Numerical Study of Smoothly Perturbed Shocks in the Newtonian Limit

Abstract

Previous weakly nonlinear analyses of strong shocks in the Newtonian limit have shown that the main characteristics of the cellular pattern of detonations, namely the network of triple points propagating in the transverse direction, are associated with nonlinear mechanisms which are inherent to the leading shock (Clavin and Denet, Phys. Rev. Lett. 88(4), 044,502, 2002; Clavin, J. Fluid Mech. 721, 324–339, 2013). Motivated by this theoretical analysis, experimental and numerical studies have been conducted on a smoothly perturbed Mach 1.5 shock in air, reflected from a sinusoidal wall of small amplitude (Jourdan et al., Shock Waves 13(6), 501–504, 2004; Denet et al., Combust Sci. Technol. 187, 296–323, 2015; Lodato et al., J. Fluid Mech. 789, 221–258, 2016). Under such flow conditions, the reflected shock is relatively weak and the Newtonian limit, used in the above mentioned analysis, is rather far from being met. Despite of this, the theoretical results concerning the nonlinear dynamics of the shock front were, for the most part, confirmed. In an effort to get closer to the conditions of the theoretical analysis, namely strong shocks in the Newtonian limit, a similar numerical analysis is performed in the present study where the incident Mach number is increased up to 5 and the specific heat ratio is decreased down to 1.15, leading to reflected shocks Mach numbers of about 3.2. This provides additional evidence about the main driving mechanism behind the structure of cellular detonations. Theoretical predictions regarding the spontaneous formation and transverse velocity of the triple points are further confirmed. In particular, significant improvements are observed in reproducing the theoretically predicted trajectories of the triple points. As a result of the increased Mach number of the reflected shock, stronger vortex sheets are formed within the shocked gases. This enables to better assess the impact of the molecular viscosity—a previously left open question—but also to highlight similarities with cellular detonations on a wider range of heat releases.