Compressible Rayleigh–Taylor Turbulence

• Compressible Rayleigh–Taylor turbulence is the nonlinear mixing process occurring when a heavier fluid overlies a lighter one, featuring significant density, pressure, and temperature variations.
• It leverages advanced numerical methods such as lattice Boltzmann, discrete Boltzmann, and data-driven LES to capture non-equilibrium effects and turbulent mixing dynamics.
• Parameter sensitivities—evident in Atwood and Mach numbers as well as forcing characteristics—lead to modified growth laws with applications in fusion, astrophysics, and geophysical flows.

Compressible Rayleigh–Taylor (RT) turbulence encompasses the nonlinear evolution and turbulent mixing that arises when a heavier fluid overlies a lighter one in the presence of acceleration, accounting for compressibility effects that manifest prominently at finite Atwood numbers, stratification, finite Mach number, and in thermodynamically non-equilibrium regimes. In contrast to the classical Boussinesq RT instability, compressible RT turbulence involves density and pressure variations, baroclinic drivers, and temperature and entropy fluctuations—core features relevant in astrophysical, fusion, and geophysical flows.

1. Governing Equations and Physical Regimes

Compressible RT turbulence is governed by the compressible Navier–Stokes–Fourier system, typically expressed as:

• Mass continuity: $$\partial_t \rho + \nabla \cdot (\rho \mathbf{u}) = 0$$
• Momentum: $$\partial_t(\rho \mathbf{u}) + \nabla \cdot (\rho \mathbf{u} \otimes \mathbf{u} + p \mathbf{I} - \boldsymbol{\Pi}) = \rho \mathbf{a}(t)$$
• Total energy: $$\partial_t E + \nabla \cdot [(E+p)\mathbf{u} - \mathbf{u} \cdot \boldsymbol{\Pi} - \mathbf{q}] = \rho \mathbf{a}(t) \cdot \mathbf{u}$$

Here, $p=\rho T$, $E=\frac12\rho |\mathbf{u}|^2 + \rho e$, $\boldsymbol{\Pi}$ is the viscous stress, and $\mathbf{q}$ the heat flux. The acceleration $\mathbf{a}(t)$ can be time-dependent to model variable forcing scenarios. Compressibility is parameterized by the Atwood number $A=(\rho_H-\rho_L)/(\rho_H+\rho_L)$ and Mach number $Ma=U/c_s$.

Key non-dimensional numbers include:

• Reynolds number $$\partial_t(\rho \mathbf{u}) + \nabla \cdot (\rho \mathbf{u} \otimes \mathbf{u} + p \mathbf{I} - \boldsymbol{\Pi}) = \rho \mathbf{a}(t)$$0
• Mach number $$\partial_t(\rho \mathbf{u}) + \nabla \cdot (\rho \mathbf{u} \otimes \mathbf{u} + p \mathbf{I} - \boldsymbol{\Pi}) = \rho \mathbf{a}(t)$$1
• Rayleigh number $$\partial_t(\rho \mathbf{u}) + \nabla \cdot (\rho \mathbf{u} \otimes \mathbf{u} + p \mathbf{I} - \boldsymbol{\Pi}) = \rho \mathbf{a}(t)$$2
• Stratification parameter and adiabatic gradient $$\partial_t(\rho \mathbf{u}) + \nabla \cdot (\rho \mathbf{u} \otimes \mathbf{u} + p \mathbf{I} - \boldsymbol{\Pi}) = \rho \mathbf{a}(t)$$3, leading to an adiabatic length $$\partial_t(\rho \mathbf{u}) + \nabla \cdot (\rho \mathbf{u} \otimes \mathbf{u} + p \mathbf{I} - \boldsymbol{\Pi}) = \rho \mathbf{a}(t)$$4 (Scagliarini et al., 2010, Scagliarini et al., 2010)

Discrete Boltzmann and lattice Boltzmann modeling frameworks, including D2Q16, D2Q37, and higher-order Hermite expansions, recover these macroscopic equations with correct thermodynamic couplings for compressible systems (Scagliarini et al., 2010, Chen et al., 2021, Scagliarini et al., 2010, Lai et al., 7 Apr 2025).

2. Non-Equilibrium Thermodynamics and Scaling Measures

Compressible RT turbulence is characterized by persistent departures from local thermodynamic equilibrium, especially in regions of strong density and temperature gradients—typically at the bubble and spike tips. Thermodynamic non-equilibrium (TNE) is quantified via kinetic moment deviations:

• Viscous-stress TNE: $$\partial_t(\rho \mathbf{u}) + \nabla \cdot (\rho \mathbf{u} \otimes \mathbf{u} + p \mathbf{I} - \boldsymbol{\Pi}) = \rho \mathbf{a}(t)$$5
• Total TNE: $$\partial_t(\rho \mathbf{u}) + \nabla \cdot (\rho \mathbf{u} \otimes \mathbf{u} + p \mathbf{I} - \boldsymbol{\Pi}) = \rho \mathbf{a}(t)$$6

Domain-averaged quantities such as $$\partial_t(\rho \mathbf{u}) + \nabla \cdot (\rho \mathbf{u} \otimes \mathbf{u} + p \mathbf{I} - \boldsymbol{\Pi}) = \rho \mathbf{a}(t)$$7 and fraction of non-equilibrium regions $$\partial_t(\rho \mathbf{u}) + \nabla \cdot (\rho \mathbf{u} \otimes \mathbf{u} + p \mathbf{I} - \boldsymbol{\Pi}) = \rho \mathbf{a}(t)$$8 provide mesoscopic metrics for turbulence intensity and mixing state (Lai et al., 7 Apr 2025, Chen et al., 2021).

