Multilevel detonation cell structures in methane-air mixtures

High-resolution numerical simulations of two-dimensional detonations in a methane-air mixture with extremely high activation energy show the formation of multiple levels of cellular structures caused by the propagation of triple-shock configurations. Two main types of these configurations were observed based on the structure of transverse waves behind the leading edge of the detonation. Collisions were observed between two triple-shock configurations with attached transverse detonations, two triple-shock configurations with inert transverse waves, and one of each of these types. These collisions give rise to the formation of highly irregular, and, in some cases incomplete, cells. Smoke foils obtained from detonation of a near-stoichiometric mixture of natural gas and air show similar results. Estimates of the width of the experimental cells qualitatively match those inferred from the calculations.     

Formation of transverse waves in oblique detonations

The structure of oblique detonation waves stabilized on a hypersonic wedge in mixtures characterized by a large activation energy is investigated via steady method of characteristics (MoC) calculations and unsteady computational flowfield simulations. The steady MoC solutions show that, after the transition from shock-induced combustion to an overdriven oblique detonation, the shock and reaction complex exhibit a spatial oscillation. The degree of overdrive required to suppress this oscillation was found to be nearly equal to the overdrive required to force a one-dimensional piston-driven detonation to be stable, demonstrating the equivalence of two-dimensional steady oblique detonations and one-dimensional unsteady detonations. Full unsteady computational simulations of the flowfield using an adaptive refinement scheme showed that these spatial oscillations are transient in nature, evolving in time into transverse waves on the leading shock front. The formation of left-running transverse waves (facing upstream) precedes the formation of right-running transverse waves (facing downstream). Both sets of waves are convected downstream away from the wedge in the supersonic flow behind the leading oblique front, such that the mechanism of instability must continuously generate new transverse waves from an initially uniform flow. Together, these waves define a cellular structure that is qualitatively similar to a normal propagating detonation.     

Dynamics of Chapman-Jouguet pulsating detonations with Chain-Branching kinetics: Fickett’s analogue and Euler equations

The present study examines the spatiotemporal nonlinear dynamics of detonations over a wide range of reaction time scales away from the neutral stability region. This is addressed by one-dimensional numerical simulations with chain-branching kinetics. Fickett’s detonation analogue and Euler’s equations were used as evolution equations. A shock-fitting solver is used to reduce CPU time. Up to four thousand five hundred simulations have been carried out. Detailed bifurcation diagrams have been generated to explore the detonation dynamics. For long/intermediate reaction time scales, away from the neutral boundary, the traditional period-doubling cascade to chaos is seen. For square wave detonations, away from the neutral stability, almost periodic oscillations are recorded. This result might have implications for the existence of a characteristic length scale, the cell size, on typical cellular detonations which have a short reaction length.     

The re-initiation of cellular detonations downstream of an inert layer

The current work aims to examine how the nature of cellular instabilities controls the re-initiation capability and dynamics of a gaseous detonation transmitting across a layer of inert (or non-detonable) gases. This canonical problem is tackled via computational analysis based on the two-dimensional, reactive Euler equations. Two different chemical kinetic models were used, a simplified two-step induction-reaction model and a detailed model for hydrogen-air. For the two-step model, cases with relatively high and low activation energies, representing highly and weakly unstable cellular detonations, respectively, are considered. For the weakly unstable case, two distinct types of re-initiation mechanisms were observed. (1) For thin inert layers, at the exit of the layer the detonation wave front has not fully decayed and thus the transverse waves are still relatively strong. Detonation re-initiation in the reactive gas downstream of the inert layer occurs at the gas compressed by the collision of the transverse waves, and thus is referred to as a cellular-instability-controlled re-initiation. (2) If an inert layer is sufficiently thick, the detonation wave front has fully decayed to a planar shock when it exits the inert layer, and re-initiation still occurs downstream as a result of planar shock compression only, which is thus referred to as a planar-shock-induced re-initiation. Between these two regimes there is a transition region where the wave front is not yet fully planar, and thus perturbations by the transverse waves still play a role in the re-initiation. For the highly unstable case, re-initiation only occurs via the cellular-instability-controlled mechanisms below a critical thickness of the inert layer. Additional simulations considering detailed chemical kinetics demonstrate that the critical re-initiation behaviors of an unstable stoichiometric mixture of hydrogen-air at 1 atm and 295 K are consistent with the finding from the two-step kinetic model for a highly unstable reactive mixture.     

