Data-Driven Turbulence Modeling

Turbulence remains a major bottleneck to accurate flow prediction in climate, aerospace, and energy systems. Industrial simulations rely on averaged turbulence models, which often struggle in flows governed by multiple interacting mechanisms. This work presents a unified, data-driven turbulence modeling framework that learns from sparse, indirect observations across diverse flow regimes. By embedding physical consistency, selecting representative training cases based on flow-feature distributions, and leveraging multi-objective learning, the framework enables a single model to adapt across flow regimes and deliver improved predictive performance over a wide range of flows.

The aviation industry, a major contributor to global climate impact, faces growing demand for more efficient and sustainable designs. Accurate prediction of turbulent flow and heat transfer remains a key challenge for conventional aerodynamic simulations, particularly in industrially relevant configurations such as jet-in-crossflow in film cooling. These flows involve strong vortex interactions that govern heat transfer and surface protection, yet remain difficult to capture with existing models. This work develops data-driven turbulence and heat-flux models to improve aerothermodynamic predictions in such complex flows, including high-pressure turbines, enabling more reliable and efficient design.

With various high-fidelity calibration data, machine learning provides promising tools to construct constitutive models. We propose a neural operator to develop nonlocal constitutive models for tensorial quantities through a vector-cloud neural network with equivariance (VCNN-e). 

We propose using an ensemble Kalman method to learn a nonlinear eddy viscosity model, represented as a tensor basis neural network, from velocity data. It is demonstrated that the turbulence model learned in one flow can predict flows in similar configurations with varying slopes. The method has been compared with adjoint-based learning method and showed improved efficiency.

When sparse online data are available (e.g., from monitoring of the system to be predicted), we use data assimilation and Bayesian inference to reduce uncertainties in RANS models. The model-form uncertainties inferred from such sparse data can be used to improve predictions on other closely related flows, e.g., those with moderate changes of Reynolds numbers and geometry configurations.

• Duraisamy, K., Iaccarino, G., and Xiao, H. Turbulence modeling in the age of data. Annual Review of Fluid Mechanics, 51, 357–377, 2019. DOI: https://doi.org/10.1146/annurev-fluid-010518-040547 Also available at: arXiv:1804.00183
• Xiao, H. and Cinnella, P. Quantification of model uncertainty in RANS simulations: A review. Progress in Aerospace Sciences, 108, 1–31, 2019. DOI: https://doi.org/10.1016/j.paerosci.2018.10.001. Also available at: arXiv:1806.10434.

• Liu, Z.-R., Wang, H.-C., Zhao, Z.-L., and Xiao, H. Toward a unified data-driven turbulence model through multi-objective learning. Submitted to National Science Review. Also avaliable at https://arxiv.org/abs/2509.17189
• Zafar, M. I., Zhou, X.-H., Roy, C. J., Stelter, D., and Xiao, H. Data-driven turbulence modeling approach for cold-wall hypersonic boundary layers. Journal of Thermophysics and Heat Transfer, 2025. DOI: https://arc.aiaa.org/doi/10.2514/1.T7127
• Zhang, X.-L., Xiao, H., Luo, X., and He, G. Ensemble Kalman method for learning turbulence models from indirect observation data. Journal of Fluid Mechanics, 949, A26, 2022. DOI: https://doi.org/10.1017/jfm.2022.744. 
• Zhou, X.-H., Han, J., and Xiao, H. Frame-independent vector-cloud neural network for nonlocal constitutive modelling on arbitrary grids. Computer Methods in Applied Mechanics and Engineering, 388, 114211, 2022. DOI: https://doi.org/10.1016/j.cma.2021.114211.
• Han, J., Zhou, X.-H., and Xiao, H. An equivariant neural operator for developing nonlocal tensorial constitutive models. Journal of Computational Physics, 488, 112243, 2023. DOI: https://doi.org/10.1016/j.jcp.2023.112243. Also available at: arXiv:2201.01287.
• Xiao, H., Wu, J.-L., Wang, J.-X., Sun, R., and Roy, C. J. Quantifying and reducing model-form uncertainties in Reynolds-averaged Navier–Stokes equations: A data-driven, physics-informed Bayesian approach. Journal of Computational Physics, 324, 115–136, 2016. DOI: https://doi.org/10.1016/j.jcp.2016.07.038.

• Xiao, H., Wu, J.-L., Wang, J.-X., Sun, R., and Roy, C. J. Quantifying model-form uncertainties in Reynolds-averaged Navier–Stokes equations: An open-box, physics-based, Bayesian approach. Proceedings of the 68th Annual Meeting of the APS Division of Fluid Dynamics, Boston, Massachusetts, USA, November 22–24, 2015.
• Wu, J.-L., Wang, J.-X., and Xiao, H. Model-form uncertainty quantification in RANS simulation of wing–body junction flow. Proceedings of the 68th Annual Meeting of the APS Division of Fluid Dynamics, Boston, Massachusetts, USA, November 22–24, 2015.