Effect of spatial discretization of energy on detonation wave propagation

Abstract

Detonation propagation in the limit of highly spatially discretized energy sources is investigated. The model of this problem begins with a medium consisting of a calorically perfect gas with a prescribed energy release per unit mass. The energy release is collected into sheet-like sources that are now embedded in an inert gas that fills the spaces between them. The release of energy in the first sheet results in a planar blast wave that propagates to the next source, which is triggered after a prescribed delay, generating a new blast, and so forth. The resulting wave dynamics as the front passes through hundreds of such sources is computationally simulated by numerically solving the governing one-dimensional Euler equations in the lab-fixed reference frame. Two different solvers are used: one with a fixed uniform grid and the other using an unstructured, adaptively refined grid enabling the limit of highly concentrated, spatially discrete sources to be examined. The two different solvers generate consistent results, agreeing within the accuracy of the measured wave speeds. The average wave speed for each simulation is measured once the wave propagation has reached a quasi-periodic solution. The effect of source delay time, source energy density, specific heat ratio, and the spatial discreteness of the sources on the wave speed is studied. Sources fixed in the lab reference frame versus sources that convect with the flow are compared as well. The average wave speed is compared to the ideal Chapman-Jouguet (CJ) speed of the equivalent homogenized media. Velocities in excess of the CJ speed are found as the sources are made increasingly discrete, with the deviation above CJ being as great as 15%. The deviation above the CJ value increases with decreasing values of specific heat ratio γ. The total energy release, delay time, and whether the sources remain lab-fixed or are convected with the flow do not have a significant influence on the deviation of the average wave speed away from CJ. A simple, ad hoc analytic model is proposed to treat the case of zero delay time (i.e., source energy released at the shock front) that exhibits qualitative agreement with the computational solutions and may explain why the deviation from CJ increases with decreasing γ. When the sources are sufficiently spread out so as to make the energy release of the media nearly continuous, the classic CJ solution is obtained for the average wave speed. Such continuous waves can also be shown to have a time-averaged structure consistent with the classical ZND structure of a detonation. In the limit of highly discrete sources, temporal averaging of the wave structure shows that the effective sonic surface does not correspond to an equilibrium state. The average state of the flow leaving the wave in this case does eventually reach the equilibrium Hugoniot, but only after the effective sonic surface has been crossed. Thus, the super-CJ waves observed in the limit of highly discretized sources can be understood as weak detonations due to the...

Propagation of gaseous detonation waves in a spatially inhomogeneous reactive medium

“…The generalized-CJ condition, i.e., a vanishing thermicity (φ = 0) at the average sonic point (u * + c * = 0), is satisfied owing to the balance between the exothermic chemical reaction and the mechanical and thermal fluctuations. The finding of this study incorporating a more realistic, state-dependent reaction model complements the previous study by Mi et al [12], verifying that the classic CJ criterion assuming a homogeneous medium based on averaged properties is not always applicable to predict the wave propagation speed in a spatially inhomogeneous system, and further suggesting that the resulting super-CJ propagation is independent of the particular energy deposition mechanism.…”

“…The forward running blast triggers the next source, so the wave propagates via a mechanism of sequentially initiated blast waves by the point sources, which can be qualitatively captured by the heuristic model based on the construction of point-source blast solutions in the Appendix of Ref. [12]. Since the variation of V avg as a function of L is between the same asymptotic limits as those of Γ, the underlying mechanisms at these limits must have an equivalent effect on the wave prop-agation.…”

“…Answering the above-mentioned question was recently attempted by Mi et al [12] In their study, the propagation speed of a detonation, resulting from a medium that consists of spatially discretized energy sources separated by regions of inert material, was examined via one-dimensional, direct numerical simulations. In the cases of highly discretized energy sources, the resulting detonation velocity was observed to be greater than the predicted CJ velocity of a homogenized medium with the same amount of energy release (for the case of ratio of a specific heat capacity γ = 1.1, the average wave speed was nearly 15% greater than the CJ speed).…”

“…1(b), via concentrating the reactant into layers (or sheets), standing perpendicular to the direction of detonation wave propagation, separated by regions of inert gas. This arrangement of discrete sources is similar to that used in [12] and can be implemented in both one-and two-dimensional simulations. Another way to discretize the reactive medium is by concentrating the reactant into infinitely long square-based prisms laying along an axis that is perpendicular to the direction of detonation wave propagation, as shown in Fig.…”

“…The first objective of this study is to examine whether the super-CJ wave propagation, which was identified in [12] for the cases with highly discrete sources, still occurs in a one-or two-dimensional gaseous detonation system with state-dependent Arrhenius kinetics. The simulation results are then analyzed via a spatio-temporal averaging procedure to further elucidate the physical mechanism that is responsible for this super-CJ wave speed.…”