Jouguet Detonation Research Articles

Second-order Godunov-type scheme for reactive flow calculations on moving meshes

The method of calculating the system of gas dynamics equations coupled with the chemical reaction equation is considered. The flow parameters are updated in whole without splitting the system into a hydrodynamical part and an ODE part. The numerical algorithm is based on the Godunov’s scheme on deforming meshes with some modification to increase the scheme-order in time and space. The variational approach is applied to generate the moving adaptive mesh. At every time step the functional of smoothness, written on the graph of the control function, is minimized. The grid-lines are condensed in the vicinity of the main solution singularities, e.g., precursor shock, fire zones, intensive transverse shocks, and slip lines, which allows resolving a fine structure of the reaction domain. The numerical examples relating to the Chapman–Jouguet detonation and unstable overdriven detonation are considered in both one and two space dimensions.

Detonation Wave Propagation in Rotational Gas Flows

This paper studies the propagation of detonation and shock waves in vortex gas flows, in which the initial pressure, density, and velocity are generally functions of the coordinate — the distance from the symmetry axis. Rotational axisymmetric flow having a transverse velocity component in addition to a nonuniform longitudinal velocity is considered. The possibility of propagation of Chapman–Jouguet detonation waves in rotating flows is analyzed. A necessary conditions for the existence of a Chapman–Jouguet wave is obtained.

Existence of Chapman–Jouguet detonation for a viscous combustion model

In [SIAM J. Appl. Math. 41 (1981) 70–93], Majda proposed a model for the interaction between chemical reactions and compressible fluid dynamics. This model is a low Mach number limit of the one-component reactive Navier–Stokes equations [SIAM J. Appl. Math. 43 (1983) 1086–1118] and was extended to the case where the diffusion coefficient is positive by Larrouturou [Nonlinear Anal. 77 (2001) 405–418]. In this paper, the existence of a one-dimensional Chapman–Jouguet detonation wave, or equivalently a heteroclinic orbit, for the extended model is proven. The proof is based on an application of topological arguments to a system of ordinary differential equations which is obtained from the partial differential equations describing the interaction.

An empirical method for predicting detonation pressure of CHNOFCl explosives

This paper describes a new method for prediction of the Chapman–Jouguet detonation pressures of CHNOFCl explosives using the heat of detonation, Q det, the number of moles of gaseous products of detonation per gram of explosive, α, and the average molecular weight of gaseous products, M. The equation has the form: P CJ=15.88 α( MQ det) 1/2 ρ 0 2−11.17, where P CJ is the Chapman–Jouguet detonation pressure and ρ 0 the loading density. Calculated P CJ by this procedure show good result with respect to measured detonation pressure for any pure or mixture of ideal and some of less ideal CHNOFCl explosives at ρ 0>0.8 g/cm 3.

Direct initiation of detonation with a multi-step reaction scheme

The problem of direct initiation of detonation, where a powerful ignition source drives a blast wave into a gaseous combustible mixture to generate a Chapman–Jouguet (CJ) detonation, is investigated numerically by using a three-step chain-branching chemical kinetic model. The reaction scheme consists sequentially of a chain-initiation and a chain-branching step, followed by a temperature-independent chain termination. The three regimes of direct initiation i.e. subcritical, critical and supercritical, are numerically simulated for planar, cylindrical and spherical geometries using the present three-step chemical kinetic model. It is shown that the use of a more detailed reaction mechanism allows a well-defined value for the critical initiation energy to be determined. The numerical results demonstrate that detonation instability plays an important role in the initiation process. The effect of curvature for cylindrical and spherical geometries has been found to enhance the instability of the detonation wave and thus influence the initiation process. The results of these simulations are also used to provide further verification of some existing theories of direct initiation of detonation. It appears that these theories are satisfactory only for stable detonation waves and start to break down for highly unstable detonations because they are based on simple blast wave theory and do not include a parameter to model the detonation instability. This study suggests that a stability parameter, such as the ratio between the induction and reaction length, should be considered and a more complex chemistry should be included in future development of a more rigorous theory for direct initiation of detonation.

