One-dimensional Detonation Research Articles

On the dynamics and linear stability of one-dimensional steady detonation waves

A detailed analysis of the dynamics and linear stability of a steady one-dimensional detonation wave propagating in a binary reactive system with an Arrhenius chemical kinetics of type is carried out. Starting from the frame of the kinetic theory, the binary reactive mixture is modelled at the mesoscopic scale by the reactive Boltzmann equation (BE), assuming hard sphere cross sections for elastic collisions and step cross sections with activation energy for reactive interactions. The corresponding hydrodynamic limit is based on a second-order non-equilibrium solution of the BE obtained in a previous paper, using the Chapman–Enskog method in a chemical regime for which the reactive interactions are less frequent than the elastic collisions. The resulting hydrodynamic governing equations are the reactive Euler equations, including a rate law which exhibits an explicit dependence on the reaction heat and forward activation energy of the chemical reaction. These equations are used to describe the spatial structure of the steady detonation wave solution and investigate how this structure varies with the reaction heat. The response of the steady solution to one-dimensional disturbances is studied using a normal-mode linear approach which leads to an initial-value problem for the state variable disturbances in the reaction zone. The stability problem is treated numerically, using an iterative shooting technique to determine the unstable modes. The analysis developed here emphasizes the influence of the chemical reaction heat and activation energy on the linear stability spectra.

Detonation characteristics of diluted liquid explosives: Mixtures of tetranitromethane with methanol

Detonation in mixtures of nitromethane with methanol as an inert (nonexplosive) diluent is studied. Ignition experiments with mixtures in steel tubes of various diameters provided information on the effect of the degree of dilution on detonability. Mass velocity profiles with a chemical spike characteristic of detonation waves were recorded at the unsteady detonation front in all mixtures studied. This made it possible to distinguish the Chapman-Jouguet state and obtain a fairly complete set of detonation parameters. The dependence of the pressure in the detonation products on the methanol concentration is determined, which is required, in particular, to find the true (absolute) limit of detonation propagation for the concentration of diluted liquid explosives using the method proposed and validated by A. N. Dremin. Some results were found to be inconsistent with one-dimensional detonation theory.

The effect of diffusion on the dynamics of unsteady detonations

AbstractThe dynamics of a one-dimensional detonation predicted by a one-step irreversible Arrhenius kinetic model are investigated in the presence of mass, momentum and energy diffusion. A study is performed in which the activation energy is varied and the length scales of diffusion and reaction are held constant. As the activation energy is increased, the system goes through a series of period-doubling events and eventually undergoes a transition to chaos. The rate at which these bifurcation points converge is calculated and shown to be in agreement with the Feigenbaum constant. Within the chaotic regime, there exist regions in which there are limit cycles consisting of a small number of oscillatory modes. When an appropriately fine grid is used to capture mass, momentum and energy diffusion, predictions are independent of the differencing scheme. Diffusion affects the behaviour of the system by delaying the onset of instability and strongly influencing the dynamics in the unstable regime. The use of the reactive Euler equations to predict detonation dynamics in the unstable and marginally stable regimes is called into question as the selected reactive and diffusive length scales are representative of real physical systems; reactive Navier–Stokes is a more appropriate model in such regimes.

Gas-phase detonation propagation in mixture composition gradients

The propagation of detonations through several fuel-air mixtures with spatially varying fuel concentrations is examined numerically. The detonations propagate through two-dimensional channels, inside of which the gradient of mixture composition is oriented normal to the direction of propagation. The simulations are performed using a two-component, single-step reaction model calibrated so that one-dimensional detonation properties of model low- and high-activation-energy mixtures are similar to those observed in a typical hydrocarbon-air mixture. In the low-activation-energy mixture, the reaction zone structure is complex, consisting of curved fuel-lean and fuel-rich detonations near the line of stoichiometry that transition to decoupled shocks and turbulent deflagrations near the channel walls where the mixture is extremely fuel-lean or fuel-rich. Reactants that are not consumed by the leading detonation combine downstream and burn in a diffusion flame. Detonation cells produced by the unstable reaction front vary in size across the channel, growing larger away from the line of stoichiometry. As the size of the channel decreases relative to the size of a detonation cell, the effect of the mixture composition gradient is lessened and cells of similar sizes form. In the high-activation-energy mixture, detonations propagate more slowly as the magnitude of the mixture composition gradient is increased and can be quenched in a large enough gradient.

