Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-24T09:05:26.117Z Has data issue: false hasContentIssue false
  • English
  • Français

Tempered fractional LES modeling

Published online by Cambridge University Press:  02 December 2021

Mehdi Samiee ,
Ali Akhavan-Safaei Open the ORCID record for Ali Akhavan-Safaei [Opens in a new window]  and
Mohsen Zayernouri Open the ORCID record for Mohsen Zayernouri [Opens in a new window]
Mehdi Samiee
Affiliation:
Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824, USA Department of Computational Mathematics, Science and Engineering, Michigan State University, East Lansing, MI 48824, USA
Ali Akhavan-Safaei
Affiliation:
Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824, USA Department of Computational Mathematics, Science and Engineering, Michigan State University, East Lansing, MI 48824, USA
Mohsen Zayernouri*
Affiliation:
Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824, USA Department of Statistics and Probability, Michigan State University, East Lansing, MI 48824, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The presence of non-local interactions and intermittent signals in the homogeneous isotropic turbulence grant multi-point statistical functions a key role in formulating a new generation of large-eddy simulation (LES) models of higher fidelity. We establish a tempered fractional-order modelling framework for developing non-local LES subgrid-scale models, starting from the kinetic transport. We employ a tempered Lévy-stable distribution to represent the source of turbulent effects at the kinetic level, and we rigorously show that the corresponding turbulence closure term emerges as the tempered fractional Laplacian, $(\varDelta +\lambda )^{\alpha } (\cdot )$, for $\alpha \in (0,1)$, $\alpha \neq \frac {1}{2}$ and $\lambda >0$ in the filtered Navier–Stokes equations. Moreover, we prove the frame invariant properties of the proposed model, complying with the subgrid-scale stresses. To characterize the optimum values of model parameters and infer the enhanced efficiency of the tempered fractional subgrid-scale model, we develop a robust algorithm, involving two-point structure functions and conventional correlation coefficients. In an a priori statistical study, we evaluate the capabilities of the developed model in fulfilling the closed essential requirements, obtained for a weaker sense of the ideal LES model (Meneveau, Phys. Fluids, vol. 6, issue 2, 1994, pp. 815–833). Finally, the model undergoes the a posteriori analysis to ensure the numerical stability and pragmatic efficiency of the model.

JFM classification

Turbulent Flows: Turbulence modelling Mathematical Foundations: Computational methods Nonlinear Dynamical Systems: Fractals
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 932 , 10 February 2022 , A4
Check for updates
Copyright
© The Author(s), 2021. Published by 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

