Study of a Model Equation in Detonation Theory: Multidimensional Effects

Abstract

Abstract. We extend the reactive Burgers equation presented in [13,5] to include multidimensional effects. Furthermore, we explain how the model can be rationally justified following the ideas of the asymptotic theory developed in [6]. The proposed model is a forced version of the unsteady small disturbance transonic flow equations. We show that for physically reasonable choices of forcing functions, traveling wave solutions akin to detonation waves exist. It is demonstrated that multidimensional effects play an important role in the stability and dynamics of the traveling waves. Numerical simulations indicate that solutions of the model tend to form multi-dimensional patterns analogous to cells in gaseous detonations.

A minimal hyperbolic system for unstable shock waves

“…The detonation speed, D, which is generally unknown for time-dependent solutions, appears explicitly in (12)(13) in the shock-attached frame. Therefore, a method is needed to compute D when solving the system numerically.…”

“…For smooth flow, we integrate the shock-evolution equation ( 26) in time simultaneously with (12)(13), approximating the velocity gradient appearing in (26) based on the following second-order approximation on the stencil {x N −1 , x N , x s }:…”

“…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.…”