Chapman-Jouguet Speed Research Articles

Two-dimensional visualization of induction zone in hydrogen detonations

The induction zone length is an important length scale for gaseous detonations, as it is correlated with other characteristic parameters such as the cell size. A good characterization of the induction zone can then lead to a better comprehension of the detonation structure. In this context, this study investigates two-dimensional (2D) visualizations of induction zones in stoichiometric hydrogen-air and hydrogen-oxygen-nitrogen-argon mixtures. The detonation speed for both mixtures is close to the Chapman Jouguet speed, and without wall losses. Planar laser-induced fluorescence (PLIF) technique was employed, by exciting nitric oxide (NO) molecules (i.e., NO-PLIF). In both mixtures, 1900 ppm of NO was added as a tracer to ensure a good signal-to-noise ratio in the von Neumann condition. The NO-PLIF images displayed a variation of the experimental induction length along the cellular cycle. Large and short induction lengths were observed for incident shock and Mach stem, respectively. A total of 2727 experimental induction lengths were extracted from the NO-PLIF images, among 79 individual experiments. Statistical analyses of these data showed that the mixture with lower activation energy exhibited less dispersion of the data compared to the mixture with higher activation energy. Non-equilibrium effects were not observed in the fluorescence profiles and induction length measurements.

Detonation wave diffraction in stoichiometric C2H4/O2 mixtures using a global four-step combustion model

In this study, we revisit the problem of detonation diffraction in mildly irregular ethylene–oxygen mixtures using high resolution numerical simulations. In particular, we focus on the re-initiation of diffracted waves in the critical regime and the role of transverse detonations on the re-establishment of the detonation. This problem is significant for characterizing detonation wave propagation as well as for the development of next-generation detonation engines. A thermochemically derived four-step combustion model that responds appropriately to the thermodynamic state behind the complex shock wave dynamics was adopted in an Euler framework. While past attempts using simplified combustion models have largely failed to predict the onset of transverse detonations near the critical limit, our simulations demonstrate that the four-step model can capture these features. Our results reveal that transverse triple point collisions are fundamental for triggering the re-initiation of the detonation and multiple modes of re-initiation exist in the critical regime. The transverse detonation initiation distance and wall reflection lengths are then compared with experimental measurements and found to be in agreement. We also demonstrate that the incident wave re-establishes at the Chapman–Jouguet speed when re-initiation occurs through the diffraction process. However, if re-initiation of a quenched detonation occurs by a transverse detonation following the reflection of the expanding wave at the boundary, the re-initiated detonation becomes overdriven along the Mach wave. Finally, it was found that the transverse detonations are Chapman–Jouguet detonations that travel in the shocked but unreacted gas.

Improving Detonability in a Small-Scale Rotating Detonation Engine Using Partial Premixing

Small-scale rotating detonation engines (RDEs) with low mass flows and high frequency have potential as small thrusters, robust flame holders, and research tools that can be quickly and cheaply modified. However, the necessarily small channel geometry and low mass flow force the detonation wave to propagate near the limits of detonability. Previous success in a small-scale, 28-mm-diameter micro-RDE, operating on nitrous oxide () and ethylene (), exhibited unstable wave behavior with propagation velocities between 47 and 56% the Chapman–Jouguet speed. These results were observed at mass flows from 25 to using an in-chamber jets-in-crossflow (JIC) injection scheme with spark plug initiation. Building upon this work, the current investigation looked to improve the detonation behavior in the micro-RDE without changing the channel geometry or resultant operability and better understand the nature of mechanisms governing the design limitations in RDEs. To improve the wave behavior, a new partially premixed JIC injection scheme was used along with a predetonator ignition device. With these changes, tighter windows of operable mass flows () and equivalence ratios () were observed, between 35 and and 0.6 and 1.3, respectively. Maps of versus before and after detonation showed that the space of conditions conducive to ignition was not isomorphic with the space of conditions that sustained operation. Increases to the injection pressure to chamber pressure ratios improved the probability of detonation but resulted in greater pressure losses than were seen in the original JIC injection scheme. Single wave detonation was achieved for the first time in this RDE and had a peak frequency of 16.6 kHz, or 66% Chapman–Jouguet speed. Increases in fill height drove increases in wave frequency. Decreased fill height estimates compared to the previous design showed that partial premixing decreased the required fill time. These results support that the limits of detonability in small RDEs can be expanded without significantly compromising the characteristic low-mass-flow and high-frequency operation inherent to their geometry.

