Chapman-Jouguet Velocity Research Articles

Low-frequency two-dimensional linear instability of plane detonation

An analytical dispersion relation describing the linear stability of a plane detonation wave to low-frequency two-dimensional disturbances with arbitrary wavenumbers is derived using a normal mode approach and a combination of high activation energy and Newtonian limit asymptotics, where the ratio of specific heats γ→1. The reaction chemistry is characterized by one-step Arrhenius kinetics. The analysis assumes a large activation energy in the plane steady-state detonation wave and a characteristic linear disturbance wavelength which is longer than the fire-zone thickness. Newtonian limit asymptotics are employed to obtain a complete analytical description of the disturbance behaviour in the induction zone of the detonation wave. The analytical dispersion relation that is derived depends on the activation energy and exhibits favourable agreement with numerical solutions of the full linear stability problem for low-frequency one- and two-dimensional disturbances, even when the activation energy is only moderate. Moreover, the dispersion relation retains vitally important characteristics of the full problem such as the one-dimensional stability of the detonation wave to low-frequency disturbances for decreasing activation energies or increasing overdrives. When two-dimensional oscillatory disturbances are considered, the analytical dispersion relation predicts a monotonic increase in the disturbance growth rate with increasing wavenumber, until a maximum growth rate is reached at a finite wavenumber. Subsequently the growth rate decays with further increases in wavenumber until the detonation becomes stable to the two-dimensional disturbance. In addition, through a new detailed analysis of the behaviour of the perturbations near the fire front, the present analysis is found to be equally valid for detonation waves travelling at the Chapman–Jouguet velocity and for detonation waves which are overdriven. It is found that in contrast to the standard imposition of a radiation or piston condition on acoustic disturbances in the equilibrium zone for overdriven waves, a compatibility condition on the perturbation jump conditions across the fire zone must be satisfied for detonation waves propagating at the Chapman–Jouguet detonation velocity. An insight into the physical mechanisms of the one- and two-dimensional linear instability is also gained, and is found to involve an intricate coupling of acoustic and entropy wave propagation within the detonation wave.

Mechanisms of detonation transmission in layered H2-O2 mixtures

When a plane detonation propagating through an explosive comes into contact with a bounding explosive, different types of diffraction patterns, which may result in the transmission of a detonation into the bounding mixture, are observed. The nature of these diffraction patterns and the mode of detonation transmission depend on the properties of the primary and bounding explosives. An experimental and analytical study of such diffractions, which are fundamental to many explosive applications, has been conducted in a two channel shock tube, using H2-O2 mixtures of different equivalence ratios as the primary and bounding or secondary explosive. The combination of mixtures was varied from rich primary / lean secondary to lean primary / rich secondary since the nature of the diffraction was found to depend on whether the Chapman-Jouguet velocity of the primary mixture,D p, was greater than or less than that of the secondary mixture,D s. Schlieren framing photographs of the different diffraction patterns were obtained and used to measure shock and oblique detonation wave angles and velocities for the different diffraction patterns, and these were compared with the results of a steady-state shock-polar solution of the diffraction problem. Two basic types of diffraction and modes of detonation reinitiation were observed. WhenD p>D s, an oblique shock connecting the primary detonation to an oblique detonation in the secondary mixture was observed. WithD p<D s, two modes of reinitiation were observed. In some cases, ignition occurs behind the Mach reflection of the shock wave, which is transmitted into the secondary mixture when the primary detonation first comes into contact with it, from the walls of the shock tube. In other cases, a detonation is initiated in the secondary mixture when the reflected shock crosses the contact surface behind the incident detonation. These observed modes of Mach stem and contact surface ignition have also been observed in numerical simulations of layered detonation interactions, as has the combined oblique-shock oblique-detonation configuration whenD p>D s. WhenD p>D s, the primary wave acts like a wedge moving into the secondary mixture with velocityD p after steady state has been reached, a configuration which also arises in oblique-detonation ramjets and hypervelocity drivers.

