Nonideal Rayleigh–Taylor mixing

Abstract

Rayleigh-Taylor mixing is a classical hydrodynamic instability that occurs when a light fluid pushes against a heavy fluid. The two main sources of nonideal behavior in Rayleigh-Taylor (RT) mixing are regularizations (physical and numerical), which produce deviations from a pure Euler equation, scale invariant formulation, and nonideal (i.e., experimental) initial conditions. The Kolmogorov theory of turbulence predicts stirring at all length scales for the Euler fluid equations without regularization. We interpret mathematical theories of existence and nonuniqueness in this context, and we provide numerical evidence for dependence of the RT mixing rate on nonideal regularizations; in other words, indeterminacy when modeled by Euler equations. Operationally, indeterminacy shows up as nonunique solutions for RT mixing, parametrized by Schmidt and Prandtl numbers, in the large Reynolds number (Euler equation) limit. Verification and validation evidence is presented for the large eddy simulation algorithm used here. Mesh convergence depends on breaking the nonuniqueness with explicit use of the laminar Schmidt and Prandtl numbers and their turbulent counterparts, defined in terms of subgrid scale models. The dependence of the mixing rate on the Schmidt and Prandtl numbers and other physical parameters will be illustrated. We demonstrate numerically the influence of initial conditions on the mixing rate. Both the dominant short wavelength initial conditions and long wavelength perturbations are observed to play a role. By examination of two classes of experiments, we observe the absence of a single universal explanation, with long and short wavelength initial conditions, and the various physical and numerical regularizations contributing in different proportions in these two different contexts.large eddy simulations | subgrid scale models | turbulence R ayleigh-Taylor (RT) instability is a classical hydrodynamic instability (1, 2) that occurs when a light fluid pushes against a heavy fluid. The density contrast is measured dimensionlessly by the Atwood number A ¼ ðρ 2 − ρ 1 Þ∕ðρ 2 þ ρ 1 Þ, with ρ i the density of fluid i. The mixing rate can be characterized by the dimensionless parameter α, defined through the equationwhere h is the penetration distance of the light fluid into the heavy and g is the acceleration. The mixing rapidly becomes turbulent, with Reynolds numbers Re ≈ 50;000 observed in typical experiments. For this reason, simulations based on compressible codes, with hyperbolic time step (CFL) restrictions, are unable to resolve the viscous length scales, and such codes are run in an under-resolved manner. Such simulations are called large eddy simulations (LES). LES require subgrid scale (SGS) models to describe the effect of the small (subgrid) scales on the large (grid-resolved) ones. Models for the Reynolds stress based on eddy viscosity are the most familiar of this type.Much of the work on RT mixing has been influenced by ideas of universality and self similarity. If α is to be a universal physical...

New directions for Rayleigh–Taylor mixing

“…We simulated the Mueschke-Andrews problem [3] with and without the long-wavelength component of the initial data, and found an increase of α by 0.023, or 27% of the experimental value 0.085 ± 0.005 due to the long-wavelength perturbations as measured experimentally. This increase does not account for the simulation values of α ≈ 0.02 or 0.03 obtained in simulation by others, nor does it allow for a factor of 2-3 variation in α due to long-wavelength perturbations.…”

“…Usually, the initial mass diffusion layer is as thin as the experimentalist is able to achieve. This leaves the dependence on the Schmidt and Prandtl numbers, which we have shown to be strictly non-zero but not large [3]. Conventional (ILES) RT simulations do not control most of these dimensionless parameters, resulting in significant variation from one ILES simulation to another, not to mention comparison to experiment.…”

“…The agreement among the three simulations and these with experiment is sufficient to assert the mesh convergence and approximate mesh independence of the results. Further mesh convergence studies were reported in [3,4].…”

Initial conditions for turbulent mixing simulations

“…• agreement for both immiscible and miscible experiments, the latter for high, moderate and low Schmidt number [8,9],…”

Euler equation existence, non-uniqueness and mesh converged statistics

“…The third is a shear layer or tangential velocity discontinuity across an interface. The FT/LES/SGS algorithm has achieved some of the most extensive validation results available, especially for RT (see [11,12] and references therein). In contrast to many codes [13], we find agreement between simulation and experiment for the dimensionless constant α that characterizes the growth rate of the outer edge of the mixing zone.…”