Turbulence Modelling of Mixing Layers under Anisotropic Strain

This study investigates the impact of anisotropic strain rates on turbulent mixing layers using the K-L RANS model, demonstrating that a transverse strain closure improves predictive accuracy over the default isotropic approach and suggesting corresponding modifications for K-$\epsilon$ and K-$\omega$ models.

The Gist

Imagine you have a bowl of two different colored liquids, like oil and water, sitting on a table. If you shake the table, the boundary between them gets messy and starts to mix. This is similar to what happens in a turbulent mixing layer in physics: two fluids of different densities are pushed together, creating a chaotic, swirling mess.

This paper is about understanding what happens when you don't just shake the table, but also stretch or squeeze the entire room where the mixing is happening.

Here is a breakdown of the paper's story, using simple analogies:

1. The Setting: Stretching the Room

In many real-world scenarios—like a star exploding (supernova) or a nuclear fusion bomb being compressed—the space where the fluids are mixing isn't just sitting still. The space itself is expanding or contracting.

• The Analogy: Imagine the mixing layer is a piece of dough being kneaded. Usually, scientists study how the dough mixes when you just push it around. But in this paper, the authors ask: "What happens if, while you are kneading, someone is also pulling the table the dough is on, stretching it lengthwise or squeezing it widthwise?"
• The Problem: The "stretching" (strain) isn't the same in all directions. If you pull a rubber band, it gets longer in one direction but thinner in the others. This is called anisotropic strain. Most computer models used to predict these mixings assume the stretching is the same in every direction (like blowing up a perfect balloon), which doesn't match reality.

2. The Tool: The "K-L" Model

To predict how the fluids mix, the authors use a computer program called the K-L turbulence model.

• The Analogy: Think of this model as a recipe book for predicting chaos. It has two main ingredients it tracks:
  1. How much energy is in the swirls (Turbulent Kinetic Energy).
  2. How big the swirls are (Turbulent Length Scale).
• The model tries to guess how big the swirls will get as the fluids mix. The tricky part is a rule in the recipe called the "bulk compression" term. This rule tells the model how the size of the swirls changes when the whole room is being squeezed or stretched.

3. The Experiment: Testing Three Different Rules

The authors ran computer simulations to see which "rule" for the bulk compression worked best when the room was being stretched in specific directions. They tested three versions of the recipe:

1. The "Average" Rule: It assumes the stretching is the same in all directions (the default setting).
2. The "Lengthwise" Rule: It assumes the size of the swirls changes based only on how much the room is stretching along the mixing direction.
3. The "Sideways" Rule: It assumes the size of the swirls changes based on how much the room is stretching across the mixing direction (perpendicular to the flow).

4. The Results: The "Sideways" Rule Wins

The authors compared their computer predictions against highly detailed, high-resolution simulations (which act like a "perfect" reference).

• The Finding: The default "Average" rule was okay, but not great. The "Lengthwise" rule actually made the predictions worse.
• The Winner: The "Sideways" Rule (using the transverse strain) was the most accurate.
• Why? The authors explain that when you stretch a mixing layer, the big "eddies" (swirls) behave differently depending on the direction. It turns out that the size of these swirls is more sensitive to how the space is changing sideways (transversely) than how it's changing lengthwise. By using the sideways stretching to adjust the size of the swirls in the recipe, the model predicted the mixing width and energy much more accurately.

5. The Bigger Picture: A New "Three-Part" Recipe

The paper also looked at how to simplify these complex equations into a "Buoyancy-Drag" model (a simpler way to think about the mixing).

• They realized that the "width of the mix" and the "size of the swirls" are actually reacting to different forces. The width stretches with the lengthwise pull, but the swirl size reacts to the sideways squeeze.
• The Conclusion: To get the best prediction, you need a model that treats these two things separately. Instead of one rule for everything, you need a three-part model that evolves the width and the swirl size independently.

Summary

In short, this paper is about fixing a computer model used to predict how fluids mix when the space around them is being distorted. The authors discovered that the standard way of calculating how the "swirls" shrink or grow was wrong for these specific conditions. By changing the rule to look at how the space is stretching sideways instead of just averaging it out, they made the model much more accurate. This helps scientists better understand complex events like stellar explosions or fusion energy experiments, where fluids are constantly being squeezed and stretched in uneven ways.

Technical Summary

1. Problem Statement

Turbulent mixing layers, such as those generated by Rayleigh-Taylor (RTI) and Richtmyer-Meshkov (RMI) instabilities, are fundamental to high-energy density physics, including inertial confinement fusion (ICF) and astrophysical phenomena like supernovae. In these scenarios, the flow often experiences anisotropic strain rates due to radial motion in convergent geometries (implosions) or divergent geometries (explosions).

Current Reynolds-Averaged Navier-Stokes (RANS) turbulence models, particularly the widely used K-L model (a two-equation model solving for turbulent kinetic energy $K$ and turbulent length scale $L$), typically rely on closure assumptions derived for isotropic strain or incompressible flows. The standard closure for the bulk compression term in the length scale transport equation assumes that the length scale evolves based on the mean isotropic strain rate (divergence of velocity divided by 3). However, in realistic anisotropic scenarios (e.g., spherical implosions), the strain rates in the radial (axial) and circumferential (transverse) directions differ significantly. The paper addresses the lack of theoretical definition and accurate modeling for how bulk compression affects the turbulent length scale under these anisotropic conditions.

