References
- Fröhlich J, Von Terzi D. Hybrid les/rans methods for the simulation of turbulent flows. Prog Aerosp Sci. 2008;44(5):349–377.
- Duraisamy K, Iaccarino G, Xiao H. Turbulence modeling in the age of data. Annu Rev Fluid Mech. 2019;51:357–377.
- Chen H, Kandasamy S, Orszag S, et al. Extended boltzmann kinetic equation for turbulent flows. Science. 2003;301(5633):633–636.
- Gatski T. Constitutive equations for turbulent flows. Theor Comp Fluid Dyn. 2004;18(5):345–369.
- Johansson A. Engineering turbulence models and their development, with emphasis on explicit algebraic Reynolds stress models. In: Oberlack M, Busse FH, editors. Theories of turbulence. Vienna: Springer; 2002. p. 253–300.
- Speziale CG. Analytical methods for the development of Reynolds-stress closures in turbulence. Annu Rev Fluid Mech. 1991;23(1):107–157.
- Brunton SL, Proctor JL, Kutz JN. Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the National Academy of Sciences. 2016. p. 201517384.
- Colabrese S, Gustavsson K, Celani A, et al. Flow navigation by smart microswimmers via reinforcement learning. Phys Rev Lett. 2017;118(15):158004.
- Jiménez J. Machine-aided turbulence theory. J Fluid Mech. 2018;854:R1.
- Raissi M, Perdikaris P, Karniadakis G. Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J Comput Phys. 2019;378:686–707.
- Ling J, Jones R, Templeton J. Machine learning strategies for systems with invariance properties. J Comput Phys. 2016;318:22–35.
- Ling J, Kurzawski A, Templeton J. Reynolds averaged turbulence modelling using deep neural networks with embedded invariance. J Fluid Mech. 2016;807:155–166.
- Tracey B, Duraisamy K, Alonso JJ. A machine learning strategy to assist turbulence model development. 53rd AIAA Aerospace Sciences Meeting. 2015, Kissimmee, United States. p. 1287. N/A. 2015.
- Wang J-X, Wu J-L, Xiao H. Physics-informed machine learning approach for reconstructing Reynolds stress modeling discrepancies based on dns data. Phys Rev Fluids. 2017;2(3).
- Wu J-L, Xiao H, Paterson E. Physics-informed machine learning approach for augmenting turbulence models: a comprehensive framework. Phys Rev Fluids. 2018;3(7):074602.
- Zhang ZJ, Duraisamy K. Machine learning methods for data-driven turbulence modeling. 22nd AIAA Computational Fluid Dynamics Conference. 2015, Dallas, United States. p. 2460.
- Pope SB. A more general effective-viscosity hypothesis. J Fluid Mech. 1975;72(2):331–340.
- Domaradzki JA, Rogallo RS. Local energy transfer and nonlocal interactions in homogeneous, isotropic turbulence. Phys Fluids A. 1990;2(3):413–426.
- Laval J, Dubrulle B, Nazarenko S. Nonlocality and intermittency in three-dimensional turbulence. Phys Fluids. 2001;13(7):1995–2012.
- Speziale CG, Eringen AC. Nonlocal fluid mechanics description of wall turbulence. Comput Math Appl. 1981;7(1):27–41.
- Hamlington P, Dahm W. Reynolds stress closure including nonlocal and nonequilibrium effects in turbulent flows. 39th AIAA Fluid Dynamics Conference. 2009, San Antonio, United States. p. 4162.
- Hamba F. Nonlocal analysis of the Reynolds stress in turbulent shear flow. Phys Fluids. 2005;17(11).
- Pope SB. Turbulent flows. Cambridge, United Kingdom: Cambridge University Press; 2001.
- Hamlington PE, Dahm WJ. Reynolds stress closure for nonequilibrium effects in turbulent flows. Phys Fluids. 2008;20(11):115101.
- Song F, Karniadakis GE. A universal fractional model of wall-turbulence. arXiv preprint arXiv:1808.10276. 2018.
- Succi S. The lattice Boltzmann equation: for complex states of flowing matter. Oxford, United Kingdom: Oxford University Press; 2018.
- Lee M, Moser RD. Direct numerical simulation of turbulent channel flow up to Reτ≈5200. J Fluid Mech. 2015;774:395–415.
- Krizhevsky A, Sutskever I, Hinton GE. Imagenet classification with deep convolutional neural networks. Advances in Neural Information Processing Systems. 2012, Lake Tahoe, United States. p. 1097–1105.
- LeCun Y, Bengio Y, Hinton G. Deep learning. Nature. 2015;521(7553):436–444.
- Silver D, Schrittwieser J, Simonyan K, et al. Mastering the game of go without human knowledge. Nature. 2017;550(7676):354–359.
- Ruder S. An overview of gradient descent optimization algorithms. arXiv preprint arXiv:1609.04747. 2016.
- Goodfellow I, Bengio Y, Courville A, et al. Deep learning. Vol. 1. Cambridge: MIT press; 2016.
- Hamlington PE, Dahm WJ. Nonlocal form of the rapid pressure-strain correlation in turbulent flows. Phys Rev E. 2009;80(4):046311.
- Li C, Zeng F. Numerical methods for fractional calculus. Boca Raton: Chapman and Hall/CRC; 2015.
- Kingma DP, Ba J. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980. 2014.
- He K, Zhang X, Ren S. Delving deep into rectifiers: surpassing human-level performance on imagenet classification. Proceedings of the IEEE International Conference on Computer Vision. 2015, Santiago, Chile. p. 1026–1034.
- Prechelt L. Early stopping-but when? In: Montavon G, Orr GB, Müller KR, editors. Neural networks: tricks of the trade. Berlin, Heidelberg: Springer; 1998. p. 55–69.
- Clevert D-A, Unterthiner T, Hochreiter S. Fast and accurate deep network learning by exponential linear units (elus). arXiv preprint arXiv:1511.07289. 2015.
- Ansumali S, Karlin IV, Succi S. Kinetic theory of turbulence modeling: smallness parameter, scaling and microscopic derivation of smagorinsky model. Phys A. 2004;338(3–4):379–394.