Subgrid-scale Flux Modeling of a Passive Scalar in Turbulent Channel Flow using Artificial Neural Network

Document Type : Research Paper


1 Assistant Professor, Faculty of Mechanical and Energy Engineering, Shahid Beheshti University, 16589-53571, Tehran, Iran

2 MSc. Student, Faculty of Mechanical and Energy Engineering, Shahid Beheshti University,16589-53571, Tehran, Iran


A deep neural network (DNN) has been developed to model the subgrid-scale (SGS) flux associated with a passive scalar in incompressible turbulent channel flow. To construct the training dataset for the DNN, a direct numerical simulation (DNS) was performed for a channel flow at the friction Reynolds number  encompassing a passive scalar transport with Prandtl number  using a pseudo-spectral in-house code. The DNS data of velocity and scalar fields was filtered to obtain the SGS scalar flux vector, , filtered scalar gradient, and filtered strain-rate tensor, which were subsequently used to train the DNN, enabling it to predict  for large-eddy simulation. A priori evaluation of the DNN’s performance in predicting  revealed a close match with filtered DNS data, demonstrating correlations of up to 98%, 79% and 85% for the three components of . Additionally, analysis of the mean SGS dissipation and its probability density function indicated promising predictions by the DNN. Notably, this study extends the applications of DNNs for predicting  to the case of turbulent channel flow.


