Variable-density mixing in buoyancy-driven turbulence

Abstract

The homogenization of a heterogeneous mixture of two pure fluids with different densities by molecular diffusion and stirring induced by buoyancy-generated motions, as occurs in the Rayleigh–Taylor (RT) instability, is studied using direct numerical simulations. The Schmidt number, Sc, is varied by a factor of 20, 0.1 ≤ Sc ≤ 2.0, and the Atwood number, A, by a factor of 10, 0.05 ≤ A ≤ 0.5. Initial-density intensities are as high as 50% of the mean density. As a consequence of differential accelerations experienced by the two fluids, substantial and important differences between the mixing in a variable-density flow, as compared to the Boussinesq approximation, are observed. In short, the pure heavy fluid mixes more slowly than the pure light fluid: an initially symmetric double delta density probability density function (PDF) is rapidly skewed and, only at long times and low density fluctuations, does it relax to a Gaussian-like PDF. The heavy–light fluid mixing process asymmetry is relevant to the nature of molecular mixing on different sides of a high-Atwood-number RT layer. Diverse mix metrics are used to examine the homogenization of the two fluids. The conventional mix parameter, θ, is mathematically related to the variance of the excess reactant of a hypothetical fast chemical reaction. Bounds relating θ and the normalized product, Ξ, are derived. It is shown that θ underpredicts the mixing, as compared to Ξ, in the central regions of an RT layer; in the edge regions, θ is larger than Ξ. The shape of the density PDF cannot be inferred from the usual mix metrics popular in applications. For example, when θ, Ξ ≥ 0.6, characteristic of the interior of a fully developed RT layer, the PDFs can have vastly different shapes. Bounds on the fluid composition using two low-order moments of the density PDF are derived. The bounds can be used as realizability conditions for low-dimensional models. For the measures studied, the tightest bounds are obtained using Ξ and mean density. The structure of the flow is also examined. It is found that, at early times, the buoyancy production term in the spectral kinetic energy equation is important at all wavenumbers and leads to anisotropy at all scales of motion. At later times, the anisotropy is confined to the largest and smallest scales: the intermediate scales are more isotropic than the small scales. In the viscous range, there is a cancellation between the viscous and nonlinear effects, and the buoyancy production leads to a persistent small-scale anisotropy.

Numerical investigation of initial condition effects on Rayleigh-Taylor instability with acceleration reversals

“…The asymmetry in Fig. 15 might be also attributed to light fluid mixing more rapidly than heavier fluid, similar to the observation reported by Livescu and Ristorcelli [52] . Since the molecular mixing increases during the deceleration period, the asymmetry becomes more pronounced by the end of deceleration period.…”

Direct numerical simulation and large-eddy simulation of stationary buoyancy-driven turbulence

“…That the anisotropy increases with wavenumber is perhaps counterintuitive, and perhaps contradictory to the notion of small-scale isotropy. This was first pointed out by Livescu & Ristorcelli (2008) and further discussed by Livescu et al (2009 (no summation), but found that, with this measure, the small scales were isotropic. A question then arises as to why the velocity-anisotropy parameter increases in the viscosity-dominated range 10 −1 < k r η < 2.…”

Numerical simulations of two-fluid turbulent mixing at large density ratios and applications to the Rayleigh–Taylor instability

“…The CFDNS code also uses a mixed spectral-compact finite differences scheme. DNS of HRT are reported in references [1,[90][91][92][93] (A up to 0.5) for the unsteady case and reference [16] for the stationary case (A up to 0.9). LES of incompressible RTI using explicit subgrid modelling were performed with the MIRANDA code in reference [94] at A = 0.5 and of stationary HRT using the stretched vortex model of Pullin [95] in reference [16] at A up to 0.9.…”

“…A comprehensive discussion of these metrics, including rigorous bounds, can be found in reference [92]. Thus, the usual mix measure θ = f 1 f 2 /F 1 F 2 depends on the mean and variance of the density.…”