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.     

Detonability limits in thin annular channels

In this paper, detonability limits in two-dimensional annular channels are investigated. Since the channel heights are small in comparison to the tube diameter, curvature effects can be neglected and the annular channels can be considered to be essentially two-dimensional. Mixtures that are highly diluted with argon are used since previous investigations seem to indicate that detonations in such mixtures are “stable” in that cellular instabilities play minor roles on the propagation of the detonation. For stable detonations where the ZND structure is valid, boundary layer effects can be modeled as a flow divergence term in the conservation of mass equation following the pioneering work of Fay [J.A. Fay, Phys. Fluids 2(3) (1959) 283–289]. Expansion due to flow divergence in the reaction zone results in a velocity deficit. There exists a maximum deficit when an eigenvalue detonation velocity can no longer be found, which can be taken as the onset of the detonability limits. Experimentally, it was found that unlike “unstable” detonations, the detonability limits for “stable” detonations are well-defined. No unstable near-limit phenomena (e.g., galloping detonations) was observed. Good agreement is found between the theoretical predictions and the experimentally obtained velocity deficits and limits in the two channel heights of 2.2 and 6.9 mm for hydrogen–oxygen and acetylene–oxygen mixtures diluted with over 50% argon. It may be concluded that at least for these special mixtures where the detonation is “stable,” the failure mechanism is due to flow divergence caused by the negative displacement thickness of the boundary layer behind the leading shock front of the detonation wave.     

Spinning detonation and velocity deficit in small diameter tubes

Detonation velocities and soot patterns of H₂/O₂ mixtures were measured in glass tubes of 3, 6 and 10 mm diameters at pressures ranging from 70 to 400 Torr and equivalence ratios of 0.5–1.5. It was confirmed that the transition from a multi-head to a spinning detonation occurred at the pressure where the cell size is equal to the length of circumference. At this transition pressure, the velocity was 95% of the C-J detonation. Stable spinning detonations were observed at wide range of initial pressures below the transition point. Detonation velocities were continuously decreasing with decreasing initial pressures in this pressure region. Spinning detonations with velocities down to 85% of the C-J detonation were observed. Those deficits in detonation velocities were well predicted by the modified ZND model with full detailed chemical kinetics. Heat and momentum losses were taking into account in this model. Validity of the modified ZND model to define the limit of detonation propagation was discussed.     

A review of direct numerical simulations of astrophysical detonations and their implications

Multi-dimensional direct numerical simulations (DNS) of astrophysical detonations in degenerate matter have revealed that the nuclear burning is typically characterized by cellular structure caused by transverse instabilities in the detonation front. Type Ia supernova modelers often use onedimensional DNS of detonations as inputs or constraints for their whole star simulations.While these one-dimensional studies are useful tools, the true nature of the detonation is multi-dimensional. The multi-dimensional structure of the burning influences the speed, stability, and the composition of the detonation and its burning products, and therefore, could have an impact on the spectra of Type Ia supernovae. Considerable effort has been expended modeling Type Ia supernovae at densities above 1×10⁷ g·cm^-3 where the complexities of turbulent burning dominate the flame propagation. However, most full star models turn the nuclear burning schemes off when the density falls below 1×10⁷ g·cm^-3 and distributed burning begins. The deflagration to detonation transition (DDT) is believed to occur at just these densities and consequently they are the densities important for studying the properties of the subsequent detonation. This work will review the status of DNS studies of detonations and their possible implications for Type Ia supernova models. It will cover the development of Detonation theory from the first simple Chapman–Jouguet (CJ) detonation models to the current models based on the time-dependent, compressible, reactive flow Euler equations of fluid dynamics.     

Effect of Cellular Instability on the Initiation of Cylindrical Detonations

The direct initiation of detonations in one-dimensional(1 D) and two-dimensional(2 D) cylindrical geometries is investigated through numerical simulations. In comparison of 1 D and 2 D simulations, it is found that cellular instability has a negative effect on the 2 D initiation and makes it more difficult to initiate a sustaining 2 D cylindrical detonation. This effect associates closely with the activation energy. For the lower activation energy,the 2 D initiation of cylindrical detonations can be achieved through a subcritical initiation way. With increasing the activation energy; the 2 D cylindrical detonation has increased difficulty in its initiation due to the presence of unreacted pockets behind the detonation front and usually requires rather larger source energy.     

