Failure and reignition of one-dimensional detonations—The high activation energy limit

The structure of failed one-dimensional detonations is derived using a high activation energy analysis. On a time scale longer than the chemical time based upon the von Neumann temperature, high activation energy one-dimensional detonations break down into a weaker shock, a contact surface separating hot burned gases from a colder, unburned mixture, and an expansion wave. While this leading order solution is unaffected by chemistry, hence self-similar, the perturbation, accounting for the chemistry, which depends upon chemical times, is not. By accounting for the chemistry, the perturbation problem determines the delay until reignition of the detonation occurs. This problem is almost identical to the problem of initiation in the region between a shock and contact surface, which is created by the collision of two shock waves. The main difference between that problem and the current analysis is that the downstream boundary condition now consists of radiating acoustics into hot burned products, at the location of the surface discontinuity. When the Newtonian limit is applied, that is, for a ratio of the specific heats approaching unity, the hot spot at which reignition occurs approaches the location of the contact surface. The time and length to reignition are then found to vary exponentially with the activation energy of the mixture. However, the Newtonian limit is not a very realistic model, because it makes the interval between a Mach number of 1/√γ and 1 disappear: in this range of Mach numbers, adding energy to a steady flow lowers the temperature, hence the reaction rate.