A Shakhov-based Bhatnagar-Gross-Krook model for polyatomic molecules and for atomic as well as polyatomic mixtures

This paper extends the Shakhov-based Bhatnagar-Gross-Krook (SBGK) model within the PICLas code to simulate polyatomic molecules and their mixtures with non-equilibrium internal degrees of freedom, demonstrating through supersonic and hypersonic test cases that it accurately captures transport properties and shock structures with improved precision over the ESBGK model compared to DSMC results.

The Gist

Imagine you are trying to predict how a swarm of bees, a cloud of dust, and a group of fireflies would move and interact as they fly through a windy canyon. Some of these creatures are simple (like the dust), while others are complex (like the bees, which have wings that flap and bodies that vibrate).

This paper is about building a better computer simulation to predict exactly how these different "creatures" (gas molecules) behave when they are flying at supersonic speeds, especially when they are mixed together and not in a perfect, calm state.

Here is the breakdown of what the authors did, using simple analogies:

1. The Problem: The "Too Slow" and the "Too Simple"

In the world of physics, there are two main ways to simulate gas:

• The "Billiard Ball" Method (DSMC): Imagine simulating every single molecule as a tiny billiard ball bouncing off others. It's incredibly accurate, like watching a high-speed camera record every collision. But it's slow. If you have a billion balls, your computer takes forever to crunch the numbers.
• The "Traffic Flow" Method (CFD): This treats gas like a smooth river of water. It's fast, but it breaks down when the gas is thin or moving very fast (like in space), because it assumes the gas is always calm and smooth.

The Goal: The authors wanted a "Goldilocks" method—something faster than the billiard balls but more accurate than the smooth river, specifically for mixtures of different gases (like Oxygen mixed with Nitrogen) where the molecules are complex (they spin and vibrate).

2. The Solution: The "Shakhov-BGK" Model

The authors upgraded an existing mathematical shortcut called the BGK model.

• The Old Shortcut (Standard BGK): Imagine a teacher telling a classroom of students, "Everyone, just relax and move toward the average speed of the class." It's a simple rule. But it has a flaw: it doesn't handle heat flow correctly. It's like a thermostat that gets the temperature right but fails to account for how heat moves through a room.
• The New Shortcut (Shakhov-BGK): The authors tweaked the teacher's rule. Now, the teacher says, "Relax toward the average speed, but also adjust your movement based on how hot or cold your neighbors are." This small tweak fixes the "heat flow" problem, making the simulation much more accurate without slowing it down.

3. The New Challenge: Complex Molecules and Mixtures

The real breakthrough in this paper is handling two new complications:

A. The "Dancing" Molecules (Polyatomic Gases)

• Simple Atoms (like Helium): These are like smooth marbles. They only move forward, backward, left, and right.
• Complex Molecules (like Nitrogen or CO2): These are like dumbbells or spinning tops. They don't just move; they spin (rotation) and vibrate (like a spring).
• The Fix: The authors updated their model to track these extra movements. They realized that when a spinning molecule hits another, it doesn't just bounce; it transfers energy into its spin and vibration. The new model accounts for this "internal energy" so the temperature calculations are correct.

B. The "Mixed Crowd" (Gas Mixtures)

• Imagine a crowd where some people are heavy sumo wrestlers (Argon) and others are lightweight jockeys (Helium).
• The Problem: When you mix them, the heavy ones and light ones interact differently than if they were all the same. Calculating how they share energy and momentum is mathematically messy.
• The Fix: The authors created a set of rules (using "collision integrals") to calculate exactly how a heavy sumo wrestler and a light jockey should bounce off each other in the simulation. They made sure the model works whether you have 2 types of gas or 3.

4. The Test Drive: The "Shockwave"

To prove their new model works, they ran two tests:

1. The Moving Walls (Couette Flow): Imagine gas trapped between two plates sliding past each other. They tested different gas mixes (pure Nitrogen, a mix of Argon/Helium, and a mix of Nitrogen gas/atoms).

