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Steady detonation in multi-dimensional flow is controlled by the chemical energy release that occurs in a subsonic elliptic flow region known as the detonation driving zone (DDZ). It is the region encompassing the detonation shock and sonic flow locus (in the frame of the detonation shock). A detonation that is strongly confined by material surrounding the explosive has the shock and sonic locus separated at the material interface. Information about the material boundary is traditionally believed to influence the DDZ structure via the subsonic flow on the boundary ahead of the sonic locus. A detonation that is weakly confined has the detonation shock and sonic locus intersecting at the material boundary. The sonic nature of the flow at the intersection point on the boundary is believed to isolate the DDZ structure from the material properties of the confinement. In this study, we examine the paths of characteristics propagating information about the confinement through the supersonic hyperbolic flow region that exists beyond the sonic locus, and determine whether these paths may impinge on the sonic locus and consequently influence the DDZ structure. Our configuration consists of a solid wall boundary deflected through a specified angle on detonation shock arrival, so that the streamline turning angle of the wall at the explosive edge is unambiguously defined. By varying the wall deflection angle from small through large values, we can systematically capture the evolution of the DDZ structure and the characteristic flow regions that influence its structure for strongly to weakly confined detonations. In all strong and weak confinement cases examined, we find that a subset of characteristics from the supersonic flow regions always impinge on the sonic locus. Limiting characteristics are identified that define the boundary between characteristics that impinge on the sonic surface and those that propagate information downstream of the sonic surface. In combination with an oblique-shock polar analysis, we show that the effects on the DDZ of characteristic impingement can be significant.

We examine asymptotically the dynamics of two-dimensional, steady detonation wave propagation and failure for a strongly confined high explosive (HE), in which the width of the explosive is small relative to the reaction zone length. An energy balance equation is derived, which shows how the longitudinal acceleration of subsonic flow behind the detonation shock is influenced both by chemical reaction and by the effects of HE boundary streamline deflection, specifically via the induced rate of change of mass flux through the detonation wave. The latter serves to either counteract or reinforce the acceleration of longitudinal flow, depending on the sign of the gradient of the boundary streamline deflection at the detonation shock. The analysis is valid for general equations of state and chemical reaction rates in the HE. The asymptotically derived form of the energy equation represents an eigenvalue problem for the determination of the steady detonation propagation speed, solved via a shooting method. We explore specific results for ideal and stiffened equations of state, along with a pressure-dependent reaction rate for which changes in the pressure exponent and reaction order are also studied. We consider the influences of both straight and curved HE boundary streamline shapes. The asymptotic analysis reveals significant physical insights into how detonation propagation and failure are affected by strong confinement.

The dynamics of steady detonation propagation in a two-dimensional, high explosive circular arc geometry are examined computationally using a reactive flow model approach. The arc is surrounded by a low impedance material confiner on its inner surface, while its outer surface is surrounded either by the low impedance confiner or by a high impedance confiner. The angular speed of the detonation and properties of the steady detonation driving zone structure, i.e. the region between the detonation shock and sonic flow locus, are examined as a function of increasing arc thickness for a fixed inner arc radius. For low impedance material confinement on the inner and outer arc surfaces, the angular speed increases monotonically with increasing arc thickness, before limiting to a constant. The limiting behaviour is found to occur when the detonation driving zone detaches from the outer arc surface, leaving a region of supersonic flow on the outer surface. Consequently, the angular speed of the detonation becomes insensitive to further increases in the arc thickness. For high impedance material confinement on the outer arc surface, the observed flow structures are significantly more complex. As the arc thickness increases, we sequentially observe regions of negative shock curvature on the detonation front, reflected shock formation downstream of the reaction zone, and eventually Mach stem formation on the detonation front. Subsequently, a region of supersonic flow develops between the detonation driving zone and the Mach stem structure. For sufficiently wide arcs, the Mach stem structure disappears. For the high impedance material confinement, the angular speed of the detonation first increases with increasing arc thickness, reaches a maximum, decreases, and then limits to a constant for sufficiently large arc thickness. The limiting angular speed is the same as that found for the low impedance confiner on the outer arc surface.

