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Data-Driven Turbulence Modeling

A complex physical system characterized by a wide range of temporal and spatial scales, turbulence is among the last unsolved problems in classical physics that affects natural and engineered systems from sub-meter to planetary scales. Current simulations of turbulent flows rely on Reynolds Averaged Navier Stokes equations with closure models.  In light of the decades-long stagnation in traditional turbulence modeling, we proposed data-driven, physics-informed methods to address this challenge in all scenarios of data availability as investigated in the projects below.

Equivariant neural operator for developing nonlocal tensorial constitutive models


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). 


  • J. Han, X.-H. Zhou, and H. Xiao. An equivariant neural operator for developing nonlocal tensorial constitutive models. Submitted to Journal of Computational Physics. Also at arXiv:2201.01287.

  • M. I. Zafar, J. Han, X.-H. Zhou, and H. Xiao. Frame invariance and scalability of neural operators for partial differential equations. Communications in Computational Physics, 32, 336-363 (2022).

  • X.-H. Zhou, J. Han, and H. Xiao, Frame-independent vector-cloud neural network for nonlocal constitutive modelling on arbitrary grids. Computer Methods in Applied Mechanics and Engineering, 388, 114211 (2022).

  • X.-H. Zhou, J. Han, and H. Xiao, Learning nonlocal constitutive models with neural networks. Computer Methods in Applied Mechanics and Engineering, 384, 113927 (2021).

Physics-informed machine learning for predictive turbulence modeling


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.


  • X.-L. Zhang, H. Xiao, X. Luo, G. He. Ensemble Kalman method for learning turbulence models from indirect observation data. Journal of Fluid Mechanics, 949(A26), 2022.

  • C. Michelén-Ströfer, H. Xiao. End-to-end differentiable learning of turbulence models from indirect observations. Theoretical and Applied Mechanics Letters, 11(4), 100280, 2021.

Data-driven, physics-informed Bayesian approach to reduce model-form uncertainties


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.


  • X.-L. Zhang, C. Michelén-Ströfer, H. Xiao. Regularized ensemble Kalman methods for inverse problems. Journal of Computational Physics, 416, 109517, 2020.

  • H. Xiao, J.-L. Wu, J.-X. Wang, R. Sun, and C. J. Roy. 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.

  • X.-L. Zhang, H. Xiao, G. He.  Assessment of regularized ensemble  Kalman inversion of turbulence quantity fields. AIAA Journal, 60(1), 3-13, 2022.

Random matrix approach for quantifying model-form uncertainties in turbulence modeling


When no data available, we proposed a random matrix approach based on maximum entropy theory to sample and propagate the model-form uncertainty according to the physical constraints (realizability) on the Reynolds stresses. 


  • H. Xiao, J.-X. Wang and R. G. Gahnem. A random matrix approach for quantifying model-form uncertainties in turbulence modeling. Computer Methods in Applied Mechanics and Engineering. 313, 941-965, 2017.

  •  J.-X. Wang, R. Sun, H. Xiao. Quantification of uncertainty in RANS models: A comparison of physics-based and random matrix theoretic approaches. International Journal of Heat and Fluid Flow, 62, 577–592, 2016. 

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