| Various regimes for unstable detonations are studied numerically using a new high resolution method. The need for such a method is established in a preliminary review of a series of conventional shock capturing methods applied to a simple model for detonations. A variety of numerical artifacts are observed resulting from the poor numerical treatment of the interaction between the leading shock and the stiff chemistry at the front; that none of those methods will meet our requirement for accurate and efficient computation of unstable detonations, especially in several space dimensions. A new method is designed to that purpose, combining a higher order Godunov scheme with conservative front tracking and adaptive mesh refinement. We use the new scheme to study the spatio-temporal structure of unstable detonations in one space dimension. In the transition regime, we observe excellent quantitative agreement between the numerical solutions and the theoretical prediction using a nonlinear asymptotic approach. Further away from the transition region, we observe multi-mode and chaotic behaviors. The numerical results provide with a qualitative explanation of the "galloping" instability mechanism observed with supersonic blunt bodies propagating in an explosive gas. In two space dimensions, we investigate the formation of unstable cells and Mach stems, a typical regime in physical experiment with reactive gas. The very complex structure within each cell is captured in detail by the numerical scheme: the computed triple points display good agreement with experimental data. We compare the observed cell spacing with the spacings computed according to various theories (linear stability and high frequency nonlinear geometric optics): the theories, based on small amplitude perturbations, fail to provide with quantitatively correct spacings for the complex, large amplitude instabilities observed in the numerical experiments. We also study two-dimensional instabilities for condensed phases. A new class of "pulsating" instabilities is observed, with fluctuations on the length scale of the reaction length. Close to the transition, the direct simulations show good agreement with the detailed spatial structure as well as the saturation and multi-mode interactions predicted by a nonlinear low frequency asymptotic approach. |