The fluid mechanics of the interaction between a porous material confiner and a steady propagating high explosive (HE) detonation in a two-dimensional slab geometry is investigated through analytical oblique wave polar analysis and multi-material numerical simulation. Two HE models are considered, broadly representing the properties of either a high- or low-detonation-speed HE, which permits studies of detonation propagating at speeds faster or slower than the confiner sound speed. The HE detonation is responsible for driving the compaction front in the confiner, while, in turn, the high material density generated in the confiner as a result of the compaction process can provide a strong confinement effect on the HE detonation structure. Polar solutions that describe the local flow interaction of the oblique HE detonation shock and equilibrium state behind an oblique compaction wave with rapid compaction relaxation rates are studied for varying initial solid volume fractions of the porous confiner. Multi-material numerical simulations are conducted to study the effect of detonation wave driven compaction in the porous confiner on both the detonation propagation speed and detonation driving zone structure. We perform a parametric study to establish how detonation confinement is influenced both by the initial solid volume fraction of the porous confiner and by the time scale of the dynamic compaction relaxation process relative to the detonation reaction time scale, for both the high- and low-detonation-speed HE models. The compaction relaxation time scale is found to have a significant influence on the confinement dynamics, with slower compaction relaxation time scales resulting in more strongly confined detonations and increased detonation speeds. The dynamics of detonation confinement by porous materials when the detonation is propagating either faster or slower than the confiner sound speed is found to be significantly different from that with solid material confiners.

We present a theory for the evolution of a one-dimensional, steady-state detonation reaction zone to a two-dimensional reaction zone, when the explosive experiences a sudden loss of side-on confinement as a boundary of the explosive is impulsively withdrawn. Our focus is on condensed-phase explosives, which we describe as having a constant adiabatic gamma equation of state and an irreversible, state-independent reaction rate. We consider two detonation models: (i) the instantaneous reaction heat release Chapman–Jouguet (CJ)-limit and (ii) the spatially resolved reaction heat-release Zel’dovich–von Neumann–Doring (ZND) model, in the limit where only a small fraction of the energy release is resolved (the SRHR-limit). Two competing rarefaction waves are generated by this loss of confinement: (i) a smooth wave coming off the full length of the withdrawn boundary and (ii) a singular fan spreading out from the point where the detonation shock and the withdrawn boundary meet. For the CJ-limit, in all cases the singular rarefaction fan eventually dominates the competition to control the steady-state behaviour. For the SRHR-limit, the spatially resolved heat release moderates this competition. When the withdrawal speed is fast, the rarefaction fan dominates; when the withdrawal speed is slower, the smooth rarefaction eventually dominates, although the flow features a fan at early times. By examining the mathematical properties of the steady two-dimensional fan-based solution, we set down a mechanism for this transition in behaviours.

We study the compressible flow dynamics of two-dimensional, steady detonation wave propagation in a high explosive (HE) confined by aluminium (Al) or stainless steel (SS), outside of which is an air layer. We examine how the thickness of the confinement affects the subsonic detonation driving zone structure (DDZ) and the detonation speed , demonstrating a strong dependence on whether the oblique shock-driven flow in the confiner is supersonic, as for SS, or subsonic, as for Al. A characteristic path analysis is used to examine the information flow from the material boundaries through the supersonic flow regions in both the HE and confiner that can impact the sonic surfaces bounding the subsonic flow regions. It is shown that the nature of gas-dynamic wave reflection off the SS–air or Al–air boundary can significantly influence the DDZ and .

