Impact of Structure-Preserving Discretizations on Compressible Wall-Bounded Turbulence of Thermally Perfect Gases

This study demonstrates that for direct numerical simulations of compressible wall-bounded turbulence in thermally perfect gases, the consistency between structure-preserving discretizations and the thermodynamic model is critical for accurately capturing turbulence statistics and mean flow properties, particularly at high Mach numbers where entropy conservation and pressure discretization coupling significantly influence dynamical fields.

The Gist

Imagine you are trying to predict the weather inside a very long, narrow tunnel where air is rushing through at incredible speeds—faster than the speed of sound, even hypersonic speeds (like a rocket re-entering the atmosphere).

This paper is about how to build the best computer simulation to predict what happens to that air, specifically when the air gets so hot that its basic properties change.

Here is the breakdown using simple analogies:

1. The Problem: The "Hot Air" Challenge

Usually, when we simulate air flow, we treat air like a simple, predictable gas (like a balloon that just gets bigger when heated). This works fine for normal speeds.

But in this study, the authors are looking at Carbon Dioxide (CO2) moving at hypersonic speeds (like on a Mars landing). At these speeds, the air gets incredibly hot.

• The Analogy: Imagine the air molecules are like a crowd of people. At normal speeds, they are just walking. At hypersonic speeds, they are running, sweating, and their behavior changes. They aren't just "hot"; their internal energy, how they bounce off each other, and how they carry heat changes in complex ways.
• The Issue: Standard computer math (discretization) often treats these hot, complex gases too simply. It's like trying to predict the behavior of a chaotic mosh pit using the rules for a quiet library. The simulation might crash, or it might give you a result that looks okay but is actually wrong.

2. The Solution: "Structure-Preserving" Math

The authors tested different mathematical recipes (discretizations) to see which one handles this chaos best. They focused on two main "laws" that the math must obey to stay accurate:

• Kinetic Energy Preservation (KEP): Think of this as a bank account. If you have \100 and you move money around, you should still have \100 at the end. Standard math sometimes accidentally "loses" or "creates" energy due to rounding errors, which ruins the simulation. These new methods ensure the energy balance is perfect, like a perfectly balanced ledger.
• Entropy Conservation (EC): This is the thermodynamic rulebook. Entropy is a measure of disorder or "messiness." In a smooth flow without shocks, the "messiness" shouldn't magically appear or disappear. The authors found that if your math doesn't respect this rule, the simulation eventually blows up or gives weird results, especially when the gas gets hot.

3. The Experiment: The "Taste Test"

The researchers ran simulations of CO2 flowing through a channel at three different speeds:

1. Supersonic (Mach 3)
2. Supersonic (Mach 4)
3. Hypersonic (Mach 5)

They compared four different mathematical "recipes":

• The "Old Reliable" (KEEP): Good for simple gases, but struggles when the gas gets hot and complex.
• The "Specialist" (Gouasmi et al.): Good at handling the "messiness" (entropy) of hot gas, but had a flaw in how it handled pressure.
• The "Hybrid" (Ranocha): A middle-ground approach.
• The "Perfect Match" (EC-TP): A new method designed specifically to handle both the energy balance and the "messiness" of hot, complex gases perfectly.

4. The Results: Why the "Perfect Match" Won

As the speed (and heat) increased, the differences between the recipes became huge.

• The "Old Reliable" (KEEP): At high speeds, it started to hallucinate. It predicted that the air pressure and temperature were fluctuating wildly and incorrectly. It was like a GPS that starts giving you directions to the wrong city when you drive too fast.
• The "Specialist" (Gouasmi): It was better than the old one, but still made small errors in how it calculated the "push" (pressure) of the gas.
• The "Perfect Match" (EC-TP): This method stayed calm and accurate. It correctly predicted how the gas behaved, how the heat moved, and how the turbulence (the swirling eddies) formed.

The Key Finding:
When you are dealing with extreme heat and speed, you cannot just use a "good enough" math formula. You need a formula that respects the physics of the specific gas you are simulating. If you ignore the fact that CO2 changes its behavior when hot, your simulation will eventually fail.

5. The Real-World Impact

Why does this matter?

• Space Exploration: When a spacecraft lands on Mars (which has a CO2 atmosphere), it hits the atmosphere at hypersonic speeds. The heat is intense.
• Safety: If engineers use the wrong math to design the heat shield, the spacecraft could burn up.
• Efficiency: Accurate simulations mean we can design better engines and vehicles without needing to build expensive physical prototypes for every test.

Summary

Think of this paper as a guide for building the ultimate calculator for hot, fast air. The authors proved that to simulate extreme conditions (like a rocket landing on Mars), you need a calculator that doesn't just do the math, but also respects the rules of thermodynamics (energy and entropy) perfectly. If you cut corners on the math, the simulation breaks. If you use their new "Structure-Preserving" method, you get a reliable, accurate picture of reality.

Technical Summary

1. Problem Statement

The paper addresses the challenges of performing Direct Numerical Simulations (DNS) of compressible turbulent channel flows at supersonic and hypersonic Mach numbers using Thermally Perfect (TP) gas models (specifically Carbon Dioxide, $CO_2$).

• Thermodynamic Complexity: Unlike Calorically Perfect (CP) gases where specific heats are constant, TP gases exhibit temperature-dependent specific heats ($c_p(T), c_v(T)$). This alters the algebraic structure of entropy relations, making standard numerical schemes designed for ideal gases potentially inconsistent or unstable.
• Numerical Instability: Standard discretizations of nonlinear convective terms in high-enthalpy regimes often suffer from numerical instabilities. While artificial dissipation (e.g., WENO schemes) can stabilize simulations, it distorts energy transfer and turbulence statistics.
• The Gap: There is a lack of rigorous assessment regarding how structure-preserving discretizations (specifically those preserving Kinetic Energy and Entropy) perform when applied to TP gases in wall-bounded turbulence, particularly at high Mach numbers where thermodynamic-dynamic coupling is strong.

