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Machine learning methods for turbulence modeling in subsonic flows over airfoils
Weiwei Zhang, Linyang Zhu, Yilang Liu, Jiaqing Kou
TL;DR
DNS remains impractical for aeronautical industries, while turbulence models require prescribed parameters determined a priori and turbulence questions remain unresolved. The paper develops a regional RBFNN-based eddy-viscosity model from three NACA0012 training cases, reporting high accuracy, generalization, and greater efficiency than the original SA model, but identifies the work as preliminary for high-Reynolds-number and complex separated flows.
Problem
DNS remains impractical in aeronautical industries, while turbulence models have limitations because some prescribed parameters must be determined a priori.
Method
The approach divides the flow into zones and uses a radial basis function neural network to model eddy viscosity from three NACA0012 turbulent-flow training cases.
Results
The proposed approach achieves high accuracy and generalization, agrees well with the SA model’s friction distribution, and is more efficient than the original SA model.
Takeaways & Limitations
The results demonstrate the promising prospect of data-driven methods for turbulence modeling.
Takeaways & Limitations
The paper is preliminary work toward modeling high-Reynolds-number turbulent flows and addresses separated flows and other complex flows as future scope.
Abstract
from arXiv · showhide
Reynolds-Averaged Navier-Stokes(RANS) method will still play a vital role in the following several decade in aerospace engineering. Although RANS models are widely used, empiricism and large discrepancies between models reduce the reliability of simulating complex flows. Therefore, in recent years, data-driven turbulence model has aroused widespread concern in fluid mechanics. Based on the experimental/numerical simulation results, this approach aims to modify or construct the turbulence model for specific purposes by machine learning technologies. In this paper, we take the results calculated by SA model as training data. Different from low Reynolds number turbulent flows, the data from high Reynolds number flows shows an apparent scaling effect, thus leading to difficulties in the data-driven modeling. In order to improve the fitting accuracy, we divided the flow field into near-wall region, wake region, and far-field region, and built individual model for every region. In this paper, we adopted the radial basis function neural network (RBFNN) and some auxiliary optimization algorithms to reconstruct a mapping function between mean variables and the eddy viscosity. Since this model reflects the relationship between local flow characteristics and turbulent eddy viscosity, it is independent on the airfoil shape and flow condition. The training data in this paper is generated from only three subsonic flow calculations of NACA0012 airfoil. By coupling the proposed approach with N-S equations, we calculated various flow cases as well as two different airfoils and showed the eddy viscosity contours, velocity profiles along the normal direction of wall and skin friction coefficient distributions, etc. Compared with the SA model, the results show a reasonable accuracy and better efficiency, which indicates the positive prospect of data-driven methods in turbulence modeling.
1. Introduction
RANS remains important for engineering applications because DNS and LES remain impractical or demanding for many aerospace simulations. However, traditional RANS models rely on prescribed empirical parameters and can disagree substantially for complex separated flows, motivating data-driven alternatives.
- DNS is impractical for aeronautical industries because its grid-resolution requirement grows exponentially with Reynolds number.
- LES simulations across the full flight envelope require HPC and algorithmic advances that may not be realized until 2030.
- RANS models remain widely used because they offer high efficiency and easy implementation, despite limitations in universality.
- Traditional models require prescribed parameters derived from specific experiments and DNS results, limiting their universality.
- Complex separated flows can produce large discrepancies between turbulence models, forcing users to choose models based on experience and problem characteristics.
- Machine learning and data-mining methods have recently been explored to model turbulence variables, reduce uncertainty, and improve or replace RANS models.
- This paper focuses on reconstructing the eddy-viscosity functional form for high-Reynolds-number airfoil flows, with lower computational cost expected because the approach does not solve PDEs.
2. Method
The method constructs a surrogate eddy-viscosity model by combining flow-region partitioning, selected local features, and a one-hidden-layer RBFNN embedded in a CFD solver.
- Model construction: The approach separates learning-machine construction from surrogate-machine coupling, where predicted eddy viscosity is passed into the CFD solver.The learning machine handles sample selection, framework design, and optimization; the surrogate machine evaluates eddy viscosity from mean-flow variables during CFD calculations.
- Modeling strategy: The model combines freestream conditions, local mean variables, derivatives, and location information to improve generalization across cases.This increases feature dimension while reducing the number of separate local models relative to a fully local strategy.
- Modeling strategy: High-Reynolds-number flows produce large wall-normal eddy-viscosity ranges, so the flow field is divided into zones to reduce outliers.The zones are defined according to normal distance from the wall, addressing scaling difficulties in direct modeling.
- Modeling strategy: Near-wall fitting receives extra weight because small eddy-viscosity errors there can produce large skin-friction discrepancies.An exponential weighting function based on the minimum wall-normal distance is introduced for this purpose.
- Artificial neural networks: A one-hidden-layer RBFNN maps selected flow features to sample eddy viscosity, using Gaussian basis functions and 80 hidden neurons.The inputs include freestream conditions, mean flow variables, and derivatives; the output is the eddy viscosity of each sample.
- Sample selection and optimization: Representative sample selection reduces redundancy and density imbalance, while optimization uses clustering and gradient-based or least-squares parameter estimation.The selected samples approximate the original sample space with fewer, more representative points; K-means clustering is used for centers.
3. Example and analysis
The study validates the RBFNN turbulence model on NACA0012 and tests its generalization to interpolated, extrapolated, and different-airfoil cases. The model generally agrees with the SA-based reference while offering improved efficiency.
- Reference validation: The baseline SA model agrees well with experimental and CFL3D results for the NACA0012 validation cases and captures RAE2822 surface behavior with a slight downstream shock movement in case 10.The RAE2822 skin-friction coefficient uses the local boundary-layer freestream condition for nondimensionalization.
- Validation setup: Three NACA0012 flow fields form the training set, while prediction cases cover interpolation and extrapolation of flow states plus two different airfoils.NACA0012 cases are sampled over Mach numbers from 0.1 to 0.5 and angles of attack from 0 to 5 degrees.
- Training-case results: For training cases, the RBFNN skin-friction coefficient is almost identical to the reference, with slight errors on the lower surface and rear upper surface.Its eddy-viscosity contour agrees well except in the wake, and larger near-wall profile discrepancies have little impact on velocity profiles.
- Efficiency and generalization: The proposed global model uses information from the whole divided zone rather than neighboring nodes and avoids solving transport equations.The paper reports better efficiency for nearly all cases, especially those with better accuracy, and good agreement for interpolated and extrapolated cases without shock waves or separations.
4. Conclusions and future work
The paper reconstructs eddy-viscosity models for subsonic attached flows using limited NACA0012 training data, achieving accurate generalization and greater efficiency than the SA model. It remains preliminary for high-Reynolds-number turbulence, especially separated and more complex flows.
- Approach: Three turbulent-flow cases over NACA0012 provide training data for an RBFNN model of eddy viscosity in subsonic attached flows.The approach is compared with the original SA model for different airfoils and flow states.
- Approach: Partitioning the flow field reduces outliers caused by large data ranges, improving accuracy in important domains.The model is built separately for different flow-field regions.
- Results: Coupling the approach with the Navier–Stokes equations achieves final convergence.
- Results: Velocity profiles and skin-friction distributions agree well with the SA model for training and prediction cases.These comparisons support the model’s accuracy and generalization across the evaluated cases.
- Results: The proposed model is more efficient than the original SA model, using a concise one-hidden-layer network and fewer iteration steps for convergence.The network contains about a hundred neurons.
- Future work: The study is preliminary for high-Reynolds-number turbulent-flow modeling; separated and other more complex flows remain for future investigation.