Nonlinear Theory of Unstable Two-Dimensional Detonation

A two-dimensional theory is presented for the structure of detonations for systems whose steady-state solutions are hydrodynamically unstable. Periodic boundary conditions are imposed in the direction transverse to the steady flow. The analysis, which depends on the assumption that the steady detonation is only “slightly” unstable, permits the time-dependent, two-dimensional flow to be described by a system of two, complex, autonomous, ordinary differential equations in the time. Both traveling-wave and standing-wave solutions are found to be possible, and their significance is discussed. The theory is applied to an ideal-gas, one-reaction system and solutions of both types are found to exist and to be stable for a variety of transverse periods within the unstable regime. With increasing heat of reaction, the magnitude of the perturbations within these waves increases rapidly.