Fractional and tempered fractional models for Reynolds-averaged Navier–Stokes equations

References

• Meerschaert MM, Sikorskii A. Stochastic models for fractional calculus. Berlin: De Gruyter; 2012. doi: 10.1515/9783110258165
   Google Scholar
• Einstein A. über die von der molekularkinetischen theorie der wärme geforderte bewegung von in ruhenden flüssigkeiten suspendierten teilchen. Ann Phys. 1905;322(8):549–560. doi: 10.1002/(ISSN)1521-3889
   Google Scholar
• Deng W, Wang X, Nie D. Functional distribution of anomalous and nonergodic diffusion: from stochastic processes to pdes. World Scientific; 2022.
   Google Scholar
• Richardson LF. Atmospheric diffusion shown on a distance-neighbour graph. Proc R Soc Lond Ser A Contain Pap Math Phys Character. 1926;110(756):709–737.
   Google Scholar
• Reynolds O. Iv. on the dynamical theory of incompressible viscous fluids and the determination of the criterion. Philos Trans Royal Soc Lond. 1895;186:123–164.
   Google Scholar
• VonNeumann J, Richtmyer RD. A method for the numerical calculation of hydrodynamic shocks. J Appl Phys. 1950;21(3):232–237. doi: 10.1063/1.1699639
   Web of Science ®Google Scholar
• Chen W. A speculative study of 2/ 3-order fractional Laplacian modeling of turbulence: some thoughts and conjectures. Chaos Interdiscip J Nonlinear Sci. 2006;16(2):023126. doi: 10.1063/1.2208452
   PubMed Web of Science ®Google Scholar
• Song F, Karniadakis G. A universal fractional model of wall-turbulence. arXiv preprint arXiv:180810276. 2018.
   Google Scholar
• Mehta PP, Pang G, Song F, et al. Discovering a universal variable-order fractional model for turbulent couette flow using a physics-informed neural network. Fract Calc Appl Anal. 2019;22(6):1675–1688. doi: 10.1515/fca-2019-0086
   Web of Science ®Google Scholar
• Kraichnan R. Eddy viscosity and diffusivity: exact formulas and approximations. Complex Syst. 1987;1(4-6):805–820.
   Google Scholar
• Hamba F. An analysis of nonlocal scalar transport in the convective boundary layer using the Green's function. J Atmos Sci. 1995;52(8):1084–1095. doi: 10.1175/1520-0469(1995)052¡1084:AAONST¿2.0.CO;2
   Web of Science ®Google Scholar
• Hamba F. Nonlocal expression for scalar flux in turbulent shear flow. Phys Fluids. 2004;16(5):1493–1508. doi: 10.1063/1.1697396
   Web of Science ®Google Scholar
• Hamba F. Nonlocal analysis of the Reynolds stress in turbulent shear flow. Phys Fluids. 2005;17(11):115102. doi: 10.1063/1.2130749
   Web of Science ®Google Scholar
• Hamba F. The mechanism of zero mean absolute vorticity state in rotating channel flow. Phys Fluids. 2006;18(12):125104. doi: 10.1063/1.2401632
   Web of Science ®Google Scholar
• Hamba F. History effect on the Reynolds stress in turbulent swirling flow. Phys Fluids. 2017;29(2):025103. doi: 10.1063/1.4976718
   Web of Science ®Google Scholar
• Epps BP, Cushman-Roisin B. Turbulence modeling via the fractional laplacian. arXiv preprint arXiv:180305286. 2018.
   Google Scholar
• Hayot F, Wagner L. A non-local modification of a lattice Boltzmann model. EPL Europhys Lett. 1996;33(6):435–440. doi: 10.1209/epl/i1996-00358-9
   Google Scholar
• Samiee M, Akhavan-Safaei A, Zayernouri M. A fractional subgrid-scale model for turbulent flows: theoretical formulation and a priori study. Phys Fluids. 2020;32(5):055102. doi: 10.1063/1.5128379
   Web of Science ®Google Scholar
• Grunwald AK. Über ‘begrente’ derivationen und deren anwedung. Z Angew Math Phys. 1867;12:441–480.