A streamline approach to two-dimensional steady non-ideal detonation: the straight streamline approximation

Detonations in non-ideal explosives tend to propagate significantly below the nominal one-dimensional detonation speed. In these cases, multi-dimensional effects within the reaction zone are important. A streamline-based approach to steady-state non-ideal detonation theory is developed. It is shown in this study that, given the streamline shapes, the two-dimensional problem reduces to an ordinary differential equation eigenvalue problem along each streamline, the solution of which determines the local shock shape that, in turn, leads to the solution of the detonation speed as a function of charge diameter. A simple approximation of straight but diverging streamlines is considered. The results of the approximate theory are compared with those of high-resolution direct numerical simulations of the problem. It is shown that the straight streamline approximation is remarkably predictive of highly non-ideal explosive diameter effects. It is even predictive of failure diameters. Given this predictive capability, one potential use of the method is in the determination of rate law parameters by fitting to data from unconfined rate stick experiments. This is illustrated by using data for ammonium nitrate fuel oil explosives.

Instability Criterion of One-Dimensional Detonation Wave with Three-Step Chain Branching Reaction Model

One-dimensional detonation waves are simulated with the three-step chain branching reaction model, and the instability criterion is studied. The ratio of the induction zone length and the reaction zone length may be used to decide the instability, and the detonation becomes unstable with the high ratio. However, the ratio is not invariable with different heat release values. The critical ratio, corresponding to the transition from the stable detonation to the unstable detonation, has a negative correlation with the heat release. An empirical relation of the Chapman—Jouguet Mach number and the length ratio is proposed as the instability criterion.

High Order Weighted Essentially Non-Oscillation Schemes for One-Dimensional Detonation Wave Simulations

High Order Weighted Essentially Non-Oscillation Schemes for One-Dimensional Detonation Wave Simulations

Applying nonlinear dynamic theory to one-dimensional pulsating detonations

The dynamical behaviour of one-dimensional pulsating detonations was investigated in detail, with the aid of nonlinear theory tools such as phase plots and correlation dimension. The period-doubling cascade, as routes to deterministic chaos, is depicted through the transformations of the shapes of the attractors. Using a correlation dimension method, the dimension of the attractors is determined and we show that the chaos within a one-dimensional pulsating detonation is deterministic.

Characteristics analysis of the one-dimensional pulsating dynamics of chain-branching detonations

The nonlinear pulsating mechanism of one-dimensional detonations was studied numerically using a simple two-step chain-branching model with separate induction and reaction zones. Numerical simulations were performed for a wide range of parameters, which revealed four distinct pulsating regimes classified according to the mechanism controlling the frequency of the pulsations. The dynamics of these regimes were clarified by reconstructing the characteristics, representing the trajectory of pressure waves and particle paths. The high and low frequency regimes of oscillation previously observed in one-step and realistic chemistry simulations were clarified. Under some parameter range, simultaneous low and high frequency pulsations were observed. A novel regime was also found with a pulsation period smaller than the induction time. It involves coupling between the acoustic and the reactivity disturbances propagating, respectively, along the C− and C0 characteristics. These are generated at successive lead shock pulsations and arrive at the reaction zone simultaneously. For all regimes, the dominating mechanism of the pulsating instability was found to be in good qualitative agreement with Toong’s phenomenological model based on the wave dynamics in a square wave reaction zone structure.

Detonation wave initiated by explosive condensation of supersaturated carbon vapor

An experimental study of the influence of condensation of supersaturated carbon vapor formed behind reflected shock waves on the process of propagation of a shock wave and formation of a detonation wave of condensation is carried out. Highly supersaturated carbon vapor was formed from thermal decay of unstable carbon suboxide C3O2 → C + 2CO behind a shock wave in mixtures containing 10–30% C3O2 in Ar. This reaction was followed by fast growth of condensed carbon particles, accompanied by heat release. Experiments have shown a considerable temperature and pressure increase in the narrow zone behind the wave front, resulting in shock wave amplification and transition to a detonation-like regime. An analysis of the kinetics and heat release in the given conditions and calculations based upon one-dimensional detonation theory have shown that in a mixture of 10% C3O2 + Ar, insufficient heat release resulted in a regime of “overdriven detonation”. In a mixture of 20% C3O2 + Ar a very good coincidence of measured values of pressure and wave velocity with calculated Chapman–Jouguet parameters is observed. In a 30% C3O2 + Ar mixture, an excess heat release caused a slow down of the effective condensation rate and a regime of “underdriven detonation” is observed.