Exploring the controlling mechanisms for gradient evolution in unsteady detonation flows

Proper characterization of the transient interaction between shock front motion and chemical energy release is the key for further understanding of unsteady detonation dynamics. We have shown that the gradients could be the adequate intermediate quantities to reflect the transient interaction between shock front motion and chemical energy release through the Lagrangian particle analysis of the simulated direct detonation initiation (DDI) process in H2–O2–Ar mixtures. Specifically, the “shock change equations” are verified to describe the direct relation between the shock front motion and the gradients immediately behind the shock. Moreover, given the time derivatives of shock speed, the gradient evolution in the induction zone can be reproduced by the gradient evolution equations that are deduced from the Euler equations, no matter if the shock front undergoes rapid deceleration or acceleration. While in the reaction zone where the heat release is significant, it is demonstrated that the evolutions of velocity, pressure, and their gradients can be described by the Zel'dovich–von Neumann–Döring model with a constant wave speed that is below the Chapman–Jouguet speed, no matter if the shock front undergoes rapid deceleration or acceleration. These two distinct controlling mechanisms are verified in both planar and spherical DDI processes, showing their general applicability. This work suggests a new perspective, in terms of gradient evolution, for further understanding the unsteady detonation dynamics.

Experimental and modeling analysis of detonation in circular arcs of the conventional high explosive PBX 9501

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.

Numerical analysis of the mean structure of gaseous detonation with dilute water spray

Two-dimensional (2-D) numerical simulations based on the Eulerian–Lagrangian method that take droplet break-up into account are conducted to clarify the mean structure of gaseous detonation laden with a dilute water spray. The premixed mixture is a slightly diluted stoichiometric hydrogen–oxygen mixture at low pressure. The simulated results are analysed via 2-D flow fields and statistical Favre spatiotemporal averaging techniques. Gaseous detonation with water droplets (WD) propagates stably with a velocity decrease compared with the dry Chapman–Jouguet speed. The mean structure of gaseous detonation with dilute water spray shares a similar structure as the one without water spray. However, the hydrodynamic thickness is changed due to the interaction with water spray. Overall interphase exchanges (mass, momentum and energy) that take place within the hydrodynamic thickness induce a decrease of the detonation velocity and lower the level of fluctuations downstream of the mean leading shock wave. Droplet break-up occurs downstream of the induction zone and in our case, the water vapour from the evaporation of water spray does not affect the reactivity of gaseous detonation. The laminar master equation for gaseous detonation laden with inert WD shows that the hydrodynamic thickness should rely on the gaseous sound speed, and works well as the working mixture is weakly unstable and its cellular structure is regular. The droplet flow regimes and break-up modes have also been determined. The characteristic lengths of detonation and interphase exchanges have been ordered under the present simulation conditions and have been shown to be intimately intertwined.

Effect of spatial inhomogeneities on detonation propagation with yielding confinement

The propagation of detonations in layers of reactive gas bounded by inert gas is simulated computationally in both homogeneous and inhomogeneous systems described by the two-dimensional Euler equations with the energy release governed by an Arrhenius rate equation. The thickness of the reactive layer is varied and the detonation velocity is recorded as the layer thickness approaches the critical value necessary for successful propagation. In homogeneous systems, as activation energy is increased, the detonation wave exhibits increasingly irregular cellular structure characteristic of the inherent multidimensional instability. The critical layer thickness necessary to observe successful propagation increases rapidly, by a factor of five, as the activation energy is increased from $$E_\mathrm{a}/RT_0 = 20$$ –30; propagation could not be observed at higher activation energies due to computational limitations. For simulations of inhomogeneous systems, the source energy is concentrated into randomly positioned squares of reactive medium embedded in inert gas; this discretization is done in such a way that the average energy content and the theoretical Chapman–Jouguet (CJ) speed remain the same. In the limit of highly discrete systems with layer thicknesses much greater than critical, velocities greater than the CJ speed are obtained, consistent with our prior results in effectively infinite width systems. In the limit of highly discretized systems wherein energy is concentrated into pockets representing 10% or less of the area of the reactive layer, the detonation is able to propagate in layers much thinner (by an order of magnitude) than the equivalent homogeneous system. The critical layer thickness increases only gradually as the activation energy is increased from $$E_\mathrm{a}/R T_0 = 20{-}55$$ , a behavior that is in sharp contrast to the homogeneous simulations. The dependence of the detonation velocity on layer thickness and the critical layer thickness is remarkably well described by a front curvature model derived from the classic, ZND-based model of Wood and Kirkwood. The results of discrete sources are discussed as a conceptual link to the behavior that is experimentally observed in cellular detonations with highly irregular cellular structure in which intense turbulent burning rapidly consumes detached pockets behind the main shock front. The fact that highly discrete systems are well described by classical, curvature-based mechanisms is offered as a possible explanation as to why curvature-based models are successful in describing heterogeneous, condensed-phase explosives.