   • Result: Their new model matched the super-slow "Billiard Ball" method almost perfectly, even with the tricky heavy/light mix.

2. The Hypersonic Cone (The Shockwave): They simulated a rocket nose cone (blunted cone) flying at hypersonic speeds (70 degrees angle). This creates a shockwave—a sudden, violent wall of compressed air and heat.

   • The Showdown: They compared their new model (SBGK) against an older, popular model (ESBGK) and the super-accurate "Billiard Ball" method.
   • The Winner: The new SBGK model was the best at predicting the shockwave. The older model (ESBGK) was a bit "blurry" at the shock front, predicting the temperature rise too early. The new model hit the "shock" with the precision of a laser, matching the super-accurate reference data.

The Bottom Line

The authors built a smarter, faster, and more accurate calculator for gas dynamics.

• Why it matters: This is crucial for designing spacecraft that enter atmospheres (like Mars or Earth), where gases are hot, moving fast, and mixed together.
• The Analogy: If the old methods were like guessing the weather by looking out a window, and the "Billiard Ball" method was like measuring every single raindrop, this new model is like having a super-accurate weather radar that is fast enough to run on a laptop but detailed enough to see the storm clouds forming.

They proved that by adding a little bit of "mathematical spice" (the Shakhov correction) and carefully tracking how complex molecules spin and vibrate, they can simulate the chaos of space flight much better than before.

Technical Summary

1. Problem Statement

Numerical simulations of fluid dynamics in space applications, micro/nanoscale flows, and vacuum technologies face significant challenges due to large density gradients ranging from the continuum to the free molecular flow regimes.

• Limitations of DSMC: While the Direct Simulation Monte Carlo (DSMC) method is highly accurate for non-equilibrium flows, it becomes computationally expensive in the transition and continuum regimes due to the need to resolve the mean free path and collision frequency.
• Limitations of Standard BGK: The Bhatnagar-Gross-Krook (BGK) model offers a computationally efficient alternative by approximating the Boltzmann collision integral with a relaxation process. However, the standard BGK model fails to reproduce the correct Prandtl number (typically predicting $Pr=1$ instead of the physical $\approx 2/3$ for monatomic gases), leading to inaccurate heat flux predictions.
• Gap in Existing Models: While the Ellipsoidal Statistical BGK (ESBGK) model corrects the Prandtl number, previous studies suggest it may not capture shock structures as precisely as the Shakhov BGK (SBGK) model. Furthermore, existing particle-based SBGK implementations often lack robust extensions for polyatomic gases (accounting for rotational and vibrational energy) and complex gas mixtures (atomic-polyatomic combinations) while maintaining conservation properties and correct transport coefficients.

2. Methodology

The authors extended the open-source particle code PICLas to implement a Shakhov BGK (SBGK) model capable of handling:

1. Polyatomic Molecules: Accounting for internal degrees of freedom (DOF), specifically rotational and vibrational modes.
2. Gas Mixtures: Simulating mixtures of atoms and molecules (e.g., N$_2$-N, Ar-He, CO$_2$-N$_2$).

Key Theoretical Components:

