Unstable Detonation Research Articles

High‐Resolution Numerical Simulation and Analysis of Mach ReflectionStructures in Detonation Waves in Low‐Pressure H2–O2–Ar Mixtures: A Summary of Results Obtained with the Adaptive Mesh Refinement Framework AMROC

Numerical simulation can be key to the understanding of the multidimensional nature of transient detonation waves. However, the accurate approximation of realistic detonations is demanding as a wide range of scales needs to be resolved. This paper describes a successful solution strategy that utilizes logically rectangular dynamically adaptive meshes. The hydrodynamic transport scheme and the treatment of the nonequilibrium reaction terms are sketched. A ghost fluid approach is integrated into the method to allow for embedded geometrically complex boundaries. Large‐scale parallel simulations of unstable detonation structures of Chapman‐Jouguet detonations in low‐pressure hydrogen‐oxygen‐argon mixtures demonstrate the efficiency of the described techniques in practice. In particular, computations of regular cellular structures in two and three space dimensions and their development under transient conditions, that is, under diffraction and for propagation through bends are presented. Some of the observed patterns are classified by shock polar analysis, and a diagram of the transition boundaries between possible Mach reflection structures is constructed.

On the Stability of the Detonation Wave Front in the High Explosive Liquid Mixture Tetranitromethane/Nitrobenzene

We performed experimental studies on the stability of the detonation wave front in mixtures of the liquids tetranitromethane (TNM) and nitrobenzene (NB). Tetranitromethane is an oxygen-rich explosive and nitrobenzene was used as a solvent or dilutant. (NB is not classed as an explosive but as an explosive would be oxygen poor and fuel rich.) The primary diagnostic was a laser velocimetry method with high temporal resolution. Data obtained were compared with the detonation parameters of the TNM/NB mixtures. In previous experimental work [1,2] it was shown that the detonation wave front in liquid explosives may be either smooth or rough. Rough detonation fronts have been reported in nitromethane, as well as nitromethane mixed with a solvent. Smooth detonation fronts have been reported in tetranitromethane. Previously, we conducted studies on the structure of the detonation wave front in liquid explosives containing tetranitromethane [3-5]. Smooth, stable fronts were recorded in pure tetranitromethane and in a 46/54 mixture of tetranitromethane and nitromethane. A pulsating, unstable detonation wave front was recorded in a 74/26 mixture of tetranitromethane and nitrobenzene. The goal of the present work is to extend our research on the structure of the detonation wave front in mixtures of tetranitromethane diluted with less energetic nitrobenzene. To this end, the following TNM/NB mixtures were studied: 95/5, 90/10, 85/15, 80/20, 74/26, and 50/50.

Propagation of near-limit gaseous detonations in small diameter tubes

In this study, detonation limits in very small diameter tubes are investigated to further the understanding of the near-limit detonation phenomenon. Three small diameter circular tubes of 1.8, 6.3, and 9.5 mm inner diameters, of 3 m length, were used to permit the near-limit detonations to be observed over long distances of 300 to 1500 tube diameters. Mixtures with high argon dilution (stable) and without dilution (unstable) are used for the experiments. For stable mixtures highly diluted with argon for which instabilities are not important and where failure is due to losses only, the limit obtained experimentally appears well to be in good agreement in comparison to that computed by the quasi-steady ZND theory with flow divergence or curvature term modeling the boundary layer effects. For unstable detonations it is suggested that suppression of the instabilities of the cellular detonation due to boundary conditions is responsible for the failure of the detonation wave. Different near-limit propagation regimes are also observed, including the spinning and galloping mode. Based on the present experimental results, an attempt is made to study an operational criterion for the propagation limits of stable and unstable detonations.

