The Instability of Steady-State Flows with a Chapman—Jouguet Detonation Wave in a Channel of Variable Cross Section

Abstract

This work investigates the stability of steady flows of ideal (inviscid and nonconductive) gas in channels with variable cross sections, in the diverging part of which a Chapman-Jouguet detonation wave (DWCJ) is located. Steady and unsteady flows are described by equations in a one-dimensional approximation with the detonation wave represented by a planar discontinuity surface, normal to the channel axis. The combustible mixture before the detonation wave and the combustion products behind it are perfect gases with constant heat capacities, which, like additive constants in terms of the internal energy and enthalpy, are different before and after the wave. Since the Mach number behind the DWCJ is equal to unity, it can stand only in the diverging part of the channel in the steady flow. Moreover, depending on the conditions at the channel exit, the flow behind it can be either super- or subsonic. In the first case, the initial Pertubation can reach the DWCJ only from the left (the gas moves from left to right), and, in the second case it can come from both sides. When investigating the stability, is it is assumed, first, that after the initial perturbation, the detonation wave, which is slightly shifted, remains a DWCJ. In this case, analysis of stability reduces to calculating the derivative of the Mach number of DWCJ from the Mach number of the steady flow in front of the wave. In this formulation, the derivative value is always such that the flow under consideration is unstable. Small perturbations of the DWCJ make it slightly overcompressed. However, even taking this into account, a steady flow with a DWCJ in a diverging channel is always unstable.