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We examine the diffraction dynamics of a two-dimensional (2D) detonation in a circular arc of the conventional HMX-based, high performance, solid explosive PBX 9501, for which the detonation reaction zone length scale is estimated to be of the order of 100–150 µm. In this configuration, a steady propagating detonation will develop, sweeping around the arc with constant angular speed. We report on results from three PBX 9501 arc experiments, exploring the variation in linear speed on the inner and outer arc surfaces for the steady wave along with the structure of the curved detonation front, as a function of varying inner surface radius and arc thickness. Comparisons of the properties of the motion of the steady wave for each arc configuration are then made with a spatially-distributed PBX 9501 reactive burn model, calibrated to detonation performance properties in a 2D planar slab geometry. We show that geometry-induced curvature of the detonation near the inner arc surface has a significant effect on the detonation motion even for conventional high explosives. We also examine the detonation driving zone structure for each arc case, and thus the subsonic regions of the flow that determine the influence of the arc geometry on the detonation propagation. In addition, streamline paths and reaction progress isolines are calculated. We conclude that a common approximation for modeling conventional high explosive detonation, wherein the shock-normal detonation speed is assumed equal to the Chapman–Jouguet speed, can lead to significant errors in describing the speed at which the detonation propagates.

We study the effects of non-parallel (diverging or converging) channel walls on flame propagation and acceleration in planar and cylindrical narrow channels, closed at the ignition end and open at the other, accounting for thermal expansion in both the zero Mach number and weakly compressible flow limits. For parallel channel walls, previous work has shown that thermal expansion induces an axial flow in the channel, which can significantly increase the propagation speed and acceleration of the flame. In this study, we consider examples of diverging/converging linear walls, although our asymptotic analysis is also valid for curved walls. The slope of the channel walls is chosen so that the magnitude of the thermal-expansion induced flow through the channel obtained for parallel walls is modified at leading-order, thereby influencing the leading-order flame propagation. For zero Mach number flows, the effect of the diverging/converging channel walls is moderate. However, for weakly compressible flows, the non-parallel walls directly affect the rate at which pressure diffuses through the channel, significantly inhibiting flame acceleration for diverging walls, whereas the flame acceleration process is enhanced for converging walls. We consider several values of the compressibility factor and channel wall slopes. We also show that the effect of a cylindrical channel geometry can act to significantly enhance flame acceleration relative to planar channels. The study reveals several new physical insights on how non-parallel channel walls can influence the ability of flames to accelerate by modifying the flow and pressure distribution induced by thermal expansion.

Detonation performance experiments and modeling are reported for the explosive PBX 9701, which is composed of 97% 3,3’-diamino-4,4’-azoxyfurazan (DAAF) and 3% vinylidene fluoride-chlorotrifluoroethylene copolymer (Kel-F 800) binder by weight. PBX 9701 is a newly developed reduced-sensitivity explosive with increased performance relative to the triaminotrinitrobenzene or TATB-based PBX 9502 while still retaining low sensitivity to mechanical insult. The first detonation performance measurements for this formulation are presented, including fron curvature rate sticks and cylinder expansion test data. Prior shock initiation data is also reviewed. These data are used to develop programmed burn (PB) and reactive burn (RB) calibrations for existing, commonly used, performance models which allow engineering calculations with PBX 9701. The calibration process involves several enhancements relative to conventional approaches, including the use of an analytical scaling correlation to speed the equation of state (EOS) calibration process. It uses a PB hydrocode-based approach and development of a new methodology to improve the consistency between the PB and RB model calibrations and associated calculations. This link is achieved by populating the RB products EOS in direct reference to the PB release isentrope and Chapman-Jouguet state and in calibrating the timing components of each model using an equivalent procedure, all in order to facilitate comparison between the two modeling approaches. Overall, PBX 9701 is seen to exhibit improved performance relative to insensitive explosives, with a trinitrotoluene (TNT) equivalence of 1.24. The detonation performance properties are found to be well captured by existing models.

