Hostname: page-component-745bb68f8f-g4j75 Total loading time: 0 Render date: 2025-01-09T01:10:23.011Z Has data issue: false hasContentIssue false
  • English
  • Français

Sub-grid scale model classification and blending through deep learning

Published online by Cambridge University Press:  14 May 2019

Romit Maulik ,
Omer San Open the ORCID record for Omer San [Opens in a new window] ,
Jamey D. Jacob  and
Christopher Crick
Romit Maulik
Affiliation:
Mechanical and Aerospace Engineering, Oklahoma State University, Stillwater, OK 74078, USA
Omer San*
Affiliation:
Mechanical and Aerospace Engineering, Oklahoma State University, Stillwater, OK 74078, USA
Jamey D. Jacob
Affiliation:
Mechanical and Aerospace Engineering, Oklahoma State University, Stillwater, OK 74078, USA
Christopher Crick
Affiliation:
Computer Science, Oklahoma State University, Stillwater, OK 74078, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this article we detail the use of machine learning for spatio-temporally dynamic turbulence model classification and hybridization for large eddy simulations (LES) of turbulence. Our predictive framework is devised around the determination of local conditional probabilities for turbulence models that have varying underlying hypotheses. As a first deployment of this learning, we classify a point on our computational grid as that which requires the functional hypothesis, the structural hypothesis or no modelling at all. This ensures that the appropriate model is specified from a priori knowledge and an efficient balance of model characteristics is obtained in a particular flow computation. In addition, we also utilize the conditional-probability predictions of the same machine learning to blend turbulence models for another hybrid closure. Our test case for the demonstration of this concept is given by Kraichnan turbulence, which exhibits a strong interplay of enstrophy and energy cascades in the wavenumber domain. Our results indicate that the proposed methods lead to robust and stable closure and may potentially be used to combine the strengths of various models for complex flow phenomena prediction.