We study the physics of steady detonation wave propagation in a two-dimensional circular arc via a Detonation Shock Dynamics (DSD) surface evolution model. The dependence of the surface angular speed and surface spatial structure on the inner arc radius ($R_{i}$), the arc thickness ($R_{e}-R_{i}$, where $R_{e}$ is the outer arc radius) and the degree of confinement on the inner and outer arc is examined. We first analyse the results for a linear $D_{n}$–$\unicode[STIX]{x1D705}$ model, in which the normal surface velocity $D_{n}=D_{CJ}(1-B\unicode[STIX]{x1D705})$, where $D_{CJ}$ is the planar Chapman–Jouguet velocity, $\unicode[STIX]{x1D705}$ is the total surface curvature and $B$ is a length scale representative of a reaction zone thickness. An asymptotic analysis assuming the ratio $B/R_{i}\ll 1$ is conducted for this model and reveals a complex surface structure as a function of the radial variation from the inner to the outer arc. For sufficiently thin arcs, where $(R_{e}-R_{i})/R_{i}=O(B/R_{i})$, the angular speed of the surface depends on the inner arc radius, the arc thickness and the inner and outer arc confinement. For thicker arcs, where $(R_{e}-R_{i})/R_{i}=O(1)$, the angular speed does not depend on the outer arc radius or the outer arc confinement to the order calculated. It is found that the leading-order angular speed depends only on $D_{CJ}$ and $R_{i}$, and corresponds to a Huygens limit (zero curvature) propagation model where $D_{n}=D_{CJ}$, assuming a constant angular speed and perfect confinement on the inner arc surface. Having the normal surface speed depend on curvature requires the insertion of a boundary layer structure near the inner arc surface. This is driven by an increase in the magnitude of the surface wave curvature as the inner arc surface is approached that is needed to meet the confinement condition on the inner arc surface. For weak inner arc confinement, the surface wave spatial variation with the radial coordinate is described by a triple-deck structure. The first-order correction to the angular speed brings in a dependence on the surface curvature through the parameter $B$, while the influence of the inner arc confinement on the angular velocity only appears in the second-order correction. For stronger inner arc confinement, the surface wave structure is described by a two-layer solution, where the effect of the confinement on the angular speed is promoted to the first-order correction. We also compare the steady-state arc solution for a PBX 9502 DSD model to an experimental two-dimensional arc geometry validation test.

Experiments were conducted to characterize the detonation phase-velocity dependence on charge thickness for two-dimensional detonation in condensed-phase explosive slabs of PBX 9501, PBX 9502 and ANFO. In combination with previous diameter-effect measurements from a cylindrical rate-stick geometry, these data permit examination of the relative scaling of detonation phase velocity between axisymmetric and two-dimensional detonation. We find that the ratio of cylinder radius ($R$) to slab thickness ($T$) at each detonation phase velocity ($D_{0}$) is such that$R(D_{0})/T(D_{0}). The variation in the$R(D_{0})/T(D_{0})$scaling is investigated with two detonation shock dynamics (DSD) models: a lower-order model relates the normal detonation velocity to local shock curvature, while a higher-order model includes the effect of front acceleration and transverse flow. The experimentally observed$R(D_{0})/T(D_{0})$(${) scaling behaviour for PBX 9501 and PBX 9502 is captured by the lower-order DSD theory, revealing that the variation in the scale factor is due to a difference in the slab and axisymmetric components of the curvature along the shock in the cylindrical geometry. The higher-order DSD theory is required to capture the observed$R(D_{0})/T(D_{0})$(${) scaling behaviour for ANFO. An asymptotic analysis of the lower-order DSD formulation describes the geometric scaling of the detonation phase velocity between the cylinder and slab geometries as the detonation phase velocity approaches the Chapman–Jouguet value.

The influence of thermal expansion on the dynamics of thick to moderately thick premixed flames (flame thickness less than or comparable to the channel height) for a variable-density flow in a narrow, rectangular channel is explored. The study is conducted within the framework of the zero-Mach-number, variable-density Navier–Stokes equations. Both adiabatic and non-adiabatic channel walls are considered. A small Péclet number asymptotic solution is developed for steady, variable-density flame propagation in the narrow channel. The dynamics of channel flames are also examined numerically for O(1) Péclet numbers in configurations which include flame propagation in a semi-closed channel from the closed to the open end of the channel, flame propagation in a semi-closed channel towards the closed end of the channel and flame propagation in an open channel in which a Poiseuille flow (flame assisting or flame opposing) is imposed at the channel inlet. Comparisons of the finite-Péclet-number dynamics are made with the behaviour of the small-Péclet-number solutions. We also compare how thermal expansion modifies the flow dynamics from those determined by a constant-density model. The small-Péclet-number variable-density solution for a flame propagating in a circular pipe is given in the Appendix.

The stability of travelling wave Chapman–Jouguet and moderately overdriven detonations of Zeldovich–von Neumann–Döring type 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.