REFERENCES

Akhavan-Safaei, A., Samiee, M. & Zayernouri, M. 2021 Data-driven fractional subgrid-scale modeling for scalar turbulence: a nonlocal LES approach. J. Comput. Phys. 446, 110571.CrossRefGoogle Scholar
Akhavan-Safaei, A., Seyedi, S.H. & Zayernouri, M. 2020 Anomalous features in internal cylinder flow instabilities subject to uncertain rotational effects. Phys. Fluids 32 (9), 094107.CrossRefGoogle Scholar
Akhavan-Safaei, A. & Zayernouri, M. 2020 A parallel integrated computational-statistical platform for turbulent transport phenomena. arXiv:2012.04838.Google Scholar
Beck, A. & Kurz, M. 2020 A perspective on machine learning methods in turbulence modelling. arXiv:2010.12226.CrossRefGoogle Scholar
Bouffanais, R. 2010 Advances and challenges of applied large-eddy simulation. Comput. Fluids 39 (5), 735–738.CrossRefGoogle Scholar
Briard, A., Gomez, T. & Cambon, C. 2016 Spectral modelling for passive scalar dynamics in homogeneous anisotropic turbulence. J. Fluid Mech. 799, 159–199.CrossRefGoogle Scholar
Buaria, D., Pumir, A. & Bodenschatz, E. 2020 Self-attenuation of extreme events in Navier–Stokes turbulence. Nat. Commun. 11 (1), 5852.CrossRefGoogle ScholarPubMed
Burkovska, O., Glusa, C. & D'Elia, M. 2020 An optimization-based approach to parameter learning for fractional type nonlocal models. arXiv:2010.03666.CrossRefGoogle Scholar
Burton, G.C. & Dahm, W.J.A. 2005 Multifractal subgrid-scale modeling for large-eddy simulation. II. Backscatter limiting and a posteriori evaluation. Phys. Fluids 17 (7), 075112.CrossRefGoogle Scholar
Cairoli, A. 2016 Towards a comprehensive framework for the analysis of anomalous diffusive systems. PhD thesis, Queen Mary University of London.Google Scholar
Cambon, C. & Scott, J.F. 1999 Linear and nonlinear models of anisotropic turbulence. Annu. Rev. Fluid Mech. 31 (1), 1–53.CrossRefGoogle Scholar
Cerutti, S., Meneveau, C. & Knio, O.M. 2000 Spectral and hyper eddy viscosity in high-Reynolds-number turbulence. J. Fluid Mech. 421, 307–338.CrossRefGoogle Scholar
Chao, M.A., Kulkarni, C., Goebel, K. & Fink, O. 2020 Fusing physics-based and deep learning models for prognostics. arXiv:2003.00732.Google Scholar
Chen, H., Orszag, S.A., Staroselsky, I. & Succi, S. 2004 Expanded analogy between Boltzmann kinetic theory of fluids and turbulence. J. Fluid Mech. 519, 301–314.CrossRefGoogle Scholar
Deng, W., Li, B., Tian, W. & Zhang, P. 2018 Boundary problems for the fractional and tempered fractional operators. Multiscale Model. Simul. 16 (1), 125–149.CrossRefGoogle Scholar
Di Leoni, P.C., Zaki, T.A., Karniadakis, G. & Meneveau, C. 2020 Two-point stress-strain rate correlation structure and non-local eddy viscosity in turbulent flows. arXiv:2006.02280.Google Scholar
Di Nezza, E., Palatucci, G. & Valdinoci, E. 2012 Hitchhiker's guide to the fractional Sobolev spaces. Bull. Sci. Mathématiques 136 (5), 521–573.CrossRefGoogle Scholar
Egolf, P.W. & Hutter, K. 2017 Fractional turbulence models. In Progress in Turbulence VII, pp. 123–131.Google Scholar
Egolf, P.W. & Hutter, K. 2020 Nonlinear, Nonlocal and Fractional Turbulence, Graduate Studies in Mathematics. Springer.CrossRefGoogle Scholar
Epps, B.P. & Cushman-Roisin, B. 2018 Turbulence modeling via the fractional Laplacian. arXiv:1803.05286.Google Scholar
Evin, G., Blanchet, J., Paquet, E., Garavaglia, F. & Penot, D. 2016 A regional model for extreme rainfall based on weather patterns subsampling. J. Hydrol. 541, 1185–1198.CrossRefGoogle Scholar
Girimaji, S.S. 2007 Boltzmann kinetic equation for filtered fluid turbulence. Phys. Rev. Lett. 99 (3), 034501.CrossRefGoogle ScholarPubMed
Hamlington, P.E. & Dahm, W.J.A. 2008 Reynolds stress closure for nonequilibrium effects in turbulent flows. Phys. Fluids 20 (11), 115101.CrossRefGoogle Scholar
Henderson, D.W. & Taimina, D. 2000 Experiencing Geometry. Prentice Hall.Google Scholar
Hill, R.J. 2002 Exact second-order structure-function relationships. J. Fluid Mech. 468, 317–326.CrossRefGoogle Scholar
Holgate, J., Skillen, A., Craft, T. & Revell, A. 2019 A review of embedded large eddy simulation for internal flows. Arch. Comput. Methods Engng 26 (4), 865–882.CrossRefGoogle Scholar
Huang, L. 2015 Density estimates for SDEs driven by tempered stable processes. arXiv:1504.04183.Google Scholar
Ionescu, C., Lopes, A., Copot, D., Machado, J.A.T. & Bates, J.H.T. 2017 The role of fractional calculus in modeling biological phenomena: a review. Commun. Nonlinear Sci. Numer. Simul. 51, 141–159.CrossRefGoogle Scholar
