutive terms, and their domain of applicability, are clarified. Deficiencies and inconsistencies in the derivation are cited, and improvements suggested. It is emphasized that the entropy inequality constrains but does not uniquely determine the phase interaction terms. The resulting flexibility is exploited to suggest improved forms for the phase interactions. These improved forms better treat the energy associated with the dynamic compacti...

Two-phase modeling of DDT: Structure of the velocity-relaxation zone

The structure of the velocity relaxation zone in a hyperbolic, nonconservative, two-phase model is examined in the limit of large drag, and in the context of the problem of deflagration-to-detonation transition in a granular explosive. The primary motivation for the study is the desire to relate the end states across the relaxation zone, which can then be treated as a discontinuity in a reduced, equivelocity model, that is computationally more efficient than its parent. In contrast to a conservative system, where end states across thin zones of rapid variation are determined principally by algebraic statements of conservation, the nonconservative character of the present system requires an explicit consideration of the structure. Starting with the minimum admissible wave speed, the structure is mapped out as the wave speed increases. Several critical wave speeds corresponding to changes in the structure are identified. The archetypal structure is partly dispersed, monotonic, and involves conventional hydr...

The shock dynamics of stable multidimensional detonation

The paper develops a description for the propagation of an unsupported, unsteady, multidimensional detonation wave for an explosive with a fully resolved reaction zone and a polytropic equation of state. The main features of the detonation are determined once the leading shock surface is known. The principal result is that the detonation velocity in the direction along the normal to the shock is the Chapman-Jouguet velocity plus a correction that is a function of the local total curvature of the shock. A specific example of unsteady propagation is discussed and the stability of a two-dimensional steady solution is examined.

Modeling two‐dimensional detonations with detonation shock dynamics

One of the principal shortcomings of the computer models that are presently used for two‐dimensional explosive engineering design is their inadequate treatment of the explosive’s detonation reaction zone. Current methods lack the resolution to both calculate the broad gas expansion region and model the thin reaction zone with reasonable detail. Recently an alternative method for modeling the reaction zone has been developed. This method applies when the radius of curvature of the shock is large compared to the reaction‐zone length. In this limit, the dynamics of the interaction between the chemical heat release and the two‐dimensional flow in the reaction zone is quasisteady. It is summarized by a relation Dn(κ), between the local normal shock velocity Dn and shock curvature κ. When this relation is combined with the kinematic surface condition (an equation that describes how disturbances move along the shock), the two‐dimensional reaction‐zone calculation is reduced to a one‐dimensional calculation.

A study of the steady‐state reaction‐zone structure of a homogeneous and a heterogeneous explosive

The two‐dimensional steady‐state reaction‐zone structure of a homogeneous and a heterogeneous explosive is studied. To do this theoretical results obtained from the Euler equations of compressible flow are combined with experimental data on steady‐state detonation shock‐wave speed and shape as a function of the explosive charge size. The theoretical results, constrained by the experiments, define an inverse problem for the chemical heat‐release function in the reaction zone which follows the shock wave; this problem is solved. The heterogeneous explosive is made from the homogeneous one by adding small quantities of other materials. Because of this, the two explosives were closely related in many respects. In spite of this, quite large differences in the detonation characteristics are observed between the two explosives, both in the wave speed as a function of charge size and in the shape of shock‐wave loci near the explosive edge. It is found that a single forward rate exponentially dependent on the inve...

A study of detonation diffraction in the ignition-and-growth model

Heterogeneous high-energy explosives are morphologically, mechanically and chemically complex. As such, their ab initio modelling, in which well-characterized phenomena at the scale of the microstructure lead to a rationally homogenized description at the much larger scale of observation, is a subject of active research but not yet a reality. An alternative approach is to construct phenomenological models, in which forms of constitutive behaviour are postulated with an eye on the perceived picture of the micro-scale phenomena, and which are strongly linked to experimental calibration. Most prominent among these is the ignition-and-growth model conceived by Lee and Tarver. The model treats the explosive as a homogeneous mixture of two distinct constituents, the unreacted explosive and the products of reaction. To each constituent is assigned an equation of state, and a single reaction-rate law is prescribed for the conversion of the explosive to products. It is assumed that the two constituents are always in pressure and temperature equilibrium. The purpose of this paper is to investigate in detail the behaviour of the model in situations where a detonation turns a corner and undergoes diffraction. A set of parameters appropriate for the explosive LX-17 is selected. The model is first examined analytically for steady, planar, one-dimensional (1D) solutions and the reaction-zone structure of Chapman–Jouguet detonations is determined. A computational study of two classes of problems is then undertaken. The first class corresponds to planar, 1D initiation by an impact, and the second to corner turning and diffraction in planar and axisymmetric geometries. The 1D initiation, although interesting in its own right, is utilized here as a means for interpretation of the two-dimensional results. It is found that there are two generic ways in which 1D detonations are initiated in the model, and that these scenarios play a part in the post-diffraction evolution as well. For the parameter set under study the model shows detonation failure, but only locally and temporarily, and does not generate sustained dead zones. The computations employ adaptive mesh refinement and are finely resolved. Results are obtained for a rigid confinement of the explosive. Compliant confinement represents its own computational challenges and is currently under study. Also under development is an extended ignition-and-growth model which takes into account observed desensitization of heterogeneous explosives by weak shocks.

Time-dependent two-dimensional detonation: the interaction of edge rarefactions with finite-length reaction zones

A theory of time-dependent two-dimensional detonation is developed for an explosive with a finite-thickness reaction zone. A representative initial–boundary-value problem is treated that illustrates how the planar shock of an initially one-dimensional detonation becomes non-planar in response to the action of an edge rarefaction that is generated at the explosives lateral surface. The solution of this time-dependent problem has a wave-hierarchy structure that at late times includes a weakly two-dimensional hyperbolic region and a fully two-dimensional parabolic region. The wave head of the rarefaction is carried by the hyperbolic region. We show that the shock locus is analytic at the wave head. The dynamics of the final approach to two-dimensional steady-state detonation is controlled by Burgers’ equation for the shock locus. We also present some results concerning the stability of the solutions to our problem.

Kinetics study of a condensed detonating explosive

We discuss a method for determining chemical–kinetic parameters for condensed explosive materials under the extreme pressure and temperature conditions of detonation. It utilizes theoretical results obtained from the Euler equations of fluid mechanics and experimental detonation data, which consist of steady‐state shock shapes and detonation velocities measured as a function of charge size. We determine Arrhenius parameters for detonating commercial‐grade nitromethane. The sensitivity of the method to various uncertainties in the input is tested. Results for the Arrhenius activation energy (EA = 92±23 kcal/mole) and vibrational factor [log10 k(sec−1) = 18±2] are larger than the gas phase values. These results are compared with those of other workers.

Shock-to-detonation transition: A model problem

The process by which a trigger ignited shockless weak detonation is transformed into a classical Zeldovich–von Neumann–Doering (ZND) detonation as the trigger wave decelerates is considered in detail. This occurs as the trigger speed passes through the Chapman–Jouguet velocity DCJ. The model that is developed describes the birth and subsequent growth of a shock in the weak detonation reaction zone. This shock ultimately sustains the shock‐ignited ZND detonation. The time‐dependent solution presented here fills one of the many gaps in the understanding of detonation flows; an understanding based almost entirely on steady wave structures.