Calculations for steady propagation of a generic ram accelerator configuration

This study describes a methodology and gives analysis to determine the steady propagation speed of a projectile fired into a gaseous mixture of fuel and oxidizer. So that numerical results can be analytically verified in limiting cases, a simple model and geometry are used: the steady supersonic flow of an in viscid, calorically perfect, ideal, reacting gas with high activation energy over an unconfined symmetric double wedge is considered. Typical predicted flow fie Ids consist of a lead inert oblique shock followed by a thick induction zone, thin reaction zone, rarefaction, and trailing oblique shock. The reaction strengthens the lead shock to the extent that the far-field wave angle is that of a Chapman-Jouguet (CJ) oblique detonation. Propagation speeds are found that give rise to shocks of such strength as to induce the reaction zone to be in a region that allows the combustion-induced thrust to balance the wave drag. For a fixed heat release greater than a critical value, two steady propagation speeds are predicted. The solution at the higher Mach number is stable to quasistatic perturbations, whereas the solution at the lower Mach number is unstable. The steady solutions are superdetonative in that the projectile velocities that give rise to force balances are greater than the CJ velocity for a detonation wave aligned normal to the direction of projectile motion. This methodology can be applied to analyze devices that have more complex geometries, such as the ram accelerator or oblique detonation wave engine.

On deviations from ideal Chapman-Jouguet detonation velocity

Experimental data shows that under certain conditions, measured detonation velocities exhibit deviations from theoretical values computed according to the Chapman-Jouguet (CJ) model. The purpose of this paper is to investigate why in some circumstances such deviations occur. A Zeldovitch-von Neuman-Doring model (ZND) taking into account the specific chemical kinetics of the mixture enables us to identify a critical parameter: the effective chemical reaction length. This parameter is mainly related to the chemical composition of the reactive mixture. It is shown that the ratio of this parameter to the dimension of the tube where the velocity measurements are carried out can explain deviations from ideal values. This simple model has been applied to a series of existing experimental data characterized by such deviations. Different initial conditions, namely, (1) mixtures near detonation limits, (2) low initial pressures, and (3) mixtures containing halogenated hydrocarbons, are investigated. All data show that the velocity deficit, Δ D cj , is an incrasing, but not necessarily linear, function of the ratio reaction length/dimension of the tube, d rea / d tube . To support this statement, a detailed one-dimensional numerical computation has been performed, taking into account both nonstationary hydrodynamics and detailed chemical kinetics. These computations end up with a detonation wave traveling at the ideal Chapman-Jouguet velocity. This suggests that the deviation from the CJ values ought to be explained by parameters taking into account the relative size of the surrounding of the detonation wave; d rea / d tube is such a parameter.

Numerical simulations of detonation transmission

The complex, dynamic shock-detonation structure formed by the glancing interaction of a primary detonation with a secondary explosive is studied by using time-dependent two-dimensional simulations and related experiments. The materials considered in the simulations are stoichiometric and lean mixtures of hydrogen and oxygen diluted with argon. Related experiments have used undiluted hydrogen and oxygen as well as other gases. For the conditions simulated: (a) the primary mixture is stoichiometric and the secondary inert; (b) both the primary and secondary mixtures are the same and stoichiometric; (c) the primary mixture is lean and the secondary is stoichiometric; and (d) the primary mixture is stoichiometric and the secondary is lean. In addition, for cases (b) and (d) , comparisons are made between simulations in which the primary mixture is overdriven and when it is a Chapman-Jouguet detonation. For the overdriven stoichiometric primary detonation interacting with the lean mixture, a complex detonation structure forms and quickly asymptotes to the detonation velocity of the primary mixture. For this same case, but when the primary detonation is initially at Chapman-Jouguet velocity, the detonation appears to die but then reignites due to a series of shock reflections and then propagates as a complex structure. The lowest velocity of the complex structure is always greater than the Chapman-Jouguet velocity of the lean mixture and it increases in time, appearing to approach the Chapman-Jouguet velocity of the stoichiometric mixture. The dynamics of this decay and reignition process are described and discussed in detail.