Effect of Chemically Inert Particles on Thermodynamic Characteristics and Detonation of a Combustible Gas

An approximate model of chemical equilibrium in heterogeneous mixtures of a combustible gas with chemically inert solid or liquid particles has been suggested. It includes explicit algebraic formulas for the calculation of the molar mass of the gas, internal energy, and heat capacities of gas-particles mixture, and ordinary differential equations for the description of isentropic compression and adiabatic index of the system. The model can be also useful for the rough estimations of thermodynamic parameters of gaseous mixtures with particles of soot. As an example of a possible application of the suggested model of chemical equilibrium, parameters of stationary one-dimensional detonation wave in gas-particles mixtures are calculated. The algorithm of estimation of detonation cell size in such heterogeneous mixtures is presented. Detonation wave parameters and cell size in the stoichiometric hydrogen-oxygen mixture with particles of W, Al2O3, and SiO2 have been calculated. The results of the calculations of detonation parameters and cell sizes are used for analysis of the method of multi-front detonation wave suppression by particles injection before the leading shock front of the wave. The minimal total mass of the particles and characteristic size of the cloud, which are necessary for detonation suppression, have been calculated. It is shown that such suppression is more effective if the particles have high heat capacity, low melting point, and high heat of melting. Among the particles under consideration, the particles of Al2O3 are the best for the detonation wave suppression.

Modelling detonation waves in condensed energetic materials: multiphase CJ conditions and multidimensional computations

A hyperbolic multiphase flow model with a single pressure and a single velocity but several temperatures is proposed to deal with the detonation dynamics of condensed energetic materials. Temperature non-equilibrium effects are mandatory in order to deal with wave propagation (shocks, detonations) in heterogeneous mixtures. The model is obtained as the asymptotic limit of a total non-equilibrium multiphase flow model in the limit of stiff mechanical relaxation only (Kapila et al. in Phys Fluids 13:3002–3024, 2001). Special attention is given to mass transfer modelling, that is obtained on the basis of entropy production analysis in each phase and in the system (Saurel et al. in J Fluid Mech 607:313–350, 2008). With the help of the shock relations given in Saurel et al. (Shock Waves 16:209–232, 2007) the model is closed and provides a generalized ZND formulation for condensed energetic materials. In particular, generalized CJ conditions are obtained. They are based on a balance between the chemical reaction energy release and internal heat exchanges among phases. Moreover, the sound speed that appears at sonic surface corresponds to the one of Wood (A textbook of sound, G. Bell and Sons LTD, London, 1930) that presents a non-monotonic behaviour versus volume fraction. Therefore, non-conventional reaction zone structure is observed. When heat exchanges are absent, the conventional ZND model with conventional CJ conditions is recovered. When heat exchanges are involved interesting features are observed. The flow behaviour presents similarities with non ideal detonations (Wood and Kirkwood in J Chem Phys 22:1920–1924, 1950) and pathological detonations (Von Neuman in Theory of detonation waves, 1942; Guenoche et al. in AIAA Prog Astron Aeronaut 75: 387–407, 1981). It also present non-conventional behaviour with detonation velocity eventually greater than the CJ one. Multidimensional resolution of the corresponding model is then addressed. This poses serious difficulties related to the presence of material interfaces and shock propagation in multiphase mixtures. The first issue is solved by an extension of the method derived in Saurel et al. (J Comput Phys 228(5):1678–1712, 2009) in the presence of heat and mass transfers. The second issue poses the difficult mathematical question of numerical approximation of non-conservative systems in the presence of shocks associated to the physical question of energy partition among phases for a multiphase shock. A novel approach is used, based on extra evolution equations used to retain the information of the material initial state. This method insures convergence in the post-shock state. Thanks to these various theoretical and numerical ingredients, one-dimensional and multidimensional unsteady detonation waves computations are done, eventually in the presence of material interfaces. Convergence of the numerical hyperbolic solver against ZND multiphase solution is reached. Material interfaces, shocks, detonations are solved with a unified formulation where the same equations are solved everywhere with the same numerical scheme.

