Effect of spatial inhomogeneities on detonation propagation with yielding confinement

Abstract

The propagation of detonations in layers of reactive gas bounded by inert gas is simulated computationally in both homogeneous and inhomogeneous systems described by the two-dimensional Euler equations with the energy release governed by an Arrhenius rate equation. The thickness of the reactive layer is varied and the detonation velocity is recorded as the layer thickness approaches the critical value necessary for successful propagation. In homogeneous systems, as activation energy is increased, the detonation wave exhibits increasingly irregular cellular structure characteristic of the inherent multidimensional instability. The critical layer thickness necessary to observe successful propagation increases rapidly, by a factor of five, as the activation energy is increased from $E_\mathrm{a}/RT_0 = 20$–30; propagation could not be observed at higher activation energies due to computational limitations. For simulations of inhomogeneous systems, the source energy is concentrated into randomly positioned squares of reactive medium embedded in inert gas; this discretization is done in such a way that the average energy content and the theoretical Chapman–Jouguet (CJ) speed remain the same. In the limit of highly discrete systems with layer thicknesses much greater than critical, velocities greater than the CJ speed are obtained, consistent with our prior results in effectively infinite width systems. In the limit of highly discretized systems wherein energy is concentrated into pockets representing 10% or less of the area of the reactive layer, the detonation is able to propagate in layers much thinner (by an order of magnitude) than the equivalent homogeneous system. The critical layer thickness increases only gradually as the activation energy is increased from $E_\mathrm{a}/R T_0 = 20{-}55$, a behavior that is in sharp contrast to the homogeneous simulations. The dependence of the detonation velocity on layer thickness and the critical layer thickness is remarkably well described by a front curvature model derived from the classic, ZND-based model of Wood and Kirkwood. The results of discrete sources are discussed as a conceptual link to the behavior that is experimentally observed in cellular detonations with highly irregular cellular structure in which intense turbulent burning rapidly consumes detached pockets behind the main shock front. The fact that highly discrete systems are well described by classical, curvature-based mechanisms is offered as a possible explanation as to why curvature-based models are successful in describing heterogeneous, condensed-phase explosives.

Appendix: Numerical convergence study

For two selected values of moderately high activation energy considered, i.e., $E_\mathrm{a}=20$ and $E_\mathrm{a}=30$, numerical convergence tests were performed for the homogeneous cases ($\varGamma =1$) and the most highly discretized cases ($\varGamma =0.01$ and $L=10$). The convergence study for the resulting critical reactive layer thickness $h^{*}$ from these cases is shown as the “go” versus “no-go” charts plotted in Figs. 9 and 10.

Fig. 9

Numerical convergence study for the result of critical reactive layer thickness $h^{*}$ with $E_\mathrm{a}=20$ for a homogeneous cases and b inhomogeneous cases with $\varGamma =0.01$ and $L=10$

Fig. 10

Numerical convergence study for the result of critical reactive layer thickness $h^{*}$ with $E_\mathrm{a}=30$ for a homogeneous cases and b inhomogeneous cases with $\varGamma =0.01$ and $L=10$

Fig. 11

Numerical convergence study for the result of critical reactive layer thickness $h^{*}$ with an inhomogeneous medium with $\varGamma =0.01$ and $L=10$ for a $E_\mathrm{a}=40$ and b $E_\mathrm{a}=55$

In Figs. 9, 10, and 11, each symbol represents a case of one or several simulations with a reactive layer thickness h at a numerical resolution in terms of the number of computational cells per the ideal half-reaction-zone length $l_{1/2}/\varDelta x$. In these figures, a circle $\circ$ indicates a “go,” i.e., a case resulting in a self-sustained propagation; a cross $\times$ indicates a “no-go,” i.e., a case of propagation failure. The dashed line indicates the boundary between “go” and “no-go” results that defines the critical thickness $h^{*}$ as a function of numerical resolution. Considering the stochastic nature of the distribution of reactive sources in a highly inhomogeneous medium, for the near-limit cases, five simulations have been performed for the same value of h. Only if all of these five simulations result in a successful wave propagation over a distance that is more than approximately 150 times the average source spacing L, then the case with the corresponding h is considered as a “go.”

For the homogeneous case with $E_\mathrm{a}=20$ shown in Fig. 9a, as the numerical resolution was increased from $l_{1/2}/\varDelta x=10$ to 20, the result of $h^{*}$ increased by approximately $9\%$. As the numerical resolution was increased from $l_{1/2}/\varDelta x=20$ to 30, there was no change greater than $\pm 4\%$ in the result of $h^{*}$. For the highly discretized case with $E_\mathrm{a}=20, \varGamma =0.01$, and $L=10$ shown in Fig. 9b, there was no change greater than the prescribed average source spacing L in the result of $h^{*}$ as the resolution was increased from $l_{1/2}/\varDelta x=5$ to 30. For $E_\mathrm{a}=30$, the homogeneous case shown in Fig. 10a, as the numerical resolution was increased from $l_{1/2}/\varDelta x=5$ to 10, the result of $h^{*}$ increased by approximately $26\%$. As the numerical resolution was increased from $l_{1/2}/\varDelta x=10$ to 20, the change in the result of $h^{*}$ was less than $\pm 1\%$. As shown in Fig. 10b for the highly discretized case with $E_\mathrm{a}=30, \varGamma =0.01$, and $L=10$, there was no change greater than the prescribed average source spacing L in the result of $h^{*}$ as the resolution was increased from $l_{1/2}/\varDelta x=5$ to 20.

For the homogeneous cases with both $E_\mathrm{a}=20$ and $E_\mathrm{a}=30$ as shown in Figs. 9a and 10a, respectively, simulations at a relatively coarse resolution result in a smaller critical thickness. This reduction in $h^{*}$ could be attributed to the fact that the effect of numerical diffusion becomes more significant when the inviscid Euler equations are solved at coarser resolutions. In a cellular detonation structure that arises from a homogeneous reactive medium, there is a large amount of reactant that is shocked by the weak parts of the leading shock and undergoes a very slow burning process. The efficiency of this slow reaction process can be significantly enhanced by numerical diffusion. Simulations at coarse resolution likely result in an artificially (numerically) enhanced energy release rate and, thus, enables a detonation wave to propagate in a thinner reactive layer.

For the large values of activation energy $E_\mathrm{a}=40$ and 55, the simulations have been performed only for the highly inhomogeneous cases with $\varGamma = 0.01$ and $L=10$. As shown in Fig. 11a for the cases with $E_\mathrm{a}=40$, simulations at three different resolutions $l_{1/2}/\varDelta x=10, 20$, and 30 resulted in the same critical thickness. For the cases with the greatest value of activation energy considered in this study $E_\mathrm{a}=55$, a minimal resolution of $l_{1/2}/\varDelta x=20$ was required to obtain numerically converged result of $h^{*}$ as shown in Fig. 11b. Therefore, the results of $h^{*}$ as a function of $E_\mathrm{a}$ presented in Fig. 8 were all based on the simulations at a resolution of $l_{1/2}/\varDelta x=10$ except the data point for the $E_\mathrm{a}=55$, highly inhomogeneous case, which was performed at a resolution of $l_{1/2}/\varDelta x=20$.