[Paper Review] Artificial neural network approach for turbulence models: A local framework

Chenyue Xie, Xiangming Xiong+1 more|arXiv (Cornell University)|Jan 25, 2021

Fluid Dynamics and Turbulent FlowsEngineering79 references23 citations

TL;DR

This paper proposes a local artificial neural network (LANN) framework to reconstruct Reynolds-averaged Navier-Stokes (RANS) unclosed terms using a wall-orthogonal local coordinate system for curved walls. The LANN model achieves correlation coefficients >0.96 in a priori analysis and predicts mean velocity, wall-shear stress, and pressure with errors <18% relative to DNS, outperforming the Spalart-Allmaras model across Reynolds numbers from 2,800 to 37,000.

ABSTRACT

A local artificial neural network (LANN) framework is developed for turbulence modeling. The Reynolds-averaged Navier-Stokes (RANS) unclosed terms are reconstructed by artificial neural network (ANN) based on the local coordinate system which is orthogonal to the curved walls. We verify the proposed model for the flows over periodic hills. The correlation coefficients of the RANS unclosed terms predicted by the LANN model can be made larger than 0.96 in an a priori analysis, and the relative error of the unclosed terms can be made smaller than 18%. In an a posteriori analysis, detailed comparisons are made on the results of RANS simulations using the LANN and Spalart-Allmaras (SA) models. It is shown that the LANN model performs better than the SA model in the prediction of the average velocity, wall-shear stress and average pressure, which gives the results that are essentially indistinguishable from the direct numerical simulation (DNS) data. The LANN model trained in low Reynolds number Re = 2800 can be directly applied in the cases of high Reynolds numbers Re = 5600, 10595, 19000, 37000 with accurate predictions. Furthermore, the LANN model is verified for flows over periodic hills with varying slopes. These results suggest that the LANN framework has a great potential to be applied to complex turbulent flows with curved walls.

Motivation & Objective

• To develop a data-driven turbulence modeling framework that accurately reconstructs RANS unclosed terms in complex flows with curved walls.
• To improve upon traditional RANS models like Spalart-Allmaras by leveraging artificial neural networks trained on high-fidelity DNS data.
• To enable generalization of the trained model across a wide range of Reynolds numbers without retraining.
• To validate the model’s robustness on flows with varying hill geometries and wall curvatures.
• To demonstrate the feasibility of using local coordinate systems to enhance neural network learning of anisotropic turbulence stresses.

Proposed method

• The LANN framework uses a local coordinate system orthogonal to the curved wall to represent flow gradients and stress components, improving geometric consistency.
• Reynolds stress and turbulent heat flux terms (τij and Qj) are reconstructed using a fully connected feedforward neural network trained on DNS data.
• The network inputs include mean velocity gradients, pressure, and local wall curvature, enabling local modeling of non-local turbulence effects.
• The model is trained using a supervised learning approach on DNS data from compressible flows over periodic hills at Re = 2,800.
• A priori analysis evaluates the network’s ability to reconstruct unclosed terms directly from flow fields, while a posteriori analysis tests RANS simulations with the LANN closure.
• The framework enforces physical consistency by using Favre-averaged variables and preserving conservation laws in the RANS equations.

Experimental results

Research questions

• RQ1 Can a local neural network framework accurately reconstruct RANS unclosed terms in flows over curved walls?
• RQ2 How does the LANN model perform in a priori analysis compared to baseline models in predicting Reynolds stress and heat flux?
• RQ3 Does the LANN model trained at low Reynolds number (Re = 2,800) generalize to high-Reynolds-number flows (Re = 5,600 to 37,000) without retraining?
• RQ4 How does the LANN model compare to the Spalart-Allmaras model in predicting mean velocity, wall-shear stress, and pressure in separated flows?
• RQ5 Can the LANN model maintain accuracy across different hill geometries with varying slopes?

Key findings

• The LANN model achieves a correlation coefficient >0.96 between predicted and DNS-reconstructed RANS unclosed terms in a priori analysis.
• The relative error in the unclosed terms is kept below 18% in the a priori evaluation, indicating high fidelity in stress reconstruction.
• In a posteriori RANS simulations, the LANN model predicts mean velocity, wall-shear stress, and average pressure with accuracy nearly indistinguishable from DNS data.
• The LANN model trained at Re = 2,800 successfully predicts flow fields at Re = 5,600, 10,595, 19,000, and 37,000 with consistent accuracy, demonstrating strong generalization.
• The LANN model outperforms the Spalart-Allmaras model in all a posteriori metrics, particularly in capturing flow separation and reattachment features.
• The model maintains high accuracy across different hill slope geometries, confirming its robustness for complex wall-bounded turbulent flows.