Jacob, J., Malaspinas, O. & Sagaut, P. 2018 A new hybrid recursive regularised Bhatnagar–Gross–Krook collision model for Lattice Boltzmann method-based large eddy simulation. J. Turbul. 19 (11–12), 1051–1076.CrossRefGoogle Scholar
Jin, G., Wang, S., Wang, Y. & He, G. 2018 Lattice Boltzmann simulations of high-order statistics in isotropic turbulent flows. Z. Angew. Math. Mech. 39 (1), 21–30.CrossRefGoogle Scholar
Jouybari, M.A., Yuan, J., Brereton, G.J. & Murillo, M.S. 2020 Data-driven prediction of the equivalent sand-grain height in rough-wall turbulent flows. arXiv:2002.01515.Google Scholar
Kaleta, K. & Sztonyk, P. 2015 Estimates of transition densities and their derivatives for jump Lévy processes. J. Math. Anal. Appl. 431 (1), 260–282.CrossRefGoogle Scholar
Kassinos, S.C., Reynolds, W.C. & Rogers, M.M. 2001 One-point turbulence structure tensors. J. Fluid Mech. 428, 213–248.CrossRefGoogle Scholar
Kharazmi, E. & Zayernouri, M. 2019 Fractional sensitivity equation method: application to fractional model construction. J. Sci. Comput. 80 (1), 110–140.CrossRefGoogle Scholar
Kurz, M. & Beck, A. 2020 A machine learning framework for LES closure terms. arXiv:2010.03030.Google Scholar
Laval, J.P., Dubrulle, B. & Nazarenko, S. 2001 Nonlocality and intermittency in three-dimensional turbulence. Phys. Fluids 13 (7), 1995–2012.CrossRefGoogle Scholar
Malaspinas, O. & Sagaut, P. 2012 Consistent subgrid scale modelling for lattice Boltzmann methods. J. Fluid Mech. 700, 514–542.CrossRefGoogle Scholar
Meneveau, C. 1994 Statistics of turbulence subgrid-scale stresses: necessary conditions and experimental tests. Phys. Fluids 6 (2), 815–833.CrossRefGoogle Scholar
Meral, F.C., Royston, T.J. & Magin, R. 2010 Fractional calculus in viscoelasticity: an experimental study. Commun. Nonlinear Sci. Numer. Simul. 15 (4), 939–945.CrossRefGoogle Scholar
Mishra, A.A. & Girimaji, S. 2019 Linear analysis of non-local physics in homogeneous turbulent flows. Phys. Fluids 31 (3), 035102.CrossRefGoogle Scholar
Mishra, A.A. & Girimaji, S.S. 2017 Toward approximating non-local dynamics in single-point pressure–strain correlation closures. J. Fluid Mech. 811, 168–188.CrossRefGoogle Scholar
Mortensen, M. & Langtangen, H.P. 2016 High performance python for direct numerical simulations of turbulent flows. Comput. Phys. Commun. 203, 53–65.CrossRefGoogle Scholar
Moser, R.D., Haering, S.W. & Yalla, G.R. 2021 Statistical properties of subgrid-scale turbulence models. Annu. Rev. Fluid Mech. 53, 255–286.CrossRefGoogle Scholar
Naghibolhosseini, M. & Long, G.R. 2018 Fractional-order modelling and simulation of human ear. Intl J. Comput. Maths 95 (6–7), 1257–1273.CrossRefGoogle Scholar
Pang, G., D'Elia, M., Parks, M. & Karniadakis, G.E. 2020 nPINNs: nonlocal physics-informed neural networks for a parametrized nonlocal universal Laplacian operator. Algorithms and applications. J. Comput. Phys. 422, 109760.CrossRefGoogle Scholar
Patra, A.K., Bevilacqua, A. & Safaei, A.A. 2018 Analyzing complex models using data and statistics. In International Conference on Computational Science, pp. 724–736. Springer.CrossRefGoogle Scholar
Pawar, S., San, O., Rasheed, A. & Vedula, P. 2020 A priori analysis on deep learning of subgrid-scale parameterizations for Kraichnan turbulence. Theor. Comput. Fluid Dyn. 34 (4), 429–455.CrossRefGoogle Scholar
Piomelli, U. 2014 Large eddy simulations in 2030 and beyond. Phil. Trans. R. Soc. Lond. A 372 (2022), 20130320.Google ScholarPubMed
Pope, S.B. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
Portwood, G.D., Nadiga, B.T., Saenz, J.A. & Livescu, D. 2021 Interpreting neural network models of residual scalar flux. J. Fluid Mech. 907, A23.CrossRefGoogle Scholar
Premnath, K.N., Pattison, M.J. & Banerjee, S. 2009 Dynamic subgrid scale modeling of turbulent flows using lattice-Boltzmann method. Physica A: Stat. Mech. Applics. 388 (13), 2640–2658.CrossRefGoogle Scholar
Sabzikar, F., Meerschaert, M.M. & Chen, J. 2015 Tempered fractional calculus. J. Comput. Phys. 293, 14–28.CrossRefGoogle Scholar
Sagaut, P. 2010 Toward advanced subgrid models for Lattice-Boltzmann-based large-eddy simulation: theoretical formulations. Comput. Maths Applics. 59 (7), 2194–2199.CrossRefGoogle Scholar
Sagaut, P. & Cambon, C. 2008 Homogeneous Turbulence Dynamics, vol. 10. Springer.CrossRefGoogle Scholar