Triggering of Detonation upon Flame Interaction with an Expansion Wave

The transition of a deflagration wave into an abruptly expanding part of a plane channel, where a quasi‐steady supersonic underexpanded jet of an unburned gas is formed, is studied for a propane–oxygen mixture using schlieren pictures. Two explosion‐initiation modes (weak and strong) are registered. In the first case, almost instantaneous onset of the detonation wave occurs when the flame front enters the expanding section; the initial velocity of this wave is approximately 1.5 times the Chapman–Jouguet detonation velocity (DCJ) and then decreases to a value corresponding to self‐sustaining detonation. In the second case, the front velocity gradually increases from ≈ 0.4DCJ to ≈ 1.0DCJ. It is established that the starting pulse triggering the transformation of turbulent combustion to explosion and detonation regimes is generated by interaction of the flame front with expansion waves, which are elements of the structure of the initial section of the jet.

The propagation mechanism of high speed turbulent deflagrations

The propagation regimes of combustion waves in a 30 cm by 30 cm square cross–sectioned tube with an obstacle array of staggered vertical cylindrical rods (with BR=0.41 and BR=0.19) are investigated. Mixtures of hydrogen, ethylene, propane, and methane with air at ambient conditions over a range of equivalence ratios are used. In contrast to the previous results obtained in circular cross–sectioned tubes, it is found that only the quasi–detonation regime and the slow turbulent deflagration regimes are observed for ethylene–air and for propane–air. The transition from the quasi–detonation regime to the slow turbulent deflagration regime occurs at $D/\lambda\simeq1$ (where D is the tube “diameter” and $\lambda$ is the detonation cell size). When $D/\lambda\gg1$ , the quasi–detonation velocities that are observed are similar to those in unobstructed smooth tubes. For hydrogen–air mixtures, it is found that there is a gradual transition from the quasi–detonation regime to the high speed turbulent deflagration regime. The high speed turbulent deflagration regime is also observed for methane–air mixtures near stoichiometric composition. This regime was previously interpreted as the “choking” regime in circular tubes with orifice plate obstacles. Presently, it is proposed that the propagation mechanism of these high speed turbulent deflagrations is similar to that of Chapman–Jouguet detonations and quasi-detonations. As well, it is observed that there exists unstable flame propagation at the lean limit where $D/\lambda\simeq1$ . The local velocity fluctuates significantly about an averaged velocity for hydrogen–air, ethylene–air, and propane–air mixtures. Unstable flame propagation is also observed for the entire range of high speed turbulent deflagrations in methane–air mixtures. It is proposed that these fluctuations are due to quenching of the combustion front due to turbulent mixing. Quenched pockets of unburned reactants are swept downstream, and the subsequent explosion serves to overdrive the combustion front. The present study indicates that the dependence on the propagation mechanisms on obstacle geometry can be exploited to elucidate the different complex mechanisms of supersonic combustion waves.

Sonic Hyperbolic Phase Transitions and Chapman-Jouguet Detonations

We prove that the Cauchy problem for an n×n system of strictly hyperbolic conservation laws in one space dimension admits a weak global solution also in presence of sonic phase boundaries. Applications to Chapman–Jouguet detonations, liquid–vapor transitions and elastodynamics are considered.

