Critical Slow Dynamics of Detonation in a Gas with Non-Uniform Initial Temperature and Composition: A Large-Activation-Energy Analysis (original) (raw)
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Mathematical Modeling in Combustion Science
We mun.rnarize some recent developments of J. B. Bdzil and D. S. Stewart's investigation into the theory of multi-dimemional, time+dependent detonation. These advances have led to the development of a theory for describing the propagation of high-order deb nation in condensed-ph=e wcploaivea. The central approximation in the theory "b that the detonation shock ia weakly curved. Specifically, we umme that Lhe radius of curvature of the deton&tion shock ia large compared to a relevzmt reaction-zone thickness. Our main Endings are: (1) the flow is mmai-steuly and nearly one dimensional along the normal to the detonation chock, and (2) the small deviation of the nonnai detonation velocity from the Chapman-Jouguet (CJ) value in generally q function of cumture. The exact functional form of the correction dependa on the equation of state (EOS) and the form of the energy-release law,
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We present an asymptotic theory for the dynamics of detonation when the radius of curvature of the detonation shock is large compared to the one-dimensional, steady, Chapman-Jouguet (CJ) detonation reaction-zone thickness. The analysis considers additional time-dependence in the slowly varying reaction zone to that considered in previous works. The detonation is assumed to have a sonic point in the reactionzone structure behind the shock, and is referred to as an eigenvalue detonation. A new, iterative method is used to calculate the eigenvalue relation, which ultimately is expressed as an intrinsic, partial differential equation (PDE) for the motion of the shock surface. Two cases are considered for an ideal equation of state. The first corresponds to a model of a condensed-phase explosive, with modest reaction rate sensitivity, and the intrinsic shock surface PDE is a relation between the normal detonation shock velocity, D,, the first normal time derivative of the normal shock velocity, D,,, and the shock curvature, IC. The second case corresponds to a gaseous explosive mixture, with the large reaction rate sensitivity of Arrhenius kinetics, and the intrinsic shock surface PDE is a relation between the normal detonation shock velocity, D,, its first and second normal time derivatives of the normal shock velocity, b,, B,, and the shock curvature, IC, and its first normal time derivative of the curvature, k. For the second case, one obtains a one-dimensional theory of pulsations of plane CJ detonation and a theory that predicts the evolution of self-sustained cellular detonation. Versions of the theory include the limits of near-CJ detonation, and when the normal detonation velocity is significantly below its CJ value. The curvature of the detonation can also be of either sign, corresponding to both diverging and converging geometries.
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Shock Waves, 1999
We present a theoretical and numerical study on the induction of adiabatic explosions by accelerated curved shocks in homogeneous explosives, and pay a special attention to critical conditions for initiation. We characterize the first stage of the decomposition process, or induction, as an initial-value problem. During induction, the reaction progress-variable remains small; the induction time is given by the runaway of the dependent variables and corresponds to a logarithmic singularity in theirs material distributions. We express these distributions as first-order expansions in the progress variable about the shock. Then, the framework of our procedure is the formal Cauchy problem for quasi-linear hyperbolic sets of first-order differential equations, such as the balance laws for adiabatic flows of inviscid fluids considered in this study. When a shock front is used as data surface, the solution to the Cauchy problem yields the flow derivatives at the shock, then the induction time, as functions of the shock normal velocity and acceleration, D n and δDn/δt, and the shock total curvature C. We next derive a necessary condition for explosion as a constraint among Dn, δDn/δt and C that ensures bounded values of the induction time. This criterion is akin to Semenov's, in the sense that the critical condition for explosion is that the heat-production rate must just exceed the heat-loss rate, here given by the volumetric expansion rate at the shock. The violation of the criterion defines a critical shock dynamics as a relationship among Dn, δDn/δt and C that generates infinite induction times. Depending on the rear-boundary conditions, which determine the shock dynamics, this event can be interpreted as either a non-initiation, or the decoupling of the shock and of the flame front induced by the shock. We illustrate our approach by a simple solution to the problem of the initiation by impact of a noncompressible piston. From the continuity constraint in the material speed and acceleration at the contact surface of the piston and the explosive, we first derive the initial shock dynamics, and then rewrite the induction time and the initiation condition in terms of the piston speed, acceleration and curvature. We compare these theoretical predictions to those of our direct numerical simulations, and to numerical results obtained by other authors, in the case of impacts on a gaseous explosive.
