Ralph Menikoff

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...

Representations of a local current algebra in nonsimply connected space and the Aharonov–Bohm effect

A recent paper established technical conditions for the construction of a class of induced representations of the nonrelativistic current group SΛK, where S is Schwartz’s space of rapidly decreasing C∞ functions, and K is a group of C∞ diffeomorphisms of Rs. Bose and Fermi N‐particle systems were recovered as unitarily inequivalent induced representations of the group by lifting the action of K on an orbit Δ⊆S′ to its universal covering space δ. For s⩾3, δ is the coordinate space for N particles, which is simply connected. In two‐dimensional space, however, the coordinate space is multiply connected, implying induced representations other than those describing the usual Bose or Fermi statistics; these are explored in the present paper. Likewise the Aharonov–Bohm effect is described by means of induced representations of the local observables, defined in a nonsimply connected region of Rs. The vector potential plays no role in this description of the Aharonov–Bohm effect.

The dynamics of bubble growth for Rayleigh-Taylor unstable interfaces

A statistical model is analyzed for the growth of bubbles in a Rayleigh–Taylor unstable interface. The model is compared to solutions of the full Euler equations for compressible two phase flow, using numerical solutions based on the method of front tracking. The front tracking method has the distinguishing feature of being a predominantly Eulerian method in which sharp interfaces are preserved with zero numerical diffusion. Various regimes in the statistical model exhibiting qualitatively distinct behavior are explored.

A molecular dynamics simulation study of elastic properties of HMX

Atomistic simulations were used to calculate the isothermal elastic properties for β-, α-, and δ-octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine (HMX). The room-temperature isotherm for each polymorph was computed in the pressure interval 0⩽p⩽10.6 GPa and was used to extract the initial isothermal bulk modulus Ko and its pressure derivative using equations of state employed previously in experimental studies of the β-HMX isotherm. The complete elastic tensor for each polymorph was calculated at room temperature and atmospheric pressure. For the case of β-HMX, the calculated elastic tensor is compared to one based on a fit to sound speed data yielding reasonably good agreement. The bulk modulus of β-HMX obtained from equation-of-state fits to the room-temperature isotherm agrees well with that determined from the complete elastic tensor and from volume fluctuations at atmospheric pressure. However, the value of Ko obtained from the isotherm is sensitive to choice of equation of state fitting form and to t...

Constituent properties of HMX needed for mesoscale simulations

Plastic-bonded explosives are heterogeneous materials. Improved burn models for weak initiation relevant to accident scenarios require a better understanding of the physics associated with the formation and growth of hot spots. Since the relevant length scale is subgrain in extent, mesoscale simulations are needed to study hot spots. Mesoscale simulations require as input constitutive properties of an explosive grain. In addition, it is essential to account for physical dissipative mechanisms since hot spots represent local peaks in the fluctuations of the temperature field. Here, constitutive properties of the explosive HMX needed for mesoscale simulations are discussed and experimental data reviewed. Because some decomposition may occur during a measurement, it is difficult to account for systematic error in the data. To get a sense of the uncertainties in material parameters, it is necessary to examine all the available data. In addition, we discuss results from molecular dynamics simulations of some properties for which experimental data are lacking.

Methods for numerical conformal mapping

Abstract Nonlinear integral equations for the boundary functions which determine conformal transformations in two dimensions are developed and analyzed. One of these equations has a nonsingular logarithmic kernel and is especially well suited for numerical computations of conformal maps including those which deal with regions having highly distorted boundaries. Numerical procedures based on interspersed Gaussian quadrature for approximating the integrals and a Newton-Raphson technique to solve the resulting nonlinear algebraic equations are described. The Newton-Raphson iteration converges reliably with very crude initial approximations. Numerical examples are given for the mapping of a half-infinite region with periodic boundary onto a half plane, with up to nine-figure accuracy for values of the map function on the boundary and for its first derivatives. The examples include regions bounded by “spike” curves characteristic of Rayleigh-Taylor instability phenomena. A differential equation is derived which relates changes in the map function to changes of the boundary. This is relevant to potential problems for regions with time-dependent boundaries. Further nonsingular integral formulas are derived for conformal mapping in a variety of geometries and for application to the boundary-value problems of potential theory.

Particle statistics from induced representations of a local current group

Representations of the nonrelativistic current group S‐K are studied in the Gel’fand–Vilenkin formalism, where S is Schwartz’ space of rapidly decreasing functions, and K is a group of diffeomorphisms of Rs. For the case of N identical particles, information about particle statistics is contained in a representation of KF (the stability group of a point F∈S′) which factors through the permutation group SN. Starting from a quasi‐invariant measure μ concentrated on a K orbit Δ in S′, together with a suitable representation of KF for F∈Δ, sufficient conditions are developed for inducing a representation of S‐K. The Hilbert space for the induced representation consists of square‐integrable functions on a covering space of Δ, which transform in accordance with a representation of KF. The Bose and Fermi N‐particle representations (on spaces of symmetric or antisymmetric wave functions) are recovered as induced representations. Under the conditions which are assumed, the following results hold: (1) A representat...

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...

Anomalous reflection of a shock wave at a fluid interface

Several wave patterns can be produced by the interaction of a shock wave with a fluid interface. We focus on the case when the shock passes from a medium of high to low acoustic impedance. Curvature of either the shock front or contact causes the flow to bifurcate from a locally self-similar quasi-stationary shock diffraction, to an unsteady anomalous reflection. This process is analogous to the transition from a regular to a Mach reflection when the reflected wave is a rarefaction instead of a shock. These bifurcations have been incorporated into a front tracking code that provides an accurate description of wave interactions. Numerical results for two illustrative cases are described; a planar shock passing over a bubble, and an expanding shock impacting a planar contact.