Mode selection in weakly unstable two-dimensional detonations

Abstract

A formulation of the reactive Euler equations in the shock-attached frame is used to study the two-dimensional instability of weakly unstable detonation through direct numerical simulation. The results are shown to agree with the predictions of linear stability analysis. Comparisons are made with linear perturbation growth rates and oscillation frequencies as a function of transverse disturbance wavelength. The perturbation eigenfunctions predicted by linear stability analysis are directly validated through numerical simulation. Three regimes of unstable behavior -linear, weakly nonlinear, and fully nonlinear -are explored and characterized in terms of the power spectrum of the normal shock velocity for a Chapman-Jouguet detonation with weak heat release. IntroductionNumerical simulations of detonation have been conducted by many researchers over the years. The earliest simulations of one-dimensional pulsating detonation were performed by Fickett and Wood in 1966 [1]. In 1978, Taki and Fujiwara [2] performed the first two-dimensional simulations of cellular detonation. As computational capacity has grown, simulations have been made that incorporate higher grid resolution and more detailed models. Despite the significant advances made, however, many fundamental problems remain to be investigated further. In particular, the physical nature of the detonation instability, nonlinear growth of the instability, physical mechanisms of cell formation and the nature of cell dynamics demand further elucidation theoretically as well as by means of accurate numerical simulation. Our present paper is concerned primarily with the latter.Recently, methods of simulating shocks and detonations in a reference frame attached to the shock surface have been developed [3,4]. Rather than attempting to capture the evolving shock front numerically, as in conventional simulations, these shock-attached frame methods map the shock surface to a fixed location in the computational domain and solve for the shock state as part of the numerical solution. These methods allow a sharp representation of the lead shock, avoid the numerical smearing inherent in shock capturing methods, and can be much more accurate than conventional methods of simulation. These methods are ideal for the study of detonation stability since they are posed in the same frame

Evolution of cellular structures on oblique detonation surfaces

“…It is nevertheless found that an equivalent resolution of 32 pts/l 1/2 is sufficient for the mixture and flow conditions used in the present study of unstable oblique detonation and hence, this resolution is used in all later simulations. Results using this same degree of resolution have also been reported in recent normal detonation studies [24,18]. Oblique detonations are overdriven, so on the direction vertical to the surface, there are fewer grids in half reaction length.…”

Numerical study on unstable surfaces of oblique detonations

“…However, it should not be concluded that this level of resolution is sufficient for obtaining quantitative values and suitable for all unstable detonation simulations. For cases with highly irregular detonation cells and for numerical studies aiming to predict the correct quantitative value of the growth rate, frequencies, cell spacing, stability boundary, etc., much higher resolutions are likely to be required (Short & Stewart 1998;Sharpe & Falle 2000a;Sharpe & Quirk 2008;Taylor, Kasimov & Stewart 2009). In fact, for very unstable detonations with highly irregular structure, the notion of 'grids per reaction zone length', i.e.…”

Numerical investigation on the induction zone structure of the oblique detonation waves

“…Previous researchers use the grid number in half reaction length to evaluate the resolution of detonation simulations, and in recent studies the resolution of 32 grids half reaction length is usually used [22,23]. Unfortunately, we use the detailed reaction mechanism, which is hard to give the half reaction length because there is no ''reaction progress'' in these mechanisms.…”