Andrew J. Higgins

Nonlinear dynamics and chaos analysis of one-dimensional pulsating detonations

To understand the nonlinear dynamical behaviour of a one-dimensional pulsating detonation, results obtained from numerical simulations of the Euler equations with simple one-step Arrhenius kinetics are analysed using basic nonlinear dynamics and chaos theory. To illustrate the transition pattern from a simple harmonic limit-cycle to a more complex irregular oscillation, a bifurcation diagram is constructed from the computational results. Evidence suggests that the route to higher instability modes may follow closely the Feigenbaum scenario of a period-doubling cascade observed in many generic nonlinear systems. Analysis of the one-dimensional pulsating detonation shows that the Feigenbaum number, defined as the ratio of intervals between successive bifurcations, appears to be in reasonable agreement with the universal value of d = 4.669. Using the concept of the largest Lyapunov exponent, the existence of chaos in a one-dimensional unsteady detonation is demonstrated.

Numerical investigation of the instability for one-dimensional Chapman–Jouguet detonations with chain-branching kinetics

The dynamics of one-dimensional Chapman–Jouguet detonations driven by chain-branching kinetics is studied using numerical simulations. The chemical kinetic model is based on a two-step reaction mechanism, consisting of a thermally neutral induction step followed by a main reaction layer, both governed by Arrhenius kinetics. Results are in agreement with previous studies that detonations become unstable when the induction zone dominates over the main reaction layer. To study the nonlinear dynamics, a bifurcation diagram is constructed from the computational results. Similar to previous results obtained with a single-step Arrhenius rate law, it is shown that the route to higher instability follows the Feigenbaum route of a period-doubling cascade. The corresponding Feigenbaum number, defined as the ratio of intervals between successive bifurcations, appears to be close to the universal value of 4.669. The present parametric analysis determines quantitatively the relevant non-dimensional parameter χ, defined as the activation energy for the induction process ϵ I multiplied by the ratio of the induction length Δ I to the reaction length Δ R . The reaction length Δ R is estimated by the inverse of the maximum thermicity (1/ max) multiplied by the Chapman–Jouguet particle velocity u CJ . An attempt is made to provide a physical explanation of this stability parameter from the coherence concept. A series of computations is carried out to obtain the neutral stability curve for one-dimensional detonation waves over a wide range of chemical parameters for the model. These results are compared with those obtained from numerical simulations using detailed chemistry for some common gaseous combustible mixtures.

Comments on criteria for direct initiation of detonation

The current status of the direct initiation problem, where a powerful source drives a blast wave into an explosive gaseous mixture to generate a Chapman–Jouguet (CJ) detonation, is critically assessed. The current theories which are most successful in estimating the critical energy required for initiation are semiempirical in nature, in that they involve an experimentally determined length–scale (typically cell size) to characterize the explosive mixture. The eventual analytic theory of initiation should be based exclusively on the constitutive properties of the explosive. To date, attempts at a comprehensive theory of initiation have invoked quenching of the reaction front by curvature or unsteadiness of the blast wave. Simple analytic models of initiation as well as numerical simulations and experiments, however, all indicate that initiation near the critical regime is the result of a reacceleration of the blast wave from a sub–CJ minimum. Hence, the criterion for initiation must take into account the amplification of the blast wave due to coherent coupling with the chemical energy release. The effect of ‘hot spots’ is also shown to have a pronounced effect in reducing the critical energy required for initiation. These results suggest directions which future investigations can pursue toward a rigorous theory of direct initiation.

The effect of particle strength on the ballistic resistance of shear thickening fluids

The response of shear thickening fluids (STFs) under ballistic impact has received considerable attention due to its field-responsive nature. While efforts have primarily focused on traditional ballistic fabrics impregnated with these fluids, the response of pure STFs to penetration has received limited attention. In the present study, the ballistic response of particle-based STFs is investigated and the effects of fluid density and particle strength on ballistic performance are isolated. It is shown that the loss of ballistic resistance in the STFs at higher impact velocities is governed by the material strength of the particles in suspension. The results illustrate the range of velocities over which these STFs may provide effective armor solutions.

Powdered Metals as Fuel for Hypersonic Ramjets

The concept of a hypersonic ramjet fueled by powdered metals (B, Al, Mg, and MgB2) is considered. Thermodynamic calculations of the combustion heat release, specific impulse, and volumetric specific impulse show that metal fuels can exceed hydrocarbon fuels in volumetric energy content and, in the case of boron, exceed conventional fuels on a mass basis as well. The refractory nature of metal fuels and their combustion products also suggest they may permit ramjets utilizing subsonic combustion to extend their operation to hypersonic Mach numbers (greater than Mach 5). The feasibility of stabilizing combustion using a pure powdered-metal fuel without the use of hydrocarbons is investigated experimentally. The ability to effectively inject powdered metal into a combustor is demonstrated using a laboratory-sca le dispersion apparatus. This apparatus is then used to measure fundamental burning characteristics of aluminum powder suspensions in air. The burning rate of micron-size aluminum air suspensions is seen to be similar to gaseous hydrocarbons in air, but the dependence of burning rate on fuel equivalence ratio is very different from gas flames. A nearly constant plateau in burning velocity obtained with fiiel-rich mixtures suggests that the combustor must operate rich for stable operation. Thermodynamic calculations show that maximum specific impulse is obtained for lean metal-air mixtures. Thus, a conceptual design of the dust combustor is proposed that uses a preburner to stabilize a fiiel-rich metal flame followed by gradual mixing of the burning suspension with a secondary airflow to obtain a high air/fuel excess ratio.