We examine, via multi-material simulation in a two-dimensional planar geometry, the effects on steady detonation propagation of the presence of a low-density intermediate layer between a condensed-phase high explosive (HE) and a high-density metallic confiner of finite thickness. Such elastomer intermediate layers are often added to eliminate air-gaps and the associated jetting effects that can arise due to machining imprecisions, or to prevent HE cracking due to environmental changes. Without an intermediate layer, the flow structure of a steady detonation/metal confiner interaction is well understood. In particular, there is no reflected wave passed into the HE due to the metal confinement. With the elastomer layer present, we find that, as the intermediate layer width increases, a complex wave interaction and communication path develops between the HE, intermediate, and metal layers. For thin intermediate layers, a shock-driven subsonic flow develops in the intermediate layer, passing information from the metal layer to the HE, with the detonation speed decreasing as the intermediate layer width increases. For wider intermediate layers, a Mach stem configuration develops in the intermediate layer, forcing a shock to be reflected into the HE. Simultaneously, a localized Prandtl-Meyer fan emerges from the intersection of the detonation shock with the HE-intermediate layer material interface. These HE structures are shown to have a substantial effect on the structure of the detonation driving zone. The Prandtl-Meyer fan becomes the dominant structure for critically large intermediate layer widths, wherein the presence of the metal layer does not affect the detonation propagation. We examine the detonation propagation speed and reaction and driving zone structure as a function of varying intermediate layer width. Two confinement metals are examined, along with two high explosive and three metal layer widths.

A surface evolution model is developed for the detonation propagation dynamics of the HMX-based conventional high explosive PBX 9501, which relates the normal surface speed Dn to its local surface curvature κ. Such surface evolution models are important for the understanding and modification of engineering design calculations for high explosive applications. We describe a series of unconfined PBX 9501 slab geometry experiments of varying thickness, and detail how the steady axial detonation speed and detonation front shape data are obtained. A merit-function based calibration process is then described that uses both the PBX 9501 thickness effect and front shape data to parametrize the D n − κ propagation law. The time-dependent PBX 9501 D n − κ surface evolution law is then applied to detonation wave propagation in two-dimensional circular arc geometries, systematically examining the effect of arc thickness, inner radius, relaxation dynamics to steady-state propagation and confinement.

We examine numerically the effect that detonation wavefront curvature has on gaseous detonation cellular structure in a circular arc geometry with rigid walls. For the model considered, a planar detonation is weakly unstable. The detonation curvature is induced by diffraction and flow divergence as the wave sweeps around the arc. We examine the effect of arc thickness, inner arc radius and activation energy on the cellular pattern. A combination of pseudo-Schlieren, pseudo-soot foil records and detonation shock and sonic flow loci are used to explore the observed detonation structures. We find that for sufficiently wide arcs, the detonation wavefront curvature renders the detonation hydrodynamically stable, i.e. laminar flow with no cells. The stable configuration consists of a broadly curved wave originating from the inner arc surface, which is typically connected to the outer arc boundary by a stable Mach stem. The stable detonation is driven by a small region of subsonic flow attached to the inner arc surface, outside of which the flow is supersonic. The initiation dynamics of a stable propagating wave case are explored. The evolution dynamics between an unstable (cellular) and stable arc thickness are also examined. Smaller inner arc radii and lower activation energies increase the range of stable solutions.

The shock motion of calculated weakly unstable cellular detonation was analyzed using concepts from velocity-curvature theory to develop new insight into the underlying physical mechanisms driving the cellular instability. The cellular cycle was shown to follow a surface in a three-dimensional coordinate system composed of the local shock velocity, curvature, and acceleration. Wavelets of the detonation shock were found to follow a velocity-curvature trajectory that was characteristic of an initially reactive wave that decays in time to a decoupled blast wave. Near the Mach stem, these trajectories were only modulated by the strength of the Mach stem at the time the wavelet was generated, indicating that the small reaction zone present in this region is the dominant factor driving the flow. Away from the Mach stem, all wavelet trajectories collapsed to a common curve in velocity-curvature space that was consistent with motion of a decaying blast wave. For the mixture studied, the apparent adherence of shock motion to a unique surface in velocity-curvature-acceleration space indicates the possible existence of an intrinsic mixture-specific relationship for cellular gaseous detonation. This relationship and analysis methodology also provides a mechanism for quantification of the shock velocity and shape fluctuations present in cellular detonation, which may provide utility for modeling detonation engineering applications such as rotating detonation engine design.

