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

This paper revisits and extends Belen'kii & Fradkin's 1965 theoretical model for variable-density Rayleigh-Taylor turbulence, demonstrating that the full similarity equation accurately captures non-Boussinesq flow features and that a mass-corrected simplified solution effectively approximates the complex mixing dynamics governed by the competition between diffusion and mass conservation.

The Gist

Imagine two fluids, one heavy (like honey) and one light (like air), sitting on top of each other. Gravity wants the heavy one to sink and the light one to rise, but they are stuck in a messy, churning battle at the interface. This is the Rayleigh-Taylor instability. As they mix, they form a turbulent "soup" where heavy spikes dive down and light bubbles float up.

For decades, scientists have tried to predict how fast this mixing layer grows. Most modern theories assume the fluids are "almost" the same density, using a simple rule of thumb. However, this paper revisits a forgotten, 60-year-old theory from 1965 by Belen'kii and Fradkin that offers a different, more accurate way to look at this chaos, especially when the density difference is huge.

Here is the breakdown of what the paper does, using simple analogies:

1. The Forgotten Recipe

The authors found an old "recipe" (a mathematical model) for how these fluids mix. The original recipe was written in Russian, was a bit messy to read, and had some typos.

• What they did: They cleaned up the recipe, translated it, and rewrote it using modern, clear language.
• The Core Idea: Instead of thinking of the mixing as a complex 3D explosion, they treated it like a one-dimensional diffusion problem. Imagine the mixing layer not as a chaotic storm, but as a single, spreading stain on a piece of paper. They modeled this "stain" spreading using a concept called turbulent diffusivity (how fast the chaos spreads).

2. The "Logarithm" vs. The "Linear" Rule

The big discovery in this paper is about how the mixing layer grows over time.

• The Old View: Most scientists thought the growth rate depended on a linear number called the Atwood number (which measures the difference between the heavy and light fluid). If the difference doubles, the mixing speed doubles.
• The New (Old) View: The 1965 model suggests the growth depends on the natural logarithm of the density ratio ($\ln R$).
  • The Analogy: Think of the Atwood number like a straight line on a graph. The logarithm is like a curve that flattens out. The paper argues that when the density difference gets huge (like comparing lead to air), the mixing doesn't speed up linearly; it slows down its growth rate, following this logarithmic curve. This matches recent computer simulations better than the old linear rule.

3. The "Heavy" and "Light" Asymmetry

When heavy and light fluids mix, they don't behave the same way.

• The Observation: The heavy fluid forms "spikes" that dive down fast, while the light fluid forms "bubbles" that rise slower.
• The Paper's Insight: The old 1965 model naturally predicts this asymmetry without needing extra adjustments. It shows that as the density difference grows, the "spikes" get much longer than the "bubbles."
• The Velocity Shift: The paper also shows that the speed of the mixing shifts toward the light fluid side.
  • The Analogy: Imagine a tug-of-war where one team is much heavier. The rope doesn't just move to the middle; the whole center of the action shifts toward the lighter team. The model captures this "shift" perfectly.

4. The "Mass Correction" Trick

The original 1965 model had a simplified version that was easy to solve but had a flaw: it violated the law of conservation of mass.

• The Problem: If you just use the simple math, it's like a balloon that magically gains or loses air as it expands. The total amount of "stuff" (mass) doesn't add up correctly.
• The Fix: The authors realized that if you simply shift the entire mixing profile slightly toward the light fluid side, the math suddenly works perfectly.
  • The Analogy: Imagine a perfectly symmetrical hill of sand (the simplified model). It looks nice, but if you weigh the sand, it's missing a little bit. If you slide the whole hill a few inches to the left, the weight balances out, and it suddenly looks exactly like the messy, real-world data.
  • This "shift" explains why the spikes grow faster than the bubbles: the diffusion of the "logarithm of density" is symmetric, but the need to save mass forces the whole structure to lean toward the light side.

5. The Bottom Line

The paper concludes that this simple, one-dimensional model from 1965 is actually a "gold mine."

• It captures all the weird, complex behaviors of high-density mixing (asymmetry, shifting velocities, logarithmic growth) that modern scientists have only recently confirmed with supercomputers.
• It suggests that the physics of this turbulence is governed by a competition between diffusion (spreading out) and mass conservation (keeping the total amount of fluid the same).

In summary: The authors dug up an old, dusty theory, dusted it off, and showed that it explains modern observations of fluid mixing better than many current theories. They proved that a simple "shift" in the math fixes the old model's errors and perfectly describes why heavy fluids dive faster than light fluids rise when they are very different in density.

Technical Summary

1. Problem Statement

The Rayleigh–Taylor (RT) instability occurs when a heavy fluid is accelerated into a lighter fluid, leading to turbulent mixing. While the late-time growth of the mixing layer height ($h$) is well-established to scale as $h \sim g t^2$, the dependence on the density ratio ($R = \rho_H / \rho_L$) remains a subject of debate in variable-density (non-Boussinesq) flows.

• Current Consensus: Most modern studies interpret results using the Boussinesq approximation, where the growth rate scales linearly with the Atwood number ($A = (R-1)/(R+1)$), i.e., $h \approx \alpha A g t^2$.
• The Discrepancy: Experimental and numerical data show that bubble growth is relatively insensitive to $A$, while spike growth increases significantly with $A$, leading to asymmetries in velocity and density profiles.
• The Gap: A seminal 1965 work by Belen'kii and Fradkin proposed a turbulent diffusivity model yielding a scaling of $h \propto (\ln R) g t^2$. However, this work was published in Russian, contained typographical errors, and was largely overlooked. Furthermore, the original analysis relied on a simplified Ordinary Differential Equation (ODE) without fully exploring the properties of the full similarity equation or comparing it against modern high-fidelity data.