Numerical study on spinning detonations

Spinning detonations in both a circular tube and a square tube are presented in order to reveal characteristics of spinning modes by using three-dimensional simulations with a detailed chemical reaction model. The present results show a feature of a single spinning detonation which was discovered in 1926. The shock patterns in both cases are similar except the pressure trail, however, the shock wave angles and the shock wave lengths are shown to be dependent on the cross section configuration of the tube. The pitch angle, the track angle, the Mach stem angle, and the incident shock angle on the tube wall in the numerical results agree well with those in the experimental ones, and they are independent of the compositions of mixture, tube diameters, and initial pressures.     

Structural characteristics of detonation expansion from a small channel to a larger one

We report on numerical simulations of the evolution of two-dimensional detonation waves that are expanded from a small channel to a larger one. In accordance with experimental data, the simulations predict three different types of evolution, namely, supercritical, critical and subcritical detonations. In a supercritical detonation, the reaction zone remains always attached to the precursor shock, whereas in a critical one it temporarily detaches and then re-attaches to the front. In the subcritical type, the extinction is permanent, i.e., the detonation quenches. The effects of the fuel’s activation energy and the channel-width ratio are studied via a parametric study. It is found that sufficiently large values of these two parameters can result to flows of the critical and even the subcritical type. Finally, three-dimensional simulations have also been performed and are briefly discussed herein.     

Energy release effect of mixture on single spinning detonation structure

Unsteady three-dimensional numerical simulation on a single spinning detonation in a circular tube are presented in order to understand the effects of energy release of the mixture on the detonation structure. Overall structures of the spinning detonations such as the shock structure around the spin head, the long pressure trail, and the track angle on the wall are not affected by these effects because they depend on the specific heat ratio of the products which has approximately a constant value. The calculated averaged detonation velocities on the symmetry axis during one cycle decrease inversely with an exponential curve to become the value lower than the CJ detonation velocity. Those for p₀ = 0.1 MPa and p₀ = 0.01 MPa become approximately 0.98 D_CJ and 0.92 D_CJ, respectively, because the energy release in the CJ state for p₀ = 0.01 MPa is 10% lower than that for p₀ = 0.1 MPa. The state of gas behind the head of spinning detonation is also evaluated by the classical oblique shock theory and equilibrium calculation by using the track angle, shock wave angle, and detonation velocity in order to compare with the present and other researcher’s numerical results. The effects of the energy release in the mixture are large on the strength of the transverse detonation.     

Statistical analysis of cellular detonation dynamics from numerical simulations: one-step chemistry

In this paper, two methods are developed for statistically analysing the nonlinear cellular dynamics from numerical simulations of gaseous detonations, one use of which is the systematic determination of detonation cell sizes from such simulations. Both these methods rely on signed vorticity records in which the individual families of transverse waves are captured independently. The first method involves an automated extraction of the main triple-point tracks from the vorticity records, allowing statistical analysis of the spacings between neighbouring tracks. The second method uses the autocorrelation function to spectrally analyse the vorticity records. These methods are then employed for a preliminary analysis of the cellular dynamics of the standard, idealized one-step chemistry model. Evidence is found for ‘cell size doubling’ bifurcations in the one-step model as the cellular dynamics become more irregular (e.g. as the activation is increased). It is also shown that the statistical models converge slowly due to systematic ‘shot-to-shot’ variation in the cellular dynamics for fixed parameters with different initial perturbations. Instead, it appears that a range of equally probable cell sizes can be obtained for given parameters.     

Critical diffraction of irregular structure detonations and their predictability from experimentally obtained D−κ data

