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

TL;DR

This work revisits the Belén’kii–Fradkin turbulent-diffusivity model for variable-density Rayleigh–Taylor turbulence by clarifying its 1D diffusion foundation, deriving a full self-similar ODE and a tractable simplified ODE, and connecting the similarity solution to physical-space quantities. The full ODE captures established RT features at finite Atwood numbers, including spike–bubble asymmetry, a shift in velocity statistics toward the light fluid, and density-based diffusion effects, while a mass-corrected simplified ODE reproduces the full solution's main behaviors with a parsimonious representation. A key result is the explicit $h_T$ scaling with density ratio, $h_T \,\sim\,K^2\,(270\ln R)\, g t^2$ for small density contrasts, and the robust emergence of $h\sim g t^2$ across regimes when the diffusion of $\ln\bar{\rho}$ and mass conservation are balanced. The findings bridge early theoretical models with modern RT observations, offering a compact framework for interpreting variable-density RT mixing and guiding Reynolds-averaged turbulence modeling in RT settings.

Abstract

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.

Figures (7)

• Figure 1: Solutions to full and simplified ODEs. (a,b) Full ODE: dimensionless diffusivity $x = (\varphi'/\varphi)^{1/2}$ and density $\varphi$; (c,d) simplified ODE: dimensionless diffusivity $\hat{x}$ and density $\hat{\varphi}$. Legend: $A=$ 0.01 (black solid), 0.2 (blue dashed), 0.5 (green dash-dotted), and 0.8 (red dotted).
• Figure 2: Normalized solution to the full ODE: (a) normalized diffusivity $x/2\lambda_T^2$, and (b) mole fraction $X$. Legend: $A=$ 0.01 (black solid), 0.2 (blue dashed), 0.5 (green dash-dotted), and 0.8 (red dotted).
• Figure 3: Comparison of the full ODE solution (lines) with DNS results at ${\rm Pe}\approx 1100$ (symbols): (a) normalized diffusivity and (b) mole fraction. Legend: $A=$ 0.01 (black solid, $\square$), 0.2 (blue dashed, $\circ$), 0.5 (green dash-dotted, $\triangle$), and 0.8 (red dotted, $\triangledown$).
• Figure 4: (a--c) Normalized solution to the simplified ODE: normalized diffusivity $\hat{x}/2\hat{\lambda}_T^2$, mole fraction $\hat{X}$, and normalized $\ln \hat{\varphi}$. Legend: $A=$ 0.01 (black solid), 0.2 (blue dashed), 0.5 (green dash-dotted), and 0.8 (red dotted). (d) Normalized displacement length of the simplified solution, computed from numerically integrating Eq. (\ref{['eq:deltanorm']}) (blue solid) and from the analytical expression (\ref{['eq:deltanormfit']}) (red dotted)
• Figure 5: Comparison of the full ODE solution ($x,X$; black dashed), simplified ODE solution ($\hat{x},\hat{X}$; red dotted), and mass-corrected solution ($\hat{x}^*,\hat{X}^*$; thick red solid) at $A=0.8$: (a) normalized diffusivity/velocity and (b) mole fraction.