This work introduces a significant advancement in turbulence modeling for computational fluid dynamics (CFD) by presenting a machine learning-based framework for adaptively blending specialized data-driven turbulence models. The primary motivation stems from the long-standing challenge of achieving generalizability and accuracy in Reynolds-Averaged Navier-Stokes (RANS) simulations across diverse, complex flow regimes. Traditional RANS models, while computationally efficient, often struggle with non-equilibrium turbulence, strong gradients, and separated flows, and data-driven models, while promising, typically specialize to narrow flow classes and may fail to generalize or even degrade performance in conditions outside their training.
The core innovation lies in its intrusive internal blending strategy. Unlike prior “external” blending approaches that combine the predictions of multiple RANS models, which often violate conservation laws and require computationally expensive multiple simulations, this method directly blends the corrective terms applied to a baseline RANS turbulence model (specifically, the k-ω SST model). By working at the level of model corrections rather than full solutions, the resulting blended model inherently satisfies the governing RANS equations, preserving physical conservation principles and requiring only a single RANS simulation for prediction.
A second key innovation is the adaptive, localized weighting mechanism. The framework employs a Random Forest Regressor (RFR) to dynamically determine the blending weights for each specialized model at every spatial location within the flow domain. This RFR is trained to map local physical flow features—such as turbulent kinetic energy, pressure gradients, and strain rates—to the optimal weights, which are derived from a Gaussian kernel that quantifies how well each specialized model performs compared to high-fidelity data in specific flow regions. This allows the blended model to intelligently select the most appropriate “expert” in real-time, based on the local flow characteristics, effectively leveraging the strengths of individual models where they are most accurate.
The foundational prior ingredients enabling this work include:
* Reynolds-Averaged Navier-Stokes (RANS) Equations and Baseline Turbulence Models: The standard k-ω SST model forms the bedrock upon which the data-driven corrections are applied and blended.
* Data-Driven Turbulence Modeling: The general concept of using machine learning to improve RANS closures, particularly the approach of augmenting baseline models with learned correction terms for the Reynolds stresses and transport equations.
* Sparse Bayesian Learning (SBL) and Symbolic Regression: These techniques, particularly the SBL-SpaRTA algorithm, are crucial for training the specialized “expert” data-driven models. These experts are pre-trained for distinct flow classes like turbulent channel flows, separated flows, and axisymmetric jets, providing the individual components that the blending framework then combines.
* Mixture-of-Experts (MoE) Concepts: While deviating significantly in its intrusive nature, the paper builds on the conceptual foundation of MoE architectures and space-dependent model aggregation techniques, which aim to combine the predictive power of multiple individual models.
* Physical Flow Features for Machine Learning: The selection of specific, physically interpretable flow features (e.g., non-dimensional strain and rotation rates, pressure gradients) is essential as inputs for the RFR to learn the local blending weights. These features allow the ML model to understand the underlying physics of different flow regions.
* Random Forest Regression: This ensemble machine learning algorithm is chosen for its robustness and ability to capture complex, non-linear relationships between the input flow features and the desired blending weights.
* High-Fidelity Data (DNS/LES/Experiments): Extensive high-fidelity data from Direct Numerical Simulations (DNS), Large Eddy Simulations (LES), and experiments are indispensable for both training the specialized expert models via symbolic regression and for training the RFR to determine the optimal blending weights.
The robust performance demonstrated across various test cases, including untrained scenarios like a NACA0012 airfoil at various angles of attack, underscores the potential of this framework to provide more generalizable, accurate, and physically consistent turbulence models for a wide range of engineering applications.