Propagation of detonation waves along a tubular bubble cluster in liquid

The dynamics are studied of axisymmetric two-dimensional detonation waves in liquid with bubbles, where the bubbles are found in the form of a tubular cluster in the axial region of a pipe and where they are arranged in the form of a ring bubble layer. It is demonstrated that such clusters may serve as a waveguide for transmitting pulsed signals which are detonation solitons. It is found that, because of precompression of a bubble cluster by waves propagating in “pure” liquid, the velocity of detonation soliton may significantly increase compared to the case of one-dimensional bubble detonation.

Effect of chemically inert particles on parameters and suppression of detonation in gases

An algorithm for calculating the parameters of a steady one-dimensional detonation wave in mixtures of a gas with chemically inert particles and estimating the detonation-cell size in such mixtures is proposed. The calculated detonation parameters and cell size in stoichiometric hydrogen-oxygen mixtures with W, Al2O3, and SiO2 particles are used to analyze the method of suppression of multifront gas detonation by injecting chemically inert particles ahead of the leading wave front. The ratio between the channel diameter and the detonation-cell size is used to estimate the limit of heterogeneous detonation in the mixtures considered. The minimum mass of particles and the characteristic cloud size necessary for detonation suppression are calculated. The effect of thermodynamic parameters of particles on the detonation suppression process is analyzed for the first time. Particles with a high specific heat and (if melting occurs) a high phase-transition heat are found to exert the most pronounced effect.

Flame acceleration and the development of detonation in fuel–oxygen mixtures at elevated temperatures and pressures

Experimental measurements of the conditions required for the development of detonation in a 7 mm tube following ignition by a low energy spark are reported. There are then compared to previous experimental propagation limit criterion using theoretical predictions of detonation cell sizes based on a one-dimensional detonation length scale computed using a detailed chemical kinetic scheme. Technical difficulties precluded direct cell size measurements. Ethylene–oxygen and hydrogen–methane–oxygen mixtures were investigated as well as methane-ammonia-oxygen, at initial pressures and temperatures in the ranges 1–7 bar and 293–540 K, respectively. The likelihood of detonation in ethylene-air mixtures in 150 mm and 50 mm pipes at ambient initial conditions is also discussed in relation to published cell width data.The results indicate that whilst detonation cell width predictions do not provide a quantitative measure of the conditions for which detonation may develop in a pipe of given diameter, for prescribed initial conditions, predicted detonation cell size data does provide useful qualitative guidance as to possible hazardous compositions, particularly if preliminary experimental safety testing is thought to be necessary.

Unsteady features on one-dimensional hydrogen-air detonations

The features of one-dimensional unsteady detonations are studied numerically using a hydrogen-air detailed chemical reaction model. A series of simulations are carried out while degree of overdrive, initial pressure, initial temperature, and equivalence ratio are varied. The oscillation modes and mechanisms of the one-dimensional detonations are discussed with reference to shock pressure histories and x-t diagrams of density distributions. As the degree of overdrive is reduced with a stoichiometric mixture of hydrogen-air at P0=0.421atm and T0=293K, a steady state appears, along with a high-frequency mode and a low-frequency mode. The oscillation mechanism of the high-frequency mode is the same as that of the regular regime of unsteady shock-induced combustion observed around a spherical projectile flying at hypersonic velocity in detonable gases. The degree of overdrive threshold between the steady and unsteady region increases monotonically with initial pressure and decreases monotonically with initial temperature. When the equivalence ratio is changed, the threshold has a minimum value around ϕ=1. We focus attention on a nondimensional effective activation energy, which is generally used for linear stability analysis. The oscillation mode depends highly on the nondimensional effective activation energy. The oscillation of the detonation front appears as the nondimensional effective activation energy goes past a threshold value of 5.2. Furthermore, we investigate the failed regime and possible reignition in this regime. In the failed regime, a detonation wave breaks up into a leading shock, a contact discontinuity, and a rarefaction wave. When the shock is weak, reignition time becomes very long. Therefore, the reignition after the failed regime is difficult to reproduce in the restricted computational domain. The reignition process in the failed regime is investigated by means of analysis consisting of integration along the point of intersection between a Rayleigh line for weak leading shock and a partially burnt Hugoniot curve. The reignition time increases dramatically with decreasing temperature behind the shock wave, when the gas condition goes past the second explosion limit. The second explosion limit is one of the characteristics of the hydrogen-air detailed chemical reaction model and does not exist in the one-step chemical reaction model. Lastly, the reignition time obtained by the analysis is compared with that obtained by the simulation results. The simulation results agree well with the analytical results.