The present work reports new experiments of detonation diffraction in a 2D channel configuration in stoichiometric mixtures of ethylene, ethane, and methane with oxygen as oxidizer. The flow field details are obtained using high-speed schlieren near the critical conditions of diffraction. The critical initial pressure for successful diffraction is reported for the ethylene, ethane and methane mixtures. The flow field details revealed that the lateral portion of the wave results in a zone of quenched ignition. The dynamics of the laterally diffracting shock front are found in good agreement with the recent model developed by Radulescu et al. (Physics of Fluids 2021). The model provides noticeable improvement over the local models using Whitham’s characteristic rule and Wescott, Bdzil and Stewart’s model for weakly curved reactive shocks. These models provide a link between the critical channel height and the critical wave curvature. The critical channel heights and global curvatures are found in very good agreement with the critical curvatures measured independently by Xiao and Radulescu (Combust. Flame 2020) in quasi-steady experiments in exponential horns for three mixtures tested. Furthermore, critical curvature data obtained by others in the literature was found to provide a good prediction of critical diffraction in 2D. These findings suggest that the critical diffraction of unstable detonations may be well predicted by a model based on the maximum curvature of the detonation front, where the latter is to be measured experimentally and account for the role of the cellular structure in the burning mechanism. This finding provides support to the view that models for unstable detonations at a meso-scale larger than the cell size, i.e., hydrodynamic average models, are meaningful.     

Multiplicity of detonation regimes in systems with a multi-peaked thermicity

The study investigates detonations with multiple quasi-steady velocities that have been observed in the past in systems with multi-peaked thermicity, using Fickett's detonation analogue. A steady-state analysis of the travelling wave predicts multiple states, however, all but the one with the highest velocity develop a singularity after the sonic point. Simulations show singularities are associated with a shock wave which overtakes all sonic points, establishing a detonation travelling at the highest of the predicted velocities. Under a certain parameter range, the steady-state detonation can have multiple sonic points and solutions. Embedded shocks can exist behind sonic points, where they link the weak and strong solutions. Sonic points whose characteristics do not diverge are found to be unstable, and to be the source of the embedded shocks. Numerical simulations show that these shocks are only quasi-stable. This is believed to be due in part to a feature of the model which permits shocks anywhere behind a sonic point.     

Blast waves in a cylindrical channel generated by the nonideal detonation of aluminum-Teflon-RDX high-density formulations

The pressure at the front and the pressure impulse of blast waves generated in a cylindrical tube by the expanding products of the nonideal detonation of low-porosity charges prepared by pressing of fine-grained powders of aluminum, Teflon, and RDX were measured. The measured parameters are compared to the same parameters of blast waves produced by the detonation of TNT charges of identical mass. The relative quantities were used to evaluate the effectiveness of blast waves with respect to those generated by TNT. Mixed compositions differing in the shape (brand) of the aluminum powder particles and the ratio between the components at 30% RDX are studied. It is shown that, for the investigated compositions, the pressure at the leading front of the wave exceeds the pressure achieved during TNT explosion on average by 10–30%, almost independently of the distance traveled along the tube in the range from 0.8 to 3.8 m. The dependence of the wave amplitude on the particle shape and aluminum content was weak. In the same range of distances, the relative pulse pressure increases strongly, from 0.5 to 2.1 and higher, mainly due to an increase in the width of the wave. This result is of interest from the point of view of achieving a high pressure impulse of the blast wave in an area remote from the charge. The obtained data suggest that RDX mainly reacts in the detonation wave, with the chemical transformation of Teflon and aluminum in the detonation wave and near-to-charge zone occurring, if at all, to a small extent. On the contrary, as the blast wave front moves through the channel, the burning of aluminum in the fluoride formed during the decomposition of Teflon provides an appreciable support to the blast wave, causing a significant increase in the pressure impulse.     

A flame technique to isolate the detonation/product interface relevant to rotating detonation engines

A novel experimental technique is proposed to study the detonation propagation in a layer of non-reacted gas weakly confined by combustion products. This problem is relevant to rotating detonation engines, where transverse detonations are confined by products of a previous rotation cycle, and other applications such as industrial safety. The experimental technique utilizes a flame ignited along the top wall in a long channel. The preferential growth of the flame along the long direction of the channel creates a finger flame and permits to create a narrow layer of unburned gas. A detonation ignited outside of this layer then propagates through the layer. This permits to conduct accurate observations of the detonation interaction with the inert gas and determine the boundary condition of the interaction. The present paper provides a proof-of-concept demonstration of the technique in a 3.4 m by 0.2 m channel, in which long finger flames were observed in ethylene-oxygen mixtures. The flame is visualized by high-speed direct luminosity over its entire travel, coupled with pressure measurements. A direct simulation of the flame growth served to supplement the experiments and evaluate the role of the induced flow by the flame growth, which gives rise to a non-uniform velocity distribution along the channel length. Detonation experiments were also performed at various layer heights in order to establish the details of the interaction. The structure was visualized using high speed Schlieren video. It was found that an inert shock always runs ahead of the detonation wave, which gives rise to a unique double shock reflection interaction.     