Shock-to-detonation transition: A model problem

The process by which a trigger ignited shockless weak detonation is transformed into a classical Zeldovich–von Neumann–Doering (ZND) detonation as the trigger wave decelerates is considered in detail. This occurs as the trigger speed passes through the Chapman–Jouguet velocity DCJ. The model that is developed describes the birth and subsequent growth of a shock in the weak detonation reaction zone. This shock ultimately sustains the shock-ignited ZND detonation. The time-dependent solution presented here fills one of the many gaps in the understanding of detonation flows; an understanding based almost entirely on steady wave structures.

Morphology of standing oblique detonation waves

Proposals have been made to utilize stabilized oblique detonation waves (ODWs) for the propulsion of hypersonic air-breathing vehicles and hypervelocity mass launchers. There exists hypersonic flight regimes where premixing of fuel and air may be desirable or unavoidable due to finite chemical induction times. Consequently, it is essential to understand under what conditions detonations may occur in order to design supersonic combustors to either avoid or utilize them efficiently. A theoretical analysis is made of supersonic flow of a combustible gas mixture past a wedge or an inclined wall, which shows that, for approach velocities roughly 25% or more greater than the Chapman-Jouguet velocity of the reactant mixture, there exists a usefully wide range of turning angles within which ODWs may be attached or stabilized. For smaller wedge angles, either an incomplete ODW, shock-induced combustion, or no combustion at all may ensue. For larger wedge angles, the wave will detach and form an overdriven normal detonation or normal shock-induced combustion wave immediately upstream of the stagnation point, decaying off axis to either a single oblique Chapman-Jouguet wave, or bifurcating to form an oblique shock followed by a shock-induced deflagration.

Unstable detonations in the near-limit regime in tubes

Unstable detonations in near-limit gaseous mixtures are investigated in five tubes of 152, 97, 74, 49, and 38 mm i.d. respectively and 11 m long, interconnected by 1980 o bends to form a continuous loop. The detonation is initiated in the largest tube by a solid explosive charge and transmits to the smaller tubes through the bends. A short section in which the wall is lined by an acoustic absorbing material is placed at the beginning of each tube immediately after the bend to damp out any transverse perturbations associated with the cellular detonation itself or induced by the propagation through the bend. In this manner, a standard “clean” initial conditionis obtained. A short section of defined wall roughness is also placed immediately after the damping section to introduce some finite transverse perturbations to assist the transition to detonation. Most of the experiments were carried out at 1 atm, with lean mixtures of H 2 , C 2 H 4 and C 3 H 8 in air, some of them with stoichiometric H 2 /O 2 mixtures diluted with argon. The results show that immediately after the damping and rough wall sections of earch tube, a dissociated “shock-reaction zone complex” propagating at approximately half the Chapman-Jouguet velocity of the mixture is obtained. It is found that in general, for a cell size over diameter ratio λ/d≤1, re-transition to detonation occurs rapidly and the detonation formed is stable. However for λ/d>1, the re-transition occurs far downstream and the phenomenon is unstable with large velocity fluctuations resembling those of galloping detonations. Even with standardized initial conditions, a λ/d criterion for detonation limits in a given mixture cannot be obtained. Highly unstable detonations are observed for concentrations up to λ/d=13. Mixtures wit different fuels or different diluents also exhibit different behaviors for the same λ/d ratio indicating that, in the near-limit regime, the detonation phenomenon cannot be characterized by a single parameter. The present results also raise the fundamental question of the existence of detonation limits since it requires a unique definition for a detonation wave itself.

The shock dynamics of stable multidimensional detonation

The paper develops a description for the propagation of an unsupported, unsteady, multidimensional detonation wave for an explosive with a fully resolved reaction zone and a polytropic equation of state. The main features of the detonation are determined once the leading shock surface is known. The principal result is that the detonation velocity in the direction along the normal to the shock is the Chapman-Jouguet velocity plus a correction that is a function of the local total curvature of the shock. A specific example of unsteady propagation is discussed and the stability of a two-dimensional steady solution is examined.