Aluminum Particle Combustion in High-Speed Detonation Products

Aluminum particles ranging from 2 to 100 μm were subjected to the flow of detonation products of a stoichiometric mixture of hydrogen and oxygen at atmospheric pressure. Luminosity emitted from the reacting particles was used to determine the reaction delay and duration. The reaction duration was found to increase as d n with n ≈ 0.5, which is more consistent with kinetically controlled reaction rather than the classical diffusion-controlled regime. Emission spectroscopy was used to estimate the combustion temperature, which was found to be well below the flow temperature. This fact also suggests combustion in the kinetic regime. Finally, the flow field was modeled with a CFD code, and the results were used to model analytically the behavior of the aluminum particles.

Effect of discreteness on heterogeneous flames: Propagation limits in regular and random particle arrays

In a system with discrete heat sources distributed in an inert, heat conducting medium, there exists two asymptotic regimes of flame propagation. When the flame thickness is much greater than the inter-particle spacing, the system approximates a homogeneous medium and the flame can be modeled as a continuum. In the other extreme, when the flame is very thin due to rapid reaction of particles, the heterogeneous flame can no longer be treated as a continuum since discrete effects become dominant. The effects of discreteness are characterized by a strong dependence on the spatial distribution of the sources. The present work investigates the effects of discreteness on flame propagation and demonstrates that these effects result in a propagation limit in the absence of losses. For a system of regularly spaced particles, this limit can be found analytically for one-, two-, and three-dimensional systems, although the flame exhibits a complex dynamic of bifurcations as it approaches this limit. Propagation of a flame beyond this limit is only possible through concentration fluctuations in a system with randomly distributed particles. Two-dimensional numerical simulations with randomly distributed particles show a strong dependence of the propagation limit on the size of the computational domain. A consequence of the random particle distribution is that the flammability limit can only be defined as a probabilistic outcome of the flame propagation simulations.

Ram Accelerators: Outstanding Issues and New Directions

An assessment of the key issues affecting the thrust and maximum velocities that can be obtained in ram accelerators is presented. The regimes of ram accelerator operation (subdetonative and superdetonative) are discussed, and simple models for thrust are compared to experimental results and found to be satisfactory. The phenomena that are responsible for the operating limits of these modes of operation are explored, and potential solutions for overcoming these limits are discussed. In particular, the possibility that flow separation may cause unstarts in both the subdetonative and superdetonative ram accelerator is shown to explain qualitatively the experimentally observed limits. A potential remedy to the unstart problem that involves modifying the geometry of the ram accelerator tube is presented. The role of the projectile material, which may react with the oxidizing environment of the propellant, also has a significant effect on superdetonative operation. Techniques to address this problem are outlined. Other novel concepts, such as the use of an explosive-lined launch tube and the laser-driven ram accelerator, are discussed as well.

Shock wave propagation in dense particle suspensions

Shock wave propagation in a multiphase suspension is studied experimentally. Particle suspensions are used as a means of obtaining a system in which there is limited initial interparticle contacts with a large degree of parametric variability. Suspensions were created in ethylene glycol at several volume fractions (41%, 48%, and 54%) of silicon carbide particles. Plate impact experiments are conducted to obtain the shock Hugoniots of the various suspensions at particle velocities in the range of 200–900 m/s. Transitions are shown to exist in the Us-up Hugoniots of the suspensions. In situ longitudinal and lateral stress measurements are made in the 48% suspension at two different impact velocities demonstrating a deviatoric stress component to the stress state in the suspension. The results are discussed in terms of the development of extensive interparticle contacts in a mechanism analogous to classical shear thickening in dense suspensions.

Ram Accelerator Operating Limits, Part 1: Identification of Limits

Operational limits of the thermally choked ram accelerator are investigated. A quasisteady, one-dimensional model of the ram accelerator predicts it should be able to operate when the projectile Mach number is sufficient to maintain supersonic flow past the projectile throat and the heat release is sufficient to stabilize a normal shock upon the projectile body, but not enough to drive the shock over the projectile throat. These limits to operation can be expressed as relations of Mach number and heat release Q, and together they define a theoretical envelope of operation in the Q-M plane. The corresponding experimental envelope was investigated by injecting projectiles at different Mach numbers into methane/oxygen/ nitrogen and hydrogen/oxygen/methane mixtures at 25 and 50 atm, respectively. The results indicated a broad range of mixtures that were able to accelerate the projectile through the Chapman-Jouguet (CJ) detonation speed of the mixture. In the more energetic mixtures, the normal shock wave surged forward and immediately unstarted the projectile as it entered the test section. Unstarts were also observed when the projectile was accelerated beyond the CJ detonation speed, but because of the projectiles long intube residence time, these unstarts are believed to have been structural, not gasdynamic in nature.