Direct numerical simulation of shock wavy-wall interaction: analysis of cellular shock structures and flow patterns

Abstract

The reflection on a wavy wall of a planar shock propagating at Mach number 1.5 in air is simulated in a two-dimensional geometry by solving the fully compressible Navier–Stokes equations. A high-order spectral difference numerical discretization is used over an unstructured mesh composed of quadrilateral elements. The shock discontinuity is handled numerically through a specific treatment, which is limited in space to a small portion of the computational cell through which the shock is travelling. In the conditions under investigation, the reflection on the wavy wall leads to a weak and smooth deformation of the shock front without singularities just after reflection. Long-living triple points (Mach stems) are spontaneously formed on the reflected shock at a finite distance from the wavy wall. They then propagate on the front in both directions and collide regularly, forming a periodic cellular pattern quite similar to that of a cellular detonation. Transverse waves, issued from the triple points, are generated in the shocked gas. As a result of their mutual interaction, a complex and strongly unsteady flow is produced in the shocked gas. The topology of the instantaneous streamline patterns is characterized by short-lived critical points that appear intermittently. Due to the compressible character of the unsteady two-dimensional flow, the topology of critical points which can be observed is more diverse than would be expected for incompressible two-dimensional flows. Some of them take the form of short-lived sources or sinks. The mechanism of formation and the dynamics of the triple points, as well as the instantaneous streamline patterns, are analysed in the present paper. The results are useful for deciphering the cellular structure of unstable detonations.