Effect of spatial discretization of energy on detonation wave propagation

Detonation propagation in the limit of highly spatially discretized energy sources is investigated. The model of this problem begins with a medium consisting of a calorically perfect gas with a prescribed energy release per unit mass. The energy release is collected into sheet-like sources that are embedded in an inert gas that fills the spaces between them. The release of energy in the first sheet results in a planar blast wave that propagates to the next source, which is triggered after a prescribed delay, generating a new blast, and so forth. The resulting wave dynamics as the front passes through hundreds of such sources is computationally simulated by numerically solving the governing one-dimensional Euler equations in the laboratory-fixed reference frame. Two different solvers are used: one with a fixed uniform grid and the other using an unstructured, adaptively refined grid enabling the limit of highly concentrated, spatially discrete sources to be examined. The two different solvers generate consistent results, agreeing within the accuracy of the measured wave speeds. The average wave speed for each simulation is measured once the wave propagation has reached a quasi-periodic solution. The effect of source delay time, source energy density, specific heat ratio and the spatial discreteness of the sources on the wave speed is studied. Sources fixed in the laboratory reference frame versus sources that convect with the flow are compared. Simulations using an Arrhenius-rate-dependent energy release are performed as well. The average wave speed is compared to the ideal Chapman–Jouguet (CJ) speed of the equivalent homogenized media. Velocities in excess of the CJ speed are found as the sources are made increasingly discrete, with the deviation above CJ being as great as 15 %. The deviation above the CJ value increases with decreasing values of specific heat ratio $\unicode[STIX]{x1D6FE}$. The total energy release, delay time and whether the sources remain laboratory-fixed or are convected with the flow do not have a significant influence on the deviation of the average wave speed away from CJ. A simple, ad hoc analytic model is proposed to treat the case of zero delay time (i.e. source energy released at the shock front) that exhibits qualitative agreement with the computational solutions and may explain why the deviation from CJ increases with decreasing $\unicode[STIX]{x1D6FE}$. When the sources are sufficiently spread out so as to make the energy release of the media nearly continuous, the classic CJ solution is obtained for the average wave speed. Such continuous waves can also be shown to have a time-averaged structure consistent with the classical Zel’dovich–von Neumann–Döring (ZND) structure of a detonation. In the limit of highly discrete sources, temporal averaging of the wave structure shows that the effective sonic surface does not correspond to an equilibrium state. The average state of the flow leaving the wave in this case does eventually reach the equilibrium Hugoniot, but only after the effective sonic surface has been crossed. Thus, the super-CJ waves observed in the limit of highly discretized sources can be understood as weak detonations due to the non-equilibrium state at the effective sonic surface. These results have implications for the validity of the CJ criterion as applied to highly unstable detonations in gases and heterogeneous detonations in condensed phase and multiphase media.

A Semi-analytic Criterion for the Spontaneous Initiation of Carbon Detonations in White Dwarfs

Abstract Despite over 40 years of active research, the nature of the white dwarf progenitors of SNe Ia remains unclear. However, in the last decade, various progenitor scenarios have highlighted the need for detonations to be the primary mechanism by which these white dwarfs are consumed, but it is unclear how these detonations are triggered. In this paper we study how detonations are spontaneously initiated due to temperature inhomogeneities, e.g., hotspots, in burning nuclear fuel in a simplified physical scenario. Following the earlier work by Zel’Dovich, we describe the physics of detonation initiation in terms of the comparison between the spontaneous wave speed and the Chapman–Jouguet speed. We develop an analytic expression for the spontaneous wave speed and utilize it to determine a semi-analytic criterion for the minimum size of a hotspot with a linear temperature gradient between a peak and base temperature for which detonations in burning carbon–oxygen material can occur. Our results suggest that spontaneous detonations may easily form under a diverse range of conditions, likely allowing a number of progenitor scenarios to initiate detonations that burn up the star.

