Highly Accurate Numerical Simulations of Pulsating One-Dimensional Detonations

Abstract

A novel, highly accurate numerical scheme based on shock-fitting coupled with high order spatial and temporal discretization is applied to a classical unsteady detonation problem to generate solutions with unprecedented accuracy. The one-dimensional reactive Euler equations for a calorically perfect mixture of ideal gases whose reaction is described by single-step irreversible Arrhenius kinetics are solved in a series of calculations in which the activation energy is varied. In contrast with nearly all known simulations of this problem, which converge at a rate no greater than first order as the spatial and temporal grid is refined, the present method is shown to converge at a rate consistent with the fifth order accuracy of the spatial discretization scheme. This high accuracy enables more precise verification of known results and prediction of heretofore unknown phenomena. To five significant figures, the scheme faithfully recovers the stability boundary, growth rates, and wave-numbers predicted by an independent linear stability theory in the stable and weakly unstable regime. As the activation energy is increased, a series of period-doubling events are predicted, and the system undergoes a transition to chaos. Consistent with general theories of non-linear dynamics, the bifurcation points are seen to converge at a rate for which the Feigenbaum constant is 4.66 ± 0.09, in close agreement with the true value of 4.669201 . . .. As activation energy is increased further, domains are identified in which the system undergoes a transition from a chaotic state back to one whose limit cycles are characterized by a small number of non-linear oscillatory modes. This result is consistent with behavior of other non-linear dynamical systems, but not typically considered in detonation dynamics.

Simulations of pulsating one-dimensional detonations with true fifth order accuracy

“…In this paper, we describe and implement a new algorithm tailored to simulate a well established model problem: one-dimensional unsteady detonation; this study gives full details of recently reported results [1]. The new predictions are many orders of magnitude more precise than those previously published; this enhanced precision enables the prediction of remarkable new bifurcation phenomena.…”

“…This scheme is a conservative flux difference method, which has been shown to be stable, and yields the proper viscosity-vanishing solution to Eqs. (1). The derivation of the difference operator LðuÞj x¼xi in Eq.…”

Exact Solutions for Two-Dimensional Reactive Flow for Verificaiton of Numerical Algorithms

“…Moreover, one can turn to studies from the shock fitting literature to find convergence rates consistent with the nominal order of the discretization method. For example, the recent study of Brooks and Powers 29 shows spectral convergence to machine accuracy on a coarse 65 × 33 grid for a blunt body re-entry problem, and the results of Henrick, et al 30 exhibit true fifth order accuracy which gives rise to predictions of detonation instabilities which agree remarkably well with those of linear stability theory.…”

“…Equations (30)(31)(32)(33) have full equivalents in Refs. 10-12, 14 and 16, albeit in slightly different notation.…”