Computation of the critical gas layer thickness by means of the ratio to detonation wave propagation

The numerical modeling method of detonation wave propagation proposed in earlier work for gas volumes of finite thickness was used in this paper with a more adequate description of the chemical kinetics to compute the critical gas layer thickness of the mixtures 2H/sub 2/ + O/sub 2/, 29% H/sub 2/ + 71% air, 23% H/sub 2/ + 77% air, and 4H/sub 2/ + O/sub 2/. To model the detonation wave propagation, two-dimensional gas dynamic equations in the Lagrange description were taken, supplemented by the chemical kinetics equation in Arrhenius form. A formula was derived for computing specific heat liberation. Results are presented of computed detonation wave velocities for different values of the effective specific heat of combustion as well as for values of the Chapman-Jouguet velocity for the mixtures.

Gaseous detonations in porous media

There are many applications with pronounced energy implications where it is desirable to increase the permeability of a porous bed. Such applications include “in situ” coal gasification and shale oil retorting and enhanced oil and gas recovery. A closely related problem is that of the breakup of a frozen mass of coal which was transported in open railroad hopper cars. This study was directed to the feasibility of using a gaseous detonation to effect an increase in permeability. Towards that end, a heavy walled pipe (1.829 m long, 0.124 m I.D., and 0.168 m O.D.) was filled with solid spheres. In separate experiments, steel spheres of 19.05 mm and 38.1 mm and ceramic spheres of 38.1 mm were used. The gaseous combustible mixtures tested included hydrogen, methane, and propane all with oxygen as the oxidizer. A range of initial pressures and mixture equivalence ratios were tested. In each case the variation of wave velocity and pressure along the tube was determined. The spheres caused the detonation velocity to be lower than the theoretical Chapman-Jouguet velocity which would be expected in an open tube of that size. Increase of the initial pressure, diameter of the spheres, and equivalence ratio from lean towards stoichiometric resulted in an increase in detonation velocity. The material of the sphere had a slight effect; the steel spheres resulting in higher velocities. An approximate one-dimensional, steady flow, analytical model which included losses in the reaction zone was developed. This expression was used to calculate the critical condition wherein the detonation would no longer propagate through the porous bed, thus predicting the sphere diameter required for quenching. Finally, some experiments were conducted wherein a large container was filled with coal, mixed with snow, and allowed to freeze. Propane-oxygen was blown through the pile and then detonated. Considerable fracturing was experienced.

Detonation of methane-air mixtures

Experimentally, the propagation of detonation waves in methane-air mixtures has been investigated. In a tube having a large length to diameter ratio steady state wave propagation was achieved. The actual propagation velocities were close to the calculated Chapman-Jouguet velocities. The limits of detonation were ascertained to be between 8 and 14.5 percent methane. A minimum initiation energy density of 9.42MJ/m2 was established for stoichiometric mixtures implying the need for rather substantial charges for the initiation of spherically symmetric systems. In all cases the detonation wave exhibited a spinning structure. The motion compensated and laser schlieren photographs, smoked foil traces, and pressure transducer records strongly support the currently accepted model for the structure of such a wave, which results in the creation of extremely high local overpressures.

The ground impulse generated by a plane fuel-air explosion with side relief

Detonations can be initiated in unconfined fuel-oxidizer clouds by blast waves of sufficient energy. The ground impulse generated by such detonations can cause considerable damage. An analytical theory is developed in the present paper, which makes it possible to predict this impulse, and experimental measurements which verify the theory are reported. A plane detonation wave propagating through a one-dimensional fuel-air cloud in contact with the ground and with the inert atmosphere at height h is considered. The wave is followed by an expansion wave which propagates from the inert gases into the combustion products, and an oblique shock is induced in the inert bounding the explosive. In computing the impulse only the region behind the detonation where the expansion is reflected from the ground is considered. The impulse per unit area is found to be the product of p 2( h C ) and a universal impulse function U s ( ξ), where p 2 is the pressure behind the detonation and C is the Chapman-Jouguet velocity, and ξ = x l is the dimensionless distance from the detonation. Experimental values of the pressure and impulse behind a plane wave were obtained using a plastic bag, 4 × 4 × 20 ft long, filled with a stoichiometric MAPP-air mixture. The detonation was initiated using an explosive initiator at one end of the bag. Pressure transducers placed on the ground plane along the center of the bags were used to determine the variation of the pressure and impulse per unit area with time. Theory and experiment were found to be in excellent agreement.