Typical temporal patterns for $$\partial_t(\rho \mathbf{u}) + \nabla \cdot (\rho \mathbf{u} \otimes \mathbf{u} + p \mathbf{I} - \boldsymbol{\Pi}) = \rho \mathbf{a}(t)$$9 and $$\partial_t E + \nabla \cdot [(E+p)\mathbf{u} - \mathbf{u} \cdot \boldsymbol{\Pi} - \mathbf{q}] = \rho \mathbf{a}(t) \cdot \mathbf{u}$$0 include an initial dip (due to diffusion), a peak during nonlinear growth (interface stretching, vortex formation), and decay at late times as dissipation dominates. Exponential and linear laws parameterize the dependence of gradients and TNE on acceleration period, amplitude, and phase—allowing predictive control over mixing and non-equilibrium intensity (Lai et al., 7 Apr 2025).

3. Mixing-Layer Dynamics, Growth Laws, and Compressibility Effects

In classical (incompressible) RT, the mixing-layer width $$\partial_t E + \nabla \cdot [(E+p)\mathbf{u} - \mathbf{u} \cdot \boldsymbol{\Pi} - \mathbf{q}] = \rho \mathbf{a}(t) \cdot \mathbf{u}$$1 grows quadratically:

$$\partial_t E + \nabla \cdot [(E+p)\mathbf{u} - \mathbf{u} \cdot \boldsymbol{\Pi} - \mathbf{q}] = \rho \mathbf{a}(t) \cdot \mathbf{u}$$2

with typical $$\partial_t E + \nabla \cdot [(E+p)\mathbf{u} - \mathbf{u} \cdot \boldsymbol{\Pi} - \mathbf{q}] = \rho \mathbf{a}(t) \cdot \mathbf{u}$$3 in the $$\partial_t E + \nabla \cdot [(E+p)\mathbf{u} - \mathbf{u} \cdot \boldsymbol{\Pi} - \mathbf{q}] = \rho \mathbf{a}(t) \cdot \mathbf{u}$$4–$$\partial_t E + \nabla \cdot [(E+p)\mathbf{u} - \mathbf{u} \cdot \boldsymbol{\Pi} - \mathbf{q}] = \rho \mathbf{a}(t) \cdot \mathbf{u}$$5 range (Scagliarini et al., 2010, Scagliarini et al., 2010). Compressibility introduces key modifications:

• Up–down asymmetry emerges: spikes (denser/cold plumes) grow faster than bubbles as Atwood number increases, with $$\partial_t E + \nabla \cdot [(E+p)\mathbf{u} - \mathbf{u} \cdot \boldsymbol{\Pi} - \mathbf{q}] = \rho \mathbf{a}(t) \cdot \mathbf{u}$$6 shifting from $$\partial_t E + \nabla \cdot [(E+p)\mathbf{u} - \mathbf{u} \cdot \boldsymbol{\Pi} - \mathbf{q}] = \rho \mathbf{a}(t) \cdot \mathbf{u}$$7 at $$\partial_t E + \nabla \cdot [(E+p)\mathbf{u} - \mathbf{u} \cdot \boldsymbol{\Pi} - \mathbf{q}] = \rho \mathbf{a}(t) \cdot \mathbf{u}$$8 to $$\partial_t E + \nabla \cdot [(E+p)\mathbf{u} - \mathbf{u} \cdot \boldsymbol{\Pi} - \mathbf{q}] = \rho \mathbf{a}(t) \cdot \mathbf{u}$$9 at $p=\rho T$0 (Scagliarini et al., 2010, Scagliarini et al., 2010).
• Finite At leads to $p=\rho T$1 pressure fluctuations, slower homogenization of rms velocities and temperatures, and persistent intermittency in high-order statistics.
• At high stratification, mixing arrests at the finite “adiabatic length” $p=\rho T$2, with the mean temperature gradient approaching the adiabatic value and buoyancy vanishing. This is modeled with phenomenological “shutoff” functions $p=\rho T$3 modifying the quadratic growth law (Scagliarini et al., 2010).

For time-varying acceleration, mixing and TNE measures oscillate in-phase with the forcing, and the timing and magnitude of mixing events can be systematically altered by tuning forcing waveform parameters (Lai et al., 7 Apr 2025).