Mode selection in weakly unstable two-dimensional detonations

A formulation of the reactive Euler equations in the shock-attached frame is used to study the two-dimensional instability of weakly unstable detonation through direct numerical simulation. The results are shown to agree with the predictions of linear stability analysis. Comparisons are made with linear perturbation growth rates and oscillation frequencies as a function of transverse disturbance wavelength. The perturbation eigenfunctions predicted by linear stability analysis are directly validated through numerical simulation. Three regimes of unstable behavior – linear, weakly nonlinear, and fully nonlinear – are explored and characterized in terms of the power spectrum of the normal shock velocity for a Chapman–Jouguet detonation with weak heat release.

Professor John H. S. Lee: The Detonation Phenomenon

The book “The Detonation Phenomenon” by Professor John H. S. Lee is intended for engineers and graduate students with backgrounds in thermodynamics and fluid dynamics. It fulfills that mission perfectly. The book is an excellent, thorough review of the basic experimental and theoretical aspects of gas phase detonation. Gas phase detonation has frequently been studied, because its three-dimensional shock wave structure can be photographed, the pressure it produces is not extreme, as opposed to that of a solid or liquid explosive, and the chemical reaction rates that control its propagation can be measured in shock tubes. Professor Lee has assembled a wonderful combination of fantastic photographs from all over the world plus basic theoretical explanations of the various phenomena involved in gas phase detonation. The book has a mechanical engineering slant, emphasizing the structures of shock and detonation waves and their interactions with surrounding materials. The book covers the discovery of detonation with pictures of the scientists who uncovered the basic properties of detonation waves. The first three chapters and part of Chapter 4 contain necessary experimental and theoretical background found in other books, but this information is presented in a very clear concise manner. Each chapter ends with closing remarks, which often contain a list of unsolved problems. The bibliographies at the end of each chapter contain most of the relevant references. The remaining chapters of “The Detonation Phenomena” are unique and brilliantly presented. Chapter 5 is a numerical analysis of unstable detonation that provides an excellent summary of several complex topics. Professor Lee systematically discusses: linear stability analysis; asymptotic modeling; direct computer simulations; and chemical effects on the stability of detonation waves. He ends Chapter 5 with its main message that theory and numerical modeling support that the experimental fact that all detonation waves are unstable and exhibit multi-dimensional wave structures. Chapter 6 is entitled “Unstable Detonations: Experimental Observations.” It begins with the 1926 discovery of “spinning detonation” and presents photographs and explanations of all of the detonation flows discovered thus far. This chapter is by far the best collection of famous photographs from many authors of detonation wave structures. Many cases of multi-headed detonation waves and the triple shock intersection points that control their propagation are illustrated. The effects of various geometries on the “cellular structures” produced by the interactions of these “triple points” are covered more completely than in other books on gaseous detonation. Chapter 6 ends with a basic discussion of how chemical energy release controls detonation wave cell size. Chapter 7 discusses the influence of boundary conditions on gaseous detonation wave propagation. The influences of rough versus smooth walls (turbulent flow) on detonation velocity and cellular structures are important engineering concerns that are covered very well in Chapter 7. An imaginative experiment developed by Professor Lee’s group was to pass a detonation wave over an acoustically absorbing material. When the triple shock intersection points of a detonation waves interacted with this material, the cellular structure of the detonation was destroyed. The detonation wave exited that section of tube as a planar shock wave followed by a deflagration wave. This experiment proved that the cellular structure was essential to gas phase detonation

Some numerical issues on simulation of detonation cell structures

The present study examines several numerical issues on simulation of detonation cell structures. Various stability regimes ranging from weakly to highly unstable detonations are considered. The analysis treats two-dimensional inviscid fluid-dynamics equations and a one-step reaction model. A series of investigations is carried out to identify numerical requirements for high-fidelity simulations of detonation cell structures. Emphasis is placed on the wave-front dynamics and evolution of cellular patterns. The effects of the preexponential factor, grid size, time step, domain length, and exit boundary condition on the cellular structure and cell size are examined systematically. The required numerical grid size is determined and compared with various length scales associated with a steady Zel’dovich-von Neumann-Doring detonation wave. A general rule for the grid-resolution requirement is proposed for the first time: a minimum of 5 grid points should be included in the heat-release zone of the corresponding steady Zel’dovich-von Neumann-Doring detonation wave, in order to achieve an accurate simulation of detonation cell structures.

