A theoretical one-dimensional model for variable-density Rayleigh-Taylor turbulence

In an early theoretical work published in 1965, Belen’kii & Fradkin proposed a turbulent diffusivity model for Rayleigh–Taylor (RT) mixing. We review its derivation and present alternative arguments leading to the same final similarity equation. The original work then introduced an approximation that led to a simplified ordinary differential equation (ODE), which was used primarily to derive the important scaling result, $h \sim (\ln R)gt^2$. Here, we extend the analysis by examining the solutions to both the full similarity ODE and the simplified ODE in detail. It is shown that the full similarity equation captures many now well-known features of non-Boussinesq RT flows, including asymmetric spike and bubble growth and a systematic shift of velocity statistics toward the light-fluid side. Comparisons of the theoretical model with numerical and experimental studies show reasonable agreement in both spatial profiles and growth trends of mixing layer heights. We further show that a global mass correction applied to the simplified solution closely approximates the full solution, highlighting that, to leading order, RT mixing is governed by the competing dynamics between diffusion of $\ln \barρ$ and mass conservation.

💡 Research Summary

The paper revisits the classic 1965 turbulent‑diffusivity model for Rayleigh–Taylor (RT) mixing proposed by Belen’kii and Fradkin, and places it on a modern theoretical footing. Starting from the one‑dimensional molecular diffusion equations for density and mass fraction, the authors derive the Reynolds‑averaged continuity, scalar, and state equations for a three‑dimensional RT flow. Assuming a high Peclet number, they drop molecular diffusion and obtain a mean‑density diffusion equation ∂ₜ ρ̄ = ∂ᵧ(Dₜ ∂ᵧ ρ̄).

A new, more transparent derivation of the turbulent diffusivity Dₜ is presented. Using a Prandtl mixing‑length ansatz (Dₜ ∼ vₜ hₜ) together with an energy‑balance argument that equates the global potential‑energy loss Δρ g h² t to the local kinetic energy ρ̄ vₜ² h, they find a local velocity scale vₜ ∼ √