Tariq D. Aslam

Simulations of pulsating one-dimensional detonations with true fifth order accuracy

A novel, highly accurate numerical scheme based on shock-fitting coupled with fifth order spatial and temporal discretizations is applied to a classical unsteady detonation problem to generate solutions with unprecedented accuracy. The one-dimensional reactive Euler equations for a calorically perfect mixture of ideal gases whose reaction is described by single-step irreversible Arrhenius kinetics are solved in a series of calculations in which the activation energy is varied. In contrast with nearly all known simulations of this problem, which converge at a rate no greater than first order as the spatial and temporal grid is refined, the present method is shown to converge at a rate consistent with the fifth order accuracy of the spatial and temporal discretization schemes. This high accuracy enables more precise verification of known results and prediction of heretofore unknown phenomena. To five significant figures, the scheme faithfully recovers the stability boundary, growth rates, and wave-numbers predicted by an independent linear stability theory in the stable and weakly unstable regime. As the activation energy is increased, a series of period-doubling events are predicted, and the system undergoes a transition to chaos. Consistent with general theories of non-linear dynamics, the bifurcation points are seen to converge at a rate for which the Feigenbaum constant is 4.66+/-0.09, in close agreement with the true value of 4.669201.... As activation energy is increased further, domains are identified in which the system undergoes a transition from a chaotic state back to one whose limit cycles are characterized by a small number of non-linear oscillatory modes. This result is consistent with behavior of other non-linear dynamical systems, but not typically considered in detonation dynamics. The period and average detonation velocity are calculated for a variety of asymptotically stable limit cycles. The average velocity for such pulsating detonations is found to be slightly greater than the Chapman-Jouguet velocity.

On sub-linear convergence for linearly degenerate waves in capturing schemes

A common attribute of capturing schemes used to find approximate solutions to the Euler equations is a sub-linear rate of convergence with respect to mesh resolution. Purely nonlinear jumps, such as shock waves produce a first-order convergence rate, but linearly degenerate discontinuous waves, where present, produce sub-linear convergence rates which eventually dominate the global rate of convergence. The classical explanation for this phenomenon investigates the behavior of the exact solution to the numerical method in combination with the finite error terms, often referred to as the modified equation. For a first-order method, the modified equation produces the hyperbolic evolution equation with second-order diffusive terms. In the frame of reference of the traveling wave, the solution of a discontinuous wave consists of a diffusive layer that grows with a rate of t^1^/^2, yielding a convergence rate of 1/2. Self-similar heuristics for higher-order discretizations produce a growth rate for the layer thickness of @Dt^1^/^(^p^+^1^) which yields an estimate for the convergence rate as p/(p+1) where p is the order of the discretization. In this paper we show that this estimated convergence rate can be derived with greater rigor for both dissipative and dispersive forms of the discrete error. In particular, the form of the analytical solution for linear modified equations can be solved exactly. These estimates and forms for the error are confirmed in a variety of demonstrations ranging from simple linear waves to multidimensional solutions of the Euler equations.

Numerical resolution of pulsating detonation waves

The canonical problem of the one-dimensional, pulsating, overdriven detonation wave has been studied for over 30 years, not only for its phenomenological relation to the evolution of multidimensional detonation instabilities, but also to provide a robust, reactive, high-speed flowfield with which to test numerical schemes. The present study examines this flowfield using high-order, essentially non-oscillatory schemes, systematically varying the level of resolution of the reaction zone, the size and retention of information in the computational domain, the initial conditions, and the order of the scheme. It is found that there can be profound differences in peak pressures as well as in the period of oscillation, not only for cases in which the reaction front is under-resolved, but for cases in which the computation is corrupted due to a too-small computational domain. Methods for estimating the required size of the computational domain to reduce costs while avoiding erroneous solutions are proposed and tested.

A Level Set Algorithm for Tracking Discontinuities in Hyperbolic Conservation Laws II: Systems of Equations

A level set algorithm for tracking discontinuities in hyperbolic conservation laws is presented. The algorithm uses a simple finite difference approach, analogous to the method of lines scheme presented in [36]. The zero of a level set function is used to specify the location of the discontinuity. Since a level set function is used to describe the front location, no extra data structures are needed to keep track of the location of the discontinuity. Also, two solution states are used at all computational nodes, one corresponding to the “real” state, and one corresponding to a “ghost node” state, analogous to the “Ghost Fluid Method” of [12]. High order pointwise convergence was demonstrated for scalar linear and nonlinear conservation laws, even at discontinuities and in multiple dimensions in the first paper of this series [3]. The solutions here are compared to standard high order shock capturing schemes, when appropriate. This paper focuses on the issues involved in tracking discontinuities in systems of conservation laws. Examples will be presented of tracking contacts and hydrodynamic shocks in inert and chemically reacting compressible flow.

Detonation shock dynamics and comparisons with direct numerical simulation

Comparisons between direct numerical simulation (DNS) of detonation and detonation shock dynamics (DSD) is made. The theory of DSD defines the motion of the detonation shock in terms of the intrinsic geometry of the shock surface, in particular for condensed phase explosives the shock normal velocity, D n , the normal acceleration, [Ddot] n , and the total curvature, κ. In particular, the properties of three intrinsic front evolution laws are studied and compared. These are (i) constant speed detonation (Huygens construction), (ii) curvature-dependent speed propagation (κ relation) and (iii) curvature- and speed-dependent acceleration ([Ddot] n –D n –κ relation). We show that it is possible to measure shock dynamics directly from simulation of the reactive Euler equations and that subsequent numerical solution of the intrinsic partial differential equation for the shock motion (e.g. a [Ddot] n –D n –κ relation) reproduces the computed shock motion with high precision.

