Stability of One-Dimensional Steady Flows with Detonation Wave in a Channel of Variable Cross-Sectional Area

The stability of one-dimensional steady flows of an ideal (inviscid and non-heat-conducting) gas in channels of variable cross section with combustion in a structurally stable detonation wave is studied. The detonation wave represents a discontinuity surface propagating normally to the axis of the channel. It was previously established that, in this formulation, steady flows with combustion in a Chapman-Jouguet detonation wave are always unstable, in contrast to combustion flows in an overcompressed detonation wave. The stability analysis of such flows is reduced to the numerical solution of an initial-boundary value problem describing the evolution of finite flow perturbations between the moving detonation wave and the minimal nozzle cross section (in the case of a sudden contraction) or the exit nozzle cross section. The problem is solved by applying a modified Godunov scheme of higher order accuracy. More specifically, a Riemann solver with switching (between the overcompressed and Chapman-Jouguet detonation waves) with an explicitly represented detonation wave is created. Examples of stable flows and flows collapsing due to large initial perturbations are given, and their dynamics with transitions to a Chapman-Jouguet detonation wave and back are computed.