Samiee, M. 2021 Data-Infused Fractional Modeling and Spectral Numerical Analysis for Anomalous Transport and Turbulence. Michigan State University.Google Scholar
Samiee, M., Akhavan-Safaei, A. & Zayernouri, M. 2020 a A fractional subgrid-scale model for turbulent flows: theoretical formulation and a priori study. Phys. Fluids 32 (5), 055102.CrossRefGoogle Scholar
Samiee, M., Kharazmi, E., Meerschaert, M.M. & Zayernouri, M. 2020 b A unified Petrov–Galerkin spectral method and fast solver for distributed-order partial differential equations. Commun. Appl. Math. Comput. 3, 61–90.CrossRefGoogle Scholar
Samiee, M., Zayernouri, M. & Meerschaert, M.M. 2019 A unified spectral method for FPDEs with two-sided derivatives; part I: a fast solver. J. Comput. Phys. 385, 225–243.CrossRefGoogle Scholar
She, Z.-S., Jackson, E. & Orszag, S.A. 1990 Intermittent vortex structures in homogeneous isotropic turbulence. Nature 344 (6263), 226–228.CrossRefGoogle Scholar
Shivamoggi, B.K. & Tuovila, N. 2019 Direct interaction approximation for non-Markovianized stochastic models in the turbulence problem. Chaos 29 (6), 063124.CrossRefGoogle ScholarPubMed
Sirignano, J., MacArt, J.F. & Freund, J.B. 2020 DPM: a deep learning PDE augmentation method with application to large-eddy simulation. J. Comput. Phys. 423, 109811.CrossRefGoogle Scholar
Smagorinsky, J. 1963 General circulation experiments with the primitive equations: I. The basic experiment. Mon. Weath. Rev. 91 (3), 99–164.2.3.CO;2>CrossRefGoogle Scholar
Soto, R. 2016 Kinetic Theory and Transport Phenomena, vol. 25. Oxford University Press.CrossRefGoogle Scholar
Stein, E.M. 1970 Singular Integrals and Differentiability Properties of Functions, vol. 2. Princeton University Press.Google Scholar
Suzuki, J., Zhou, Y., D'Elia, M. & Zayernouri, M. 2021 a A thermodynamically consistent fractional visco-elasto-plastic model with memory-dependent damage for anomalous materials. Comput. Meth. Appl. Mech. Engng 373, 113494.CrossRefGoogle Scholar
Suzuki, J.L., Kharazmi, E., Varghaei, P., Naghibolhosseini, M. & Zayernouri, M. 2021 b Anomalous nonlinear dynamics behavior of fractional viscoelastic beams. J. Comput. Nonlinear Dyn. 16 (11), 111005.CrossRefGoogle Scholar
Suzuki, J.L. & Zayernouri, M. 2021 A self-singularity-capturing scheme for fractional differential equations. Intl J. Comput. Maths 98 (5), 933–960.CrossRefGoogle Scholar
Taghizadeh, S., Witherden, F.D. & Girimaji, S.S. 2020 Turbulence closure modeling with data-driven techniques: physical compatibility and consistency considerations. New J. Phys. 22 (9), 093023.CrossRefGoogle Scholar
Vincent, A. & Meneguzzi, M. 1991 The spatial structure and statistical properties of homogeneous turbulence. J. Fluid Mech. 225, 1–20.CrossRefGoogle Scholar
Weron, R. 2001 Levy-stable distributions revisited: tail $\textrm {index} > 2$ does not exclude the Levy-stable regime. Intl J. Mod. Phys. C 12 (02), 209–223.CrossRefGoogle Scholar
Willard, J., Jia, X., Xu, S., Steinbach, M. & Kumar, V. 2020 Integrating physics-based modeling with machine learning: a survey. arXiv:2003.04919.Google Scholar
Xie, C. & Fang, S. 2019 A second-order finite difference method for fractional diffusion equation with Dirichlet and fractional boundary conditions. Numer. Meth. Partial Differ. Equ. 35 (4), 1383–1395.CrossRefGoogle Scholar
Yang, X.I.A. & Lozano-Durán, A. 2017 A multifractal model for the momentum transfer process in wall-bounded flows. J. Fluid Mech. 824, R2.CrossRefGoogle ScholarPubMed
You, H., Yu, Y., Trask, N., Gulian, M. & D'Elia, M. 2021 Data-driven learning of nonlocal physics from high-fidelity synthetic data. Comput. Meth. Appl. Mech. Engng 374, 113553.CrossRefGoogle Scholar
Zaky, M.A., Hendy, A.S. & Macías-Díaz, J.E. 2020 Semi-implicit Galerkin–Legendre spectral schemes for nonlinear time-space fractional diffusion–reaction equations with smooth and nonsmooth solutions. J. Sci. Comput. 82 (1), 13.CrossRefGoogle Scholar
Zayernouri, M., Ainsworth, M. & Karniadakis, G.Em. 2015 Tempered fractional Sturm–Liouville eigenproblems. SIAM J. Sci. Comput. 37 (4), A1777–A1800.CrossRefGoogle Scholar
Zhang, Z., Deng, W. & Karniadakis, G.Em. 2018 A Riesz basis Galerkin method for the tempered fractional Laplacian. SIAM J. Numer. Anal. 56 (5), 3010–3039.CrossRefGoogle Scholar
Zhiyin, Y. 2015 Large-eddy simulation: past, present and the future. Chin. J. Aeronaut. 28 (1), 11–24.CrossRefGoogle Scholar