We examine the evolution of cellular detonation patterns in a two-dimensional channel with yielding confinement on one side. It is shown that in a narrow channel of fixed width the number of cells first increase with decreasing level of confinement. Subsequently, with increasingly weaker confinement, the cells then grow in size and the total number of cells in the channel decreases. For sufficiently weak confinement, the flow becomes laminar with no detonation cells. We examine the relative importance of two fluid mechanisms underlying the observed evolution: global curvature of the detonation shock front due to induced flow divergence caused by the yielding confinement, and energy loss associated with transverse shock wave transmission to the confining material. In order to determine which effect is dominant, we compare two types of numerical calculations. One involves specialized calculations in which the explosive boundary, along which impermeable flow conditions are applied, is deflected through a range of specified angles upon detonation arrival. This set-up mimics the effect of yielding confinement in terms of induced flow divergence, but removes the transverse wave energy loss that would otherwise occur due to wave transmission into the confiner material. The second involves multi-material simulations which can account for transverse wave energy loss into the confining material. Shock polar theory is used to select confiner densities in the multi-material calculations that provide equivalent material interface deflection angles at the detonation shock to the angles imposed in the deflected solid wall calculations. We determine that the induced global curvature of the wave primarily drives both the evolution of the cellular pattern and eventual stabilization of the detonation front, characterized by laminar flow solutions. In wider channels, we show that the detonation front will likely remain unstable even for very weak confinement, as the mean curvature of the front only becomes significant near the edge of the explosive domain.

We describe the details of an unconfined insensitive high explosive (PBX 9502) circular arc section experiment, in which, after a transient period, a detonation sweeps around the arc with constant angular speed. The arc section is sufficiently wide that the flow along the centerline of the arc section remains two-dimensional. Data includes time-of-arrival diagnostics of the detonation along the centerline inner and outer arc surfaces, which is used to obtain the angular speed of the steadily rotating detonation. We also obtain the lead shock shape of the detonation as it sweeps around the arc. Reactive burn model simulations of the PBX 9502 arc experiment are then conducted to establish the structure of the detonation driving zone, i.e. the region enclosed between the detonation shock and flow sonic locus (in the frame of the steady rotating detonation). It is only the energy released in this zone which determines the speed at which the steady detonation sweeps around the arc. We show that the sonic flow locus of the detonation driving zone largely lies at the end of, or within, the fast reaction stage of the PBX 9502 detonation, with the largest section of the detonation driving zone lying close to the inner arc surface. We also demonstrate that the reactive burn model provides a good prediction of both the angular speed of the detonation wave and the curved detonation front shape.

A series of detonation performance experiments are described for the plastic-bonded high explosive (HE) PBX 9012, nominally composed of 90% octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazine (HMX) and 10% polymeric binder vinylidene-hexafluoropropylene copolymer (Viton VTR-5883), by weight. The experiments provide information on the PBX 9012’s detonation propagation properties and metal pushing characteristics. Although this HE formulation has been in use for 50 years as a booster (compositionally similar to LX-07), our new data represent the first full detonation performance characterization for the HE (along with recently reported shock initiation experiments). Here, we compare the new PBX 9012 data with the well-studied PBX 9501 (95% HMX and 5% polymeric binder), establishing how the 5% difference in HMX content between the explosives is manifested in the performance measurements. A scaling argument based on the known energetic material content in a large selection of high HMX content formulations is also given to help explain the observed results. Finally, the accumulated PBX 9012 data is used to generate performance models using both programmed and reactive burn techniques, providing an additional avenue to compare PBX 9012 to the more energetic PBX 9501.

