Transition to longitudinal instability of detonation waves is generically associated with Hopf bifurcation to time-periodic galloping solutions

TL;DR

This paper demonstrates that the transition to longitudinal instability in strong detonation waves is typically linked to a Hopf bifurcation leading to time-periodic galloping solutions, supported by stability analysis and numerical verification.

Contribution

It provides the first complete nonlinear stability analysis for strong detonations in reacting Navier--Stokes equations, identifying bifurcation conditions via Evans function and linking instability to Hopf bifurcation.

Findings

Transition to instability associated with Hopf bifurcation.

Numerical verification of stability and bifurcation conditions.

First nonlinear stability result for strong detonations as amplitude approaches zero.

Abstract

We show that transition to longitudinal instability of strong detonation solutions of reactive compressible Navier--Stokes equations is generically associated with Hopf bifurcation to nearby time-periodic "galloping", or "pulsating", solutions, in agreement with physical and numerical observation. In the process, we determine readily numerically verifiable stability and bifurcation conditions in terms of an associated Evans function, and obtain the first complete nonlinear stability result for strong detonations of the reacting Navier--Stokes equations, in the limit as amplitude (hence also heat release) goes to zero. The analysis is by pointwise semigroup techniques introduced by the authors and collaborators in previous works.