Numerical study of three-dimensional detonation wave dynamics in a circular tube

The three-dimensional structures of a detonation wave propagating in a circular tube were investigated using a one-step irreversible Arrhenius kinetics model. A series of parametric studies were carried out to investigate the different modes of cell structure formation by changing the pre-exponential factor. Maximum pressure trace was recorded along the tube wall to investigate the detonation cell structures. The unsteady results obtained in three dimensions revealed the generation mechanism of the wave front structures of two-, three- and four-cell mode detonations. A six-cell mode detonation could be obtained using a finer grid. With the increase in pre-exponential factor, it was found that the number of detonation cells is increased while the cell size is reduced accordingly. In all the multi-cell modes, the detonation wave structures and smoked-foil records on the wall are formed by the propagation of transverse waves along the wall in clockwise and counter-clockwise directions, while the slapping wave moves in the radial direction. The presence of the slapping wave further strengthens the wave interactions in three-dimensional simulation. Comparison with two-dimensional simulation confirms the effect of the slapping wave in the radial direction. As a result, the detonation wave front structures changes from the polygonal shape to the multi-bladed fan shape, periodically.     

Experimental characterization of RDE combustor flowfield using linear channel

An experimental study was conducted to characterize fundamental behavior of detonation waves propagating across an array of reactant jets inside a narrow channel, which simulated an unwrapped rotating detonation engine (RDE) configuration. Several key flow features in an ethylene-oxygen combustor were explored by sending detonation waves across reactant jets entering into cold bounding gas as well as hot combustion products. In this setup, ethylene and oxygen were injected separately into each recessed injector tube, while a total of 15 injectors were used to establish a partially premixed reactant jet array. The results revealed various details of transient flowfield, including a complex detonation wave front leading a curved oblique shock wave, the unsteady production of transverse waves at the edge of the reactant jets, and the onset of suppressed reactant jets re-entering the combustor following a detonation wave passage. The visualization images showed a complex, multidimensional, and highly irregular detonation wave front. It appeared non-uniform mixing of reactant jets lead to dynamic transverse wave structure. The refreshed reactant jets evolving in the wake of the detonation wave were severely distorted, indicating the effect of dynamic flowfield and rapid pressure change. The results suggest that the mixing between the fuel and oxidizer, as well as the mixing between the fresh reactants and the background products, should affect the stability of the RDE combustor processes.     

Generation of blast waves in a channel by nonideal detonation of aluminum-enriched high-density compositions

The results of experimental studies of the nonideal detonation of high-density, high-energy aluminum-ammonium perchlorate-organic fuel-HE compositions and of the blast waves it generates in a channel filled with air are presented. Aluminum-enriched compositions have high densities (up to 2 g/cm³) and high heats of explosion, nearly twice that for TNT. The studies were performed to work out scientific fundamentals of controlling nonideal detonation and to explore the possibility of creating new high-energy high-density formulations with an enhanced fugacity effect. The factors that enable controlling the nonideal detonation of such charges were determined. It was demonstrated that, at RDX contents above 15%, the detonation velocity increases linearly with the charge density while the critical detonation diameter decreases. Adjusting the density, HE content, ratio of the components makes it possible to vary the detonation velocity in high-density charges over a wide range, from 4 to 7 km/s. The experimental data were compared to the thermodynamically calculated velocity of ideal detonation. For the compositions under study, the pressure- time histories of the blast wave generated in a cylindrical tube by the expanding detonation products at different distances from the charge were measured. The results were compared to analogous data obtained under the same conditions for the detonation of the same mass of TNT (100 g). The parameters of blast waves generated by the test compositions are markedly superior to those characteristic of TNT: the pressure at the leading front of the wave and pressure impulse at a given distance from the charge were found to be 1.5–2.0 (or even more) times those observed for TNT. The TNT equivalency at pressures 30–60 atm has similar values. The TNT equivalencies in pressure and pressure impulse depend nonmonotonically on the distance from the charge, so far unclear why. It was established that the interaction between excess fuel and air oxygen during the expansion of detonation products contributes little to supporting the blast wave.