Session HE: Rayleigh-Taylor Instabilities

3-D Simulations to Investigate Initial Condition Effects on the Growth of Rayleigh-Taylor Mixing

The effect of initial conditions on the growth rate of turbulent Rayleigh--Taylor (RT) mixing has been studied using carefully formulated numerical simulations. An integrated large-eddy simulation (ILES) using a finite-volume technique was employed to solve the three-dimensional incompressible Euler equations with numerical dissipation. The initial conditions were chosen to test the dependence of the RT growth parameters (\textit{$\alpha $}$_{b}$,\textit{ $\alpha $}$_{s})$ on variations in (a) the spectral bandwidth, (b) the spectral shape, and (c) discrete banded spectra. Our findings support the notion that the overall growth of the RT mixing is strongly dependent on initial conditions. Variation in spectral shapes and bandwidths are found to have a complex effect of the late time development of the RT mixing layer and raises the question of whether we can design RT transition and turbulence based on our choice of initial conditions. In addition, our results provide a useful database for the initialization and development of closures describing RT transition and turbulence.   

Exploring Rayleigh-Taylor Initial Conditions Using a New LES Moment Closure

The effects of initial conditions on the initial transient and long-time development of self-similarity in a Rayleigh-Taylor mixing layer are explored using a new, moment-closure based, large-eddy simulation (LES). The model is derived using moment closure techniques analogous to those developed for Reynolds-averaged Navier Stokes (RANS) of variable density turbulence. Instead of using a scaling equation to obtain a RANS length scale, an LES filter width is chosen just small enough to resolve the large turbulence structures. This model can reproduce the evolution of the large turbulent structures, with good results in both two and three-dimensional simulations. In this presentation, results using the new LES model are compared to DNS predictions as well as theoretical scaling laws. Two regimes are examined: first, the initial transient and approach to self-similarity, and second, the late-time quadratic growth. The interplay of the initial conditions for the resolved and modeled fields is also examined. Finally, some conclusions are made regarding the initial condition problem, as well as the suitability of the new closure approach for this type of investigation.   

Large Atwood number, miscible-liquid experiments and simulations on the Rayleigh-Taylor instability

Experiments and numerical simulations are presented in which an incompressible system of two miscible liquids is accelerated to produce the Rayleigh-Taylor instability. In the experiment, the initially stable, stratified liquid combination is accelerated on a vertical rail system. Either a rectangular or square plexiglass tank, which encloses the liquids, is affixed to a test sled on the rail system. The test sled is then pulled downward, using a system of weights and pulleys, at a rate greater than that of gravity. This produces the upward body force that drives the instability. The resulting fluid flows are visualized with backlit imaging using an LED backlight in conjunction with a monochrome high-speed video camera, both of which are affixed to the test sled. Initial perturbations are either unforced and allowed to progress from thermal background noise or forced by vertically oscillating the liquid combination to produce Faraday internal waves. The results from both of these experimental setups are compared to numerical simulations performed using the CFD code Miranda. Good agreement between the experiment and the simulation is obtained.   

Experimental study of Rayleigh-Taylor instability at moderate to high Atwood numbers

The Rayleigh-Taylor instability is studied experimentally at a moderate Atwood number of 0.46 and a high Atwood number of $\sim $ 1. Two 2-D single mode sinusoidal initial conditions are examined: a single wavelength protruding from a flat interface and a periodic waveform, both with the same initial amplitude and wavelengths. The experiments are performed using a magnetorheological (MR) fluid, composed of 4.5 micron spherical iron particles suspended in hexane with a small amount of oleic acid used as a surfactant. A discontinuous, membrane-less, and initially static interface is created by magnetically immobilizing the MR fluid with the desired shape on the interface, which is then coupled with either water or air, depending on the desired Atwood number. The resulting well defined interface shape can be quickly released by removal of the magnetic field, allowing the instability to develop. The temporal growth of the bubbles and spikes is observed with a high speed X-Ray radiography system. The late time growth rates obtained from these experiments are compared with published analytical, experimental, and numerical results.   

Experimental Measurement of the Density Fluctuation PDF for small Atwood, high Schmidt number, Rayleigh-Taylor mixing

The experimental measurement of the probability density function (pdf) of density fluctuations in a Rayleigh-Taylor small Atwood water channel facility with high Schmidt number is reported. In the experiments, molecular mixing is measured by a phenolphthalein chemical indicator that reacts, turning from transparent to pink, when the heavy (salty \& acidic) and light (fresh \& alkali) water streams mix together. The degree of molecular mixing is determined from the relationship between amount of the chemical reaction formed and the density variance $\bar{\rho'^2}$. By measuring the concentration of the reaction product by backlit optical technique for various initial pH differentials, a detailed pdf of the density fluctuation has been obtained. The shape of the density fluctuation pdf as well as future research will be discussed.   