LATERALLY PROPAGATING DETONATIONS IN THIN HELIUM LAYERS ON ACCRETING WHITE DWARFS

Theoretical work has shown that intermediate mass (0.01Msun<M_He<0.1Msun) Helium shells will unstably ignite on the accreting white dwarf (WD) in an AM CVn binary. For more massive (M>0.8Msun) WDs, these helium shells can be dense enough (5x10^5 g/cc) that the convectively burning region runs away on a timescale comparable to the sound travel time across the shell; raising the possibility for an explosive outcome. The nature of the explosion (i.e. deflagration or detonation) remains ambiguous. In the case of detonation, this causes a laterally propagating front whose properties in these geometrically thin and low density shells we begin to study here. Our calculations show that the radial expansion time of <0.1 s leads to incomplete helium burning, in agreement with recent work by Sim and collaborators, but that the nuclear energy released is still adequate to realize a self-sustaining detonation propagating laterally at slower than the Chapman-Jouguet speed. Our simulations resolve the subsonic region behind the front and are consistent with a direct computation of the reaction structure from the shock strength. The ashes are typically He rich, and consist of predominantly Ti-44, Cr-48, along with a small amount of Fe-52, with very little Ni-56 and with significant Ca-40 in carbon-enriched layers. If this helium detonation results in a Type Ia Supernova, its spectral signatures would appear for the first few days after explosion. (abridged)

Modelling phase transition in metastable liquids: application to cavitating and flashing flows

A hyperbolic two-phase flow model involving five partial differential equations is constructed for liquid–gas interface modelling. The model is able to deal with interfaces of simple contact where normal velocity and pressure are continuous as well as transition fronts where heat and mass transfer occur, involving pressure and velocity jumps. These fronts correspond to extra waves in the system. The model involves two temperatures and entropies but a single pressure and a single velocity. The closure is achieved by two equations of state that reproduce the phase diagram when equilibrium is reached. Relaxation toward equilibrium is achieved by temperature and chemical potential relaxation terms whose kinetics is considered infinitely fast at specific locations only, typically at evaporation fronts. Thus, metastable states are involved for locations far from these fronts. Computational results are compared to the experimental ones. Computed and measured front speeds are of the same order of magnitude and the same tendency of increasing front speed with initial temperature is reported. Moreover, the limit case of evaporation fronts propagating in highly metastable liquids with the Chapman–Jouguet speed is recovered as an expansion wave of the present model in the limit of stiff thermal and chemical relaxation.

Shock initiation of explosives: the idealized condensed-phase model

Many current models of condensed-phase explosives employ reaction rate law models where the form of the rate has a power-law dependence on pressure (i.e. proportional to pn where n is an adjustable parameter). Here, shock-induced ignition is investigated using a simple model of this form. In particular, the solutions are contrasted with those from Arrhenius rate law models as studied previously. A large n asymptotic analysis is first performed, which shows that in this limit the evolution begins with an induction stage, followed by a sequence of pressure runaways, resulting in a forward propagating, decelerating, shockless supersonic reaction wave (a weak detonation). The theory predicts secondary shock and super-detonation formation once the weak detonation reaches the Chapman–Jouguet speed. However, it is found that secondary shock formation does not occur until the weak detonation has reached a point close to the initiating shock, whereas for Arrhenius rate laws the shock forms closer to the piston. Numerical simulations are then conducted for O(1) values of n, and it is shown that the idealized condensed-phase model can qualitatively describe a wide range of experimentally observed behaviours, from growth mainly at the shock, to smooth growth of a pressure pulse behind the shock, to cases where a secondary shock and possibly a super-detonation form. The numerics are used to reveal the different evolutionary mechanisms for each of these cases. However, the evolution is found to be sensitive to n, with the whole range of behaviours covered by varying n from about 3 to 5. The simulations also confirm the predictions of the theory that pressure-dependent rate laws are unable to describe homogeneous explosive scenarios where a super-detonation forms very close to the point of initial runaway.

