Modeling flows with curved detonation waves

The important role of the shape of the front during detonation wave propagation in gas mixtures was substantiated by K. I. Shchelkin during construction of the theory of spinning detonation. Subsequently, a unique relationship between the curvature of the front and detonation wave parameters has been repeatedly confirmed in experiments, including for condensed high explosives (HEs). The existence of this relationship formed the basis of the theory of the dynamics of the detonation front which had been developed by the end of the 20th century. This paper presents the results of a study of detonation front propagation in cylindrical samples of a low-sensitivity HE of different diameters with one-point and plane-wave initiation. A unique relationship between the detonation velocity and the curvature of the detonation wave front has been found. Ordinary differential equations describing two-dimensional steady-state detonation front profiles for HE charges in the form of a plate, a cylinder, and a ring were derived assuming that the detonation velocity depends on the curvature of the front. It was taken into account that the boundary angle between the normal to the front and the HE edge is unique for each combination of HE and liner material. It was found that the same detonation front profile corresponds to several combinations of liner material and the determining size of the charge (plate thickness, radius of the cylinder or the inner radius of the ring). A comparison of experimental front profiles near the edges of HE charges for these combinations provides data on the dependence of detonation velocity on the curvature of the front at low velocities corresponding to shock-induced detonation regimes. Analysis of previously obtained data for detonating ring charges of low-sensitivity HEs shows that as the detonation velocity decreases, the total front curvature tends to a limit of about 0.05 mm−1, i.e., of the order of the inverse critical diameter. The limit of the front curvature allows predicting the critical detonation diameter.

Mach reflection of detonation waves is an important and complex phenomenon in gaseous detonation field, which usually happens when a detonation wave propagates in a convergent channel. The Mach reflection also has been extensively studied by previous researchers. Meltzer [1] used smoked foils to record cellular structure of the Mach reflection in 2H2–O2–XAr. Akbar [2] used Schlieren to obtain photographs of the Mach reflection in C2H2–2.5O2–14Ar and 2H2–O2–XAr, and also measured χ between trajectory formed by Mach triple point and wedge surface. Actually, the trajectory of Mach triple point in detonation wave is not a straight line but a curve, which has been noticed by Trotsyuk [3]. Therefore, A discrepancy of the Mach reflection between shock wave and detonation wave is that the Mach reflection of shock wave is self-similar, but not with detonation wave. According to analysis of Shepherd [4], due to a finite thickness of detonation front and movement of transverse waves, the presence of characteristic length of induction and reaction zones and instability of transverse waves renders the self-similarity is invalid for the Mach reflection of detonation wave. A more detailed discussion [5] of the failure of the self-similarity for Mach reflection of detonation wave is that the trajectory of Mach triple point is no longer a straight line starting from instead of a fluctuant curve. Therefore, curvature of the trajectory cannot be overlooked and χ angle between the trajectory and the apex of wedge always changes in whole process of the Mach reflection in detonation wave. Therefore, one purpose in this study is to find tendency of the trajectory of the Mach reflection against the three-shock theory established by von-Neumann [6]. Another characteristic of the trajectory is about fluctuation because of the movement of transverse waves in Mach stem and incident detonation front. So it is necessary to study the effect of track angle formed by transverse wave movement in the end.

The shock dynamics of multidimensional condensed and gas-phase detonations

In the design of explosive systems the generic problem that one must consider is the propagation of a well-developed detonation wave sweeping through an explosive charge with a complex shape. At a given instant of time the lead detonation shock is a surface that occupies a region of the explosive and has a dimension that is characteristic of the explosive device, typically on the scale of meters. The detonation shock is powered by a detonation reaction zone, sitting immediately behind the shock, which is on the scale of 1 millimeter or less. Thus, the ratio of the reaction zone thickness to the device dimension is of the order of 1/1,000 or less. This scale disparity can lead to great difficulties in computing three-dimensional detonation dynamics. An attack on the dilemma for the computation of detonation systems has lead to the invention of sub-scale models for a propagating detonation front that they refer to herein as program burn models. The program burn model seeks not to resolve the fine scale of the reaction zone in the sense of a DNS simulation. The goal of a program burn simulation is to resolve the hydrodynamics in the inert product gases on a grid much coarser than that required to resolve a physical reaction zone. The authors first show that traditional program burn algorithms for detonation hydrocodes used for explosive design are inconsistent and yield incorrect shock dynamic behavior. To overcome these inconsistencies, they are developing a new class of program burn models based on detonation shock dynamic (DSD) theory. It is hoped that this new class will yield a consistent and robust algorithm which reflects the correct shock dynamic behavior.

