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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.

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.

The process of planar detonation ignition, induced by a constant-velocity piston or equivalently by a shock reflected from a stationary wall, is investigated using high-resolution one-dimensional numerical simulations. The standard one-step model with Arrhenius kinetics, which models thermally sensitive explosives, is employed. Emphasis is on comparing and contrasting the results of the finite activation temperature simulations with high activation temperature asymptotic predictions and previous simulations. During the induction phase, it is shown that the asymptotic results give qualitatively good predictions. However, for parameters representative of gaseous explosives, subsequent to thermal runaway at the piston and the formation of a reaction wave, the high activation temperature asymptotic theory is qualitatively incorrect for moderately high activation temperatures. It is shown that the results are very sensitive to the value of the activation temperature, especially the distance from the piston at which a secondary shock forms and the degree of unsteadiness in the reaction wave which moves away from the piston. The dependence of the ignition evolution on the other parameters (initial shock Mach number, heat of reaction and polytropic index) is also investigated. It is shown that qualitative predictions regarding the dependence of the ignition evolution on each of the parameters can be elucidated from finite activation temperature homogeneous explosion calculations together with the high activation temperature asymptotic shock ignition results. It is found that for sufficiently strong initiating shocks the ignition evolution is qualitatively different from cases studied previously in that no secondary shock forms. For a high polytropic index, corresponding to a simple equation of state model for condensed phase explosives, the results are in much better qualitative agreement with the asymptotic theory.

We examine the structure and oscillatory instability of low Peclet number, non-premixed edge-flames in a fixed rectangular channel, closed at one end, with constant side-wall mass injection, one su...

We have described, for the first time, a fully coupled low Mach number numerical algorithm which can be used to investigate the combustion of composite propellants. The code uses a body fitted grid along the moving interface by means of a mapping technique. Specifically, the surface is assumed to be single valued, thus allowing the use of a mapping function that effectively maps the propagating corrugated surface into a stationary flat surface. The transformed connection conditions are then applied along the stationary flat surface, allowing second-order one-sided derivatives to be used. This has proven to be a very robust and efficient way to treat the surface and the connection conditions. Numerical tests are performed and the scheme is shown to be second-order accurate in the spatial directions as well as in time. Selected results are presented for heterogeneous propellants.

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.

The spatial structure and temporal evolution of a nondiffusive hot spot are examined in a confined, planar enclosure. The situation is modelled by the reactive Euler equations and single-step Arrhenius kinetics with a large (scaled) activation temperature 1/e The initial state of the medium is taken to be one of uniform pressure but with a prescribed thermal gradient of order e across the domain. The induction equations, which govern order e disturbances to the initial state, are analyzed in the limit of small α, where α measures the width of the domain relative to the acoustic length based on a characteristic reaction time at the initial state. The solution displays a sequence of finite-time singularities as the hot spot grows in strength and shrinks in size in a spatially homogeneous pressure environment. Ultimately, in a region of exponentially small size, significant pressure gradients appear as acoustic and reaction times become comparable and the hot-spot structure is found to obey the full, but previously analyzed, Clarke equations.

The steady structures and the dynamics of oscillatory motions of strained, premixed edge-flames are studied in a planar rectangular channel, closed at one end, with side-wall mass injection. The injection velocities are either constant or temperature-flux dependent. The motivation for this study is recent experiments conducted by Berghout et al (2000 Proc. Combust. Inst. 28 911–7) in a rectangular crack of a homogeneous propellant. The strained edge-flame dynamics are obtained by numerical solution of the zero-Mach-number, variable density Navier–Stokes equations, assuming a one-step reaction. Important parameters are found to be the Reynolds number Re based on the injection velocity and surface separation, the Lewis number Le, and the strain-dependent Damkohler number. We show the first theoretical illustration of edge-flame oscillations in a premixed environment, which are found to occur in near-limit conditions, induced, for example, by decreasing the mass injection surface separation distance leading ...

The stability of an overdriven detonation wave for a one–step Arrhenius reaction in an ideal gas is examined in the limits of a finite detonation Mach number, small modified heat release γ−1)β and large activation energy θ, limits that are relevant to the generation of the regular cellular detonation structures observed experimentally in highly diluted mixtures. Here, γ is the specific heat ratio and β is the formation energy. The limiting case, β = 0, corresponds to a finite Mach number reactionless shock, which in an ideal gas is always stable. Nevertheless, for small but finite values of (γ−1)β, we will show that detonation instabilities can arise when perturbations in the reaction rate generated by a small disturbance at the shock front are sufficient in magnitude to balance the corresponding acoustic fluctuations that are also generated. These acoustic disturbances are also those that govern the stability of a reactionless shock (β = 0). Such reaction–rate perturbations are only generated for sufficiently large activation energies, and it is the precise magnitude of the activation energy that leads to detonation instability that we identify here for various choices of β and γ. In particular, subject to the restriction on the modified heat release (γ−1)β ≪ 1, three new weakly exothermic ordered limits are identified that are relevant to detonation instability: (1) θ(γ−1) = O (1) and (γ−1) = O (1); (2) θ(γ−1) = O (1) and (γ−1) = o (1); and (3) θ(γ−1) ≫ 1 and (γ−1) = o (1).

