Numerical simulations of pulsating detonations: I. Nonlinear stability of steady detonations

Abstract

Very-long-time numerical simulations of an idealized pulsating detonation with one irreversible reaction having an Arrhenius form are performed using a hierarchical adaptive second-order Godunov-type scheme. The initial data are given by the steady solution and the truncation error produces the perturbation to trigger the instability. The detonation is allowed to run for thousands of half-reaction times of the underlying steady wave to ensure that the final amplitudes and periods of the nonlinear oscillations are achieved. Thorough resolution studies are performed for various representative regimes of the instability. It is shown that to obtain quantitatively good solutions over 50 numerical grid points in the half-reaction length of the steady detonation are required, while to obtain a converged solution over 100 points are required, even near the stability boundary. This is much higher resolution than has generally been used in previous papers in either one or two dimensions. Resolutions of less than approximately 20 points per half-reaction length give very poor predictions of the periods and amplitudes near the stability boundary or entirely spurious solutions for more unstable detonations. The evolution of the converged solutions as the activation energy increases, and the detonation becomes more unstable, is also investigated.