4. Numerical and Modeling Methodologies

Several advanced techniques have been developed for compressible RT turbulence:

• Lattice Boltzmann Methods (LBM): Accurately recover thermodynamically consistent compressible dynamics, handle large Atwood number and stratification, and capture growth rates, arrest effects, and turbulence statistics (Scagliarini et al., 2010, Scagliarini et al., 2010).
• Kinetic and Discrete Boltzmann Models: Resolve TNE effects and allow for rigorous quantification of mesoscopic non-equilibrium, including the effects of specific-heat ratio $p=\rho T$4, which amplifies TNE and accelerates vortex formation as $p=\rho T$5 decreases (Chen et al., 2021, Lai et al., 7 Apr 2025).
• Data-Driven Large-Eddy Simulation (LES): Fourier Neural Operator (FNO) models, trained on filtered DNS data, outperform classical LES subgrid closures (VGM, DSM, ILES) in predicting kinetic energy, mixing-width evolution, concentration and temperature fluctuations, and velocity divergence for 3D compressible RT at $p=\rho T$6, $p=\rho T$7, $p=\rho T$8–$p=\rho T$9 (Luo et al., 2024). FNO achieves $E=\frac12\rho |\mathbf{u}|^2 + \rho e$0 accuracy on bubble/spike mixing-widths and delivers order-of-magnitude computational speedups over traditional methods.
• Multiscale Decomposition: Separation into a compressible bulk (coarse grid) and incompressible–irrotational interface dynamics (“higher-order z-model”) yields speedups of $E=\frac12\rho |\mathbf{u}|^2 + \rho e$1–$E=\frac12\rho |\mathbf{u}|^2 + \rho e$2 relative to fully resolved compressible Euler simulations, with close agreement on classical RT/RM benchmarks (Ramani et al., 2019).

5. Influence of System Parameters and Forcing

The dynamics of compressible RT turbulence are highly sensitive to:

• Atwood Number ($E=\frac12\rho |\mathbf{u}|^2 + \rho e$3): Increases amplify asymmetry, pressure fluctuations, and decrease the rate of mixing-layer homogenization.
• Specific-Heat Ratio ($E=\frac12\rho |\mathbf{u}|^2 + \rho e$4): Lower $E=\frac12\rho |\mathbf{u}|^2 + \rho e$5 (higher compressibility) increases TNE intensities, proportion of non-equilibrium regions, and advances the onset of vortical breakdown (Chen et al., 2021).
• Time-Dependent Forcing: By varying acceleration period ($E=\frac12\rho |\mathbf{u}|^2 + \rho e$6), amplitude ($E=\frac12\rho |\mathbf{u}|^2 + \rho e$7), and phase ($E=\frac12\rho |\mathbf{u}|^2 + \rho e$8), the onset, duration, and intensity of mixing can be delayed or enhanced, following explicit exponential and linear scaling laws for density gradients and TNE measures (Lai et al., 7 Apr 2025).

Example scaling laws from (Lai et al., 7 Apr 2025):

Parameter	Density Gradient	TNE Strength ($E=\frac12\rho |\mathbf{u}|^2 + \rho e$9, $\boldsymbol{\Pi}$0)
$\boldsymbol{\Pi}$1 @ $\boldsymbol{\Pi}$2	$\boldsymbol{\Pi}$3	$\boldsymbol{\Pi}$4 / $\boldsymbol{\Pi}$5
$\boldsymbol{\Pi}$6 (peak)	$\boldsymbol{\Pi}$7	$\boldsymbol{\Pi}$8 / $\boldsymbol{\Pi}$9
$\mathbf{q}$0 @ $\mathbf{q}$1	Piecewise linear	Piecewise linear/quadratic

These data establish quantitative frameworks for regulating RT mixing in engineering and astrophysical applications.

6. Applications, Implications, and Open Directions

Compressible RT turbulence is central to inertial confinement fusion (where controlling the growth of RT-induced mixing is critical), core-collapse supernovae and stellar interiors (where stratified, compressible plumes dominate), and aerospace/atmospheric flows. The ability to modulate RT turbulence via time-dependent acceleration or by tuning fluid properties (e.g., $\mathbf{q}$2) yields potential for flow control strategies in fusion or laboratory experiments (Lai et al., 7 Apr 2025, Scagliarini et al., 2010).

Future research challenges include:

• Detailed three-dimensional high-Reynolds-number studies with variable Prandtl and Mach numbers (Scagliarini et al., 2010, Luo et al., 2024).
• Extension of data-driven LES and operator-learning frameworks to more complex baroclinic and multi-material interfaces.
• Comprehensive characterizations of intermittency, small-scale cascade properties, and the role of non-equilibrium in transition and late-time mixing.
• Analytical closure and prediction of adiabatically arrested mixing regimes and their dependence on compressibility, stratification, and drive waveform.

Compressibility in the RT context thus fundamentally alters turbulence development, saturation, and energy partition, mandating kinetic, mesoscale, and data-driven approaches for predictive accuracy and computational efficiency.