Exact Solution for Multidimensional Compressible Reactive Flow for Verifying Numerical Algorithms

A new exact solution of an oblique detonation is developed for the supersonic irrotational flow of an inviscid calorically perfect ideal gas, which undergoes a one-step, irreversible, exothermic, zero activation energy reaction as it passes through a straight shock over a curved wedge. The solution gives expressions for the velocity, pressure, density, temperature, and position as parametric functions of a variable characterizing the extent of reaction. For Chapman-Jouguet solutions, an explicit form with dependency on distance is obtained in terms of the Lambert W function. As the simple model employed is a rational limit of models used in the computational simulation of complex supersonic reactive flows, the solution can serve as a benchmark for mathematical verification of general computational algorithms. An example of such a verification is given by comparing the predictions a modern shock-capturing code to those of the full exact solution. The realized spatial convergence rate is 0.779, far less than the fifth-order accuracy that the chosen algorithm would exhibit for smooth flows, but consistent with the predictions of all shock-capturing codes, which never converge with greater than first-order accuracy for flows with embedded discontinuities.

Stability of detonations for an idealized condensed-phase model

The stability of travelling wave Chapman-Jouguet and moderately overdriven detonations of Zeldovich-von Neumann-Dtype is formulated for a general system that incorporates the idealized gas and condensed-phase (liquid or solid) detonation models. The general model consists of a two-component mixture with a one-step irreversible reaction between reactant and product. The reaction rate has both temperature and pressure sensitivities and has a variable reaction order. The idealized condensed-phase model assumes a pressure-sensitive reaction rate, a constant-γ caloric equation of state for an ideal fluid, with the isentropic derivative γ =3 , and invokes the strong shock limit. A linear stability analysis of the steady, planar, ZND detonation wave for the general model is conducted using a normal- mode approach. An asymptotic analysis of the eigenmode structure at the end of the reaction zone is conducted, and spatial boundedness (closure) conditions formally derived, whose precise form depends on the magnitude of the detonation overdrive and reaction order. A scaling analysis of the transonic flow region for Chapman- Jouguet detonations is also studied to illustrate the validity of the linearization for Chapman-Jouguet detonations. Neutral stability boundaries are calculated for the idealized condensed-phase model for one- and two-dimensional perturbations. Comparisons of the growth rates and frequencies predicted by the normal-mode analysis for an unstable detonation are made with a numerical solution of the reactive Euler equations. The numerical calculations are conducted using a new, high-order algorithm that employs a shock-fitting strategy, an approach that has significant advantages over standard shock-capturing methods for calculating unstable detonations. For the idealized condensed-phase model, nonlinear numerical solutions are also obtained to study the long-time behaviour of one- and two-dimensional unstable Chapman-Jouguet ZND waves.

Application of detonation shock dynamics to the propagation of a detonation in nitromethane in a packed inert particle bed

A multidimensional implementation of DSD, formulated with the level set method, is applied to track the propagation of a detonation wave in a heterogeneous explosive consisting of an array of inert cylindrical obstacles with a liquid explosive in the interstitial space. With the Huygens assumption, the average detonation velocity through the explosive is less than that for the liquid explosive alone, due to the increased path length. When the normal detonation velocity is assumed to depend on front curvature, there is an additional, smaller reduction in the detonation velocity, which depends on the cylinder material. The detonation velocity deficits obtained in the computations are of the same order as those observed experimentally for a heterogeneous explosive consisting of a packed bed of spherical inert beads saturated with sensitized nitromethane. The DSD computations are relevant to the experimental results in the large-bead limit in which the pore dimension is large enough to support the propagation...

The reactants equation of state for the tri-amino-tri-nitro-benzene (TATB) based explosive PBX 9502

The response of high explosives (HEs), due to mechanical and/or thermal insults, is of great importance for both safety and performance. A major component of how an HE responds to these stimuli stems from its reactant equation of state (EOS). Here, the tri-amino-tri-nitro-benzene based explosive PBX 9502 is investigated by examining recent experiments. Furthermore, a complete thermal EOS is calibrated based on the functional form devised by Wescott, Stewart, and Davis [J. Appl. Phys. 98, 053514 (2005)]. It is found, by comparing to earlier calibrations, that a variety of thermodynamic data are needed to sufficiently constrain the EOS response over a wide range of thermodynamic state space. Included in the calibration presented here is the specific heat as a function of temperature, isobaric thermal expansion, and shock Hugoniot response. As validation of the resulting model, isothermal compression and isentropic compression are compared with recent experiments.

PROTON RADIOGRAPHY OF PBX 9502 DETONATION SHOCK DYNAMICS CONFINEMENT SANDWICH TEST

Recent results utilizing proton radiography (P‐Rad) during the detonation of the high explosive PBX 9502 are presented. Specifically, the effects of confinement of the detonation are examined in the LANL detonation confinement sandwich geometry. The resulting detonation velocity and detonation shock shape are measured. In addition, proton radiography allows one to image the reflected shocks through the detonation products. Comparisons are made with detonation shock dynamics (DSD) and the reactive flow model Ignition and Growth ( I&G) for the lead detonation shock and detonation velocity. In addition, predictions of reflected shocks are made with the reactive flow model.