Theoretical and Numerical Structure for Unstable One-Dimensional Detonations

Abstract

The spatio-temporal structure of unstable detonations in a single space dimension is studied through a combination of numerical and asymptotic methods. A new high resolution numerical method for computing unstable detonations is developed. This method combines the piecewise parabolic method (PPM) with conservative shock tracking and adaptive mesh refinement. A new nonlinear asymptotic theory for the spatio-temporal growth of instabilities is also developed. This asymptotic theory involves a nonclassical “Hopf bifurcation”, because resonant acoustic scattering states with exponential growth in space cross the imaginary axis and become nonlinear eigenmodes in a complex free-boundary problem for a nonlinear hyperbolic equation. An interplay between the asymptotic theory and numerical simulation is used to elucidate the spatio-temporal mechanisms of nonlinear stability near the transition boundary; in particular, a quantitative-qualitative explanation is developed for the experimentally observed instabilities for supersonic blunt bodies advancing into appropriate reactive mixtures. The new numerical method is also used to predict regimes of multimode and chaotic pulsation instabilities. This numerical method is tested thoroughly on a classical unstable detonation problem. Along the way, a surprising number of nonphysical numerical artifacts are documented for other high-resolution methods applied to the classical test problem. This illustrates the need for the high-resolution scheme developed here.