Detonation propagation, decay, and reinitiation in nonuniform gaseous mixtures
27th Symposium (international) on Combustion, 1998
Experiments on the behavior of detonation waves in nonuniform mixtures are presented. The situation studied was the propagation of a detonation wave from a driver mixture of variable length through a concentration gradient of variable width into a less reactive acceptor mixture. The effect of the gradient on the transmission process were studied. A detonation tube of 174 mm id. was used. The tube was initially divided by a fast-opening, stretched rubber diaphragm. Stoichiometric hydrogen/air mixture was used in the driver section. Hydrogen/air mixtures (14.0–19.0% H2) were used as acceptor mixtures. Natural diffusion was used to create a concentration gradient between two mixtures. It was shown that the behavior of detonations at concentration gradients depends significantly on the sharpness of the gradient. For relatively sharp gradients a detonation always decays in the nonuniform region. It can be reinitiated downstream in the acceptor mixture, if the driver length is large enough for a particular acceptor mixture. For relatively smooth gradients, detonation is able to propagate through without decay. The boundary between these cases is defined only by the value of sensitivity gradient for a particular pair of driver and acceptor mixtures. The critical value of the gradient depends strongly on the difference in energy content of driver and acceptor mixtures. The more overdriven is the detonation in the driver mixture compared to that in the acceptor, the sharper gradient is necessary for detonation decay. The order of magnitude of critical values of the gradient shows that evolution of the cellular structure may play a role effecting conditions for detonation decay at concentration gradients.
Propagation laws for steady curved detonations with chain-branching kinetics
Journal of Fluid Mechanics, 2003
An extension to the theory of detonation shock dynamics is made and new propagation laws are derived for steady, near-CJ (Chapman-Jouguet), weakly curved detonations for a chain-branching reaction model having two components. The first is a thermally neutral induction stage governed by an Arrhenius reaction with a large activation energy, which terminates at a location called the transition interface, where instantaneous conversion of fuel into an intermediate species (chain radical) occurs. The second is an exothermic main reaction layer (or chain-recombination zone) having a temperature-independent reaction rate. We make an ansatz that the shock curvature is sufficiently large to have a leading-order influence on the induction zone structure, whereupon it is shown that multi-dimensional effects must necessarily be accounted for in the main reaction layer. Only for exactly cylindrical or spherical waves can such multi-dimensional effects be omitted. A requirement that the main reaction layer structure pass smoothly through a sonic plane leads to a propagation law for the detonation front: a relationship between the detonation velocity, the shock curvature and various shock arclength derivatives of the position of the transition interface. For exactly cylindrically or spherically expanding waves, a multi-valued detonation velocity-curvature relationship is found, similar to that found previously for a state-sensitive one-step reaction. The change in this relationship is investigated as the ratio of the length of the main reaction layer to the induction layer is changed. We also discuss the implications of chain-branching reaction kinetics for the prediction of critical detonation initiation energy based on detonation-velocity curvature laws. Finally several calculations that illustrate the important effect that arclength and transverse flow variations may have on the steady propagation of non-planar detonation fronts are presented. Such variations may be important for the propagation of cellular gaseous detonation fronts and for the axial propagation of detonations in a cylindrical stick of condensed-phase explosive. We also show that the arclength variations provide a formal mechanism for the existence of steady non-planar detonation fronts having converging sections, a possibility ruled out for simple irreversible one-step reaction mechanisms where only diverging steady waves are admissible.