JFM classification

Mathematical Foundations: Computational methods Turbulent Flows: Turbulence modelling
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 870 , 10 July 2019 , pp. 784 - 812
Copyright
© 2019 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bardina, J., Ferziger, J. H. & Reynolds, W. C.1980 Improved subgrid-scale models for large-eddy simulation. AIAA Paper 1980-1357.Google Scholar
Beck, A. D., Flad, D. G. & Munz, C.-D.2018 Neural networks for data-based turbulence models. arXiv:1806.04482.Google Scholar
Berselli, L. C., Iliescu, T. & Layton, W. J. 2006 Mathematics of Large Eddy Simulation of Turbulent Flows. Springer.Google Scholar
Bishop, C. M. 2006 Pattern Recognition and Machine Learning (Information Science and Statistics). Springer.Google Scholar
Boffetta, G. & Ecke, R. E. 2012 Two-dimensional turbulence. Annu. Rev. Fluid Mech. 44, 427–451.Google Scholar
Canuto, V. M. & Cheng, Y. 1997 Determination of the Smagorinsky–Lilly constant CS. Phys. Fluids 9 (5), 1368–1378.Google Scholar
Cushman-Roisin, B. & Beckers, J.-M. 2011 Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects. Academic Press.Google Scholar
Duraisamy, K., Iaccarino, G. & Xiao, H. 2019 Turbulence modeling in the age of data. Annu. Rev. Fluid Mech. 51, 357–377.Google Scholar
Frisch, U. 1995 Turbulence. Cambridge University Press.Google Scholar
Fukami, K., Fukagata, K. & Taira, K.2018 Super-resolution reconstruction of turbulent flows with machine learning. arXiv:1811.11328.Google Scholar
Gamahara, M. & Hattori, Y. 2017 Searching for turbulence models by artificial neural network. Phys. Rev. Fluids 2 (5), 054604.Google Scholar
Germano, M. 2015 The similarity subgrid stresses associated to the approximate Van Cittert deconvolutions. Phys. Fluids 27 (3), 035111.Google Scholar
Germano, M., Piomelli, U., Moin, P. & Cabot, W. H. 1991 A dynamic subgrid-scale eddy viscosity model. Phys. Fluids 3 (7), 1760–1765.Google Scholar
Grossmann, S. & Mertens, P. 1992 Structure functions in two-dimensional turbulence. Z. Phys. B. Con. Mat. 88 (1), 105–116.Google Scholar
Guermond, J.-L., Oden, J. T. & Prudhomme, S. 2004 Mathematical perspectives on large eddy simulation models for turbulent flows. J. Math. Fluid Mech. 6 (2), 194–248.Google Scholar
Habisreutinger, M. A., Bouffanais, R., Leriche, E. & Deville, M. O. 2007 A coupled approximate deconvolution and dynamic mixed scale model for large-eddy simulation. J. Comput. Phys. 224 (1), 241–266.Google Scholar
Hennigh, O.2017 Lat-net: compressing lattice Boltzmann flow simulations using deep neural networks. arXiv:1705.09036.Google Scholar
Hickel, S., Egerer, C. P. & Larsson, J. 2014 Subgrid-scale modeling for implicit large eddy simulation of compressible flows and shock-turbulence interaction. Phys. Fluids 26 (10), 106101.Google Scholar
Hornik, K., Stinchcombe, M. & White, H. 1989 Multilayer feedforward networks are universal approximators. Neural Netw. 2 (5), 359–366.Google Scholar
King, R., Hennigh, O., Mohan, A. & Chertkov, M.2018 From deep to physics-informed learning of turbulence: diagnostics. arXiv:1810.07785.Google Scholar
King, R. N., Hamlington, P. E. & Dahm, W. J. 2016 Autonomic closure for turbulence simulations. Phys. Rev. E 93 (3), 031301.Google Scholar
Kolmogorov, A. N. 1941 The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. In Dokl. Akad. Nauk SSSR, vol. 30, pp. 301–305. JSTOR.Google Scholar
Kraichnan, R. H. 1967 Inertial ranges in two-dimensional turbulence. Phys. Fluids 10 (7), 1417–1423.Google Scholar
LaBryer, A., Attar, P. J. & Vedula, P. 2015 A framework for large eddy simulation of Burgers turbulence based upon spatial and temporal statistical information. Phys. Fluids 27 (3), 035116.Google Scholar
Langford, J. A. & Moser, R. D. 1999 Optimal LES formulations for isotropic turbulence. J. Fluid Mech. 398, 321–346.Google Scholar
Lapeyre, C. J., Misdariis, A., Cazard, N., Veynante, D. & Poinsot, T. 2019 Training convolutional neural networks to estimate turbulent sub-grid scale reaction rates. Combust. Flame 203, 255–264.Google Scholar
Layton, W. & Lewandowski, R. 2003 A simple and stable scale-similarity model for large eddy simulation: energy balance and existence of weak solutions. Appl. Maths Lett. 16 (8), 1205–1209.Google Scholar
Leith, C. E. 1968 Diffusion approximation for two-dimensional turbulence. Phys. Fluids 11 (3), 671–672.Google Scholar
Lilly, D. K. 1992 A proposed modification of the Germano subgrid-scale closure method. Phys. Fluids 4 (3), 633–635.Google Scholar
Ling, J., Jones, R. & Templeton, J. 2016a Machine learning strategies for systems with invariance properties. J. Comput. Phys. 318, 22–35.Google Scholar
Ling, J. & Kurzawski, A.2017 Data-driven adaptive physics modeling for turbulence simulations. AIAA Paper 2017-3627.Google Scholar
Ling, J., Kurzawski, A. & Templeton, J. 2016b Reynolds averaged turbulence modelling using deep neural networks with embedded invariance. J. Fluid Mech. 807, 155–166.Google Scholar
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.Google Scholar
Mathew, J., Lechner, R., Foysi, H., Sesterhenn, J. & Friedrich, R. 2003 An explicit filtering method for large eddy simulation of compressible flows. Phys. Fluids 15 (8), 2279–2289.Google Scholar
Maulik, R. & San, O. 2017a A dynamic framework for functional parameterisations of the eddy viscosity coefficient in two-dimensional turbulence. Intl J. Comput. Fluid Dyn. 31 (2), 69–92.Google Scholar
Maulik, R. & San, O. 2017b A neural network approach for the blind deconvolution of turbulent flows. J. Fluid Mech. 831, 151–181.Google Scholar