• Target Distribution Function: The model uses a single relaxation term per species. The target distribution is factored into translational, rotational, and vibrational components.
  • Atoms: Only translational DOF.
  • Molecules: Translational, rotational (modeled via Maxwell-Boltzmann), and vibrational (modeled as quantized harmonic oscillators).
• Shakhov Correction: To correct the Prandtl number, the translational target distribution is modified by a heat flux term:
  $$f^{SBGK,tr}_s = f^M_s \left[ 1 + (1 - \alpha Pr) \frac{c \cdot q}{5 n (k_B T_{tr,rel})^2 / m_s} \left( \frac{|c|^2}{k_B T_{tr,rel}/m_s} - 5 \right) \right]$$
  Where $q$ is the heat flux vector, $Pr$ is the target Prandtl number, and $\alpha$ is a model parameter derived from mass fractions and internal DOFs.
• Transport Properties:
  • Viscosity & Thermal Conductivity: Determined using the first approximation of transport properties based on collision integrals (Chapman-Enskog expansion) rather than simple mixing rules like Wilke's, to ensure higher accuracy for mixtures.
  • Relaxation Frequencies: The relaxation of internal energies (rotation/vibration) follows the Landau-Teller form, utilizing collision numbers ($Z$) and characteristic collision times derived from the Variable Hard Sphere (VHS) model.
• Conservation: The model is designed to satisfy mass, momentum, and energy conservation principles, ensuring the formulation reduces to the single-species case when all species are identical.

3. Key Contributions

1. Unified SBGK Framework: Development of a particle-based SBGK model that simultaneously handles atomic, diatomic, and polyatomic species in mixtures, including quantized vibrational states.
2. Derivation of Model Parameters: Explicit derivation of the parameter $\alpha$ and the Shakhov correction factor ($Pr_S$) to ensure the model reproduces the physical Prandtl number of the mixture ($Pr_{phys}$) via the relation $Pr_S = \alpha Pr_{phys}$.
3. Implementation in PICLas: Integration of the model into an open-source particle code, utilizing an acceptance-rejection sampling technique for the Shakhov distribution.
4. Validation against DSMC and ESBGK: Comprehensive verification against the "gold standard" DSMC method and the established ESBGK model.

4. Results

The model was validated using two primary test cases:

A. Supersonic Couette Flow

• Cases: Pure N$_2$, Ar-He mixture, and N$_2$-N mixture.
• Findings:
  • The SBGK model showed excellent agreement with DSMC for pure N$_2$ and the N$_2$-N mixture regarding translational and rotational temperatures.
  • For the Ar-He mixture (large mass ratio), a slight deviation in the Prandtl number approximation was observed compared to DSMC, consistent with previous findings for ESBGK. This is attributed to the difficulty in approximating effective transport properties for gases with vastly different masses.

B. Hypersonic Flow over a 70° Blunted Cone

• Cases: Pure N$_2$, CO$_2$-N$_2$ mixture (polyatomic-diatomic), and N$_2$-O$_2$-N mixture (three-species).
• Findings:
  • Shock Capturing: The SBGK model captured the shock front significantly more precisely than the ESBGK model. The ESBGK model predicted an earlier onset of temperature rise, resulting in a broader, less accurate shock region.
  • Temperature Profiles: SBGK results for translational, rotational, and vibrational temperatures along the stagnation streamline matched DSMC reference data very closely, particularly in the non-equilibrium shock region.
  • Heat Flux: Excellent agreement was observed between SBGK, ESBGK, DSMC, and experimental data for surface heat flux on both the windward and rear surfaces of the cone.
  • Mixtures: The model successfully handled complex mixtures (e.g., CO$_2$-N$_2$) with multiple internal degrees of freedom, maintaining accuracy in vibrational relaxation.

5. Significance

• Computational Efficiency: The SBGK model offers a computationally cheaper alternative to DSMC for low-Knudsen number regimes while retaining the ability to model non-equilibrium effects (internal energy relaxation) which standard CFD cannot easily do.
• Superior Shock Resolution: The study demonstrates that the Shakhov-based approach is superior to the Ellipsoidal Statistical (ESBGK) approach in resolving shock structures, a critical feature for hypersonic re-entry simulations.
• Versatility: The successful extension to polyatomic mixtures and the use of rigorous collision integral methods for transport properties make this model a robust candidate for hybrid coupling strategies (coupling DSMC in rarefied regions with SBGK in continuum regions) for multi-scale space flight problems.
• Future Outlook: The authors note that the next step is the implementation of chemical reactions into the model, which would further enhance its utility for high-enthalpy atmospheric entry simulations.