Main Subjects

[1]        S. A. Orszag, "Analytical Theories of Turbulence," Journal of Fluid Mechanics, Vol. 41, No. 2, pp. 363-386, 1970, doi:
[2]        P. Sagaut, Large Eddy Simulation for Incompressible Flows: An Introduction. Springer Science & Business Media, 2005,
[3]        S. L. Brunton, B. R. Noack, and P. Koumoutsakos, "Machine Learning for Fluid Mechanics," Annual review of fluid mechanics, Vol. 52, pp. 477-508, 2020, doi:
[4]        K. Duraisamy, G. Iaccarino, and H. Xiao, "Turbulence Modeling in the Age of Data," Annual review of fluid mechanics, Vol. 51, pp. 357-377, 2019, doi:
[5]        M. Gamahara and Y. Hattori, "Searching for Turbulence Models by Artificial Neural Network," Physical Review Fluids, Vol. 2, No. 5, p. 054604, 2017, doi:
[6]        Z. Wang, K. Luo, D. Li, J. Tan, and J. Fan, "Investigations of Data-Driven Closure for Subgrid-Scale Stress in Large-Eddy Simulation," Physics of Fluids, Vol. 30, No. 12, 2018, doi:
[7]        A. Vollant, G. Balarac, and C. Corre, "Subgrid-Scale Scalar Flux Modelling Based on Optimal Estimation Theory and Machine-Learning Procedures," Journal of Turbulence, Vol. 18, No. 9, pp. 854-878, 2017, doi:
[8]        P. M. Milani, J. Ling, and J. K. Eaton, "Turbulent Scalar Flux in Inclined Jets in Crossflow: Counter Gradient Transport and Deep Learning Modelling," Journal of Fluid Mechanics, Vol. 906, p. A27, 2021, doi:
[9]        M. Bode, M. Gauding, K. Kleinheinz, and H. Pitsch, "Deep Learning at Scale for Subgrid Modeling in Turbulent Flows: Regression and Reconstruction," in International Conference on High Performance Computing, 2019: Springer, pp. 541-560, doi:
[10]      H. Frezat, G. Balarac, J. Le Sommer, R. Fablet, and R. Lguensat, "Physical Invariance in Neural Networks for Subgrid-Scale Scalar Flux Modeling," Physical Review Fluids, Vol. 6, No. 2, p. 024607, 2021, doi:
[11]      A. Akhavan-Safaei and M. Zayernouri, "Deep Learning of Subgrid-Scale Dynamics in Scalar Turbulence and Generalization to Other Transport Regimes," Journal of Machine Learning for Modeling and Computing, Vol. 5, No. 1, 2024, doi:
[12]      Z. Warhaft, "Passive Scalars in Turbulent Flows," Annual Review of Fluid Mechanics, Vol. 32, No. 1, pp. 203-240, 2000, doi:
[13]      M. Chevalier, P. Schlatter, A. Lundbladh, and D. S. Henningson, Simson: A Pseudo-Spectral Solver for Incompressible Boundary Layer Flows. 2007. [Online]. Available:
[14]      J. Kim, P. Moin, and R. Moser, "Turbulence Statistics in Fully Developed Channel Flow at Low Reynolds Number," Journal of fluid mechanics, Vol. 177, pp. 133-166, 1987, doi:
[15]      M. Y. Hussaini and T. A. Zang, "Spectral Methods in Fluid Dynamics," Annual review of fluid mechanics, Vol. 19, No. 1, pp. 339-367, 1987.
[16]      A. Rasam, G. Brethouwer, and A. V. Johansson, "An Explicit Algebraic Model for the Subgrid-Scale Passive Scalar Flux," Journal of Fluid Mechanics, Vol. 721, pp. 541-577, 2013, doi:
[17]      A. Rasam, G. Brethouwer, P. Schlatter, Q. Li, and A. V. Johansson, "Effects of Modelling, Resolution and Anisotropy of Subgrid-Scales on Large Eddy Simulations of Channel Flow," Journal of turbulence, No. 12, p. N10, 2011, doi:
[18]      R. D. Moser, J. Kim, and N. N. Mansour, "Direct Numerical Simulation of Turbulent Channel Flow up to Re Τ= 590," Physics of fluids, Vol. 11, No. 4, pp. 943-945, 1999, doi:
[19]      N. Kasagi and O. Iida, "Progress in Direct Numerical Simulation of Turbulent Heat Transfer," Proceedings of the 5th ASME/JSME Joint Thermal Engineering Conference, pp. 15-19, 1999, doi:
[20]      J. Park and H. Choi, "Toward Neural-Network-Based Large Eddy Simulation: Application to Turbulent Channel Flow," Journal of Fluid Mechanics, Vol. 914, p. A16, 2021, doi:
[21]      P. C. Di Leoni, T. A. Zaki, G. Karniadakis, and C. Meneveau, "Two-Point Stress–Strain-Rate Correlation Structure and Non-Local Eddy Viscosity in Turbulent Flows," Journal of Fluid Mechanics, vol. 914, p. A6, 2021, doi:
[22]      N. Park, S. Lee, J. Lee, and H. Choi, "A Dynamic Subgrid-Scale Eddy Viscosity Model with a Global Model Coefficient," Physics of Fluids, Vol. 18, No. 12, 2006, doi:
[23]      S. Völker, R. D. Moser, and P. Venugopal, "Optimal Large Eddy Simulation of Turbulent Channel Flow Based on Direct Numerical Simulation Statistical Data," Physics of Fluids, Vol. 14, No. 10, pp. 3675-3691, 2002, doi:
[24]      M. Abadi et al., "Tensorflow}: A System for {Large-Scale} Machine Learning," in 12th USENIX symposium on operating systems design and implementation (OSDI 16), 2016, pp. 265-283, doi:
[25]      S. Haykin, Neural Networks and Learning Machines, 3/E. Pearson Education India, 2009. [Online]. Available:
[26]      D. P. Kingma and J. Ba, "Adam: A Method for Stochastic Optimization," arXiv preprint arXiv:1412.6980, 2014, doi:
[27]      A. Vela-Martín, "Subgrid-Scale Models of Isotropic Turbulence Need Not Produce Energy Backscatter," Journal of Fluid Mechanics, Vol. 937, p. A14, 2022, doi:
[28]      A. Rasam, G. Brethouwer, and A. Johansson, "A Stochastic Extension of the Explicit Algebraic Subgrid-Scale Models," Physics of fluids, Vol. 26, No. 5, 2014, doi: