Skip to main content

Subgrid modelling for two-dimensional turbulence using neural networks

  • R. Maulik (a1), O. San (a1), A. Rasheed (a2) and P. Vedula (a3)

In this investigation, a data-driven turbulence closure framework is introduced and deployed for the subgrid modelling of Kraichnan turbulence. The novelty of the proposed method lies in the fact that snapshots from high-fidelity numerical data are used to inform artificial neural networks for predicting the turbulence source term through localized grid-resolved information. In particular, our proposed methodology successfully establishes a map between inputs given by stencils of the vorticity and the streamfunction along with information from two well-known eddy-viscosity kernels. Through this we predict the subgrid vorticity forcing in a temporally and spatially dynamic fashion. Our study is both a priori and a posteriori in nature. In the former, we present an extensive hyper-parameter optimization analysis in addition to learning quantification through probability-density-function-based validation of subgrid predictions. In the latter, we analyse the performance of our framework for flow evolution in a classical decaying two-dimensional turbulence test case in the presence of errors related to temporal and spatial discretization. Statistical assessments in the form of angle-averaged kinetic energy spectra demonstrate the promise of the proposed methodology for subgrid quantity inference. In addition, it is also observed that some measure of a posteriori error must be considered during optimal model selection for greater accuracy. The results in this article thus represent a promising development in the formalization of a framework for generation of heuristic-free turbulence closures from data.