The results of 14 tests examining the behavior of aluminum (Al) confiners driven by non-ideal ANFO detonation in a cylinder test configuration are presented. In each test, the measured detonation phase velocity is slower than the aluminum sound speed. Thus, in the detonation reference frame, the flow in the Al is both shockless and subsonic. The tests involve: 3-in. inner diameter (ID) cylinders with Al wall thicknesses of 1/4, 3/8, 1/2, 1 and 2 in.; a 4-in. ID cylinder with a 1/2-in. Al wall thickness; and 6-in. ID cylinders with Al wall thicknesses of 1/2, 1 and 2 in. The ANFO detonation velocity is seen to increase with increasing wall thickness for both the 3- and 6-in. ID tests, with no limiting velocity reached for the wall thicknesses used. The motion of the outer Al wall due to precursor elastic waves in the Al running ahead of the detonation is also measured at various axial locations along the cylinders. It is found that the magnitude of the outer wall motion due to the precursor elastic waves is small, while the associated wall motion is unsteady and decays in amplitude as the elastic disturbances move further ahead of the detonation front. The variations in the expansion history of the main outer wall motion of the cylinders are presented for increasing wall thickness at fixed ID, and for increasing cylinder inner diameter at a fixed wall thickness. Finally, we also explore the existence of a geometric similarity scaling of the wall expansion history for three geometrically scaled tests (3- and 6-in. ID cylinders with 1/4- and 1/2-in. walls, 3- and 6-in. ID cylinders with 1/2- and 1-in. walls and 3- and 6-in. ID cylinders with 1- and 2-in. walls respectively). We find that the wall velocity histories for each of the three scaled tests, when plotted directly against time relative to start of main motion of the wall, are similar over a certain range of wall velocities without any geometric based rescaling in time. The range of wall velocities where the overlap occurs increases as the ratio of the wall thickness to inner diameter decreases. This is in contrast to ideal high explosives, where the outer wall velocity histories are only similar when the geometric scale factor (in this case a factor of 2) is applied to the wall velocity motion.

The evolution of the normal detonation shock velocity (Dn) with local shock curvature (κ) is experimentally and numerically examined along entire evolving fronts of a weakly unstable cellular detonation cycle with the intention of extending the understanding of cellular evolution dynamics. As expected, a single velocity–curvature relation is not recovered due to the unsteady evolution of the cell. However, geometric features of the Dn–κ evolution during a cell cycle reveal some new details of the mechanisms driving cellular detonation. On the cell centerline, the local shock velocity and curvature monotonically decrease throughout the cellular cycle. Off centerline, a larger range of wavefront curvature was exhibited in expanding cells as compared to shrinking ones, indicating that most curvature variation in a detonation cell occurs near the Mach stem. In normal shock velocity–curvature space, the cell dynamics can be mapped to three features that are characteristic of (feature 1) a detonation with a spatially short reaction zone, (feature 2) a transitional regime of shock and reaction zone decoupling, and (feature 3) a diffracting inert blast wave. New, growing cells predominately exhibited features 1 and 2, while decaying cells only exhibited feature 3. The portions of all profiles with normal velocities below the Chapman–Jouguet velocity were characteristic of inert blast propagation, indicating the possibility that exceeding this velocity may be a necessary condition for the existence of shock and reaction zone coupling. In this inert blast regime, Dn and κ vary spatially across the wave front so each segment is not geometrically cylindrical, but when accumulated, the Dn–κ data map out a straight line, indicating elements of self-similar flow for each stage in the cell cycle.

Detonations in explosive mixtures of ammonium-nitrate-fuel-oil (ANFO) confined by aluminum allow for transport of detonation energy ahead of the detonation front due to the aluminum sound-speed exceeding the detonation velocity. The net effect of this energy transport on the detonation is unclear. It could enhance the detonation by precompressing the explosive near the wall. Alternatively, it could decrease the explosive performance by crushing porosity required for initiation by shock compression or destroying confinement ahead of the detonation. At present, these phenomena are not well understood. But with slowly detonating, non-ideal high explosive (NIHE) systems becoming increasing prevalent, proper understanding and prediction of the performance of these metal-confined NIHE systems is desirable. Experiments are discussed that measured the effect of ANFO detonation energy transported upstream of the front by a 76-mm-inner-diameter aluminum confining tube. Detonation velocity, detonation front-shape, and aluminum response are recorded as a function of confiner wall thickness and length. Detonation shape profiles display little curvature near the confining surface, which is attributed to energy transported upstream modifying the flow. Average detonation velocities were seen to increase with increasing confiner thickness, while wavefront curvature decreased due to the stiffer, subsonic confinement. Significant radial sidewall tube motion was observed immediately ahead of the detonation. Axial motion was also detected, which interfered with the front-shape measurements in some cases. It was concluded that the confiner was able to transport energy ahead of the detonation and that this transport has a definite effect on the detonation by modifying its characteristic shape.

