Role of gasdynamic wave fluctuations in shock induced ignition and transition to detonation: the hotspot cascade mechanism

The last stage of the deflagration-to-detonation transition (DDT) is complex due to the simultaneous role of compressible turbulent transport and auto-ignition phenomena [1–3]. In the present study, we focus solely on the gasdynamic effects, which may generate hotspots and facilitate the transition to detonation. We hence address the question of the role of mechanical fluctuations of different wave strengths and frequencies on the DDT process in the absence of diffusive (laminar or turbulent) phenomena. The problem we address is the classical shock induced ignition problem of a piston driven shock wave into a reactive medium and its subsequent transition to detonation. At time zero, a piston acquires a finite velocity, driving a shock into the initially quiescent reactive gas. The ensuing exothermicity triggered by the lead shock couples with the dynamics of the gas and eventually forms a detonation wave. In the absence of fluctuations, this problem is relatively well understood, as a significant effort has been devoted to clarify the coupling between gasdynamics and reactive phenomena in simple chemical models (one, two and three steps) in numerical simulations and analysis exploiting certain limiting conditions using asymptotic methods [4–8]. Previous work has identified that the most critical phenomenon in the acceleration process is the Zeldovich-Lee gradient mechanism, captured analytically in the work of Short and Sharpe [6, 7] of ignition in arbitrary gradients of induction delay. In the problem of shock induced ignition, the gradient set-up by the leading shock always gives rise to a subsonic fast flame, which may accelerate if the induction time gradient was smaller. In the present study, we allow the piston to have a non-constant speed. Its modulation at arbitrary frequencies and amplitudes drives a series of non-linear gas dynamic disturbances. We study their effect on the ignition process and transition to detonation. Note that the 1D problem solved is also equivalent to a shock induced ignition behind a reflected shock, the disturbances accounting for non-ideal effects such as gasdynamic waves generated at the boundaries by shear layers or flow-boundary layer interactions; see, for example the work of Khokhlov [9].