Realizability-Constrained Machine Learning for Turbulence Closures in Wake Flows

This paper proposes a residual- and realizability-filtered gene expression programming framework that integrates physical constraints directly into the CFD solution loop to efficiently discover stable, physically consistent turbulence closure models for wake flows, achieving significant reductions in computational cost and non-realizable outcomes while demonstrating robust generalization across diverse geometries.

The Gist

Imagine you are trying to teach a robot how to predict how smoke swirls behind a car or a boat. This is a problem called "turbulence modeling." Scientists use complex math (simulations) to do this, but the standard math they use is like a blunt instrument—it often gets the details wrong, especially in the messy wake behind an object.

To fix this, the researchers in this paper used Machine Learning to invent a new, smarter math formula. However, teaching a machine to invent physics formulas is tricky. If you let the machine run wild, it often creates formulas that look good on paper but cause the computer simulation to crash, freeze, or produce "ghosts" (results that break the laws of physics).

Here is how the paper solves this problem, explained simply:

1. The Problem: The "Wild Child" Learner

Think of the machine learning process as a teacher trying to train 256 students (candidate formulas) to solve a puzzle.

• The Old Way (Baseline): The teacher lets every student work on the puzzle for a long time (thousands of steps). If a student's answer causes the classroom to explode (the computer crashes) or if they write down a number that is physically impossible (like negative energy), the teacher only finds out after the student has wasted hours of work. This is incredibly slow and expensive.
• The Issue: Many of these "bad" formulas are mathematically unstable or "unrealizable" (they break the rules of reality), but the computer doesn't know this until it's too late.

2. The Solution: The "Three-Stage Security Check"

The authors created a new system called RR-GEP. Instead of letting every student work until the end, they installed a strict security checkpoint with three gates. If a student fails a gate, they are kicked out immediately, saving time and energy.

• Gate 1: The "Is it Exploding?" Check (Residual Check)
  Imagine a student is solving a math problem. If their numbers start jumping wildly or getting huge, the teacher stops them immediately. This catches formulas that cause the computer to crash.
• Gate 2: The "Are You Improving?" Check (Convergence Check)
  If the numbers aren't exploding, the teacher asks: "Are you getting closer to the answer?" If the student is stuck in a loop, making no progress, they are sent home. This stops formulas that just waste time without solving anything.
• Gate 3: The "Does it Make Sense?" Check (Realizability Check)
  This is the most important new feature. Even if a student is solving the math correctly and not crashing, their answer might still be impossible in the real world.
  • The Analogy: Imagine a student says, "The wind is blowing at 100 mph, but the air has negative weight." The math might be right, but the physics is wrong.
  • The researchers use a special map (called a Barycentric Map) to check if the student's answer fits inside the "Triangle of Reality." If the answer falls outside this triangle, it's rejected instantly. This ensures the new formula respects the fundamental laws of physics.

3. The Results: Faster and Smarter

By using this three-stage filter, the researchers achieved some impressive results:

• Speed: They cut the time needed to train the AI by 42%. They stopped wasting time on formulas that were doomed to fail.
• Quality: In the old method, nearly 60% of the final formulas were physically impossible ("unrealizable"). In their new method, they reduced this to less than 2%.
• Performance: The formulas they found were not just stable; they were actually better at predicting the "wake" (the swirling air/water) behind objects. They predicted the size of the swirling zone more accurately than the old standard methods.

4. Does it work on other things?

The researchers trained the AI on a simple circular cylinder (like a pipe sticking out of water). Then, they tested it on completely different shapes:

• A rectangular cylinder (like a brick).
• An airfoil (the wing of an airplane).
• A submarine shape (DARPA Suboff).

Even though the AI was only trained on the round pipe, it successfully predicted the wake for the brick, the wing, and the submarine. It didn't just memorize the pipe; it learned the underlying rules of how turbulence works, and it kept those rules "real" (physically possible) in all these new situations.

Summary

The paper presents a new way to teach computers to invent physics formulas. Instead of letting the computer guess blindly and hoping it doesn't crash, they put up three "guardrails." These guardrails stop the computer from wasting time on bad ideas and ensure that every final formula it invents obeys the laws of physics. This makes the process faster, cheaper, and much more reliable for predicting how fluids move around objects.

Technical Summary

Technical Summary: Realizability-Constrained Machine Learning for Turbulence Closures in Wake Flows

Problem Statement
Computational Fluid Dynamics (CFD)-driven machine learning frameworks, particularly those utilizing symbolic regression via Gene Expression Programming (GEP), offer a promising pathway for discovering explicit, physics-interpretable turbulence models. However, these frameworks face significant limitations regarding numerical instability and physical admissibility. During the training process, thousands of candidate closure models are sequentially embedded into Reynolds-Averaged Navier–Stokes (RANS) solvers. A non-negligible fraction of these candidates leads to solver divergence, residual stagnation, or unphysical growth of turbulence quantities. Such unstable behaviors are often detected only after thousands of iterations, resulting in substantial wasted computational resources. Furthermore, existing frameworks rarely enforce "realizability"—the requirement that modeled Reynolds stresses remain physically admissible (e.g., positive turbulent kinetic energy, bounded anisotropy)—during the discovery process. Consequently, learned closures may violate fundamental physical constraints even if they appear to improve mean-flow agreement with reference data, compromising robustness and generalization.

Methodology
The authors propose a residual- and realizability-filtered CFD-driven framework integrated within a GEP paradigm. This approach introduces a multi-stage supervisory layer directly into the CFD solution loop to enable the early identification and rejection of unstable and non-realizable candidate models. The methodology consists of three sequential filtering stages applied at specific iteration counts ($N_1$ and $N_2$) before full convergence ($N_3$) is attempted:

1. Stage 1 (Absolute Residual): After $N_1$ iterations, candidates are rejected if the maximum normalized residual of the governing equations exceeds a prescribed threshold ($\epsilon_1 \approx 10^{-1}$). This filters out models causing immediate divergence.
2. Stage 2 (Residual Reduction Ratio): Candidates passing Stage 1 are evaluated at $N_2$ iterations. They are rejected if the residual reduction ratio ($\Gamma$) between $N_1$ and $N_2$ fails to meet a minimum improvement factor, or if the solution stagnates. This ensures only models demonstrating a consistent decay in residuals proceed.
3. Stage 3 (Realizability Enforcement): For models passing the residual checks, a barycentric-map-based constraint is applied. The eigenvalues of the normalized anisotropy tensor are computed, and the resulting state is mapped onto a barycentric triangle representing limiting turbulence states (1C, 2C, 3C). Candidates are rejected if their barycentric weights violate convexity constraints, allowing for a small tolerance ($\epsilon_3$) scaled by the effective residual to account for incomplete convergence.

The framework was trained on a canonical circular cylinder wake at $Re=3900$. The cost function minimized the discrepancy between RANS and Large Eddy Simulation (LES) streamwise velocity profiles in the wake, without explicit penalty terms for stability or realizability, as these were strictly enforced via the filtering stages. The resulting models were Explicit Algebraic Reynolds Stress Models (EARSM) applied specifically in the wake region, while the Boussinesq approximation was retained elsewhere.

Key Contributions

• Development of a Filtered GEP Framework: The paper introduces a novel "Realizability Residual-filtered GEP" (RR-GEP) algorithm that integrates stability and realizability checks directly into the evolutionary loop, distinguishing it from "frozen" or standard CFD-driven approaches.
• Efficiency Gains: By terminating unproductive candidates early, the framework significantly reduces the computational cost of model discovery.
• Physical Admissibility: The method explicitly enforces realizability constraints during training, ensuring that discovered models yield physically consistent Reynolds stress states.
• Generalization Validation: The study validates the robustness of the discovered models across diverse geometries and flow regimes, including a rectangular cylinder, a NACA 0012 airfoil, and an axisymmetric DARPA Suboff body, at Reynolds numbers ranging from $10^3$ to $10^6$.

Results

• Computational Efficiency: The RR-GEP framework achieved a 42.3% reduction in computational cost compared to the baseline CFD-driven GEP (B-GEP). The model rejection rate was 37.8% on average for RR-GEP, with 47.3% of candidates rejected in the first generation alone.
• Realizability Improvement: The filtering strategy drastically reduced the prevalence of non-realizable models at convergence. While 58.4% of converged models in the B-GEP framework were non-realizable, this figure dropped to 1.7% for the RR-GEP framework.
• Predictive Accuracy: The RR-GEP models improved wake predictions for the training case (cylinder), reducing the separation bubble length from $x/D \approx 4.5$ (baseline RANS) to $2.25$, closely matching LES data. The models also demonstrated improved prediction of Reynolds shear stress and turbulent kinetic energy.
• Robustness and Generalization: The models trained on the circular cylinder successfully generalized to unseen geometries (rectangular cylinder, airfoil, Suboff) and significantly higher Reynolds numbers without losing realizability. The models maintained bounded coefficient behavior and physically consistent anisotropy trajectories across all test cases.
• Model Characteristics: The filtering process led to models with significantly reduced and tightly bounded scalar coefficients compared to the baseline. For instance, the average magnitude of the coefficient $|\zeta_1|$ decreased from $\approx 2.73$ (B-GEP) to $\approx 1.76$ (RR-GEP), indicating a preference for simpler, more controlled functional forms that avoid the amplification of unphysical stresses.

Significance
The paper claims that incorporating realizability and stability constraints directly within CFD-driven learning enables the efficient discovery of turbulence models that are both numerically robust and physically consistent. By filtering out unstable and non-realizable candidates early, the framework reduces the computational burden of model discovery while guiding the search toward physically admissible solutions. The results demonstrate that this approach offers a scalable pathway for developing reliable data-driven closures that generalize well across diverse flow configurations and operating conditions, addressing a critical gap in current machine learning-based turbulence modeling efforts.