Study of a model equation in detonation theory: multidimensional effects

TL;DR

This paper extends a reactive Burgers equation to include multidimensional effects, demonstrating that such effects influence the stability and pattern formation of detonation waves through analytical and numerical methods.

Contribution

It introduces a multidimensional extension of the reactive Burgers equation and provides a rational justification based on asymptotic theory, highlighting the role of multidimensional effects in detonation dynamics.

Findings

Multidimensional effects significantly influence detonation wave stability.

Traveling wave solutions resemble physical detonation waves.

Numerical simulations show formation of multi-dimensional cellular patterns.

Abstract

We extend the reactive Burgers equation presented in Kasimov et al. Phys. Rev. Lett., 110 (2013) and Faria et al. SIAM J. Appl. Maths, 74 (2014), to include multidimensional effects. Furthermore, we explain how the model can be rationally justified following the ideas of the asymptotic theory developed in Faria et al. JFM (2015). 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.