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.

The evolution from a linear temperature gradient to a detonation is investigated for combustible materials whose chemistry is governed by chain-branching kinetics, using a combination of high-activation-temperature asymptotics and numerical simulations. A two-step chemical model is used, which captures the main properties of detonations in chain-branching fuels. The first step is a thermally neutral induction time, representing chain initiation and branching, which has a temperature-sensitive Arrhenius form of the reaction rate. At the end of the induction time is a transition point where the fuel is instantaneously converted into chain-radicals. The second step is the main exothermic reaction, representing chain termination, assumed to be temperature insensitive. Emphasis is on comparing and contrasting the results with previous studies that used simple one-step kinetics. It is shown that the largest temperature gradient for which a detonation can be successfully ignited depends on the heat release rate of the main reaction. The slower the heat release compared to the initial induction time, the shallower the gradient has to be for successful ignition. For example, when the rate of heat release is moderate or slow on the initial induction time scale, it was found that the path of the transition point marking the end of the induction stage should move supersonically, in which case its speed is determined only by the initial temperature gradient. For steeper gradients such that the transition point propagates subsonically from the outset, the rate of heat release must be very high for a detonation to be ignited. Detonation ignition for the two-step case apparently does not involve the formation of secondary shocks, unlike some cases when one-step kinetics is used.

A nonlinear evolution equation for pulsating Chapman-Jouguet detonations with chain-branching kinetics is derived. We consider a model reaction system having two components: a thermally neutral chain-branching induction zone governed by an Arrhenius reaction, terminating at a location where conversion of fuel into chain radical occurs; and a longer exothermic main reaction layer or chain-recombination zone having a temperature-independent reaction rate. The evolution equation is derived under the assumptions of a large activation energy in the induction zone and a slow evolution time based on the particle transit time through the induction zone, and is autonomous and second-order in time in the shock velocity perturbation. It describes both stable and unstable solutions, the latter leading to stable periodic limit cycles, as the ratio of the length of the chain-recombination zone to chain-induction zone, the exothermicity of reaction, and the specific heats ratio are varied

We examine small deviations from axial symmetry in a solid-propellant rocket motor, and describe a ‘bath-tub-vortex’ effect, in which substantial axial vorticity is generated in a neighbourhood of the chamber centreline. The unperturbed flow field is essentially inviscid at modest Reynolds numbers, even at the chamber walls, as has long been known, but the inviscid perturbed flow is singular at the centreline, and viscous terms are required to regularize it. We examine perturbations sufficiently small that a linear analysis is valid everywhere (εRe small, where ε is a measure of the perturbation amplitude and Re is a Reynolds number), and larger perturbations in which a nonlinear patch is created near the centreline of radius O(√ε). Our results provide an explanation of swirl experimentally observed by others, and a cautionary note for those concerned with numerical simulations of these flows, whether laminar or turbulent.

The stability of an overdriven planar detonation wave is examined for a one-step Arrhenius reaction model with an order-one post-shock temperature-scaled activation energy θ in the limit of a small post-shock temperature-scaled heat release β. The ratio of specific heats, γ is taken such that (γ−1)=O(1). Under these assumptions, which cover a wide range of realistic physical situations, the steady detonation structure can be evaluated explicitly, with the reactant mass fraction described by an exponentially decaying function. The analytical representation of the steady structure allows a normal-mode description of the stability behaviour to be obtained via a two-term asymptotic expansion in β. The resulting dispersion relation predicts that for a finite overdrive f, the detonation is always stable to two-dimensional disturbances. For large overdrives, the identification of regimes of stability or instability is found to depend on a choice of distinguished limit between the heat release β and the detonation propagation Mach number D*. Regimes of instability are found to be characterized by the presence of a single unstable oscillatory mode over a finite range of wavenumbers.