Sensitization of hydrocarbon-oxygen mixtures to detonation via cool-flame oxidation

The effect of cool flame partial oxidation on the detonation sensitivity of hydrocarbons was experimentally investigated. Sensitivity to detonation was quantified by measuring the detonation cell-size using the smoked-foil technique. A rich pentane oxygen mixture was preheated in a pebble bed before filling a heated glass detonation tube to sub-atmospheric pressure. Cool flame reaction, monitored by a thin K-type thermocouple, occurred in the detonation tube after a known time interval as determined by the tube temperature. The mixture was ignited by a weak spark and onset of detonation was monitored using a streak camera. A smoked foil was inserted in the far end of the tube (opposite to ignition) to permit the measurement of the cell size of a well-developed detonation. The results show that the cell pattern becomes very regular at high temperature but the average cell size practically does not change. However, when the mixture was detonated while undergoing the cool flame reaction, a significant reduction of the cell-size was obtained (as large as 50%). The sensitizing effect was found to occur only in the initial stage of the cool flame reaction. When the mixture was ignited a few hundreds of milliseconds after the beginning of the coolmore » flame, the mixture was desensitized and the cell size increased. The explanation of the sensitizing effect of the cool flame reaction was investigated by using a chemical kinetic model to simulate the cool flame reaction and identify the chemical species that may be responsible for the observed results. By taking snapshots of these chemical concentrations during the simulated cool flame, these species were used as reactants in a zero-dimensional code to compute the induction kinetics for a Chapman Jouguet detonation. The numerically computed induction times closely follow the experimentally observed cell sizes and confirm that the sensitizing effect of the cool flame reaction may be attributed to the presence of free radicals and peroxides associated with the beginning of the cool flame process. However, these radicals are consumed as the cool flame reaction proceeds and the mixture becomes insensitive again.« less

Chapman-Jouguet Oblique Detonation Structure Around Hypersonic Projectiles

How to generate a steady-state detonation around a hypersonic projectile in stoichiometric hydrogen-oxygen premixed gases is studied. The speed of the hypersonic projectiles was beyond the Chapman ‐Jouguet(C‐J) detonation speed. The e owe eld around the projectile was visualized by using a gate intensie ed charge-coupled device camera(single-frame schlieren picturesand OHradical self-emissionimages ). Threeparameters are varied:1 ) the projectile e ight length from diaphragm rupture location, 2 )the initial pressure of mixture below/above the critical pressureforsteady detonation initiation around a hypersonicprojectile, and3 )theprojectilespeed.Atthepressure condition below the criteria, the detonation structure is composed of three different shock and detonation waves, which appear just after a diaphragm rupture and evolve in time: an overdriven bow detonation wave, a strong detonation wave, and a diffracted shock wave. Their propagation after diaphragm rupture is investigated, and it is found that the C ‐J detonation wave moved away from the projectile and only a reactive bow shock wave remained around projectile far away from the diaphragm. At the pressure condition above the criteria, a steady oblique detonation wave was generated around the projectile as soon as the projectile broke the diaphragm. In this steady-state detonation-wave case, with respect to the e ow Mach number behind the wave front, the whole detonation wavewas divided into fourparts: 1 )strong overdriven detonation wave, 2 ) weak overdriven detonation wave, 3) quasi-C‐J detonation wave, and 4 ) C‐J detonation wave. It has been found that a rarefaction wave is generated at the projectile shoulder and that curvature of the wave has a signie cant effect on the structure of the detonation wave.