   Google Scholar
• Podlubny I. Fractional differential equations. Academic; 1999.
   Google Scholar
• Diethelm K. The analysis of fractional differential equations. Berlin: Springer-Verlag; 2010. (lecture notes in mathematics; Vol. 2004).
   Google Scholar
• Oldham K, Spanier J. The fractional calculus theory and applications of differentiation and integration to arbitrary order. Elsevier; 1974.
   Google Scholar
• Caputo M. Linear models of dissipation whose Q is almost frequency independent–II. Geophys J Int. 1967;13(5):529–539. doi: 10.1111/j.1365-246X.1967.tb02303.x
   Web of Science ®Google Scholar
• Avsarkisov V, Hoyas S, Oberlack M, et al. Turbulent plane couette flow at moderately high reynolds number. J Fluid Mech. 2014;751:R1.
   Web of Science ®Google Scholar
• Lee M, Moser RD. Direct numerical simulation of turbulent channel flow up to Reτ≈5200. J Fluid Mech. 2015;774:395–415. doi: 10.1017/jfm.2015.268
   Web of Science ®Google Scholar
• Tennekes H, Lumley JL. A first course in turbulence. MIT Press; 1972.
   Google Scholar
• Wu X, Moin P. A direct numerical simulation study on the mean velocity characteristics in turbulent pipe flow. J Fluid Mech. 2008;608:81–112. doi: 10.1017/S0022112008002085
   Web of Science ®Google Scholar
• Batchelor GK. Expressions for some common vector differential quantities in orthogonal curvilinear co-ordinate systems. Cambridge University Press; 2000. Cambridge Mathematical Library; p. 598–603.
   Google Scholar
• El Khoury G, Schlatter P, Noorani A, et al. Direct numerical simulation of turbulent pipe flow at moderately high Reynolds numbers. Flow Turbul Combust. 2013;91(3):475–495. doi: 10.1007/s10494-013-9482-8
   Web of Science ®Google Scholar
• Raissi M, Perdikaris P, Karniadakis GE. 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. doi: 10.1016/j.jcp.2018.10.045
   Web of Science ®Google Scholar
• Pang G, Lu L, Karniadakis GE. fPINNs: fractional physics-informed neural networks. SIAM J Sci Comput. 2019;41(4):A2603–A2626. doi: 10.1137/18M1229845
   Web of Science ®Google Scholar
• Jagtap AD, Kawaguchi K, Karniadakis GE. Adaptive activation functions accelerate convergence in deep and physics-informed neural networks. J Comput Phys. 2020;404:109136. doi: 10.1016/j.jcp.2019.109136
   Web of Science ®Google Scholar
• Yang Q, Liu F, Turner I. Numerical methods for fractional partial differential equations with Riesz space fractional derivatives. Appl Math Model. 2010;34(1):200–218. doi: 10.1016/j.apm.2009.04.006
   Web of Science ®Google Scholar
• Coles D. The law of the wake in the turbulent boundary layer. J Fluid Mech. 1956;1(2):191–226. doi: 10.1017/S0022112056000135
   Web of Science ®Google Scholar
• Spalding DB. A single formula for the ‘Law of the wall’. J Appl Mech. 1961;28(3):455–458. doi: 10.1115/1.3641728
   Google Scholar
• Zagarola MV, Smits AJ. Mean-flow scaling of turbulent pipe flow. J Fluid Mech. 1998;373:33–79. doi: 10.1017/S0022112098002419
   Web of Science ®Google Scholar
• Mantegna RN, Stanley HE. Stochastic process with ultraslow convergence to a gaussian: the truncated lévy flight. Phys Rev Lett. 1994;73(22):2946–2949. doi: 10.1103/PhysRevLett.73.2946
   PubMed Web of Science ®Google Scholar
• Koponen I. Analytic approach to the problem of convergence of truncated Lévy flights towards the Gaussian stochastic process. Phys Rev E. 1995;52(1):1197–1199. doi: 10.1103/PhysRevE.52.1197
   PubMed Web of Science ®Google Scholar
• Rosiński J. Tempering stable processes. Stoch Process Their Appl. 2007;117(6):677–707. doi: 10.1016/j.spa.2006.10.003
   Web of Science ®Google Scholar
• Cartea A. Fluid limit of the continuous-time random walk with general Lévy jump distribution functions. Phys Rev E. 2007 Oct;76(4):041105. doi: 10.1103/PhysRevE.76.041105
   Web of Science ®Google Scholar
• Wu X, Deng W, Barkai E. Tempered fractional Feynman-Kac equation: theory and examples. Phys Rev E. 2016;93(3):032151. doi: 10.1103/PhysRevE.93.032151
   PubMed Web of Science ®Google Scholar
• Sabzikar F, Meerschaert MM, Chen J. Tempered fractional calculus. J Comput Phys. 2015;293:14–28. doi: 10.1016/j.jcp.2014.04.024Fractional PDEs; https://www.sciencedirect.com/science/article/pii/S0021999114002873.