2. Methodology

The authors revisit, clarify, and extend the Belen'kii and Fradkin (1965) model using a rigorous one-dimensional (1D) framework.

Mathematical Framework:

• Governing Equations: The authors start with the Reynolds-averaged Navier-Stokes (RANS) equations for variable-density flow. They derive a mean density equation analogous to a 1D diffusion problem:
  $$\frac{\partial \bar{\rho}}{\partial t} = \frac{\partial}{\partial y} \left( D_t \frac{\partial \bar{\rho}}{\partial y} \right)$$
• Turbulent Diffusivity Model: They derive a specific form for turbulent diffusivity ($D_t$) based on Prandtl mixing length theory and energy scaling arguments. The model posits:
  $$D_t^* = K \left( \frac{1}{\bar{\rho}} \frac{\partial \bar{\rho}}{\partial y} \right)^{1/2} g^{1/2} h_t^2$$
  where $h_t$ is the turbulence height scale.
• Self-Similar Solution: By introducing similarity variables, the partial differential equation is reduced to a non-linear ODE.
  • Full ODE: A third-order non-linear ODE (reduced to a first-order ODE for a transformed variable $x$) that captures the full physics.
  • Simplified ODE: An analytical approximation derived by neglecting a higher-order term ($x^3$) valid for small density ratios ($R \lesssim 4$).

Analysis Strategy:

• Numerical Solution: The full ODE is solved numerically for a wide range of Atwood numbers ($A \in [0, 0.8]$).
• Analytical Solution: The simplified ODE is solved analytically to recover the original $\ln R$ scaling.
• Mass Correction: The authors identify that the simplified ODE violates global mass conservation due to symmetry assumptions. They propose a "mass correction" (spatially shifting the solution) to align the simplified model with physical reality.
• Validation: Theoretical profiles (density, diffusivity, growth rates) are compared against Direct Numerical Simulation (DNS) data and experimental results from various literature sources.

3. Key Contributions

1. Revival and Clarification: The paper provides a clean, modern derivation of the 1965 Belen'kii and Fradkin model, correcting notation and clarifying assumptions that were previously obscured.
2. Full ODE Analysis: Unlike the original work, this paper solves the full similarity equation (not just the approximation). It demonstrates that the full model inherently captures complex non-Boussinesq features without ad-hoc adjustments.
3. Mechanism for Asymmetry: The authors provide a mechanistic explanation for the observed asymmetry between spikes and bubbles. They show that the natural variable for turbulent mixing in variable-density flows is $\ln \bar{\rho}$. The diffusion of $\ln \bar{\rho}$ is symmetric, but the exponential mapping back to density ($\bar{\rho}$) combined with the constraint of mass conservation forces a spatial shift toward the light-fluid side.
4. Mass-Corrected Simplified Model: They demonstrate that the simple analytical solution (parabolic diffusivity) can accurately reproduce the full solution if a global mass correction (spatial shift) is applied. This offers a computationally cheap, closed-form approximation for turbulence modeling.
5. Scaling Law Validation: The study validates the $\ln R$ scaling for mixing layer height, showing it acts as a "median" estimate that fits the scatter of existing experimental and numerical data better than the linear $A$ scaling at high density ratios.

4. Key Results

• Mixing Layer Height Scaling: The total mixing layer height scales as $h \propto (\ln R) g t^2$. While deviations occur at extreme $A$, this scaling remains consistent with the scatter in available DNS and experimental data up to $A \approx 0.84$.
• Spike vs. Bubble Asymmetry:
  • The model predicts that spike heights ($h_s$) grow faster than bubble heights ($h_b$) as $A$ increases.
  • The ratio $h_s/h_b$ increases with $A$, consistent with empirical observations.
  • The growth parameter $\alpha_s$ increases with $A$, while $\alpha_b$ remains relatively constant (or deviates less), explaining why the Boussinesq approximation fails for spikes at high $A$.
• Profile Shifts:
  • Velocity/Diffusivity: The turbulent diffusivity (and velocity) profiles shift systematically toward the light-fluid side as $A$ increases.
  • Density: The mole fraction profiles collapse well when normalized, but the physical boundaries (spike/bubble tips) become asymmetric ($\lambda_s > \lambda_b$).
• Comparison with Data: The theoretical profiles for normalized diffusivity and mole fraction show reasonable agreement with DNS data (e.g., from Goh et al., Livescu et al.), particularly in the core of the mixing layer. Discrepancies are mostly confined to the edges where molecular diffusion (neglected in the model) becomes relevant.

5. Significance

• Theoretical Insight: The paper resolves a long-standing ambiguity in RT turbulence by showing that the competing dynamics between the diffusion of $\ln \bar{\rho}$ (which is symmetric) and mass conservation (which requires a shift) are the fundamental drivers of non-Boussinesq asymmetry.
• Practical Utility: The "mass-corrected" simplified solution provides a set of closed-form expressions for density and diffusivity profiles. This is highly valuable for developing Reynolds-averaged turbulence models (RANS) where computational efficiency is critical.
• Re-evaluation of Scaling: The work challenges the dominance of the linear Atwood number scaling ($A$) in favor of the logarithmic density ratio scaling ($\ln R$) for variable-density RT flows, suggesting that the latter is a more robust theoretical foundation for high-density-ratio applications (e.g., inertial confinement fusion, astrophysics).

In summary, Goh and Blanquart demonstrate that a simple 1D theoretical framework, originally proposed in 1965, contains the essential physics to explain modern observations of variable-density RT turbulence, provided the model is solved fully or corrected for mass conservation.