A key issue for detonation wave simulations with a reactive burn model is the resolution needed for the reaction zone. The reaction-zone width is a physical length scale. For plastic-bonded explosives, it is small (on the order of 100 microns) but affects important detonation properties such as the detonation speed. Even with adaptive mesh refinement, a well resolved reaction zone is computationally expensive. Here we consider the trade-off between computational expense and accuracy when the resolution of the reaction zone is varied for simulations with the xRage code of PBX 9501 with the SURF burn model and of PBX 9502 with the SURFplus burn model. For assessing shock initiation, the numerical Pop plot is simulated, and for assessing propagating detonation waves, the cylinder test is used as a test problem. The reaction-zone resolution is varied from a cell size of 6.25 to 200 microns, and key properties of the solutions are compared. With 6 micron resolution, the numerical solution is near mesh converged and can be used to judge the accuracy of the coarser resolution simulations. For the propagating detonation wave in the cylinder test, at 200 microns the detonation speed is low by a couple of per cent. In terms of cpu-hours, the finest resolution is about a factor of 100 more expensive than the coarsest resolution.

There is a definite inconsistency between the classical ZND theory of detonation and contemporary experimental observations and attempts to model real detonation waves. Nevertheless, the classical onedimensional model of detonation is still extensively used in interpreting measurements because of its simplicity and physical clarity. This naturally raises the questions, what does actually the classical model represent and how one can relate the real multidimensional wave structure to an effective onedimensional detonation? The answer to these questions is extremely important from the practical point of view, because it defines how simple the solution of the criticality problems in detonation could be (i.e. whether one has to solve the very cumbersome three-dimensional nonsteady gasdynamic problem with detailed kinetics of the chemical reaction comprising a few hundreds of elementary steps to predict such parameters as critical initiation energy, critical distances, and concentrations, or the solution of a simplified onedimensional gasdynamic problem with global chemical kinetic equations would suffice?). The present communication discusses the following topics with the aim of answering partly the above questions. (1) An analysis of the detailed kinetic calculations of the heat evolution rate in some fuel-air mixtures and comparison of these results with experiments reveals that (a) in the majority of detonable mixtures heat release can be described by global equations with an accuracy quite sufficient for assessing the critical detonation parameters, (b) chemical kinetics shows no so-called recombination zones with an appreciable heat evolution that follow the main reaction zone, (c) the ratio between the induction and the explosion (within which the major fraction of the stored energy is evolved) times exceeds unity under the conditions corresponding to detonation waves in fuel-air mixtures, which contradicts the shock tube measurements. All these results can be reasonably explained by the fact that the chemical reaction in detonation waves (in shock waves as well) proceeds in a gas with a highly inhomogeneous distribution of the parameters (temperatures, pressures, and particle velocity) which gives rise to the so-called hot-spot mechanism of chemical reactions in gaseous and two-phase media. It is shown that the hot spot ignition is the main reason of instability of detonation waves. Nonsynchronous mixture ignition at different points produces the effect of extended heat evolution zone and reduces the effective induction zone. Results of measurements of averaged and local parameters behind detonation waves in gases support the idea that the effective heat evolution rate in the onedimensional representation of the wave is much more complicated than that inherent in the chemical reaction alone, because it contains also a significant contribution of the kinetic and thermodynamic energy redistribution within the major reaction zone and downstream of it, which makes the onedimensional structure of detonation waves deviate significantly from the classical pattern leading to nonmonotonic heat release behind the shock front (the possibility of appearance of two or more effective sonic planes and heat rlease maxima). However, a onedimensional analysis of marginal detonation waves has demonstrated that critical parameters can be estimated quite accurately within simplified models, which is attributable to the fact that the criticality is associated with local termination of the reaction within narrow stream tubes where the flow pattern resembles closely the ZND non-CJ detonation wave structure. Unlike the chemical reaction zone in detonations far away from the limit, which is shorter than the detonation cell size, the reaction zone in marginal detonations is comparable with the cell size. The results of onedimensional calculations of critical diameters and minimal energies of direct initiation of detonation are compared with experim

The non-standard Hamiltonian system, also referred to as a partial Hamiltonian system in the literature, of the form q ˙ i = ∂ H ∂ p i , p ˙ i = − ∂ H ∂ q i + Γ i ( t , q i , p i ) ${\dot q^i} = \frac{{\partial H}}{{\partial {p_i}}},{\text{ }}{\dot p^i} = - \frac{{\partial H}}{{\partial {q_i}}} + {\Gamma ^i}(t,{\text{ }}{q^i},{\text{ }}{p_i})$ appears widely in economics, physics, mechanics, and other fields. The non-standard (partial) Hamiltonian systems arise from physical Hamiltonian structures as well as from artificial Hamiltonian structures. We introduce the term ‘artificial Hamiltonian’ for the Hamiltonian of a model having no physical structure. We provide here explicitly the notion of an artificial Hamiltonian for dynamical systems of ordinary differential equations (ODEs). Also, we show that every system of second-order ODEs can be expressed as a non-standard (partial) Hamiltonian system of first-order ODEs by introducing an artificial Hamiltonian. This notion of an artificial Hamiltonian gives a new way to solve dynamical systems of first-order ODEs and systems of second-order ODEs that can be expressed as a non-standard (partial) Hamiltonian system by using the known techniques applicable to the non-standard Hamiltonian systems. We employ the proposed notion to solve dynamical systems of first-order ODEs arising in epidemics.