Two-dimensional cellular structure of a kinetically unstable detonation front

Based on previously published results on the detonation of gaseous and liquid explosives, an explanation is given to the formation of the two-dimensional cellular structure of the detonation front of some gas mixtures undergoing a two-step exothermic transformation at the wave front and suggestions are proposed for the mechanism of development of the two-dimensional cellular structure in the case of detonation transformation of gas mixtures with one-step chemical kinetics.

Experimental study on transmission of an overdriven detonation wave from propane/oxygen to propane/air

Two sets of experiments were performed to achieve a strong overdriven state in a weaker mixture by propagating an overdriven detonation wave via a deflagration-to-detonation transition (DDT) process. First, preliminary experiments with a propane/oxygen mixture were used to evaluate the attenuation of the overdriven detonation wave in the DDT process. Next, experiments were performed wherein a propane/oxygen mixture was separated from a propane/air mixture by a thin diaphragm to observe the transmission of an overdriven detonation wave. Based on the characteristic relations, a simple wave intersection model was used to calculate the state of the transmitted detonation wave. The results showed that a rarefaction effect must be included to ensure that there is no overestimate of the post-transmission wave properties when the incident detonation wave is overdriven. The strength of the incident overdriven detonation wave plays an important role in the wave transmission process. The experimental results showed that a transmitted overdriven detonation wave occurs instantaneously with a strong incident overdriven detonation wave. The near-CJ state of the incident wave leads to a transmitted shock wave, and then the transition to the overdriven detonation wave occurs downstream. The attenuation process for the overdriven detonation wave decaying to a near-CJ state occurs in all tests. After the attenuation process, an unstable detonation wave was observed in most tests. This may be attributed to the increase in the cell width in the attenuation process that exceeds the detonability cell width limit.

Stability of detonations for an idealized condensed-phase model

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 hydrodynamic structure of unstable cellular detonations

The study analyses the cellular reaction zone structure of unstable methane–oxygen detonations, which are characterized by large hydrodynamic fluctuations and unreacted pockets with a fine structure. Complementary series of experiments and numerical simulations are presented, which illustrate the important role of hydrodynamic instabilities and diffusive phenomena in dictating the global reaction rate in detonations. The quantitative comparison between experiment and numerics also permits identification of the current limitations of numerical simulations in capturing these effects. Simulations are also performed for parameters corresponding to weakly unstable cellular detonations, which are used for comparison and validation. The numerical and experimental results are used to guide the formulation of a stochastic steady one-dimensional representation for such detonation waves. The numerically obtained flow fields were Favre-averaged in time and space. The resulting one-dimensional profiles for the mean values and fluctuations reveal the two important length scales, the first being associated with the chemical exothermicity and the second (the ‘hydrodynamic thickness’) with the slower dissipation of the hydrodynamic fluctuations, which govern the location of the average sonic surface. This second length scale is found to be much longer than that predicted by one-dimensional reaction zone calculations.

Head-on Collision of a Detonation with a Planar Shock Wave

The phenomenon that occurs when a Chapman–Jouguet (CJ) detonation collides with a shock wave is discussed. Assuming a one-dimensional steady wave configuration analogous to a planar shock–shock frontal interaction, analytical solutions of the Rankine–Hugoniot relationships for the transmitted detonation and the transmitted shock are obtained by matching the pressure and particle velocity at the contact surface. The analytical results indicate that there exist three possible regions of solutions, i.e. the transmitted detonation can have either strong, weak or CJ solution, depending on the incident detonation and shock strengths. On the other hand, if we impose the transmitted detonation to have a CJ solution followed by a rarefaction fan, the boundary conditions are also satisfied at the contact surface. The existence of these multiple solutions is verified by an experimental investigation. It is found that the experimental results agree well with those predicted by the second wave interaction model and that the transmitted detonation is a CJ detonation. Unsteady numerical simulations of the reactive Euler equations with both simple one-step Arrhenius kinetic and chain-branching kinetic models are also carried out to look at the transient phenomena and at the influence of a finite reaction thickness of a detonation wave on the problem of head-on collision with a shock. From all the computational results, a relaxation process consisting of a quasi-steady period and an overshoot for the transmitted detonation subsequent to the head-on collisions can be observed, followed by the asymptotic decay to a CJ detonation as predicted theoretically. For unstable pulsating detonations, it is found that, due to the increase in the thermodynamic state of the reactive mixture caused by the shock, the transmitted pulsating detonation can become more stable with smaller amplitude and period oscillation. These observations are in good agreement with experimental evidence obtained from smoked foils where there is a significant decrease in the detonation cell size after a region of relaxation when the detonation collides head-on with a shock wave.

