Study of a Model Equation in Detonation Theory

Abstract

Here we analyze properties of an equation that we previously proposed to model the dynamics of unstable detonation waves [A. R. Kasimov, L. M. Faria, and R. R. Rosales, Model for shock wave chaos, Phys. Rev. Lett., 110 (2013), 104104]. The equation is utIt describes a detonation shock at x = 0 with the reaction zone in x < 0. We investigate the nature of the steady-state solutions of this nonlocal hyperbolic balance law, the linear stability of these solutions, and the nonlinear dynamics. We establish the existence of instability followed by a cascade of period-doubling bifurcations leading to chaos. Introduction.A detonation is a shock wave that propagates in a reactive medium where exothermic chemical reactions are ignited as a result of the heating by the shock compression. The energy released in these reactions, in turn, feeds back to the shock in the form of compression waves and thus sustains the shock motion. The dynamics of such shock-reaction coupling is highly nonlinear due to the sensitivity of the chemical reactions to temperature, making the problem significantly more challenging than shock dynamics in nonreactive media. A steady planar detonation wave is rarely observed in experiments. Complex time-dependent and multidimensional structures tend to develop [14,26]. Numerical simulations of the equations of reactive gas dynamics are able to reproduce at a qualitative level the complex structures observed in experiments (see, e.g., [40,1,30]). However, obtaining physical insights into the basic mechanisms of the instability requires simplified modeling and remains challenging.In one dimension, the instabilities of the reactive shock wave manifest themselves in the form of a "galloping detonation" [15,14], wherein the shock speed oscillates around its steady value. It has been shown through extensive numerical experiments that as the activation energy, E, a parameter in the equations measuring the temperature sensitivity of the chemical reactions, is varied, the shock speed transitions from a constant to an oscillatory function. Further increase of E leads to a period-doubling bifurcation cascade, which ultimately results in the shock moving at a chaotic speed [29,20]. The mechanism for such instabilities is still not completely understood.In this paper, we show that the model introduced in [23], which consists of a single nonlocal partial differential equation (PDE), is capable of reproducing the complexity observed in one-dimensional simulations of reactive Euler equations. The model possesses traveling wave solutions precisely analogous to the ZND theory (named after Zel'dovich [44], von Neumann [41], and Döring [6], who independently developed the

Study of a Model Equation in Detonation Theory: Multidimensional Effects

“…In this section, we investigate the one-dimensional traveling wave solutions of (2.3-2.4), as well as their stability properties. Note that, since in the one-dimensional case the proposed system reduces to equation (2.1), the traveling wave solutions are identical to those found in [5]. For completeness, we briefly repeat the discussion here.…”

“…The importance of this latter development stems from the fact that unsteadiness is a common feature of gaseous detonations. Along the same lines of analog modeling, in [13,5] we have shown that even a scalar forced Burgers equation contains all the ingredients necessary to reproduce the complexity of one-dimensional detonations, including complicated chaotic solutions. Importantly, all the prior work with analog models is limited to one-dimensional descriptions.…”

“…Therefore, it cannot capture the important effects played by transverse shock waves, which lead to the generation of cellular patterns in gaseous detonations [9,14]. The purpose of this paper is to extend our existing one-dimensional model [13,5] by introducing a multi-dimensional analog of detonations. With this extended model we show, through stability analysis and numerical simulations, that some multi-dimensional detonation properties are amenable to descriptions significantly simpler than the reactive Euler equations.…”

Detailed simulation of the pulsating detonation wave in the shock-attached frame

“…It was also shown in [4] using detonation analogies that the resolution of the chemical reaction zone associated with a pulsating DW is a challenging problem even in the one-dimensional statement because the size of the reaction zone varies by orders of magnitude in the process of pulsations due to the occurrence of local explosions behind the LW front. The issue of a sufficient level of resolution of the chemical reaction zone in numerical experiments on the DW propagation using shock capturing schemes remains open, and the number of papers devoted to this problem is steadily increasing (e.g., see the references in [3]).…”

A minimal hyperbolic system for unstable shock waves

“…The Fickett model [13] has subsequently motivated various extensions and modifications [34,43,25,10,12,9]. The principal aim of all of these works is to identify a minimal model that is capable of reproducing the rich set of dynamical properties of the full system of the reactive Euler equations.…”