A study of hydrogen azide detonation with heat transfer at the wall

The detonation of gaseous HN 3 was studied in tubes of 1, 2, 5 and 10 mm i.d. in 1–25 Torr pressure range. Detonation velocities were measured by different techniques and wall heat fluxes were deduced from the temperature recordings given by thin-film platinum resistance gauges. The propagation and stability of the wave were studied down to the critical conditions of detonation quenching. For the lowest pressures and smallest diameters, the detonation velocity D ф is about 30% lower than D ∞, the wave velocity independant of wall effect. D ∞ is some percents higher than D CJ , the Chapman-Jouguet velocity. The velocity deficit D ∞ − D ф is discussed in relation to the classical Fay's theory. The features of the temperature-time curves are analyzed in the whole range of pressure and diameter and especially near detonation limits. The heat transfer rates behind the detonation wave are computed from wall temperature recordings and compared to theoretical predictions provided by Mirels' theory. For the largest diameter and the highest pressures, experimental and theoretical values are in good agreement if a laminar regime in the boundary layer is assumed. In the narrowest tubes, near limit pressures, the temperature gradients behind the detonation wave are too large to use the ZND model and the Mirels' theory.

Do Weak Detonation Waves Exist?

The interaction of a with a boundary gas results in a lateral shock moving into the boundary. Certain explosive-boundary gas combinations produce a detached shock with a velocity about one-half of the Chapman-Jouguet velocity. The existence of this detonation wave gave rise to speculation that a weak might exist. To investigate this possibility, a series of experiments were conducted using explosive-boundary gas combinations that formed a detached shock, thus a sub-ChapmanJouguet wave. Results using H2-O2 mixtures bounded by H2 or He showed a traveling at a velocity approximately 83% of the Chapman-Jouguet velocity. Experiments with a CH 4-O2 mixture bounded by He resulted in a traveling at approximately 44% of the Chapman-Jouguet velocity. Although the evidence is not conclusive, it is believed that the sub-Chapman-Jouguet is a quenched decaying to a quasisteady value corresponding to the sound speed in the boundary gas.

A study of spherical detonation in mixtures of methane and oxygen diluted by nitrogen

A study has been made of spherical detonation waves in methane/oxygen/nitrogen mixtures. Using various masses M of solid explosive (Tetryl) as initiator, the detonatability limit in the (x,M) plane at atmospheric pressure has been established for mixtures of composition CH4+202+xN2with 1.4 <or=x<or=5.53. The results cast doubt on a claim to have demonstrated spherical detonation in stoichiometric methane/air mixtures. Measured detonation velocities were all close to calculated Chapman Jouguet velocities.

Blast wave initiation of gaseous and heterogeneous cylindrical detonation waves

Research is described which investigates the initiation, transition and quasi-steady propagation processes associated with blast initiated, cylindrical detonation waves. Experiments were conducted in a specially designed sectored shock tube, wherein cylindrical blast waves were generated by a condensed explosive-electric detonator combination. Both gaseous and heterogeneous fuel-air mixtures were studied, with the former being considered in more detail. A full volumetric range of methyl acetylene, propane, propadiene (MAPP)-air mixtures, at standard conditions, was examined in the gas-phase experiments. The two-phase experiments were restricted to a single fuel-air mixture for a monodisperse spray of 384 μm kerosene drops. For all mixtures tested, a wide range of blast initiation energies was used so that blast waves of different strengths impacted the combustible mixture. It was found that a certain minimum, or critical, blast initiator energy (Ecrit) was required before detonation was experienced. For energies less than Ecrit the blast wave continued to weaken with distance and tended towards an acoustic wave. For energies much greater than Ecrit (say by an order of magnitude) the blast wave decayed directly to a velocity in excess of or at the Chapman-Jouguet velocity for the mixture used. For energy levels below this but still greater than Ecrit, the detonation velocity observed was sub-Chapman-Jouguet, with the velocity being dependent on the actual energy level used. Presumably, given sufficient distance, these waves would eventually accelerate to full Chapman-Jouguet velocity. The twophase experiments exhibited behavior similar to the gaseous. Results are presented which pertain to the wave transition behavior just described, detonation velocity, critical energy, detonation limits, and the transition distance for blast wave to detonation wave behavior.

