Impact of transverse strain on linear, transitional and self-similar turbulent mixing layers

This paper investigates the impact of transverse strain on turbulent mixing layers across linear, transitional, and self-similar regimes, revealing that while transverse compression amplifies instability growth in the linear regime, it paradoxically suppresses growth in the transitional-to-turbulent regime by altering shear production and turbulent kinetic energy distribution, ultimately allowing for width prediction via an adjusted buoyancy-drag model.

The Gist

The Big Picture: Mixing Fluids in a Squeeze

Imagine you have a jar with two liquids: heavy syrup at the bottom and light oil on top. If you shake the jar, they start to mix, creating messy swirls and fingers of fluid reaching into each other. In physics, this is called a turbulent mixing layer.

This paper studies what happens to that mixing when you squeeze or stretch the jar while it's shaking.

• Squeezing (Compression): Like pressing the sides of the jar inward.
• Stretching (Expansion): Like pulling the sides of the jar outward.

The researchers wanted to know: Does squeezing the jar make the fluids mix faster or slower? Does stretching it help or hurt?

The Two Main Characters: The "Baby" and the "Adult"

The paper looks at the mixing process in two different stages, which behave very differently:

1. The Linear Regime (The "Baby" Stage):

   • What it is: The very beginning, where the fluids just start to wiggle. The waves are small and smooth.
   • The Analogy: Imagine a calm pond where you drop a pebble. The ripples are small and predictable.
   • The Finding: When you squeeze (compress) the jar, the ripples get bigger and grow faster. It's like squeezing a spring; the energy gets concentrated, and the instability amplifies.
   • The Result: Compression = Faster mixing growth (at the start).

2. The Self-Similar/Turbulent Regime (The "Adult" Stage):

   • What it is: The later stage where the fluids are fully chaotic, churning, and mixing like a stormy ocean.
   • The Analogy: Imagine a blender full of smoothie. It's already a chaotic mess.
   • The Finding: This is where it gets surprising! When you squeeze the jar now, the mixing actually slows down slightly. When you stretch the jar, the mixing speeds up.
   • Why? This is the opposite of what happens to the "baby" stage.

Why the "Adult" Stage Acts Differently: The Energy Budget

To understand why squeezing slows down the adult mixing, think of the fluid's energy like a bank account.

• Shear Production (The Deposit): When you squeeze the jar sideways, it creates friction and turbulence in the sideways direction. It's like the jar is "depositing" extra energy into the side-to-side motion of the fluid.
• The Catch (The Withdrawal): However, squeezing the jar also shrinks the space the fluid can move in. Imagine the fluid is trying to run a marathon, but the track gets narrower. The "turbulent length scale" (the size of the swirling eddies) gets smaller.
• The Result: When the swirls get smaller, they burn up energy much faster (dissipation). Even though the squeeze added some energy (deposit), the shrinking space burned it up even faster (withdrawal).
  • Net Effect: The fluid has less energy to push the heavy and light fluids apart, so the mixing layer grows slower.

The "Rubber Band" Analogy

Think of the mixing layer as a rubber band being stretched or compressed.

• In the Linear (Baby) Stage: If you compress the rubber band, the tension builds up instantly, and the instability shoots forward.
• In the Turbulent (Adult) Stage: If you compress the rubber band, you are also cramping the space where the "knots" (eddies) can wiggle. The knots get so small and tight that they snap apart (dissipate) before they can do much work. The system becomes "stiff" and less efficient at mixing.

What About the "Mixing" Quality?

The paper also looked at how thoroughly the fluids mix (homogeneity), not just how wide the layer gets.

• Squeezing (Compression): Makes the fluids mix better together. It's like kneading dough; the pressure forces the ingredients to blend into a more uniform paste. The fluid becomes more "homogeneous."
• Stretching (Expansion): Makes the fluids mix worse. It's like pulling the dough apart; you get long, thin strands that are less blended.

The "Recipe" for Prediction

The researchers tried to create a mathematical "recipe" (a model) to predict these behaviors.

