Rayleigh–Taylor instability in cylindrical geometry with compressible fluids

Abstract

A linear stability analysis of the Rayleigh–Taylor instability (RTI) between two ideal inviscid immiscible compressible fluids in cylindrical geometry is performed. Three-dimensional (3D) cylindrical as well as two-dimensional (2D) axisymmetric and circular unperturbed interfaces are considered and compared to the Cartesian cases with planar interface. Focuses are on the effects of compressibility, geometry, and differences between the convergent (gravity acting inward) and divergent (gravity acting outward) cases on the early instability growth. Compressibility can be characterized by two independent parameters—a static Mach number based on the isothermal sound speed and the ratio of specific heats. For a steady initial unperturbed state, these have opposite influence, stabilization and destabilization, on the instability growth, similar to the Cartesian case [D. Livescu, Phys. Fluids 16, 118 (2004)]. The instability is found to grow faster in the 3D cylindrical than in the Cartesian case in the convergent configuration but slower in the divergent configuration. In general, the direction of gravity has a profound influence in the cylindrical cases but marginal for planar interface. For the 3D cylindrical case, instability grows faster in the convergent than in the divergent arrangement. Similar results are obtained for the 2D axisymmetric case. However, as the flow transitions from the 3D cylindrical to the 2D circular case, the results above can be qualitatively different depending on the Atwood number, interface radius, and compressibility parameters. Thus, 2D circular calculations of RTI growth do not seem to be a good model for the fully 3D cylindrical case.