In-vessel hydrogen deflagration and detonation in ITER-FEAT

Hydrogen–air mixtures in the vacuum vessel of international thermonuclear experimental reactors (ITER) may be produced by accidents. The basic safety approach is to limit the amount of hydrogen to keep concentrations below flammability limit. For ITER, 5-kg hydrogen uniformly mixed in the vacuum vessel with air at 1.0 bar would lead to concentrations below the flammability limit of 4 vol.%. However, in the case of air ingress into a hydrogen-containing vessel, the hydrogen concentration unavoidably passes through the flammability and detonability range. Also the local formation of burnable clouds is of concern. The problem is to determine the amount of hydrogen allowed to deflagrate or detonate in the vessel without impairing the radioactive confinement function of the vessel and its secondary enclosures. First, the vacuum vessel has to withstand the static load of a deflagration, which pressurizes the vessel for a substantial time period. The final adiabatic pressure after detonation is practically equal to the deflagration pressure because there is only a small difference in burn completeness. Second, in case of a detonation, the vessel has to withstand the dynamic load of the high peak pressure of very short duration during reflection of the detonation wave at the vessel wall. To determine the static load, the maximum deflagration pressures during air ingress transients are evaluated for completely mixed gases and local clouds of flammable mixtures. To characterize the dynamic load, a model for the momentum transferred to the wall by a reflected detonation wave is developed. It results in the simple rule, that the momentum of the wave is at maximum 13% of the total theoretical momentum defined by the total mass of gas in the vessel moving with the Chapman–Jouguet (C–J) detonation front velocity [A.H. Shapiro, The Dynamics and Thermodynamics of Compressible Fluid Flow, vol. 1, Ronald Press, 1953, p. 210]. The momentum transferred to the wall is about twice the momentum of the impacting wave due to reflection on the wall. The results for the ITER limit of 5 kg show a maximum deflagration pressure below the 2 bar design limit of the vessel and a detonation momentum, which is within comparable design loads of the vessel and its support structure, such as for the dropping of a blanket module from top of the vessel to the bottom.

A nonlinear evolution equation for pulsating Chapman–Jouguet detonations with chain-branching kinetics

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

Two-dimensional numerical simulations of idealized detonations

In this paper we use a second–order Godunov scheme to perform one–dimensional time–dependent numerical simulations of an idealized Chapman–Jouguet detonation having an Arrhenius form of reaction rate. The evolution of the longitudinal instability is explored for varying activation temperatures and compared to predictions of a linear stability analysis of the steady detonation. We show that, for large enough activation temperature, the detonation propagates in a series of failures followed by reignition, which can lead to the formation of many large pockets of partly burnt fuel. These results are in disagreement with the previous results of He and Lee, although we find that we can reproduce their results when too coarse a numerical grid is used.

Experimental and Predicted Detonation Parameters for Liquid-Phase H2O2/H2O Mixtures

We report the measured failure diameter and a detonation speed for the liquid explosive mixture 90.5/9.5 wt % H2O2/H2O confined in thick seamless 304 stainless steel tubing and fired at ca. 30 °C. The detonation speed datum and knowledge of the diameter effect curves of other liquid explosives are used to estimate the infinite-medium (i.e., planewave) detonation speed of this material. Ambient condition sound speed measurements and the universal liquid Hugoniot form are used to obtain the unreacted Hugoniot of the 90.5/9.5 wt % H2O2/H2O mixture. This Hugoniot and the Rayleigh line obtained from the estimated infinite-medium detonation wave speed are used to predict the von Neumann spike pressure of the material. Estimates of the fully reacted (i.e., products) Hugoniot, the Chapman−Jouguet (CJ) detonation pressure, and CJ detonation speed for other H2O2/H2O mixtures are obtained using the CHEETAH equilibrium thermochemical code. Predictions of the infinite-medium detonation speed and the von Neumann spike ...

Existence of Solutions for Steady Detonation of Gas Suspensions

The existence of solutions of the traveling–wave type is studied for a system of equations that describes a one–dimensional motion of a suspension of evaporating particles in a viscous and heat–conducting chemically reacting gas. Using topological methods, it is shown that solutions corresponding to weak, strong, and Chapman—Jouguet detonation exist under certain restrictions on energy release and mass transfer.

One-dimensional numerical simulations of idealized detonations

In this paper we use a second–order Godunov scheme to perform one–dimensional time–dependent numerical simulations of an idealized Chapman–Jouguet detonation having an Arrhenius form of reaction rate. The evolution of the longitudinal instability is explored for varying activation temperatures and compared to predictions of a linear stability analysis of the steady detonation. We show that, for large enough activation temperature, the detonation propagates in a series of failures followed by reignition, which can lead to the formation of many large pockets of partly burnt fuel. These results are in disagreement with the previous results of He and Lee, although we find that we can reproduce their results when too coarse a numerical grid is used.