Corresponding author
Email address for correspondence:
Hide All
Arakawa, A. 1966 Computational design for long-term numerical integration of the equations of fluid motion: two-dimensional incompressible flow. Part I. J. Comput. Phys. 1 (1), 119143.
Beck, A. D., Flad, D. G. & Munz, C.-D.2018 Neural networks for data-based turbulence models. arXiv:1806.04482.
Berselli, L. C., Iliescu, T. & Layton, W. J. 2005 Mathematics of Large Eddy Simulation of Turbulent Flows. Springer.
Canuto, V. M. & Cheng, Y. 1997 Determination of the Smagorinsky–Lilly constant C S . Phys. Fluids 9 (5), 13681378.
Cohen, K., Siegel, S., McLaughlin, T. & Gillies, E. 2003 Feedback control of a cylinder wake low-dimensional model. AIAA J. 41 (7), 13891391.
Cushman-Roisin, B. & Beckers, J.-M. 2011 Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects, vol. 101. Academic.
Duraisamy, K., Iaccarino, G. & Xiao, H.2018 Turbulence modeling in the age of data. arXiv:1804.00183.
Eden, C. & Greatbatch, R. J. 2008 Towards a mesoscale eddy closure. Ocean Model. 20 (3), 223239.
Faller, W. E. & Schreck, S. J. 1997 Unsteady fluid mechanics applications of neural networks. J. Aircraft 34 (1), 4855.
Fox-Kemper, B., Danabasoglu, G., Ferrari, R., Griffies, S. M., Hallberg, R. W., Holland, M. M., Maltrud, M. E., Peacock, S. & Samuels, B. L. 2011 Parameterization of mixed layer eddies. III. Implementation and impact in global ocean climate simulations. Ocean Model. 39 (1–2), 6178.
Frederiksen, J. S., O’Kane, T. J. & Zidikheri, M. J. 2013 Subgrid modelling for geophysical flows. Phil. Trans. R. Soc. Lond. A 371 (1982), 20120166.
Frederiksen, J. S. & Zidikheri, M. J. 2016 Theoretical comparison of subgrid turbulence in atmospheric and oceanic quasi-geostrophic models. Nonlinear Process. Geophys. 23 (2), 95105.
Galperin, B. & Orszag, S. A. 1993 Large Eddy Simulation of Complex Engineering and Geophysical Flows. Cambridge University Press.
Gamahara, M. & Hattori, Y. 2017 Searching for turbulence models by artificial neural network. Phys. Rev. Fluids 2 (5), 054604.
Germano, M., Piomelli, U., Moin, P. & Cabot, W. H. 1991 A dynamic subgrid-scale eddy viscosity model. Phys. Fluids 3 (7), 17601765.
Ghosal, S., Lund, T. S., Moin, P. & Akselvoll, K. 1995 A dynamic localization model for large-eddy simulation of turbulent flows. J. Fluid Mech. 286, 229255.
King, R. N., Hamlington, P. E. & Dahm, W. J. 2016 Autonomic closure for turbulence simulations. Phys. Rev. E 93 (3), 031301.
Kingma, D. P. & Ba, J.2014 Adam: a method for stochastic optimization. arXiv:1412.6980.
Kraichnan, R. H. 1967 Inertial ranges in two-dimensional turbulence. Phys. Fluids 10 (7), 14171423.
Kutz, J. N. 2017 Deep learning in fluid dynamics. J. Fluid Mech. 814, 14.
Langford, J. A. & Moser, R. D. 1999 Optimal LES formulations for isotropic turbulence. J. Fluid Mech. 398, 321346.
Leith, C. E. 1968 Diffusion approximation for two-dimensional turbulence. Phys. Fluids 11 (3), 671672.
Ling, J., Kurzawski, A. & Templeton, J. 2016 Reynolds averaged turbulence modelling using deep neural networks with embedded invariance. J. Fluid Mech. 807, 155166.
Ling, J. & Templeton, J. 2015 Evaluation of machine learning algorithms for prediction of regions of high Reynolds averaged Navier–Stokes uncertainty. Phys. Fluids 27 (8), 085103.
Mannarino, A. & Mantegazza, P. 2014 Nonlinear aeroelastic reduced order modeling by recurrent neural networks. J. Fluid. Struct. 48, 103121.
Mansfield, J. R., Knio, O. M. & Meneveau, C. 1998 A dynamic LES scheme for the vorticity transport equation: formulation and a priori tests. J. Comput. Phys. 145 (2), 693730.
Marshall, J. S. & Beninati, M. L. 2003 Analysis of subgrid-scale torque for large-eddy simulation of turbulence. AIAA J. 41 (10), 18751881.
Maulik, R. & San, O. 2017a A neural network approach for the blind deconvolution of turbulent flows. J. Fluid Mech. 831, 151181.
Maulik, R. & San, O. 2017b A stable and scale-aware dynamic modeling framework for subgrid-scale parameterizations of two-dimensional turbulence. Comput. Fluids 158, 1138.
Milano, M. & Koumoutsakos, P. 2002 Neural network modeling for near wall turbulent flow. J. Comput. Phys. 182 (1), 126.
Mohan, A. T. & Gaitonde, D. V.2018 A deep learning based approach to reduced order modeling for turbulent flow control using LSTM neural networks. arXiv:1804.09269.
Moser, R. D., Malaya, N. P., Chang, H., Zandonade, P. S., Vedula, P., Bhattacharya, A. & Haselbacher, A. 2009 Theoretically based optimal large-eddy simulation. Phys. Fluids 21 (10), 105104.
Parish, E. J. & Duraisamy, K. 2016 A paradigm for data-driven predictive modeling using field inversion and machine learning. J. Comput. Phys. 305, 758774.
Pathak, J., Wikner, A., Fussell, R., Chandra, S., Hunt, B. R., Girvan, M. & Ott, E. 2018 Hybrid forecasting of chaotic processes: using machine learning in conjunction with a knowledge-based model. Chaos 28 (4), 041101.
Piomelli, U., Cabot, W. H., Moin, P. & Lee, S. 1991 Subgrid-scale backscatter in turbulent and transitional flows. Phys. Fluids 3 (7), 17661771.
Raissi, M. & Karniadakis, G. E. 2018 Hidden physics models: machine learning of nonlinear partial differential equations. J. Comput. Phys. 357, 125141.
Sagaut, P. 2006 Large Eddy Simulation for Incompressible Flows: An Introduction. Springer.
San, O. & Maulik, R. 2018 Neural network closures for nonlinear model order reduction. Adv. Comput. Math.; doi:10.1007/s10444-018-9590-z.
San, O. & Staples, A. E. 2012 High-order methods for decaying two-dimensional homogeneous isotropic turbulence. Comput. Fluids 63, 105127.
San, O., Staples, A. E. & Iliescu, T. 2013 Approximate deconvolution large eddy simulation of a stratified two-layer quasigeostrophic ocean model. Ocean Model. 63, 120.
Sarghini, F., De Felice, G. & Santini, S. 2003 Neural networks based subgrid scale modeling in large eddy simulations. Comput. Fluids 32 (1), 97108.
Schaeffer, H. 2017 Learning partial differential equations via data discovery and sparse optimization. Proc. R. Soc. Lond. A 473 (2197), 20160446.
Singh, A. P., Medida, S. & Duraisamy, K. 2017 Machine-learning-augmented predictive modeling of turbulent separated flows over airfoils. AIAA J. 55 (7), 22152227.
Smagorinsky, J. 1963 General circulation experiments with the primitive equations. I. The basic experiment. Mon. Weath. Rev. 91 (3), 99164.
Tracey, B. D., Duraisamy, K. & Alonso, J. J. 2015 A machine learning strategy to assist turbulence model development. In 53rd AIAA Aerospace Sciences Meeting; 5–9 January 2015, Paper no: 2015-1287, American Institute of Aeronautics and Astronautics SciTech Forum, Kissimmee, FL.
Vorobev, A. & Zikanov, O. 2008 Smagorinsky constant in LES modeling of anisotropic MHD turbulence. Theor. Comput. Fluid Dyn. 22 (3–4), 317325.
Vreman, A. W. 2004 An eddy-viscosity subgrid-scale model for turbulent shear flow: algebraic theory and applications. Phys. Fluids 16 (10), 36703681.
Wan, Z. Y., Vlachas, P., Koumoutsakos, P. & Sapsis, T. 2018 Data-assisted reduced-order modeling of extreme events in complex dynamical systems. PloS One 13 (5), e0197704.
Wang, J.-X., Wu, J., Ling, J., Iaccarino, G. & Xiao, H.2017a A comprehensive physics-informed machine learning framework for predictive turbulence modeling. arXiv:1701.07102.
Wang, J.-X., Wu, J.-L. & Xiao, H. 2017b Physics-informed machine learning approach for reconstructing Reynolds stress modeling discrepancies based on DNS data. Phys. Rev. Fluids 2 (3), 034603.
Weatheritt, J. & Sandberg, R. D. 2017 Hybrid Reynolds-averaged/large-eddy simulation methodology from symbolic regression: formulation and application. AIAA J. 55 (11), 37343746.
Wu, J.-L., Xiao, H. & Paterson, E.2018a Data-driven augmentation of turbulence models with physics-informed machine learning. arXiv:1801.02762.
Wu, J.-L., Xiao, H. & Paterson, E. 2018b Physics-informed machine learning approach for augmenting turbulence models: a comprehensive framework. Phys. Rev. Fluids 3 (7), 074602.
Xiao, H., Wu, J.-L., Wang, J.-X., Sun, R. & Roy, C. J. 2016 Quantifying and reducing model-form uncertainties in Reynolds-averaged Navier–Stokes simulations: a data-driven, physics-informed Bayesian approach. J. Comput. Phys. 324, 115136.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

JFM classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed