On the two-dimensional instability of square-wave detonations

In order to elucidate the basic mechanisms responsible for the cells develo ping on detonation fronts, the two-dimensional instability of planar Chapma n-Jouguet detonations is studied by the use of a square-wave model. It has been shown that there are two main instability mechanisms: the high sensiti vity of the heat release rate to temperature and the hydrodynamic effect ar ising from the interface (reaction front) subject to the influence of the d eflection of streamlines across the perturbed shock. Since for ordinary gas eous detonations the density across the shock is very large, the instabilit y related to this pure hydrodynamic effect is very strong. An exact dispers ion relation is derived and it is shown that the instability of square-wave detonations is described by a differential-difference equation of advanced type, which has a set of an infinite number of unstable solution branches with the growth rate increasing with the number of modes. The nonlinear sol ution shows that the discontinuity develops from smooth initial data within a finite time. Therefore the results obtained from square-wave detonations cannot be applied directly to describe the observed instability patterns. By carrying out a Taylor development of the terms with advanced time, a dis persion relation of a polynomial form is obtained and the high-frequency in stability is eliminated. On the basis of the linear results with a regular reaction model, a phenomenological nonlinear equation of the fourth order f or the position of the detonation surface is obtained by carrying out a Tay lor series development. One-dimensional numerical solution of the phenomeno logical nonlinear equation shows typical nonlinear phenomena such as period ic oscillation, period doubling and dynamic quenching.