Analysis of Numerical Simulations of Detonation Diffraction

Abstract

We investigate the problem of a self-sustaining detonation wave diffracting from a tube into an unconfined space through an abrupt area change. The expansion associated with a detonation transitioning from planar to spherical or cylindrical symmetry can cause the detonation to fail. The dominant factors in determining if failure occurs are the combustible mixture composition, initial thermodynamic state, and confining geometry dimensions. In the simplest concept of detonation failure, the decoupling of the reaction zone from the shock front is due to competition of flow processes, unsteadiness and spatial gradients, with chemical processes. We examine the specific mechanisms for detonation failure in diffraction of an initially stable (no transverse waves) detonation around a 90° corner. We perform two-dimensional numerical simulations in a Cartesian (planar) geometry for an ideal gas with irreversible, one-step Arrhenius chemistry. The convective flux is computed by Roe's approximate Riemann solver with Glaister's implementation for a general equation of state and an extension for multi-species gases in chemical non-equilibrium. The °ow solver was linked to the GrACE (Grid Hierarchy Adaptive Computational Engine) library, which provides a grid-adaptive parallel environment for structured patch integrators. The shock position and curvature are tracked as a function of time along axes of interest, and snapshots of the entire flow field are available for numerical analysis. We have examined three aspects of the diffraction problem. First, we have investigated the propagation of disturbances, created by the diffraction, along the detonation front. Second, we have studied the evolution of the wavefront at the centerline of the channel. Third, we have examined what happens to the portion of the wave that propagates along the wall, away from the corner. By tracking the advance of the transverse spatial gradient in the unperturbed region behind the shock, we have shown that the leading edge of these disturbances corresponds to the trajectories of acoustic waves propagating inside the reaction zone. The trajectory of the head of the disturbance is almost a straight line, confirming that the use of the Skews' construction for the corner turning problem can be appropriate in the case of reactive flow when the correct sound speed is used. At the centerline, the competition between energy release, unsteadiness and expansion waves is crucial in determining whether the detonation will fail (sub-critical regime) or will successfully diffract through the area change (super-critical regime). Simplified criteria for critical diffraction models are based on the hypothesis of blast-like decay at the centerline. A first result from our simulations is the analysis of the time evolution of the front at the centerline. We observe a transient build-up of curvature and shock deceleration and we reduce these data through scaling by the width of the channel and the half-reaction zone length. The initial transient is followed by one or more oscillations in curvature and shock acceleration. In the simplest case of sub-critical diffraction (decoupling at the centerline without re-initiation), we observe a monotonic decrease of curvature from a peak value. We derive an ordinary differential equation in curvature from two-dimensional shock kinematics and, by evaluating each term numerically, we show that the cylindrical blast decay is an asymptotic solution of this equation. At the wall, the activation energy Eₐ plays a major role in decoupling the shock from the reaction zone as soon as the detonation front starts to turn the corner. With zero activation energy, the effect of an abrupt area change is simply to decrease the strength of the shock, and the reaction zone length continuously stretches from the centerline to the wall. As Eₐ increases above a critical value, we observe an immediate decoupling at the wall, propagating along the detonation front toward the undisturbed centerline flow. We study this effect in a series of simulations where the only parameter varied is the activation energy and the half-reaction length is kept constant. For values of Ea small enough, the decoupling is stopped before reaching the centerline and local re-ignition occurs at an intermediate location along the front. Re-ignition propagates back toward the wall, and the detonation speed at the wall tends to recover the Chapman-Jouguet value. For values of Eₐ high enough, the decoupling reaches the centerline, and whether the detonation is successful or not is essentially a function of the width of the channel. For sufficiently small channel widths, the detonation will fail. By studying the time evolution of the detonation speed at the wall and snapshots of the flow field, we were able to bracket the critical value of activation energy that separates these two different behaviors.

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