Initial-value problem for small disturbances in an idealized one-dimensional detonation

The solution of the initial-value problem for linearized reactive Euler equations is presented as an expansion into modes of discrete and continuous spectra in the case of one-dimensional perturbations. The result provides a convenient tool to predict initial amplitudes of the unstable modes depending on the initial perturbation. Examples of initial perturbations introduced into the quiescent gas and into the reaction zone are discussed. It is shown that the discrete spectrum stemming from the solution of the initial-value problem for an idealized one-reaction detonation is equivalent to the spectrum of the conventional normal-mode approach.

A numerical study of a detonation wave in detonation transmission tubing

In this paper we formulate a one-dimensional detonation wave model for detonation transmission tubing. The reaction is described by a temperature-independent single-step kinetic equation and the resulting equations are a system of hyperbolic conservation laws. The initial boundary value problem is then solved by using the TVD MacCormack method. The effects of heat loss to the tube wall and of the tube expansions have also been considered. The result obtained shows that the heat loss and the tube expansion have a major effect on the pressure prediction.

Simulations of pulsating one-dimensional detonations with true fifth order accuracy

A novel, highly accurate numerical scheme based on shock-fitting coupled with fifth order spatial and temporal discretizations is applied to a classical unsteady detonation problem to generate solutions with unprecedented accuracy. The one-dimensional reactive Euler equations for a calorically perfect mixture of ideal gases whose reaction is described by single-step irreversible Arrhenius kinetics are solved in a series of calculations in which the activation energy is varied. In contrast with nearly all known simulations of this problem, which converge at a rate no greater than first order as the spatial and temporal grid is refined, the present method is shown to converge at a rate consistent with the fifth order accuracy of the spatial and temporal discretization schemes. This high accuracy enables more precise verification of known results and prediction of heretofore unknown phenomena. To five significant figures, the scheme faithfully recovers the stability boundary, growth rates, and wave-numbers predicted by an independent linear stability theory in the stable and weakly unstable regime. As the activation energy is increased, a series of period-doubling events are predicted, and the system undergoes a transition to chaos. Consistent with general theories of non-linear dynamics, the bifurcation points are seen to converge at a rate for which the Feigenbaum constant is 4.66 ± 0.09, in close agreement with the true value of 4.669201…. As activation energy is increased further, domains are identified in which the system undergoes a transition from a chaotic state back to one whose limit cycles are characterized by a small number of non-linear oscillatory modes. This result is consistent with behavior of other non-linear dynamical systems, but not typically considered in detonation dynamics. The period and average detonation velocity are calculated for a variety of asymptotically stable limit cycles. The average velocity for such pulsating detonations is found to be slightly greater than the Chapman–Jouguet velocity.

DDT in fuel–air mixtures at elevated temperatures and pressures

An experimental study was carried out to investigate flame acceleration and deflagration-to-detonation transition (DDT) in fuel–air mixtures at initial temperatures up to 573 K and pressures up to 2 atm. The fuels investigated include hydrogen, ethylene, acetylene and JP-10 aviation fuel. The experiments were performed in a 3.1-m long, 10-cm inner-diameter heated detonation tube equipped with equally spaced orifice plates. Ionization probes were used to measure the flame time-of-arrival from which the average flame velocity versus propagation distance could be obtained. The DDT composition limits and the distance required for the flame to transition to detonation were obtained from this flame velocity data. The correlation developed by Veser et al. (run-up distance to supersonic flames in obstacle-laden tubes. In the proceedings of the 4th International Symposium on Hazards, Prevention and Mitigation of Industrial Explosions, France (2002)) for the flame choking distance proved to work very well for correlating the detonation run-up distance measured in the present study. The only exception was for the hydrogen–air data at elevated initial temperatures which tended to fall outside the scatter of the hydrocarbon mixture data. The DDT limits obtained at room temperature were found to follow the classical d/λ = 1 correlation, where d is the orifice plate diameter and λ is the detonation cell size. Deviations found for the high-temperature data could be attributed to the one-dimensional ZND detonation structure model used to predict the detonation cell size for the DDT limit mixtures. This simple model was used in place of actual experimental data not currently available.