• They found that the old recipes (models) used for squeezing jars didn't work for the chaotic "adult" stage.
• They created a new recipe that accounts for the changing size of the "ripples" (wavelengths).
• The Secret Ingredient: They realized that to predict how fast the mixing layer grows, you have to adjust the "drag" (resistance) based on how much the jar is stretching or shrinking at that exact moment.

Summary for the General Audience

1. Start: When mixing starts (small waves), squeezing the container makes the waves grow faster.
2. Finish: When mixing is fully chaotic (turbulence), squeezing the container actually makes the overall growth slower, because the turbulence gets "cramped" and burns out its energy.
3. Quality: Squeezing makes the fluids blend more evenly (better mixing), while stretching makes them stay more separated.
4. The Takeaway: You can't use the same rules for a calm pond and a stormy ocean. What helps a small ripple grow might actually slow down a full-blown storm.

This research helps scientists understand complex events like supernova explosions (where stars collapse and explode) and nuclear fusion (where fuel is squeezed to create energy), ensuring their computer simulations are accurate.

Technical Summary

1. Problem Statement

The paper investigates the influence of transverse strain rates on the development of turbulent mixing layers driven by the Richtmyer–Meshkov Instability (RMI) and Rayleigh–Taylor Instability (RTI).

• Context: In convergent geometries (cylindrical or spherical), such as those found in inertial confinement fusion (ICF) and supernovae, mixing layers experience compression and convergence. These effects are traditionally quantified by compression rates and convergence rates.
• Gap: While the linear regime of these instabilities in convergent geometries is well-described by Bell–Plesset effects, the behavior in the transitional and self-similar turbulent regimes is less understood. Specifically, the distinct roles of axial strain (stretching/compressing the interface normal) versus transverse strain (convergence/expansion in the plane of the interface) are not fully isolated in planar simulations.
• Objective: To decouple the effects of transverse strain from convergence by applying transverse strain rates in planar geometry, thereby creating an alternative framework to study how convergence affects mixing layer growth, turbulence statistics, and self-similarity.

2. Methodology

The study employs a multi-stage approach combining analytical modeling and high-fidelity numerical simulations.

A. Analytical Modeling (Linear Regime)

• Derivation: A linearized potential flow model was derived for planar geometry incorporating both axial ($\bar{S}_{11}$) and transverse ($\bar{S}_{22}$) strain rates.
• Equivalence: The derived differential equation for amplitude growth was shown to be mathematically equivalent to the Bell–Plesset models used for spherical and cylindrical geometries, validating the use of planar strain rates to simulate convergent effects in the linear regime.
• Strain Profiles: Two strain rate profiles were tested:
  1. Constant Velocity: Domain length changes linearly ($L(t) = L_0 + V_0 t$).
  2. Constant Strain Rate: Domain length changes exponentially ($L(t) = L_0 e^{St}$), requiring a potential forcing term to isolate strain effects from pressure-driven acceleration.

B. Numerical Simulations

• Linear Regime (2D): Single-mode RMI simulations were conducted using the FLAMENCO code (finite-volume, 5th-order spatial accuracy). The simulations used a 3:1 density ratio and validated the linear model against various strain rates (compression and expansion).
• Transitional/Self-Similar Regime (3D): Implicit Large Eddy Simulations (ILES) were performed using the quarter-scale $\theta$-group case (Thornber et al., 2017).
  • Setup: Multi-mode narrowband RMI mixing layer.
  • Strain Application: Transverse strain rates were applied at $\tau=1$ (transitioning phase) using an Arbitrary Lagrangian-Eulerian (ALE) moving mesh scheme.
  • Cases: Eight strain cases (4 constant velocity, 4 constant strain rate) covering both compressive ($\hat{S} < 0$) and expansive ($\hat{S} > 0$) scenarios.
• Turbulence Modeling: A modified Reynolds Stress Transport model (LRR-IP) and a buoyancy-drag model were developed to interpret the simulation results, specifically addressing shear production and dissipation scaling.