Maulik, R. & San, O. 2017c A stable and scale-aware dynamic modeling framework for subgrid-scale parameterizations of two-dimensional turbulence. Comput. Fluids 158, 11–38.Google Scholar
Maulik, R., San, O., Rasheed, A. & Vedula, P. 2018 Data-driven deconvolution for large eddy simulations of Kraichnan turbulence. Phys. Fluids 30 (12), 125109.Google Scholar
Maulik, R., San, O., Rasheed, A. & Vedula, P. 2019 Subgrid modelling for two-dimensional turbulence using neural networks. J. Fluid Mech. 858, 122–144.Google Scholar
McWilliams, J. C. 1990 The vortices of two-dimensional turbulence. J. Fluid Mech. 219, 361–385.Google Scholar
Milano, M. & Koumoutsakos, P. 2002 Neural network modeling for near wall turbulent flow. J. Comput. Phys. 182 (1), 1–26.Google Scholar
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.Google Scholar
Pearson, B. & Fox-Kemper, B. 2018 Log-normal turbulence dissipation in global ocean models. Phys. Rev. Lett. 120 (9), 094501.Google Scholar
Pearson, B., Fox-Kemper, B., Bachman, S. & Bryan, F. 2017 Evaluation of scale-aware subgrid mesoscale eddy models in a global eddy-rich model. Ocean Model. 115, 42–58.Google Scholar
Pope, S. B. 2004 Ten questions concerning the large-eddy simulation of turbulent flows. New J. Phys. 6, 35.Google Scholar
Sagaut, P. 2006 Large Eddy Simulation For Incompressible Flows: An Introduction. Springer.Google Scholar
San, O. 2014 A dynamic eddy-viscosity closure model for large eddy simulations of two-dimensional decaying turbulence. Intl J. Comput. Fluid Dyn. 28 (6–10), 363–382.Google Scholar
San, O. & Staples, A. E. 2012 High-order methods for decaying two-dimensional homogeneous isotropic turbulence. Comput. Fluids 63, 105–127.Google Scholar
San, O., Staples, A. E., Wang, Z. & Iliescu, T. 2011 Approximate deconvolution large eddy simulation of a barotropic ocean circulation model. Ocean Model. 40 (2), 120–132.Google Scholar
San, O. & Vedula, P. 2018 Generalized deconvolution procedure for structural modeling of turbulence. J. Sci. Comput. 75 (2), 1187–1206.Google Scholar
Sarghini, F., De Felice, G. & Santini, S. 2003 Neural networks based subgrid scale modeling in large eddy simulations. Comput. Fluids 32 (1), 97–108.Google Scholar
Singh, A. P. & Duraisamy, K. 2016 Using field inversion to quantify functional errors in turbulence closures. Phys. Fluids 28 (4), 045110.Google Scholar
Singh, A. P., Medida, S. & Duraisamy, K. 2017 Machine-learning-augmented predictive modeling of turbulent separated flows over airfoils. AIAA J. 55 (7), 2215–2227.Google Scholar
Smagorinsky, J. 1963 General circulation experiments with the primitive equations: I. The basic experiment. Mon. Weath. Rev. 91 (3), 99–164.Google Scholar
Sotgiu, C., Weigand, B. & Semmler, K. 2018 A turbulent heat flux prediction framework based on tensor representation theory and machine learning. Intl Commun. Heat Mass Transfer 95, 74–79.Google Scholar
Stolz, S. & Adams, N. A. 1999 An approximate deconvolution procedure for large-eddy simulation. Phys. Fluids 11 (7), 1699–1701.Google Scholar
Tabeling, P. 2002 Two-dimensional turbulence: a physicist approach. Phys. Rep. 362 (1), 1–62.Google Scholar
Tracey, B. D., Duraisamy, K. & Alonso, J. J.2015 A machine learning strategy to assist turbulence model development. AIAA Paper 2015-1287.Google Scholar
Vollant, A., Balarac, G. & Corre, C. 2017 Subgrid-scale scalar flux modelling based on optimal estimation theory and machine-learning procedures. J. Turbul. 18 (9), 854–878.Google Scholar
Vorobev, A. & Zikanov, O. 2008 Smagorinsky constant in LES modeling of anisotropic MHD turbulence. J. Theor. Comput. Fluid Dyn. 22 (3–4), 317–325.Google Scholar
Vreman, A. W. 2004 An eddy-viscosity subgrid-scale model for turbulent shear flow: algebraic theory and applications. Phys. Fluids 16 (10), 3670–3681.Google Scholar
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.Google Scholar
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.Google Scholar
Wang, Z., Luo, K., Li, D., Tan, J. & Fan, J. 2018 Investigations of data-driven closure for subgrid-scale stress in large-eddy simulation. Phys. Fluids 30 (12), 125101.Google Scholar
Weatheritt, J. & Sandberg, R. 2016 A novel evolutionary algorithm applied to algebraic modifications of the RANS stress–strain relationship. J. Comput. Phys. 325, 22–37.Google Scholar
Weatheritt, J. & Sandberg, R. D. 2017a The development of algebraic stress models using a novel evolutionary algorithm. Intl J. Heat Fluid Flow 68, 298–318.Google Scholar
Weatheritt, J. & Sandberg, R. D. 2017b Hybrid Reynolds-averaged/large-eddy simulation methodology from symbolic regression: formulation and application. AIAA J. 55 (11), 3734–3746.Google Scholar
Wu, J.-L., Xiao, H. & Paterson, E.2018 Data-driven augmentation of turbulence models with physics-informed machine learning. Preprint, arXiv:1801.02762.Google Scholar
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, 115–136.Google Scholar
Yu, C., Xiao, Z. & Li, X. 2016 Dynamic optimization methodology based on subgrid-scale dissipation for large eddy simulation. Phys. Fluids 28 (1), 015113.Google Scholar
Zhang, Z., Song, X.-d., Ye, S.-r., Wang, Y.-w., Huang, C.-g., An, Y.-r. & Chen, Y.-s. 2019 Application of deep learning method to Reynolds stress models of channel flow based on reduced-order modeling of DNS data. J. Hydrodyn. 31, 1–8.Google Scholar
Zhu, L., Zhang, W., Kou, J. & Liu, Y. 2019 Machine learning methods for turbulence modeling in subsonic flows around airfoils. Phys. Fluids 31 (1), 015105.Google Scholar