Simulation of Shock-Initiated Ignition

The scenario of detonative ignition in shocked mixture is quite relevant to hydrogen safety, because hydrogen is prone to detonation and shock reflections may result in deflagration to detonation transition. However, even in one dimension, simulation of ignition between a contact surface or a flame and a shock moving into a combustible mixture is difficult because of the mathematical singularity present in the initial condition. Indeed, initially, as the shock starts moving into reactive mixture, the region filled with reactive mixture has zero thickness. Thus, on a fixed grid, the number of grid points between the shock and the contact surface increases as the shock moves away from the latter. Staircasing (the resulting plots will be functions composed of sets of equally spaced jumps of equal length) takes place and it will be amplified by the chemistry which is very sensitive to temperature, leading to unreliable results. In the current work, the formulation is transformed, using time and length over time as the independent variables. This frame of reference corresponds to the self-similar formulation in which the non-reactive problem remains stationary. Thus the initial singularity is removed and the initial process is well-resolved. The numerical solution uses an Essentially Non-Oscillatory algorithm, which is adequate not only for the early part of the process, but also for the latter part, when chemistry leads to appearance of a shock and eventually a detonation wave is formed.