Analytic and computational studies on micro-propulsion and micro-detonics

Based on a general theory of detonation waves with an embedded sonic locus that we have previously developed, we carry out asymptotic analysis of weakly curved slowly varying detonation waves and show that the theory predicts the phenomenon of detonation ignition and failure. The analysis is not restricted to near ChapmanJouguet detonation speeds and is capable of predicting quasi-steady, normal detonation shock speed versus curvature (D − κ) curves with multiple turning points. An evolution equation that retains the shock acceleration, Ḋ, namely a Ḋ−D−κ relation is rationally derived which describes the dynamics of pre-existing detonation waves. The solutions of the equation for spherical detonation are shown to reproduce the ignition/failure phenomenon observed in both numerical simulations of blast wave initiation and in experiments. A single-step chemical reaction described by one progress variable is employed, but the kinetics is sufficiently general and is not restricted to Arrhenius form, although most specific calculations are performed for Arrhenius kinetics. As an example, we calculate critical energies of direct initiation for hydrogenoxygen mixtures and find close agreement with available experimental data. (3) D.S. Stewart & A.R. Kasimov 2006 Theory of detonation with an embedded sonic locus. SIAM J. Appl. Math. 66, 384-407. Abstract: A steady planar self-sustained detonation has a sonic surface in the reaction zone that resides behind the lead shock. In this work we address the problem of generalizing sonic conditions for a three-dimensional unsteady selfsustained detonation wave. The conditions are proposed to be the characteristic compatibility conditions on the exceptional surface of the governing hyperbolic system of reactive Euler equations. Two equations are derived that are necessary to determine the motion of both the lead shock and the sonic surface. Detonation with an embedded sonic locus is thus treated as a two-front phenomenon: a reaction zone whose domain of influence is bounded by two surfaces, the lead shock surface and the trailing characteristic surface. The geometry of the two surfaces plays an important role in the underlying dynamics. We also discuss how the sonic conditions of detonation stability theory and detonation shock dynamics can be obtained as special cases of the general sonic conditions. (4) A.R. Kasimov and D.S. Stewart 2005 Theory of direct initiation of gaseous detonations and comparison with experiment TAM Report No. 1043 UILU ENG2004-6004 ISSN 0073-5264 Abstract: In this work we discuss the application of an evolution equation that we have developed for the dynamics of a slowly evolving weakly-curved detonation to a problem of direct detonation initiation. Despite the relative simplicity of the theory, it successfully explains basic features of the initiation process which are observed in experiments and numerical simulations. Moreover, the theory allows one to calculate initiation energies based on the explosive chemical and thermodynamic properties only, without having to invoke significant modeling assumptions. The evolution equation exhibits the competing effects of the exothermic heat release, curvature, and shock acceleration. The detonation dynamics during the initiation depends on the relative strength of the heat release and flow divergence, resulting in successful initiation of self-sustained detonation if the heat release is sufficiently stronger than divergence or in failure if otherwise. Using global kinetic data from Caltech detonation database, which are derived from detailed chemical calculations, we have calculated critical initiation energies of 4 M. Short & D.S. Stewart spherical detonation for hydrogen-oxygen, hydrogen-air, and ethylene-air mixtures at various equivalence ratios and found a very good agreement with recent experimental data. (5) A. R. Kasimov 2004 Theory of Instability and Nonlinear Evolution of Selfsustained Detonation Wave. Ph.D. thesis, Theoretical and Applied Mechanics, May 2004. Abstract: Linear stability properties and nonlinear dynamics of self-sustained detonations is investigated by means of asymptotic analysis and numerical simulations. The normal-mode linear stability analysis is carried out for gaseous detonations propagating in cylindrical tubes. By comparison of the stability predictions with experiments, it is shown that the instability plays a fundamental role in the onset of spin detonation. We derive far-field closure conditions for unsteady and multi-dimensional detonation waves in arbitrary explosive media as intrinsic properties of the reactive Euler equations in the embedded sonic surface, which is a characteristic surface. The conditions generalize previously known sonic conditions for self-sustained detonations. We investigate the nature of the sonic conditions numerically with a new numerical technique for solving the Euler equations and demonstrate that the sonic locus is a characteristic surface and an information boundary that isolates the reaction zone from the far-field flow. Starting with the general formulation, we derive an asymptotic evolution equation for self-sustained detonations in the limits of slow-time variation and weak curvature and find that the equation predicts ignition and failure of detonations. Based on the evolution equation, we formulate a theory of direct initiation of gaseous detonation that can predict critical conditions from first principles; we show that the theoretically predicted critical energies are in close agreement with experiment. The ignition theory is also extended to explosives with arbitrary equation of state. With the general conditions at the sonic locus available, we formulate the stability problem for high-explosive detonations described by non-ideal equation of state and calculate stability characteristics of detonation in PBX-9502 and nitromethane. (b ) Reseach summary: M. Short (1) M. Short, J.B. Bdzil & I. I. Anguelova Stability of Chapman?Jouguet detonations for a stiffened-gas model of condensed-phase explosives. J. Fluid Mech. 2006, 552, pp. 299-309. Abstract: The analysis of the linear stability of a planar Chapman-Jouguet detonation wave is reformulated for an arbitrary caloric (incomplete) equation of state in an attempt to better represent the stability properties of detonations in condensed-phase explosives. Calculations are performed on a “stiffened-gas” equation of state which allows us to prescribe a finite detonation Mach number while simultaneously allowing for a detonation shock pressure that is substantially larger than the ambient pressure. We show that the effect of increasing the ambient sound speed in the material, for a given detonation speed, has a stabilizing effect on the detonation. We also show that the presence of the slow reaction stage, a feature of detonations in certain types of energetic materials, where the detonation structure is characterized by a fast reaction stage behind the detonation shock followed by a slow reaction stage, tends to have a destabilizing effect. The analysis of the linear stability of a planar Chapman-Jouguet detonation wave is reformulated for an arbitrary caloric (incomplete) equation of state in an attempt to better represent the stability properties of detonations in condensed-phase explosives. Calculations are performed on a “stiffened-gas” equation of state which allows us to prescribe a finite detonation Mach number while simultaneously allowing for a detonation shock pressure that is substantially larger than the ambient pressure. We show that the effect of increasing the ambient sound speed in the material, for a given detonation speed, has a stabilizing effect on the detonation. We also show that the presence of the slow reaction stage, a feature of detonations in certain types of energetic materials, where the detonation structure is characterized by a fast reaction stage behind the detonation shock followed by a slow reaction stage, tends to have a destabilizing effect.