Initial moments and parameterizing transition for Rayleigh–Taylor unstable stochastic interfaces

Abstract

A stochastic form of linear stability analysis is used to construct initial conditions for variable density moment closures for flows due to the unstable interface between two different density fluids. The material interface can be driven by a mechanism, such as gravity, a pressure gradient (Rayleigh–Taylor), or by a shock (Richtmyer–Meskoff). Initial moments for second moment closure strategies that carry the variables k, ⟨u_iu_j ⟩, R_ij , ϵ, and a_i for the hydrodynamical field and ⟨ρ²⟩, ⟨ρv⟩, and ϵ_ρ, ϵ_ρv for the material field are derived. The method produces a one-to-one correspondence between the initial conditions of a statistical model and the initial energy production – crucially important to the early time problem that sets up the late time problem and is not addressed in models for these flows. The analysis is applied to the Rayleigh–Taylor transition and turbulence problem. Several scalings emerge from the analysis that parameterize the initial condition dependence of RT turbulence on statistics of the interfacial morphology. Computations are conducted to assess the importance of nondimensional parameters on the transition process. An interfacial Reynolds number, involving two statistically defined length scales, collapses the simulation data exceptionally well and is thus a parameter that can be used to effect the transition to turbulence. About 97% of the variability of the turbulence and bulk Reynolds numbers is explained by the initial Taylor–Reynolds number, Re _λ0. The initial Taylor–Reynolds number, Re _λ0, includes two length scales of the interface, δ is the root mean square (rms) thickness and κis rms zero-crossing wavenumber. Minimizing δ²/κ appearing in Re _λ0 stabilizes the interface and reduces the speed of transition to a turbulent flow. Maximimizing δ²/κ destabilizes the interface and increases the rate of transition to a developed turbulent flow and may be of use in reducing the computational costs of transition in the direct numerical simulations of fully developed Rayleigh–Taylor turbulence.

LANL Report:

• LA-UR-09-07231

Keywords:

• rayleigh–Taylor
• turbulence
• initial conditions

Acknowledgements

This paper was made possible by funding from the laboratory-directed research and development program at Los Alamos National Laboratory through directed research project number 20090058DR.