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A New Hypothesis on the Anisotropic Reynolds Stress Tensor for Turbulent ...

By László Könözsy

The topic is rapidly evolving and, therefore, there is no text as yet. Thus, your best bet would be to refer to the latest papers from journals and conference publications. Also, I am afraid that the primary emphasis in this early literature may be on simpler, canonical flows as against complex turbulent flows. Here are a few papers that you might begin with. Hope they are helpful.

You may also keep in mind that more advanced frameworks beyond the CNNs are now being applied in the context of turbulent flows.

Broader reviews:

1. Duraisamy, K., Iaccarino, G. & Xiao, H. 2019 Turbulence modeling in the age of data. Annu. Rev. Fluid Mech. 51 (1), 357–377.

2. Duraisamy K. Perspectives on machine learning-augmented Reynolds-averaged and large eddy simulation models of turbulence. Phys Rev Fluids. 2021;6(5):050504.

Research articles:

3. Kim, J. and Lee, C. (2020). Prediction of turbulent heat transfer using convolutional neural networks. Journal of Fluid Mechanics, 882, A18. doi:10.1017/jfm.2019.814

4. Ocáriz Borde, H. S., Sondak, D., and Protopapas, P. (2021). Convolutional neural network models and interpretability for the anisotropic Reynolds stress tensor in turbulent one-dimensional flows, Journal of Turbulence, DOI: 10.1080/14685248.2021.1999459

5. Razizadeh, O. and Yakovenko, S. N. (2020). Implementation of Convolutional Neural Network to Enhance Turbulence Models for Channel Flows. 2020 Science and Artificial Intelligence conference. https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=9303178

6. Fang, R., Sondak, D., Protopapas, P., and Succi, S. (2020) Neural network models for the anisotropic Reynolds stress tensor in turbulent channel flow, Journal of Turbulence, 21:9-10, 525-543, DOI: 10.1080/14685248.2019.1706742

7. Moghaddam, A. A. and Sadaghiyani, A. (2018) A deep learning framework for turbulence modeling using data assimilation and feature extraction. https://arxiv.org/pdf/1802.06106.pdf

8. Chang, C.-W., Fang, J., and Dinh, N. T. (2020) Reynolds-Averaged Turbulence Modeling Using Deep Learning with Local Flow Features: An Empirical Approach, Nuclear Science and Engineering, 194:8-9, 650-664, DOI: 10.1080/00295639.2020.1712928

9. J. Ling, A. Kurzawski, and J. Templeton, “ Reynolds averaged turbulence modelling using deep neural networks with embedded invariance,” J. Fluid Mech. 807, 155 (2016). https://doi.org/10.1017/jfm.2016.615

10. Beck, A., Flad, D. & Munz, C. 2019 Deep neural networks for data-driven LES closure models. J. Comput. Phys. 398, 108910.

11. T Nakamura, K Fukami, K Hasegawa, Y Nabae, and K Fukagata. (2021) Convolutional neural network and long short-term memory based reduced order surrogate for minimal turbulent channel flow. Physics of Fluids 33 (2), 025116.

Deep Learning of Ian Goodfellow y Yoshua Bengio is a good book!