High-fidelity numerical simulations of compressible turbulence and mixing generated by hydrodynamic instabilities.

摘要

High-speed flows are prone to hydrodynamic interfacial instabilities that evolve to turbulence, thereby intensely mixing different fluids and dissipating energy. The lack of knowledge of these phenomena has impeded progress in a variety of disciplines. In science, a full understanding of mixing between heavy and light elements after the collapse of a supernova and between adjacent layers of different density in geophysical (atmospheric and oceanic) flows remains lacking. In engineering, the inability to achieve ignition in inertial fusion and efficient combustion constitute further examples of this lack of basic understanding of turbulent mixing. In this work, my goal is to develop accurate and efficient numerical schemes and employ them to study compressible turbulence and mixing generated by interactions between shocked (Richtmyer-Meshkov) and accelerated (Rayleigh-Taylor) interfaces, which play important roles in high-energy-density physics environments.;To accomplish my goal, a hybrid high-order central/discontinuity-capturing finite difference scheme is first presented. The underlying principle is that, to accurately and efficiently represent both broadband motions and discontinuities, non-dissipative methods are used where the solution is smooth, while the more expensive and dissipative capturing schemes are applied near discontinuous regions. Thus, an accurate numerical sensor is developed to discriminate between smooth regions, shocks and material discontinuities, which all require a different treatment. The interface capturing approach is extended to central differences, such that smooth distributions of varying specific heats ratio can be simulated without generating spurious pressure oscillations. I verified and validated this approach against a stringent suite of problems including shocks, interfaces, turbulence and two-dimensional single-mode Richtmyer-Meshkov instability simulations. The three-dimensional code is shown to scale well up to 4000 cores.;Using a novel set-up, I perform direct numerical simulations of freely decaying turbulent multi-material mixing starting from an unperturbed material interface between two fluids in a pre-existing isotropic turbulent velocity field in the presence and absence of gravity. In the absence of gravity, the energy dissipation rate is matched in each fluid, such that anisotropy in the initial set-up solely comes from the density gradient. At large scales, the mixing region grows self-similarly after an initial transient period; a one-dimensional turbulence diffusion model in conjunction with Prandtl's mixing length theory is applied to describe the growth of the mixing region. In this regime, the growth of the mixing regions scales as time to the power of 2/7 for Batchelor turbulence, as predicted by energy budget arguments for large Reynolds numbers. At small scales, flow isotropy and intermittency are measured. Results suggest that a large density ratio between the two fluids is required to produce anisotropy at the Taylor microscale, while the flow remains isotropic at the dissipation (Kolmogorov) scales.;Having identified the role of density gradient alone, I revisit the problem in the presence of gravity in a Rayleigh-Taylor unstable configuration. Now, the baroclinic vorticity due to the gravitational field provides energy that drives the initially decaying turbulent field. Flow dynamics are characterized by the two important competing time scales of the problem, corresponding to the decay of the initial turbulent field and the Rayleigh-Taylor development. The resulting turbulence is found to be anisotropic across all scales. The velocity field is highly intermittent at the bubble and spike fronts.