Thin reaction zone and distributed reaction zone regimes in turbulent premixed methane/air flames: Scalar distributions and correlations

At the 2011 APS conference, Terrones, Burkett and Morris published an experiment primarily designed to allow examination of the propagation of a detonation front in a 3-dimensional charge of PBX9502 insensitive high explosive (IHE). The charge is confined by a cylindrical steel shell, has an elliptical tin liner, and is line-initiated along its length. The detonation wave must propagate around the inner hollow region and converge on the opposite side. The Detonation Shock Dynamics (DSD) model allows for the calculation of detonation propagation in a region of explosive using a selection of material input parameters, amongst which is the D(K) relation that governs how the local detonation velocity varies as a function of wave curvature. In this paper, experimental data are compared to calculations using the 2D DSD and newly-developed 3D DSD codes at AWE with a variety of D(K) relations.

Measurements of a CJ‐detonation wave in PBX 9501 with a VISAR technique have shown a classical ZND profile for the reaction zone. This is compatible with one‐dimensional simulations using realistic equations of state and an Arrhenius reaction rate fit to available data from other experiments. Moreover, the reaction zone width is less than the average grain size in the PBX. In contrast to initiation, which requires hot spots, the reaction rate from the bulk shock temperature is sufficiently high for propagating a detonation wave. This raises questions with burn models used for both ignition and propagation of detonation waves.

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.

10 - Applications of Systems of Ordinary Differential Equations

Rate-ratio asymptotic analysis of the influence of addition of carbon monoxide on the structure and mechanisms of extinction of nonpremixed methane flames with comparison to experiments

Measurements are described of the reflexion and transmission of 3 cm wavelength microwaves b y detonation waves in stoichiometric oxyhydrogen seeded with nitrogen, and oxyacetylene mixtures. Conductivity measurementsare also obtained by means of two types of d.c. probe, termed ‘axial’ and ‘gap ’ probe respectively. The observed reflexion coefficient in the oxyhydrogen waves is approximately tenfold greater than the calculated value assuming a step profile and equilibrium ionization at the detonation front. The character of the gap probe records indicates that regions of higher ionization occur along the detonation tube axis at locations which correspond to the intersection points o f the oblique shock waves generated b y the wavefront and the w all boundary layer. It is shown that the enhanced power reflexion can be explained in terms of these regions. Microwave absorption and d.c. axial probe measurements give values of conductivity in reasonable agreement with theory, assuming the absence of OH ions. The measured values o f reflected pow er and conductivity in oxyacetylene waves show no anomaly for initial gas pressures in the range 760 to roughly 300 m m H g. A t lower pressures the reflexion coefficient is w ell in excess o f the calculated value and both m icrow ave and d.c. probes exhibit a frontal ionization peak. This is in accord with the observation, by other investigators, o f chemiluminescence in the reaction zone of waves in organic fuel mixtures and chemi-ionization process in the reaction zone of flames. A value for the ionelectron recombination rate constant is derived which is in agreement with that observed in flames.

JWL`s and 1-D Look-up tables are shown to work for ``one-track`` experiments like cylinder shots and the expanding sphere. They fail for ``many-track`` experiments like the compressed sphere. As long as the one-track experiment has dimensions larger than the explosive`s reaction zone and the explosive is near-ideal, a general JWL with R{sub 1} = 4.5 and R{sub 2} = 1.5 can be constructed, with both {omega} and E{sub o} being calculated from thermochemical codes. These general JWL`s allow comparison between various explosives plus recalculation of the JWL for different densities. The Bigplate experiment complements the cylinder test by providing continuous oblique angles of shock incidence from 0{degrees} to 70{degrees}. Explosive reaction zone lengths are determined from metal plate thicknesses, extrapolated run-to-detonation distances, radius size effects and detonation front curvature. Simple theories of the cylinder test, Bigplate, the cylinder size effect and detonation front curvature are given. The detonation front lag at the cylinder edge is shown to be proportional to the half-power of the reaction zone length. By calibrating for wall blow-out, a full set of reaction zone lengths from PETN to ANFO are obtained. The 1800--2100 K freezing effect is shown to be caused by rapid cooling of the product gases. Compiled comparative data for about 80 explosives is listed. Ten Chapters plus an Appendix.