Direct numerical simulations and a self-similar analysis of the single-fluid Boussinesq Rayleigh–Taylor instability and transition to turbulence are used to investigate Rayleigh–Taylor turbulence. The Schmidt, Atwood and bulk Reynolds numbers are $Sc\,{=}\,1$, $A\,{=}\,0.01$, $Re \,{\le}\, 3000$. High-Reynolds-number moment self-similarity, consistent with the the energy cascade interpretation of dissipation, is used to analyse the DNS results. The mixing layer width obeys a differential equation with solution $h(t;C_o,h_0)\,{=}\,\fourth C_o Agt^2+ \sqrt{AgC_o}h^{1/2}_0 t+h_0$; the result for $h(t;C_o,h_0)$ is a rigorous consequence of only one ansatz, self-similarity. It indicates an intermediate time regime in which the growth is linear and the importance of a virtual origin. At long time the well-known $h \sim \fourth C_o Agt^2$ scaling dominates. The self-similar analysis indicates that the asymptotic growth rate is not universal. The scalings of the second-order moments, their dissipations, and production–dissipation ratios, are obtained and compared to the DNS. The flow is not self-similar in a conventional sense – there is no single length scale that scales the flow. The moment similarity method produces three different scalings for the turbulence energy-containing length scale, $\ell$, the Taylor microscale, $\la$, and the Kolmogorov dissipation scale, $\eta$. The DNS and the self-similar analysis are in accord showing $\ell \,{\sim}\, Agt^2$, $\la \,{\sim}\, t^{1/2}$ and $\eta \,{\sim}\, (({A^2g^2}/{\nu^3})t)^{-1/4}$ achieving self-similar behaviour within three initial eddy turnovers of the inception of the turbulence growth phase at bulk Reynolds numbers in the range of ${\it Re}\,=\,800$–1000 depending on initial conditions. A picture of a turbulence in which the largest scales grow, asymptotically, as $t^2$ and the smallest scales decrease as $t^{-1/4}$, emerges. As a consequence the bandwidth of the turbulence spectrum grows as $t^{9/4}$ and is consistent with the $R_t^{3/4}$ Kolmogorov scaling law of fully developed stationary turbulent flows. While not all moments are consistent, especially the dissipations and higher-order moments in the edge regions, with the self-similar results it appears possible to conclude that: (i) the turbulence length scales evolve as a power of $h(t;C_o,h_0)$; (ii) $\al$, as demonstrated mathematically for self-similar Rayleigh–Taylor turbulence and numerically by the DNS, is not a universal constant; (iii) there is statistically significant correlation between decreasing $\alpha$ and lower low-wavenumber loading of the initial spectrum.

Buoyancy-generated motions in an unstably stratified medium composed of two incompressible miscible fluids with different densities, as occurs in the variable-density Rayleigh–Taylor instability, are examined using direct numerical simulations. The non-equilibrium homogeneous buoyantly driven problem is proposed as a unit problem for variable density turbulence to study: (i) the nature of variable density turbulence, (ii) the transition to turbulence and the generation of turbulence by the conversion of potential to kinetic energy; (iii) the role of non-Boussinesq effects; and (iv) a parameterization of the initial conditions by a static Reynolds number. Simulations are performed for Atwood numbers up to 0.5 with root mean square density up to 50% of the mean density and Schmidt numbers, 0.1 ≤ Sc ≤ 2. The benchmark problem has been designed to have the largest mass flux possible and is, in this configuration, the maximally unstable non-equilibrium flow possible. It is found that the mass flux, owing to its central role in the conversion of potential to kinetic energy, is probably the single most important dynamical quantity to predict in lower-dimensional models. Other primary findings include the evolution of the mean pressure gradient: during the non-Boussinesq portions of the flow, the evolution of the mean pressure gradient is non-hydrostatic (as opposed to a Boussinesq fluid) and is set by the evolution of the specific volume pressure gradient correlation. To obtain the numerical solution, a new pressure projection algorithm which treats the pressure step exactly, useful for simulations of non-solenoidal velocity flows, has been constructed.

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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.

A closure for the compressible portion of the pressure-strain covariance is developed. It is shown that, within the context of a pressure-strain closure assumption linear in the Reynolds stresses, an expression for the pressure-dilatation can be used to construct a representation for the pressure-strain. Additional closures for the unclosed terms in the Favre-Reynolds stress equations involving the mean acceleration are also constructed. The closures accommodate compressibility corrections depending on the magnitude of the turbulent Mach number, the mean density gradient, the mean pressure gradient, the mean dilatation, and, of course, the mean velocity gradients. The effects of the compressibility corrections are consistent with current DNS results. Using the compressible pressure-strain and mean acceleration closures in the Favre-Reynolds stress equations an algebraic closure for the Favre-Reynolds stresses is constructed. Noteworthy is the fact that, in the absence of mean velocity gradients, the mean density gradient produces Favre-Reynolds stresses in accelerating mean flows. Computations of the mixing layer using the compressible closures developed are described. Full Reynolds stress closure and two-equation algebraic models are compared to laboratory data. The mixing layer configuration computations are compared to laboratory data; since the laboratory data for the turbulence stresses is inconsistent, this comparison is inconclusive. Comparisons for the spread rate reduction indicate a sizable decrease in the mixing layer growth rate.