Detonation initiation in unconfined fuel-air mixtures

The critical energy threshold for detonation initiation in unconfined gaseous fuel-air mixtures has been determined as a function of fuel concentration in air for a hydrocarbon fuel, which consists of methylacetylene, propadiene, propane, and propylene. A similar function was found for 6% by volume n-propyl nitrate and 94% hydrocarbon fuel mixture. Detonation limits of the hydrocarbon-air mixture were found to be 3.3 and 9.8%. The minimum critical energy threshold for high explosive blast wave initiation of the hydrocarbon-air mixtures was found to be 19 kcals at stoichiometric. The n-propyl nitrate/hydrocarbon fuel was detonable over the range 2.8–10.1% with a minimum critical energy threshold of 12 kcals. Propylene oxide-air mixtures were detonable with a minimum critical energy threshold of 4 kcals, and the propylene oxide was detonable in a much broader mixture range than the hydrocarbon or hydrocarbon/ n-propyl nitrate fuels. Detonation velocities were measured to be 1830 ± 43 m/sec in 5.0% hydrocarbon fuel-air mixtures, which is in good agreement with calculated Chapman-Jouguet velocities. Measured detonation pressures of 250 ± 20 psi were lower than the calculated pressures of 270 psi. The experimentally measured radii at which wave velocities from spherical initiating charges decayed to Chapman-Jouguet velocities were approximately the same as the calculated critical radii r∗. Several interesting detonation wave initiation and propagation phenomena were observed on high-speed film and recorded by pressure and impulse measurements.

Heterogeneous detonation supported by fuel fogs or films

Experimental results on the propagation of detonations in decane fogs (∼2μ) and detonations along layers of decane or hexadecane on the walls of tubes filled with gaseous oxygen are presented. The effect of fuel-to-oxygen mixture ratio and tube diameter on the detonation velocity is investigated. Self-sustaining detonation of decane fogs propagates at near Chapman-Jouguet velocities. The thickness of the reaction zone is estimated from the detonation velocity in various size tubes to be 4 times the thickness of gas-phase detonations. The process of transition to detonation in fogs is found to be similar to gaseous detonations. Self-sustaining detonation of a layer of decane or hexadecane on the walls of a tube containing oxygen propagates at 2.2 to 1.0 mm/μsec depending upon the mixture ratio and tube diameter. This corresponds to a Mach number of 6.75 to 3.1, respectively. No rich limit exists; rather, a plateau velocity is reached where an increase in the thickness of the fuel layer does not influence the propagation velocity. Under certain conditions, frozen layers of hexadecane exhibit detonation velocities nearly twice as fast as that observed when the layer is in liquid form.

Plane Steady Navier-Stokes Detonations of Oxyozone

Structure of plane steady detonation waves propagating with Chapman-Jouguet velocity is analysed in consideration of transport phenomena. For rich mixtures of oxyozone, only O 3 + X →O 2 +O+ X and O+O 3 →O 2 +O 2 have to be considered in the analysis as associated elementary reactions, where numerical integrations can be carried out from hot boundary to cold boundary singularities since the hot boundary constitutes a nodal singularity due to the first-order character of reaction. Detonation solutions are found out for arbitrarily lean mixtures. For lean mixtures, on the other hand, the inclusion of the recombination reaction O+O 2 + X →O 3 + X becomes necessary and consequently the reaction order near the hot boundary is raised from the first to the second. The examination of such hot boundary presents non-existence of the integration curve satisfying both boundary conditions.