   PubMed Web of Science ®Google Scholar
• Mehta PP. Fractional models of Reynolds-averaged Navier–Stokes equations for turbulent flows. arXiv preprint arXiv:210503646. 2021.
   Google Scholar
• Mehta PP, Karniadakis G. Defining a new variable-order tempered fractional derivative for modeling turbulent flows. 25${}^{\mathrm{th}}$ Int Congr Theor Appl Mech Book Abstr. 2021;1235–1236. https://iutam.org/publications/ictam-proceedings/ictam_2020/.
   Google Scholar
• Prandtl L. 7. bericht über untersuchungen zur ausgebildeten turbulenz. ZAMM J Appl Math Mech / Z Angew Math Mech. 1925;5(2):136–139. doi: 10.1002/zamm.v5.2
   Google Scholar
• Jones WP, Launder BE. The prediction of laminarization with a two-equation model of turbulence. Int J Heat Mass Transf. 1972;15(2):301–314. doi: 10.1016/0017-9310(72)90076-2
   Web of Science ®Google Scholar
• Launder BE, Sharma BI. Application of the energy-dissipation model of turbulence to the calculation of flow near a spinning disc. Lett Heat Mass Transf. 1974;1(2):131–137. doi: 10.1016/0094-4548(74)90150-7
   Google Scholar
• Launder B, Spalding D. The numerical computation of turbulent flows. Comput Methods Appl Mech Eng. 1974;3(2):269–289. doi: 10.1016/0045-7825(74)90029-2https://www.sciencedirect.com/science/article/pii/0045782574900292.
   Google Scholar
• Wilcox D. Reassessment of the scale-determining equation for advanced turbulence models. AIAA J. 1988;26(11):1299–1310. doi: 10.2514/3.10041
   Google Scholar
• Menter F. Zonal two equation kw turbulence models for aerodynamic flows. In: 23rd fluid dynamics, plasmadynamics, and lasers conference; 1993. p. 2906.
   Google Scholar
• Craft T, Launder B, Suga K. Development and application of a cubic eddy-viscosity model of turbulence. Int J Heat Fluid Flow. 1996;17(2):108–115. doi: 10.1016/0142-727X(95)00079-6
   Web of Science ®Google Scholar
• Craft TJ, Graham LJW, Launder BE. Impinging jet studies for turbulence model assessment–II. An examination of the performance of four turbulence models. Int J Heat Mass Transf. 1993;36(10):2685–2697. doi: 10.1016/S0017-9310(05)80205-4
   Web of Science ®Google Scholar
• Speziale CG. On nonlinear k−l and k−ϵ models of turbulence. J Fluid Mech. 1987;178(2):459–475. doi: 10.1017/S0022112087001319
   Google Scholar
• Oldroyd JG, Wilson AH. On the formulation of rheological equations of state. Proc Roy Soc Lond Ser A Math Phys Sci. 1950;200(1063):523–541.
   Web of Science ®Google Scholar