2. Methodology

The authors conducted a systematic investigation using one-dimensional RANS simulations to isolate the effects of strain direction on mixing layer evolution.

• Numerical Framework: The simulations utilized the K-L turbulence model implemented in the FLAMENCO code. The model solves transport equations for Favre-averaged turbulent kinetic energy ($\tilde{K}$) and turbulent length scale ($L$).
• Test Cases: The study was based on the "quarter-scale $\theta$-group" Richtmyer-Meshkov instability case (heavy-to-light configuration, Atwood number $A_t=0.5$, shock Mach number $M_a \approx 1.84$).
• Strain Application: To isolate the effects of anisotropy, uniform normal strain rates were applied to the simulation domain in two distinct directions:
  • Axial Strain ($S_A$): Strain applied in the direction of mixing (normal to the interface).
  • Transverse Strain ($S_T$): Strain applied in the directions parallel to the interface (homogeneous plane).
  • Both constant velocity and constant strain rate profiles were tested.
• Closure Variations: Three different closures for the bulk compression term ($\bar{\rho} L S_\phi$) in the length scale equation were compared:
  1. Isotropic Closure (Default): Scales $L$ with the mean strain rate ($S_I = \nabla \cdot \mathbf{u} / 3$).
  2. Axial Closure: Scales $L$ with the axial strain rate ($S_A$).
  3. Transverse Closure: Scales $L$ with the transverse strain rate ($S_T$).
• Validation: The RANS results were compared against high-fidelity Implicit Large Eddy Simulation (ILES) data from previous studies (Pascoe et al.) which served as the ground truth.

3. Key Contributions

• Identification of Closure Deficiencies: The paper demonstrates that the standard isotropic closure assumption fails to accurately predict the evolution of turbulent mixing layers under anisotropic strain, particularly regarding the turbulent kinetic energy (TKE) and integral width.
• Proposal of Transverse Closure: The authors propose and validate a new closure strategy where the turbulent length scale evolution under bulk compression is driven by the transverse strain rate ($S_T$) rather than the mean isotropic strain.
• Derivation of a Three-Equation Buoyancy-Drag Model: Through self-similarity analysis, the authors derived a simplified buoyancy-drag model. They found that a single length scale cannot simultaneously capture the scaling of the integral width (which follows axial strain) and the turbulent length scale (which follows transverse strain). Consequently, they developed a three-equation model that evolves the integral width and turbulent length scale separately.
• Generalization to Other Models: The study extends the findings to other popular two-equation models (K-$\epsilon$ and K-$\omega$), deriving equivalent modifications to their bulk compression terms based on the transverse strain closure.

4. Results

• Integral Width ($W$):
  • Under axial strain, the mixing layer width is directly stretched or compressed by the mean flow velocity gradient. All closures captured this trend, but the transverse closure provided the most accurate prediction of the turbulent growth rate, minimizing the Mean Absolute Percentage Error (MAPE) by a factor of four compared to the isotropic closure.
  • Under transverse strain, the isotropic closure showed almost no variation in width (incorrectly canceling shear production and compression effects), while the transverse closure correctly predicted the slight increase in growth rate observed in ILES data.
• Turbulent Kinetic Energy (TKE):
  • The transverse closure significantly outperformed the other methods in predicting TKE.
  • The axial closure performed the worst, often predicting the wrong sign of change for TKE under expansion.
  • The isotropic closure overestimated the influence of strain on TKE.
• Self-Similarity Analysis:
  • The analysis revealed that under anisotropic strain, the ratio between the turbulent length scale peak and the integral width ($\beta_W$) is not constant, violating the standard self-similarity assumption used in many K-L calibrations.
  • This necessitated the development of the three-equation buoyancy-drag model, where the integral width scales with axial strain ($S_A$) and the turbulent length scale scales with transverse strain ($S_T$).
• Model Equivalence: The derived modifications for the K-L model were successfully mapped to the K-$\epsilon$ and K-$\omega$ models, suggesting that the bulk compression term in these models should also be adjusted to account for transverse strain rather than just mean divergence.

5. Significance

This work provides a critical advancement in the modeling of compressible turbulent mixing in high-energy density applications.

• Improved Predictive Capability: By correcting the bulk compression closure to account for anisotropy, the K-L model (and its equivalents) can now more accurately predict mixing layer growth and TKE in implosion/explosion scenarios, which are central to Inertial Confinement Fusion (ICF) and astrophysical simulations (e.g., supernovae).
• Physical Insight: The study clarifies the distinct physical roles of axial and transverse strains: axial strain primarily drives the geometric stretching of the mixing layer, while transverse strain governs the evolution of the turbulent eddies (length scale) and dissipation.
• Practical Application: The proposed modifications are relatively simple to implement in existing RANS codes but offer substantial improvements in accuracy without the computational cost of LES or DNS. The derived three-equation buoyancy-drag model offers a computationally efficient tool for engineering applications where full RANS simulations are too expensive.

In conclusion, the paper argues that for anisotropic strain environments, the turbulent length scale should be modeled as evolving with the transverse strain rate, a finding that challenges the long-standing assumption of isotropic scaling in standard turbulence closures.