Vorticity Dynamics of Hydrodynamic Instabilities Occurring at Material Interfaces: Application to High-Energy-Density Systems

Abstract

Inertia-dominated hydrodynamic instabilities at material interfaces are ubiquitous phenomena observed in nature and man-made applications, spanning core collapse supernovae, inertial confinement fusion, supersonic combustion, and cavitation bubble collapse. When subjected to accelerations, perturbations along an interface may grow due to the Rayleigh-Taylor (RT) or Richtmyer-Meshkov (RM) instability, while in the presence of shear, they may grow due to the Kelvin-Helmholtz (KH) instability. The main focus of this thesis is the RM instability.

The RM instability occurs when a perturbed interface separating two fluids of different densities is impulsively accelerated, e.g., by the passage of a shock wave. During the interaction of the shock with the interface, baroclinic vorticity is generated along the interface due to the misalignment between the density and pressure gradients, thus leading to perturbation growth. The subsequent interface evolution can be described using vorticity dynamics. Although the early stage of vorticity deposition along the interface is relatively well understood, the late-time vorticity dynamics and their effects on the interface evolution are less well known. Our objective is to understand the role of vorticity dynamics in the late-time evolution of RM-type problems. To examine the vorticity dynamics of the RM instability, we implement a vortex-sheet model allowing us to isolate the different contributions of vorticity production in the evolution of the interface.

We first use the vortex-sheet model to understand the relative importance between RM and KH in the evolution of perturbations subjected to an oblique shock under high-energy-density (HED) conditions. At early times, the perturbation growth is dominated by the impulsive acceleration of the shock (RM), as evidenced by our proposed scaling accounting for the normal and tangential components of the shock. At later times, the perturbation growth is modulated by the positive and negative vorticity generated by the shear and the decompression due to the arrival of the rarefaction produced by laser turn off. As the tilt angle is increased, the onset of the shear-dominated dynamics occurs earlier and becomes more pronounced. We further demonstrate the ability of the vortex-sheet model to reproduce roll-up dynamics for non-zero Atwood numbers by comparing to past laser-driven HED experiments.

We then explain the mechanisms of vorticity generation in the late-time evolution of the single-mode RM instability. In particular, we explore the generation of secondary opposite-sign vorticity occurring inside the roll-ups as the interface spirals inward. We show that, in the case of a zero Atwood number, opposite-sign vorticity never develops. In this case, the vorticity distribution along the interface is only governed by the rate of change of the sheet surface. Near the vortex core, the rate of change of the sheet surface alternates between positive and negative values, indicating that the interface near the vortex core undergoes a series of contractions and expansions, thus giving rise to oscillations in the corresponding sheet strength. In the case of small Atwood numbers, performing a vorticity budget suggests that opposite-sign vorticity is generated by the nonlinear vorticity advection along the interface. To quantify the amount of opposite-sign vorticity generated along the interface, we consider positive and negative circulations, and their dependence of the strength of the incident shock and the Atwood number. We show that opposite-sign circulation behaves according to a power law in time and that the interface evolution scales in time with respect to the shock Mach number.