Cell-like structure of unstable oblique detonation wave from high-resolution numerical simulation

A comprehensive numerical study was carried out to investigate the unsteady cell-like structures of oblique detonation waves (ODWs) for a fixed Mach 7 inlet flow over a wedge of 30° turning angle. The effects of grid resolution and activation energy were examined systematically at a dimensionless heat addition of 10. The ODW front remains stable for a low activation energy regardless of grid resolution, but becomes unstable for a high activation energy featuring a cell-like wave front structure. Similar to the situation with an ordinary normal detonation wave (NDW), a continuous increase in the activation energy eventually causes the wave-front oscillation to transit from a regular to an irregular pattern. The wave structure of an unstable ODW, however, differs considerably from that of a NDW. Under the present flow condition, triple points and transverse waves propagate downstream, and the numerical smoke-foil record exhibits traces of triple points that rarely intersect with each other. Several instability-driving mechanisms were conjectured from the highly refined results. Since the reaction front behind a shock wave can be easily destabilized by disturbance inherent in the flowfield, the ODW front becomes unstable and displays cell-like structures due to the local pressure oscillations and/or the reflected shock waves originating from the triple points. The combined effects of various instability sources give rise to a highly unstable and complex flow structure behind an unstable ODW front.

Structure of unstable gaseous detonation waves

Structure of unstable gaseous detonation waves

Cell structure and stability of detonations with a pressure-dependent chain-branching reaction rate model

We examine detonation waves with a four-step chain-branching reaction model that exhibits explosion limits close to the two lower limits of hydrogen–oxygen chemistry. The reaction model consists of a chain-initiation step and a chain-branching step, both temperature-dependent with Arrhenius kinetics, followed by two pressure-dependent termination steps. Increasing the chain-branching activation energy or the overdrive shortens the reaction length in the ZND wavelength and leads to more unstable detonations, according to multi-dimensional linear stability analysis. Corresponding numerical simulations show that detonations with weak chain-branching reactions have a wave structure similar to those with a single-step reaction; strong chain-branching detonations show distinct keystone features. Keystone regions are bounded by a discontinuity in reactivity across the shear layers emanating from the triple points at the intersection of the transverse waves and the main front. Especially in the strong case, chain-branching occurs within a thin front at the back side of the keystone figure, or immediately behind Mach stems.

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 structure with pressure-dependent chain-branching kinetics

We study multi-dimensional stability and perform high resolution two-dimensional numerical simulations of detonations with a four-step chain-branching reaction model. The reaction model is designed to approximate hydrogen chemistry. It consists of a chain-initiation step and a chain-branching step, both temperature-dependent with Arrhenius kinetics, followed by two pressure-dependent termination steps. Increasing the chain-branching activation energy shortens the ZND reaction length and leads to more unstable detonations, according to the stability analysis. Computations with four values of the chain-branching activation energy are performed both in narrow and wide channels. In the wider channel, all cases studied show distinct keystone-shaped regions, associated with substantial differences in reactivity across the shear layer hence of the time and distance until chain-branching takes place. As the chain-branching activation energy increases, cells take a shorter time to form, and the ratio cell length over width decreases. The cell size is dominated by the longer unstable wavelength even when high frequency modes are more unstable, but cells appear earlier in narrow channels than in wider ones. Initially, the cellular structure looks weaker, and the cell size is dominated by the shorter, more unstable wavelength, but eventually, it adapts to the longest unstable wavelength still consistent with the domain width.