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 both Chapman-Jouguet and overdriven square-wave detonations is investigated in the limit of a large detonation Mach number and a ratio of specific heats close to one.

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.

The one- and two-dimensional linear stability of a plane detonation wave characterized by a one-step Arrhenius chemical reaction is studied for large activation energies using a normal mode analysis based on the approach of Lee and Stewart [J. Fluid Mech., 216 (1990), p. 103]. It is shown that for one-dimensional disturbances, a low-frequency oscillatory mode present for moderate activation energies bifurcates into a slowly evolving nonoscillatory mode and a faster-evolving nonoscillatory mode as the activation energy is increased. It is also shown that for large activation energies, the stability spectrum consists of a large number of unstable one-dimensional modes, as predicted by the asymptotic analysis of Buckmaster and Neves [Phys. Fluids, 31 (1988), p. 3571], possessing a maximum growth rate at very high frequencies. For nonplanar disturbances, it is found that as the wavenumber increases, the two nonoscillatory modes present for zero wavenumber collapse into a single oscillatory unstable mode befor...

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.

As the latest version of the fast-tube Detonation Shock Dynamics (DSD) solver is linked with the Los Alamos Lagrangian hydrocode, verification problems from a 2006 DSD report (LA-14277 [1]) have been duplicated with some of the verification criteria changed to more quantitative ones. The observed error convergence is as good as or better than reported in [1], quite possibly due to the careful treatment of floating point numbers to ensure that their precision level is maintained throughout the code. This report duplicates the three sample verification problems in LA-14277 [1] using the Los Alamos ASC Lagrangian hydrocode (FLAG), official release of 3.2 Alpha6 with a few modifications. This version of FLAG is linked with the latest fast-tube Detonation Shock Dynamics (DSD) version beta 2 solver released in 2011 as part of the LanlDSD software product [2]. New verification criteria are used for the arcwave problem where two specific locations are chosen for burn arrival time comparison. For this report FLAG's internal driver code prepares the distance function ({Psi}) and material ID fields from its hydro setup, instead of the stand-alone driver that is being utilized by the other LANL hydrocodes currently interfaced to LanlDSD. As it is implemented in versionmore » 3.2 Alpha6, the {Psi} and material ID fields and other parameters are passed from FLAG to the DSD solver directly, and the burn table is directly passed back to FLAG as part of the calling arguments. The burn-front arrival time 'exact' solutions, mentioned in the sequel for the rate-stick and 'arc-wave' problems, are computed using a pair of special-purpose Fortran codes provided by Aslam [3]. In each case an ansatz for the form of the solution is made in which the radius from the detonator center point is used as the independent space coordinate. This leads to a simplified, problem-specific, 1D form of the governing equation. This equation is solved using 2nd-order spatial differencing and the forward Euler method on a very fine temporal and geometric mesh. The boundary conditions are handled exactly at the correct location, with second order accuracy. Care has been taken to ensure that this solution is fully converged. Most other technical details are omitted here as they are comprehensively discussed in [1].« less

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.

Spatially homogeneous thermal explosions governed by a three-step chain-branching kinetic model are described in the asymptotic limit of large activation energy for a range of chain-branching cross-over temperatures. The model consists of a sequence of chain-initiation, chain-branching and chain-termination steps. Temperature-sensitive Arrhenius kinetics is employed for the initiation and branching steps, while the termination step has a temperature-independent rate. Marked distinctions in structure arise as the magnitude of the chain-branching cross-over temperature is increased relative to the initial system temperature. Attention is focused on the regime where the chain-branching cross-over temperature is close to the initial system temperature. Results are also obtained for the fast and slow explosion limits where the two temperatures are not close.

This article outlines a number of basic notions concerning the peculiarities of detonation initiation in mixtures described by onedimensional reactive Euler equations having a thermally-sensitive rate of chemical reaction. It also mentions some strange properties of such equations and touches on a few matters remaining unresolved.

In this article, a theoretical picture is presented of a sequence of events that ensue when a homogeneous reactive medium is subjected to conditions that make it detonate. For example, it may be struck by a strong shock wave or it may be subjected to a suddenly increased (initial) temperature. It is assumed that the chemistry is thermally sensitive and that the temperatures involved are high enough to generate a significant degree of reactivity Typically one finds that a form of reaction-wave is generated. If initial temperatures are sufficiently widespread (as is automatically the case after shock impingement) this reaction wave travels supersonically with respect to the sound speed at every point of its structure. It therefore assumes the asymptotic form of a quasi-steady ‘weak’ detonation—involving no shock wave in its local structure. Only as this wave slows down sufficiently to produce a sfightly subsonic flow relative to the wave at some point do characteristics begin to travel faster than the wave, quickly giving rise to a shock-wave. As this shock progressively moves through the wave, it transforms the structure from a weak to a ‘strong’ Zel’dovich, Von Neuman, Doring detonation wave.

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.