The dynamics of non-premixed edge-flames, including the generation of cellular structures, in an unsteady, symmetric counterflow are examined for positive rates of strain. A one-step reaction is assumed, ν Y F + ν X O → ν p P , in which the oxidizer Lewis number is 1. For a variety of Damkohler numbers, we examine the edge-flame evolution for two values of the fuel Lewis number Le Y , 0.3 and 1, and two values of the initial mixture fraction γ, 0.36 and 1, representing fuel lean and stoichiometric supply conditions. For Le Y = 0.3 and γ = 0.36 , unsteady forcing can convert non-cellular edge-flames into ones containing various characteristics of near- or sub-limit cellular structures, including drifting, splitting and stationary flame strings. The transition regimes between the different edge-flame structures are examined as a function of the amplitude and frequency of the strain rate variations in the unsteady counterflow and also as a function of the instantaneous and equivalent strain rate functions. For Le Y = 0.3 and γ = 1 , while no cellular edge-flames can be generated for steady counterflows, we show that cellular structures can be observed in the presence of unsteady forcing. For Le Y = 1 and γ = 1 , it is shown that unsteady forcing can significantly modify the mean propagation speeds of both ignition and failure waves. Finally, the quenching boundaries of two-dimensional edge-flames induced by the unsteady counterflow are examined for Le Y = 0.3 , γ = 0.36 and Le Y = 1 , γ = 1 .

Shock-induced ignition is considered for a three-step chain-branching mechanism at large activation energies. The nature of the ignition process depends critically on the magnitude of the initial post-shock temperature relative to the chain-branching cross-over temperature. For the limits examined, the induction region is preceded by an exponentially weak initiation zone that provides relevant initial conditions for the induction phase. When the initial post-shock temperature is sufficiently close to the chain-branching temperature, ignition is characterized by logarithmic singular behaviors in the pressure and temperature perturbations and the structure has some similarities with the one-step chemistry problem. However, for larger initial post-shock temperatures, the logarithmic singularities are replaced by linear temporal growth. Suitable non-linear Clarke equations are deduced for both of these cases and numerical solutions are presented.

A general normal-mode linear stability formulation of steady planar detonation waves is presented that is valid both for an arbitrary equation of state and for multi-step, multi-species chemical kinetics. The general formulation can be used for many purposes, including an examination of gaseous detonation stability with complex reaction kinetics in which the individual reacting species have variable thermochemical properties. In the present paper, we consider two cases that could not be obtained by previous one-step chemistry, polytropic gas formulations: the first concerns the effect of a difference in heat capacities between product and fuel species, as well as a possible mole change, in a single-step irreversible reaction. The second examines the effects of exothermic or endothermic heat release/absorption in the chain-initiation stage of a model three-step reaction.

We provide the first theoretical demonstration of the existence of quasi-one-dimensional, quasi-steady, self-sustaining convergent detonation waves. These occur in systems where, in the planar wave, the rate of heat release by chemical reaction reaches a maximum at a point of incomplete reaction. The case examined in the present paper is that for a two-step sequential reaction, with the second stage endothermic. We construct detonation velocity against curvature relationships for converging waves, and compare these theoretical curves with direct numerical simulations of imploding detonations in cylindrical and spherical geometries. We also comment on the one-dimensional stability of imploding and diverging detonation fronts governed by the two-step model.

We examine the cellular instabilities of laminar non-premixed diffusion flames that arise in a polycrystalline alumina microburner with a channel wall gap of dimension 0.75 mm. Changes in the flame structure are observed as a function of the fuel type (H 2 , CH 4 , and C 3 H 8 ) and diluent. The oxidizer is O 2 /inert. In contrast to previous observations on laminar diffusion flame instabilities, the current instabilities occur in the direction of flow above the splitter plate, and only occur for the heavier fuel types. They are not observed in a H 2 –O 2 mixture, which will only support a continuous laminar flame inside our burner, regardless of the initial mixture strength and whether or not the flame is in near-quenching conditions. The only exception is when helium is added to the H 2 –O 2 mixture, raising the effective Lewis numbers of both components.