3. Key Contributions

1. Strain Rate Framework: Proposed a unified framework where transverse strain rates in planar geometry serve as a proxy for convergence rates in convergent geometries, allowing for the isolation of specific strain effects.
2. Linear vs. Turbulent Divergence: Demonstrated a fundamental reversal in the effect of transverse compression on mixing layer growth between the linear and turbulent regimes.
3. Modified Buoyancy-Drag Model: Developed a method to predict mixing layer width under transverse strain by adjusting the effective drag lengthscale to scale with the time-varying mean wavelength ($\bar{\lambda}(t)$) rather than the initial wavelength.
4. Turbulence Anisotropy & Dissipation: Identified that transverse compression increases shear production in the transverse direction but simultaneously increases the dissipation rate due to a reduced turbulent lengthscale, leading to complex interactions in turbulent kinetic energy (TKE) budgets.

4. Key Results

A. Linear Regime

• Amplification: Transverse compression amplifies the instability growth rate, while expansion inhibits it.
• Model Accuracy: The derived linear potential flow model accurately predicts growth rates for expansion cases. For compression cases, the model remains accurate as long as the amplitude $a$ remains small relative to the time-varying wavelength $\lambda(t)$ ($a < 0.1\lambda(t)$).
• Mechanism: The growth rate scales with the instantaneous wavenumber, effectively treating the impulse as applied to the current, compressed wavelength.

B. Transitional and Self-Similar Regime (ILES)

• Growth Rate Reversal: Contrary to the linear regime, transverse compression leads to a decrease in the mixing layer growth rate (integral width), while expansion leads to a slight increase.
  • Cause: While compression increases Transverse TKE via shear production, it reduces the turbulent lengthscale, significantly increasing the dissipation rate. This enhanced dissipation counteracts the shear production, reducing the axial TKE available for mixing layer growth.
• Mixing and Homogeneity:
  • Compression: Increases molecular mixing (homogeneity) and mixed mass. The mixing layer becomes more "mixed" despite growing slower.
  • Expansion: Decreases mixing and mixed mass.
• Anisotropy:
  • Expansion: Increases anisotropy (TKE becomes more biased toward the axial direction) as transverse energy is depleted.
  • Compression: Drives the flow toward isotropy by amplifying transverse TKE components.
• Enstrophy: Compression increases enstrophy in the transverse plane ($y-z$), correlating with increased mixing, while expansion decreases it.

C. Modeling Success

• Buoyancy-Drag Model: The standard buoyancy-drag model failed to predict growth under strain when using the initial wavelength. However, by scaling the effective drag lengthscale with the time-varying wavelength ($l_{eff} \propto \bar{\lambda}(t)$), the model successfully predicted the mixing layer width and bubble/spike heights for both strained and unstrained cases.
• TKE Budget: A corrected TKE model incorporating strain-dependent dissipation (scaling with the geometric mean of expansion factors) successfully matched ILES results, highlighting the critical role of dissipation scaling over simple shear production.

5. Significance

• ICF and Astrophysics: The findings provide a more accurate physical basis for modeling mixing in convergent geometries (like ICF implosions). The result that compression reduces turbulent growth in the non-linear regime (despite amplifying linear growth) suggests that convergence effects might be less detrimental to ICF performance than previously assumed if only linear models are considered, or conversely, that the mixing behavior is more complex and requires non-linear corrections.
• Modeling Paradigm: The study establishes that transverse strain in planar simulations is a valid and powerful tool for isolating convergence effects, bypassing the numerical complexities of spherical/cylindrical meshes.
• Turbulence Theory: The work highlights the non-trivial interplay between shear production, turbulent lengthscale modification, and dissipation in compressible, strained flows, challenging standard assumptions that shear production always leads to increased mixing layer growth.

In conclusion, the paper demonstrates that while transverse compression accelerates linear instability growth, it ultimately suppresses the growth of the turbulent mixing layer width by enhancing dissipation, while simultaneously driving the flow toward a more isotropic and highly mixed state.