Asymptotic analysis of the structure of a steady planardetonation: Review and extension

The structure of a steady planar Chapman–Jouguet detonation, which is supported by a direct first‐order one‐step irreversible exothermic unimolecular reaction, subject to Arrhenius kinetics, is examined. Solutions are studied, by means of a limit‐process‐expansion analysis, valid for Λ, proportional to the ratio of the reaction rate to the flow rate, going to zero, and for β, proportional to the ratio of the activation temperature to the maximum flow temperature, going to infinity, with the product Λβ1/2 going to zero. The results, essentially in agreement with the Zeldovich–von Neumann–Doring model, show that the detonation consists of (1) a three‐region upstream shock‐like zone, wherein convection and diffusion dominate; (2) an exponentially thicker five‐region downstream deflagration‐like zone, wherein convection and reaction dominate; and (3) a transition zone, intermediate to the upstream and downstream zones, wherein convection, diffusion, and reaction are of the same order of magnitude. It is in this transition zone that the ideal Neumann state is most closely approached.

Cellular detonation stability. Part 1. A normal-mode linear analysis

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

Linear stability of idealized detonations

In this paper we describe a new normal‐modes approach to the linear stability problem of an idealized detonation having an Arrhenius form of the reaction rate, with emphasis on Chapman–Jouguet detonations. We determine analytical asymptotic solutions of the ordinary differential equations for the linearized perturbations, which are used as initial conditions for the integration of these equations. A shooting method is employed to solve the eigenvalue problem. We compare with, and extend, the previous results of Lee & Stewart in the case of one‐dimensional perturbations, and show that the growth rates of the unstable spectrum are dominated by a real eigenvalue for large activation temperatures. We also present the dependence of the unstable spectrum on the heat of reaction. Dispersion relations are given for multidimensional perturbations.

On the Critical Conditions for the Initiation of a Detonation in a Nonuniformly Perturbed Reactive Fluid

The critical conditions for the initiation of a strong Zeldovich--von Neumann--Döring (ZND) detonation in a nonuniformly perturbed reactive fluid are derived using a high activation-energy asymptotic analysis. The initial disturbances are taken to have amplitudes of the order of the small inverse activation energy, with a characteristic wavelength of the order of an acoustic scale, and consist of initial temperature, concentration, pressure, velocity, or density perturbations. The detonation initiation mechanism is based on the work of Dold and Kapila, which describes how ignition of a reactive fluid leads to the generation of a supersonic, shockless, weak detonation. A transition from the weak detonation to a ZND detonation occurs if the initial disturbance is sufficiently strong to induce a gradient of thermal ignition times that cause the weak detonation to slow to the Chapman--Jouguet detonation velocity. Underpinning the whole process is the ability to calculate the path of the weak detonation. In Dold and Kapila's theory this has to be done numerically. Here, the path of the weak detonation, and thus the precise location and time of a transition to strong detonation, is derived analytically through a long-wavelength analysis of the induction zone of the explosion by assuming the initial disturbances vary slowly on the characteristic acoustic scale. Comparisons of the analytically derived results with exact numerical solutions of the induction zone evolution, the thermal runaway, and the weak detonation path, arising from initial disturbances which vary explicitly on the acoustic scale, are shown to be excellent. Moreover, this analysis provides a theoretically based understanding of how each type of initial disturbance, be it an entropy disturbance involving initial temperature nonuniformities or an acoustic disturbance involving initial velocity and pressure fluctuations, influence both the location and the time of the initial point of thermal runaway in the fluid, and the subsequent generation and the speed of the weak detonation. In combination with the analysis of Dold and Kapila, the theory contained herein gives a completely analytical description of how a ZND detonation is generated from an initially perturbed reactive fluid. An extension of the analysis to determine the path of a weak detonation arising from an initial nonuniformity in a three-dimensional geometry is also presented. Finally, as a by-product of the present analysis, a rational derivation of Zeldovich's notion of a constant volume spontaneous flame is given; moreover, it results in a significant improvement on Zeldovich's results.