Flame acceleration and the transition to detonation of stoichiometric ethylene/oxygen in microscale tubes

Flame propagation in capillary tubes with smooth circular cross-sections and diameters of 0.5, 1.0, and 2.0 mm are investigated using high-speed photography. Flames were found to propagate and accelerate to detonation speed in stoichiometric ethylene and oxygen mixtures initially at room temperature in all three tube diameters. Ignition occurs at the midpoint along the length of the tube. We observe for the first time transition to detonation in micro-tubes. Detonation was observed with both spark and hot-wire ignition. Tubes with larger diameters take longer to transition to detonation. In fact, transition distance scales with the diameter in our 1.0 and 2.0 mm cases with spark ignition. Flame structures are observed for various stages of the process. Three types of flame propagation modes were observed in the 0.5 mm tube with spark ignition: (a) acceleration to Chapman–Jouguet (CJ) detonation speed followed by constant CJ wave propagation, (b) acceleration to CJ speed, followed by the detonation wave failure, and (c) flame acceleration to a constant speed below the CJ speed of approximately 1600 m/s. The current detonation mechanism observed in capillary tubes is applicable to predetonators for pulsed detonation, micro propulsion devices, safety issues, and addresses fundamental issues raised by recent theoretical and numerical analyses.

On the structure and accuracy of programmed burn

Accurate computation of the evolution of a (typically curved) detonation front in a complex geometry, and of the state behind it, is a practical problem in the design of devices that use high-energy explosives. Direct numerical simulations are infeasible: accuracy demands high resolution of the smallest scale (the reaction zone), which is typically several orders of magnitude smaller than the device scale. Programmed burn is an engineering alternative that has been shown to produce acceptable results at a fraction of the cost. The underlying algorithm prescribes the trajectory of the detonation front a priori and replaces the actual reaction zone by a mock up that is a few computational cells thick and in which the reaction rate is taken to be a constant. The state of the explosive at the end of the reaction zone is thereby computed at a relatively modest cost, and the bulk of the computational effort is reserved for the region behind the reaction zone wherein the products of detonation perform useful work. The reasons for the remarkable fidelity to which the physical situation is captured by the programmed burn are not well-understood. This investigation, aimed at achieving such an understanding, considers a model problem for a steady, curved detonation propagating down a rate stick. It examines the pseudo-reaction-zone structure of the programmed burn, studies the sensitivity of the state of the reaction products to the choice of the reaction zone length, and compares the results to those for the actual, physical reaction zone. Conclusions are drawn as to the causes behind the success of the programmed-burn algorithm. The analysis is based on the asymptotic limits of small front curvature and small departures from the Chapman–Jouguet speed. Results are presented for ideal as well as nonideal explosives.

Steady Non-ideal Detonations in Cylindrical Sticks of Explosives

Numerical simulations of detonations in cylindrical rate-sticks of highly non-ideal explosives are performed, using a simple model with a weakly pressure-dependent rate law and a pseudo-polytropic equation of state. Some numerical issues with such simulations are investigated, and it is shown that very high resolution (hundreds of points in the reaction zone) are required for highly accurate (converged) solutions. High-resolution simulations are then used to investigate thequalitative dependences of the detonation driving zone structure on the diameter and degree of confinement of the explosive charge. The simulation results are used to show that, given the radius of curvature of the shock at the charge axis, the steady detonation speed and the axial solution are accurately predicted by a quasi-one-dimensional theory, even for cases where the detonation propagates at speeds significantly below the Chapman-Jouguet speed. Given reaction rate and equation of state models, this quasi-one-dimensional theory offers a significant improvement to Wood-Kirkwood theories currently used in industry

Mechanisms of detonation formation due to a temperature gradient

Emergence of a detonation in a homogeneous, exothermically reacting medium can be deemed to occur in two phases. The first phase processes the medium so as to create conditions ripe for the onset of detonation. The actual events leading up to preconditioning may vary from one experiment to the next, but typically, at the end of this stage the medium is hot and in a state of nonuniformity. The second phase consists of the actual formation of the detonation wave via chemico-gasdynamic interactions. This paper considers an idealized medium with simple, rate-sensitive kinetics for which the preconditioned state is modelled as one with an initially prescribed linear gradient of temperature. Accurate and well-resolved numerical computations are carrried out to determine the mode of detonation formation as a function of the size of the initial gradient. For shallow gradients, the result is a decelerating supersonic reaction wave, a weak detonation, whose trajectory is dictated by the initial temperature profile, with only weak intervention from hydrodynamics. If the domain is long enough, or the gradient less shallow, the wave slows down to the Chapman–Jouguet speed and undergoes a swift transition to the ZND structure. For sharp gradients, gasdynamic nonlinearity plays a much stronger role. Now the path to detonation is through an accelerating pulse that runs ahead of the reaction wave and rearranges the induction-time distribution there to one that bears little resemblance to that corresponding to the initial temperature gradient. The pulse amplifies and steepens, transforming itself into a complex consisting of a lead shock, an induction zone, and a following fast deflagration. As the pulse advances, its three constituent entities attain progressively higher levels of mutual coherence, to emerge as a ZND detonation. For initial gradients that are intermediate in size, aspects of both the extreme scenarios appear in the path to detonation. The novel aspect of this study resides in the fact that it is guided by, and its results are compared with, existing asymptotic analyses of detonation evolution.