Reaction zones in highly unstable detonations

Experimental images of detonation fronts are made for several fuel-oxidizer mixtures, including hydrocarbon–air systems. Schlieren and planar laser induced fluorescence techniques are used to image both the shock configurations and the OH reaction front structure in a single experiment. The experiments are carried out in a narrow rectangular channel. The degree of instability of detonation fronts in different mixtures is evaluated by comparing calculated mixture parameters with the longitudinal neutral stability curve. The images reveal that the structure of the front increases dramatically in complexity as the mixture parameters move away from the neutral stability curve into the unstable region. Of the mixtures studied, nitrogen-diluted hydrocarbon mixtures are predicted to be the most unstable, and these show the greatest degree of wrinkling in the shock and OH fronts, with distortion occurring over a wide range of spatial scales. In the most unstable cases, separation of the shock and OH front occurs, and localized explosions in these regions are observed in a high-speed schlieren movie. This is in dramatic contrast to the weakly unstable waves that have smooth reaction fronts and quasi-steady reaction zones with no evidence of localized explosions. A key feature of highly unstable waves is very fine scale wrinkling of the OH and shock fronts, which is absent in the low-activation energy cases. This may be due to the superposition of cellular structures with a wide range of cell sizes. In contrast to soot foils, images of the OH front have a more stochastic appearance, and organized cellular structure is not as apparent.

Investigation of deflagrations and detonations in pipes and flame arresters by high-speed framing

Abstract A widespread method to avoid damage from incidental gas explosions is to use flame arresters. The new European Standard EN 12874 provides guidance for testing such flame arresters. Nevertheless, there are still unsolved problems as regards the effects of flame speed and transient explosion pressure, tests of unstable detonations and statistics of the deflagration-to-detonation transition (DDT). To better understand these phenomena, a transparent test set-up (pipe and model arrester) was built and the flame propagation was directly recorded using high-speed cameras. Flame velocity and flame pressure were determined in each phase of the explosion. Time-resolved observations of the flame propagation inside the flame arrester provided direct insight into the quenching potential and the flame transmission process.

Pulsating instability of detonations with a two-step chain-branching reaction model: theory and numerics

The nonlinear dynamics of Chapman–Jouguet pulsating detonations are studied both numerically and asymptotically for a two-step reaction model having separate induction and main heat release layers. For a sufficiently long main heat release layer, relative to the length of the induction zone, stable one-dimensional detonations are shown to be possible. As the extent of the main reaction layer is decreased, the detonation becomes unstable, illustrating a range of dynamical states including limit-cycle oscillations, period-doubled and four-period solutions. Keeping all other parameters fixed, it is also shown that detonations may be stabilized by increasing the reaction order in the main heat release layer. A comparison of these numerical results with a recently derived nonlinear evolution equation, obtained in the asymptotic limit of a long main reaction zone, is also conducted. In particular, the numerical solutions support the finding from the analytical analysis that a bifurcation boundary between stable and unstable detonations may be found when the ratio of the length of the main heat release layer to that of the induction zone layer is O(1/ε), where ε (≪1) is the inverse activation energy in the induction zone.

Instability threshold of gaseous detonations

The spectrum of linear modes governing the multidimensional instabilities of gaseous detonations is revisited by combining a numerical analysis with new analytical results. In view of recent develop`ments in nonlinear analyses for describing the cellular structure of weakly unstable detonation fronts, particular attention is paid to the neighbourhood of the instability threshold. A first objective is to check the validity domain of the analytical results and to investigate to what extent they are useful when approaching the self-sustained regime (Chapman–Jouguet conditions). A second objective is to study how the multidimensional instabilities are influenced by multiple-step chemistry. The roles of the induction period and of the stiffness of the exothermic runaway will be investigated separately.