We examine edge-flames in the context of symmetric counterflows of fresh mixture, and examine their dynamics when the rate of strain is varied periodically in time. For a Lewis number of 1, extinction boundaries and zero (in the mean) edge propagation speed boundaries are constructed in the forcing amplitude—Damkohler number plane. Forcing can turn advancing edges into retreating edges. For a Lewis number of 0.3, the fundamental distinction is between flames that display cellular structures and flames that do not. Forcing can convert a non-cellular flame into one with cells; and it can strongly affect the dynamics of cellular structures that exist in the absence of forcing.

We discuss edge-flames in the context of the plane counterflow of air and nitrogen-diluted hydrogen. For sufficiently dilute hydrogen streams, strong cellular instabilities arise which affect the dynamics of the edge-flames and, in the long term, give rise to residual cellular structures. Depending on the Damkohler number, solutions are constructed corresponding to: an advancing edge trailing a smooth flame; an advancing edge trailing a stationary periodic cellular structure, a warp of flame-strings; a finite array of flame-strings which drift apart from each other at a decreasing rate; a single stationary flame-string; dual stationary flame-strings; triple stationary flame-strings. Many of these solutions are sublimit in the sense that they exist for Damkohler numbers smaller than the one-dimensional quenching value defined for the counterflow configuration. The connection of these solutions to the linear stability of the one-dimensional system is discussed, and evidence of a subcritical bifurcation is presented. Comparisons are made with experimental results obtained by Ronney.

The linear stability of a planar detonation wave with a three-step chain-branching reaction is studied by a normal mode approach. The reaction model consists of a chain-initiation step and a chain-branching step governed by Arrhenius kinetics, with a chain-termination step which is independent of temperature. It mimics the essential reaction dynamics of a real chain-branching chemical system. The linear stability of the steady detonation wave to two-dimensional disturbances is studied with the chain-branching crossover temperature, i.e., the temperature at which chain-branching and chain-termination rates are equal, as a bifurcation parameter. This parameter determines the ratio of the length of the chain-branching induction zone to the chain-termination zone within the steady detonation wave. The effect of linear transverse disturbances is considered for two values of the chain-branching crossover temperature: in one the planar steady detonation wave is stable to one-dimensional disturbances, while in the other it is unstable to such disturbances.

Premixed flames are highly sensitive to changes in pressure. It is found that following the passage of a shock wave, a conventional diffusion-reaction deflagration can accelerate to become a fast convection-reaction driven flame that itself can further markedly accelerate. An induction zone is formed between the shock and the combustion wave. This is a relatively long region in which small changes in the dependent variables occur. Treating these as asymptotic perturbations, an intermediate zone that connects the rear of the induction zone to the front of the fast flame must also be considered. Matching conditions between the zones and boundary conditions at the shock and piston are calculated and the evolution of the shock wave, induction zone, and fast flame system is investigated. Across the induction zone there is strong acoustic coupling such that the resulting behavior of the combined system leads to thermal runaway. The time for this blow up to occur is investigated as a function of initial mass flux through the fast flame and initial shock Mach number. It is also shown that there is the possibility of hot spots occurring in the induction zone which then leads to secondary ignition ahead of the fast flame. We have also investigated the effect on primary and secondary ignition of the activation energy of the simple one-step chemistry used in this work.

Some surprising effects are seen in studying numerically the evolution of a propagating pulse of pressure in a medium reacting via a one-step exothermic Arrhenius reaction. The length and amplitude of the pulse are taken to be large enough for steepening effects to be important and for enhanced reaction to lead to a substantial reduction in ignition time. The evolution proceeds through a repeated sequence of similar stages involving: shock-formation and growth; ignition behind the shock; and the generation of another propagating pressure pulse. Substantial unsteady behavior is seen to be engendered by the entropy released through shock formation. A number of unsteady reignitions are seen to culminate in a pressure-peak, substantially higher than the von Neumann spike of a Chapman-Jouget wave, during the formation of a transient overdriven detonation; this decays subsequently towards a Chapman-Jouget state. It is conjectured that this sort of evolution may well be generic to ignition via a range of pressure-pulses in state-sensitive systems. A saturation of, or relative reduction in, the reaction's thermal sensitivity ultimately prevents the reignition process after shock-formation from happening quickly enough to continue its repetition. As such, the behavior should be strongly dependent on the nature of the chemical model and is likely to be modified significantly by changes in the chemical mechanism.