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.

Researcher(s): Jinlong Wu, Carlos Michelén-Ströfer, Dr. Jianxun Wang

Collaborator(s): Dr. Chris Roy, Dr. Julia Ling, Dr. Roger Gahnem

Physics-informed machine learning for predictive turbulence modeling

When offline data from high-fidelity simulations and experiments are available, we proposed a physics-informed machine learning approach to learning the discrepancies in the Reynolds stress and to improved the predictive capabilities of the RANS models.



  • J.-X. J.-L. Wu, H. Xiao, E. G. Paterson. Physics-informed machine learning approach for augmenting turbulence models: A comprehensive framework. Physical Review Fluids. 3, 074602, 1-28, 2018.
  • J.-X. Wang, J.-L. Wu, H. Xiao. Physics informed machine learning approach for reconstructing Reynolds stress modeling discrepancies based on DNS data. Physical Review Fluids 2(3), 034603, 2017.
  • J.-L. Wu, R. Sun, S. Laizet H. Xiao. Representation of Reynolds stress perturbations with application in machine-learning-assisted turbulence modeling. Computer Methods in Applied Mechanics and Engineering. In Press, 2018. DOI: 10.1016/j.cma.2018.09.010
  • J.-L. Wu, J.-X. Wang, H. Xiao, J. Ling. A priori assessment of prediction confidence for data-driven turbulence modeling. Flow, Turbulence and Combustion. 99(1), 25-46, 2017.

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.



  • 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.
  • J.-L. Wu, J.-X. Wang, and H. Xiao. A Bayesian calibration-prediction method for reducing model-form uncertainties with application in RANS simulations. Flow, Turbulence and Combustion, 97, 761-786, 2016.
  • J.-X. Wang, J.-L. Wu, and H. Xiao. Incorporating prior knowledge for quantifying and reducing model-form uncertainty in RANS simulations. International Journal of Uncertainty Quantification, 6 (2), 109-126, 2016.

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. 

Applications in other complex physical systems

While the focus of these data-driven physics-informed approach is on turbulent flows, the framework is general enough for other complex physical systems. We have successfully use them in the inference of tsunami characteristics (wave height and water depth) from sediment deposits and in the inference of the forces on a porous disk based on wake velocity measurements. 


  • J.-X. Wang and H. Xiao. Data-driven CFD modeling of turbulent flows through complex structures. International Journal of Heat and Fluid Flow, 62, 138–149, 2016.
  • J.-X. Wang, H. Tang, H. Xiao, and R. Weiss. Inferring tsunami flow depth and flow speed from sediment deposits based on ensemble Kalman filtering. Geophysical Journal International. 212(1),646–658 2018.
  • H. Tang, J.-X. Wang, R. Weiss and H. Xiao. TSUFLIND-EnKF: Inversion of tsunami flow depth and flow speed from deposits with quantified uncertainties. Marine Geology. 396, 15-26, 2018.