Shock formation ahead of hot spots

Deflagration to detonation transition requires formation of a shock at least as strong as Chapman-Jouguet. Chemistry ahead of a flame, in the finite zone of reactive mixture behind the precursor shock, or, more generally, behind a source of an unburned mixture, always results in a hot spot appearing ahead of the flame. Diffusion-driven flames propagate at negligible speeds compared with the speed of sound and can thus be approximated as contact surfaces. Downstream, the hot spot triggers a weak detonation that slows down to the Chapman-Jouguet speed and quickly becomes a strong wave. Eventually, this wave reaches the contact surface: chemistry stops and a reflection moves upstream. If the source delivers fresh mixture at a velocity below a specific threshold, then the hot spot does not initially trigger a combustion wave on its upstream side, but only an inert shock. This shock, however, warms up the mixture, resulting in a much shorter ignition time. If ignition behind this second shock occurs before the reflection of the downstream wave on the contact surface reaches the location of this second ignition point, a second hot spot forms, on these much shorter time and length scales. This second hot spot triggers a second identical chain of events, which triggers a third one, and so forth, resulting in an embedded sequence of ignition and shock formation. Each of these appears slightly further downstream, on increasingly shorter scales, triggering the next events in the sequence. Eventually, a sequence of shock mergers results in a strong Chapman-Jouguet wave, completing transition to detonation. The condition for reflection to take longer than ignition yields a transition criterion. The analysis assumes high activation energy and a ratio of specific heat close to unity, and the transition criterion appears as a specific distinguished limit.

One-dimensional nonlinear stability of pathological detonations

In this paper we perform high-resolution one-dimensional time-dependent numerical simulations of detonations for which the underlying steady planar waves are of the pathological type. Pathological detonations are possible when there are endothermic or dissipative effects in the system. We consider a system with two consecutive irreversible reactions A→B→C, with an Arrhenius form of the reaction rates and the second reaction endothermic. The self-sustaining steady planar detonation then travels at the minimum speed, which is faster than the Chapman–Jouguet speed, and has an internal frozen sonic point at which the thermicity vanishes. The flow downstream of this sonic point is supersonic if the detonation is unsupported or subsonic if the detonation is supported, the two cases having very different detonation wave structures. We compare and contrast the long-time nonlinear behaviour of the unsupported and supported pathological detonations. We show that the stability of the supported and unsupported steady waves can be quite different, even near the stability boundary. Indeed, the unsupported detonation can easily fail while the supported wave propagates as a pulsating detonation. We also consider overdriven detonations for the system. We show that, in agreement with a linear stability analysis, the stability of the steady wave is very sensitive to the degree of overdrive near the pathological detonation speed, and that increasing the overdrive can destabilize the wave, in contrast to systems where the self-sustaining wave is the Chapman–Jouguet detonation.

An Adaptive Finite Element Method for Unsteady Convection-Dominated Flows with Stiff Source Terms

We consider nonstationary convection-dominated flows with stiff source terms. As a unified approach to such problems a combined finite element method in space and time with streamline diffusion is examined numerically. It has good shock-capturing features and is implicit and L-stable. Moreover, being a Galerkin method, it admits residual-based weighted a posteriori error estimates of optimal order. We avoid the use of global stability constants by actually solving the dual problem. This in turn leads to efficient and mathematically rigorous mesh refinement strategies where the streamline diffusion parameter is used for optimizing the resulting adaptive scheme. All theoreticalresults are substantiated by numerical test examples. For a physical application, we turn to detonation waves at moderate relaxation times in one space dimension. The finite element method well reproduces the Chapman Jouguet speed of detonations and their wave structure. Our error estimator proves accurate and sensitive to parameters, at least on time intervals of moderate length. The adaptive scheme based on this estimator produces fairly accurate results which appear to be the best available with the present strategy. All ingredients of the numerical scheme can be extended in a natural way to higher space dimensions.