A detailed investigation of the hydrodynamic stability to transverse linear disturbances of a steady, one-dimensional detonation in an ideal gas undergoing an irreversible, unimolecular reaction with an Arrhenius rate constant is conducted via a normal-mode analysis. The method of solution is an iterative shooting technique which integrates between the detonation shock and the reaction equilibrium point. Variations in the disturbance growth rates and frequencies with transverse wavenumber, together with two-dimensional neutral stability curves and boundaries for all unstable low- and higher frequency modes, are obtained for varying detonation bifurcation parameters. These include the detonation overdrive, chemical heat release and reaction activation energy. Spatial perturbation eigenfunction behaviour and phase and group velocities are also obtained for selected sets of unstable modes. Results are presented for both Chapman–Jouguet and overdriven detonation velocities. Comparisons between the earlier pointwise determination of stability and interpolated neutral stability boundaries obtained by Erpenbeck are made. Possible physical mechanisms which govern the wavenumber selection underlying the initial onset of either regular or irregular cell patterns are also discussed.

An analytical dispersion relation describing the linear stability of a plane detonation wave to low-frequency two-dimensional disturbances with arbitrary wavenumbers is derived using a normal mode approach and a combination of high activation energy and Newtonian limit asymptotics, where the ratio of specific heats γ→1. The reaction chemistry is characterized by one-step Arrhenius kinetics. The analysis assumes a large activation energy in the plane steady-state detonation wave and a characteristic linear disturbance wavelength which is longer than the fire-zone thickness. Newtonian limit asymptotics are employed to obtain a complete analytical description of the disturbance behaviour in the induction zone of the detonation wave. The analytical dispersion relation that is derived depends on the activation energy and exhibits favourable agreement with numerical solutions of the full linear stability problem for low-frequency one- and two-dimensional disturbances, even when the activation energy is only moderate. Moreover, the dispersion relation retains vitally important characteristics of the full problem such as the one-dimensional stability of the detonation wave to low-frequency disturbances for decreasing activation energies or increasing overdrives. When two-dimensional oscillatory disturbances are considered, the analytical dispersion relation predicts a monotonic increase in the disturbance growth rate with increasing wavenumber, until a maximum growth rate is reached at a finite wavenumber. Subsequently the growth rate decays with further increases in wavenumber until the detonation becomes stable to the two-dimensional disturbance. In addition, through a new detailed analysis of the behaviour of the perturbations near the fire front, the present analysis is found to be equally valid for detonation waves travelling at the Chapman–Jouguet velocity and for detonation waves which are overdriven. It is found that in contrast to the standard imposition of a radiation or piston condition on acoustic disturbances in the equilibrium zone for overdriven waves, a compatibility condition on the perturbation jump conditions across the fire zone must be satisfied for detonation waves propagating at the Chapman–Jouguet detonation velocity. An insight into the physical mechanisms of the one- and two-dimensional linear instability is also gained, and is found to involve an intricate coupling of acoustic and entropy wave propagation within the detonation wave.

The nonlinear stability of a pulsating detonation wave driven by a three-step chain-branching reaction is studied. The reaction model consists sequentially of a chain-initiation step and a chain-branching step, both governed by Arrhenius kinetics, followed by a temperature-independent chain-termination step. The model mimics the essential dynamics of a real chain-branching chemical system, but is sufficiently idealized that a theoretical analysis of the instability is possible. We introduce as a bifurcation parameter the chain-branching cross-over temperature (TB), which is the temperature at which the chain-branching and chain-termination rates are equal. In the steady detonation structure, this parameter controls the ratio of the chain-branching induction length to the length of the recombination zone. When TB is at the lower end of the range studied, the steady detonation structure, which is dominated by the temperature-independent recombination zone, is found to be stable. Increasing TB increases the length of the chain-branching induction region relative to the length of the recombination zone, and a critical value of TB is reached where the detonation becomes unstable, with the detonation shock pressure evolving as a single-mode low-frequency pulsating oscillation. This single-mode nonlinear oscillation becomes progressively less stable as TB is increased further, persisting as the long-term dynamical behaviour for a significant range of TB before eventually undergoing a period-doubling bifurcation to a two-mode oscillation. Further increases in TB lead to a chaotic behaviour, where the detonation shock pressure history consists of a sequence of substantive discontinuous jumps, followed by lower-amplitude continuous oscillations. Finally, for further increases in TB a detonability limit is reached, where during the early onset of the detonation instability, the detonation shock temperature drops below the chain-branching cross-over temperature causing the wave to quench.

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.