DataDriven 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 submeter to planetary scales. Current simulations of turbulent flows rely on Reynolds Averaged Navier Stokes equations with closure models. In light of the decadeslong stagnation in traditional turbulence modeling, we proposed datadriven, physicsinformed 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 highfidelity 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 vectorcloud neural network with equivariance (VCNNe).
Publications:

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, 336363 (2022).

X.H. Zhou, J. Han, and H. Xiao, Frameindependent vectorcloud 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).
Physicsinformed 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 adjointbased learning method and showed improved efficiency.
Publications:

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. MichelénStröfer, H. Xiao. Endtoend differentiable learning of turbulence models from indirect observations. Theoretical and Applied Mechanics Letters, 11(4), 100280, 2021.
Datadriven, physicsinformed Bayesian approach to reduce modelform 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 modelform 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.
Publications:

X.L. Zhang, C. MichelénStröfer, 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 modelform uncertainties in Reynolds averaged Navier–Stokes equations: A datadriven, physicsinformed Bayesian approach. Journal of Computational Physics, 324, 115136, 2016.

X.L. Zhang, H. Xiao, G. He. Assessment of regularized ensemble Kalman inversion of turbulence quantity fields. AIAA Journal, 60(1), 313, 2022.
Random matrix approach for quantifying modelform uncertainties in turbulence modeling
When no data available, we proposed a random matrix approach based on maximum entropy theory to sample and propagate the modelform uncertainty according to the physical constraints (realizability) on the Reynolds stresses.
Publications:

H. Xiao, J.X. Wang and R. G. Gahnem. A random matrix approach for quantifying modelform uncertainties in turbulence modeling. Computer Methods in Applied Mechanics and Engineering. 313, 941965, 2017.

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