The growth rate of Rayleigh-Taylor turbulence depends on the large scale structures of the mixing

The growth rate $\alpha_n$ of a turbulent Rayleigh-Taylor (RT) mixing layer is defined such that the mixing layer width $L(t)=\alpha_n\,A\,g(t)\,t^2$, where $A$ is the Atwood number and $g(t)\sim t^n$ is the time history of the acceleration. We will show that the ensemble averaged growth rate of Rayleigh-Taylor can be inferred theoretically from first principle assuming a low Atwood mixing, analyticity of large scale turbulent spectra (for small $k$ the spectra behave like $E(k)\sim k^p$) and self-similarity at late time. The expression of $\alpha_n$ depends on the value of $n$ and $p$. Although it can be counter intuitive, the evolution of the mixing zone width is proved to depend most importantly upon what happens at the center of the mixing zone.   

A Posteriori Tests of an A Priori Optimized Turbulence Model for Small and Large Schmidt Number Rayleigh-Taylor Mixing

Data from a $1152 \times 720 \times 1280$ direct numerical simulation (DNS) of a buoyancy-driven hot/cold water channel experiment is used to construct an optimized four-equation turbulence model for Rayleigh--Taylor mixing. The transport equations for the turbulent kinetic energy and its dissipation rate and of mass fraction variance and its dissipation rate are closed a priori by minimizing the $L_2$-norm between the exact unclosed terms and their gradient-diffusion or scale-similarity closures. The model is tested a posteriori by applying the model to both the $Sc = 7$ hot/cold water experiment and to a $Sc \sim 10^{3}$ salt/fresh water experiment. It is shown that the mixing layer growth and molecular mixing parameters measured from both experiments are well-predicted by the model. The dependence of the predictions on different initialization times of the model, as well on choosing constant late-time values of the parameters (rather than Reynolds number-dependent parameters), are discussed.   

High Reynolds number Rayleigh-Taylor turbulence

The turbulence generated in the variable density Rayleigh-Taylor mixing layer is studied using the fully resolved $3072^3$ simulation of Cabot and Cook, Nature Phys. 2006. A comprehensive study of the budgets for the kinetic energy, mass flux, and density specific volume covariance equations is undertaken. It is found that only the large scale quantities, but not the small scale quantities, reach self-similarity. Hypotheses for the variable density turbulent transport necessary to close the second moment equations are studied. The integral length scale does not follow the $\tilde{k}^{3/2}/\epsilon$ scaling. This is due to the non-equilibrium nature of the flow and the fact that $\epsilon$, a small scale quantity, does not have the self-similar scaling. As a consequence, the popular eddy viscosity expression $\tilde{k}^{2}/\epsilon$ does not model the turbulent transport in any of the moment equations. An integral length scale, based on the layer width, does scale the turbulent transport using a gradient transport hypothesis; that integral scale is a global quantity and does not lead to pointwise local closure. Despite the fact that the intermediate scales are nearly isotropic, the small scales have a persistent anisotropy; this is due to a cancellation between the viscous and nonlinear effects, so that the anisotropic buoyancy production remains important at the smallest scales. Various asymmetries in the mixing layer, not seen in the Boussinesq case, are also identified and explained.   

Advances in Reynolds-Averaged Modeling of Transport and Mixing in High Reynolds Number Rayleigh-Taylor Turbulent Flows

Data from the Cabot and Cook [Nature Physics $2$, $562$ (2006)] $3072^3$ direct numerical simulation (DNS) of Rayleigh--Taylor flow is used to construct the profiles of the terms in the exact turbulent kinetic energy and its dissipation rate and the exact density variance and its dissipation rate transport equations across the mixing layer during the flow evolution. An $L_2$-norm minimization procedure is then applied to each dynamically important unclosed term and its gradient-diffusion or scale-similarity closure to determine the ``optimal'' model parameters as a function of Reynolds number. Using these dynamic model parameters, it is shown in a priori comparisons between the unclosed terms and their corresponding models that these closures provide good approximations to the turbulence production, dissipation, and diffusion mechanisms in large Reynolds number Rayleigh--Taylor mixing. In addition, models of the Reynolds stress tensor taking into account the anisotropic turbulence production mechanisms are discussed.   

Instability phenomena in stratified, particle-laden flow

When a layer of particle-laden water is placed above clear water of different temperature and salinity, various instabilities may arise. Depending on whether the top layer is warmer or colder than the bottom layer, distinct nonlinear convection patterns (``fingering'' vs. ``leaking'') have been reported from experiments (Parsons et al., 2001). We present linear stability results for such situations, with a focus on the role of particle settling. The effect of the settling velocity on the temporal instability growth rates is investigated in combination with various salinity and temperature distributions. The nonlinear evolution of the resulting instability structures is studied via DNS. Comparison between linear analysis, DNS and experimental literature links the occurrence of ``leaking'' (``fingering'') to the presence (absence) of a double-diffusive linear instability.