Rayleigh-Taylor turbulence with mean shear and interface tension

Abstract

Rayleigh-Taylor (RT) turbulence originates at the interface between two fluid layers in the presence of an unstable density stratification \citep{rayleigh1882investigation,taylor1950instability,fermi1953taylor}. In nature, RT turbulence is usually influenced by additional factors. Firstly, the two fluid layers may move at different velocities in the horizontal direction so that a mean shear compounds the unstable density profile. Secondly, the two fluids may be immiscible, implying that interface tension prevents their intermixing at a molecular level. This dissertation aims to provide a better comprehension of RT turbulence in the presence of these two additional factors: mean shear and interface tension. In the first part of this dissertation, we analyze compound shear and RT turbulence. Depending on the relative strength of shear versus buoyancy, the turbulence and mixing may be dominated by the former or the latter. In Chapter 2, we prove theoretically the existence of a cross-over time at which the turbulence transitions from a shear- to a buoyancy-dominated state \citep{brizzolara2021transition}. Using the Buckingham-$\Pi$ theorem, we derive a time-independent control parameter -- the Reynolds number at the cross-over time -- that determines whether shear-dominated turbulence exists or is suppressed at early times. We validate our theory through six direct numerical simulations of an unstable stratified temporal shear layer at an unprecedented Reynolds number. Our simulations, combined with the experimental data available in the literature, suggests that, as the cross-over Reynolds number increases, the non-dimensional transitional time asymptotes to a constant, universal value. This observation has practical relevance in the context of environmental flows such as in atmospheric turbulence, where the values of the Reynolds number are usually much larger than the ones achievable in simulations or laboratory experiments. In Chapter 3, we examine the combined shear and RT turbulence mixing layer from the local entrainment perspective. Our goal is to gain insight into the role of buoyancy and shear in shaping the structure of the outer boundaries of the turbulent layer, the so called turbulent/non-turbulent interface. To this end, we derive a new equation that allows a multi-scale quantification of each individual contribution to the total entrainemnt flux, namely the viscous, inviscid, baroclinic, and subfilter contribution. This is done by defining the entrainment velocity as the rate at which an isosurface of filtered enstrophy propagates relative to the filtered velocity field \citep{brizzolara2023entrainment}. Our data demonstrates that the entrainment flux is constant across scales either in the shear- or buoyancy-dominated regime. A proper non-dimensionalization of the entrainment flux shows that the entrainment rate at all scales matches the classical values obtained through the analysis of the mixing layer bulk expansion. In the second part of the thesis (Chapter 4), we analyze theoretically, experimentally and numerically RT turbulence between a pair of immiscible fluids, to shed light on the emulsification process undepinning this phenomenon. Our study unveils a unique turbulent state that originates at the interface between two fluids due to the interaction between interface tension and inertial forces \citep{brizzolara2022unveiling}. Here, the non-linear coupling of multiple capillary waves excited at the Hinze scale by Kolmogorov turbulence generates a chaotic state known as weak turbulence. We demonstrate that our discovery has the potential to improve the reliability of hydrocarbon biodegradation models for oceanic oil spills.