2. Methodology

The authors conducted DNS of a pressure-driven turbulent channel flow using the open-source, GPU-accelerated solver STREAmS 2.1.

• Governing Equations: The 3D Navier-Stokes equations for mass, momentum, and total energy were solved for a viscous, heat-conducting fluid.
• Thermodynamic Model: A Thermally Perfect gas model was used for $CO_2$, where pressure follows $p = \rho RT$, but internal energy and entropy are derived from temperature-dependent specific heats (using 7-coefficient polynomial fittings).
• Flow Configuration:
  • Geometry: Periodic channel flow with no-slip walls.
  • Regimes: Supersonic ($M_b = 3, 4$) and Hypersonic ($M_b = 5$).
  • Grid: High-resolution meshes (up to $816 \times 356 \times 512$ points) with sixth-order spatial accuracy for convective and diffusive terms.
  • Time Integration: Third-order low-storage explicit Runge-Kutta with adaptive time stepping (CFL $\approx 0.6$).
• Numerical Schemes Compared:
  The study compared four specific formulations to isolate the effects of Kinetic Energy Preservation (KEP) and Entropy Conservation (EC):
  1. KEEP (Kuya et al.): A widely used scheme that is KEP but only approximately EC (using arithmetic averages). It serves as the baseline.
  2. Ranocha: Exactly EC and KEP for CP gases; tested here to see if it holds for TP gases.
  3. Gouasmi et al.: Exactly EC for TP gases but uses a specific pressure discretization ($P_{\rho u} \propto R_\rho / (1/T)$) known to potentially spoil kinetic energy balance in certain contexts.
  4. EC-TP (Proposed/Adapted): The primary focus. An exactly Entropy-Conservative and Kinetic-Energy-Preserving scheme specifically derived for TP gases using logarithmic means and asymptotic expansions to handle singularities.

3. Key Contributions

• First High-Enthalpy TP DNS with Exact EC: This work presents the first DNS of high-enthalpy channel flow using a Thermally Perfect gas model with a discretization that is exactly entropy-conservative and kinetic-energy-preserving.
• Decoupling Numerical Effects: The study successfully disentangles the impact of entropy consistency (convective term discretization) from pressure term discretization. It demonstrates that while entropy conservation is crucial for stability, the specific treatment of the pressure term significantly affects turbulence statistics.
• Validation of Structure-Preserving Schemes: It proves that for TP gases, maintaining the discrete consistency with the Gibbs relation and the equation of state is essential for long-term robustness, especially as Mach numbers increase.

4. Results

The simulations revealed distinct differences between the numerical schemes, which became more pronounced as the Mach number increased:

• Mean Flow Statistics:
  • All schemes produced similar velocity profiles in the viscous sublayer.
  • In the logarithmic region, the KEEP scheme overpredicted the Trettel-Larsson transformed velocity, particularly at $M_b=5$.
  • The EC-TP scheme provided the most consistent results with theoretical expectations.
• Turbulent Fluctuations (Reynolds Stresses):
  • KEEP exhibited abnormally high peaks in streamwise Reynolds stress ($\tau'_{11}$), especially in the hypersonic case ($M_b=5$), indicating a failure to correctly model energy transfer.
  • Gouasmi et al. showed improved stability over KEEP but still overestimated Reynolds stresses compared to EC-TP, suggesting that entropy conservation alone is insufficient if the pressure term is not optimally discretized.
  • EC-TP yielded the most accurate and physically consistent Reynolds stress profiles.
• Thermodynamic Fluctuations:
  • Non-EC schemes (KEEP, Ranocha) showed systematic overprediction of thermodynamic fluctuations (pressure and density).
  • Crucially, non-EC schemes exhibited a "drift" in thermodynamic variables over time, leading to statistical non-stationarity and potential simulation blow-up on coarser grids.
  • EC-TP maintained a statistically stationary state throughout the simulation duration.
• Mach Number Dependence:
  • Discrepancies between schemes were negligible at $M_b=3$ but grew significantly at $M_b=5$. This confirms that the coupling between thermodynamics and dynamics intensifies with compressibility, making structure-preserving discretizations increasingly critical.

5. Significance and Conclusions

• Necessity of EC for TP Gases: The study concludes that for high-enthalpy flows involving Thermally Perfect gases, entropy-conservative discretizations are not merely beneficial but necessary for reliable, long-term simulations. Standard schemes designed for ideal gases fail to maintain stability and accuracy when specific heats vary with temperature.
• Role of Pressure Discretization: While entropy conservation is the primary driver of stability, the specific discretization of the pressure term is vital for accuracy. The EC-TP scheme, which combines exact entropy conservation with a consistent pressure treatment, outperformed all others.
• Robustness without Artificial Dissipation: The EC-TP scheme allowed for fully central-difference simulations (without WENO shock-capturing) even at hypersonic speeds, preserving the structural integrity of the turbulence without the excessive dissipation typical of hybrid schemes.
• Future Outlook: The authors suggest that future work should extend these structure-preserving concepts to viscous terms to achieve fully entropy-stable formulations and apply them to multi-species mixtures and boundary layers.

In summary, this paper establishes that numerical consistency with the thermodynamic model is a prerequisite for accurate DNS of high-speed compressible turbulence, and that Entropy-Conservative, Kinetic-Energy-Preserving (EC-KEP) schemes are the superior choice for Thermally Perfect gases.