Geodesic
Numerical modeling of the two-dimensional Rayleigh-Benard convection
Matematičeskie zametki SVFU, Tome 24 (2017) no. 1, pp. 87-98 Cet article a éte moissonné depuis la source Math-Net.Ru

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We study Rayleigh-Bénard convection which is a type of natural convection occurring in a plane horizontal layer of viscous fluid heated from below and cooled from above, in which the fluid develops a regular pattern of convection cells known as Benard cells. The process of rotation is described by a system of nonlinear differential Oberbeck-Boussinesq equations. As convection parameters, the Rayleigh number and the Prandtl number are taken. The system is solved using the finite element method by FEniCS. We obtain numerical results for varying Rayleigh numbers and study the dependence of the Nusselt number on the Rayleigh number.
Keywords: natural convection, Oberbeck-Boussinesq approximation, finite element method, convection cells. 
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V. V. Grigoriev; P. E. Zakharov. Numerical modeling of the two-dimensional Rayleigh-Benard convection. Matematičeskie zametki SVFU, Tome 24 (2017) no. 1, pp. 87-98. http://geodesic.mathdoc.fr/item/SVFU_2017_24_1_a7/

[1] Getling A. V., Rayleigh–Bénard Convection: Structures and Dynamics, Adv. Ser. Nonlinear Dyn, 11, World Sci., Singapore, 1998 | DOI | MR

[2] Kolmychkov V. V., Mazhorova O. S., Popov Yu. P., Fedoseev E. E., “On numerical modeling of the Rayleigh–Bénard convection”, preprint Keldysh Inst. Appl. Math., 2007

[3] Getling A.V., “Formation of spatial structures in Rayleigh–Bénard convection”, Sov. Phys. Usp., 34:9 (1991), 737–776 | DOI | MR

[4] Zakinyan R. G., Sukhov S. A., and Larchenko I. N., “Mathematical modeling of the heat convection”, Contemporary Problems in Science and Education, 6 (2011) available online at www.science-education.ru/100-5016

[5] Chillà F., Schumacher J., “New perspectives in turbulent Rayleigh-Bénard convection”, Europ. Phys. J.(E), 35:7 (2012), 1–25 | MR

[6] Spiegelman M., “Myths and methods in modeling”, Columbia Univ. Course Lecture Notes, 2004 Available online at www.ldeo.columbia.edu/~mspieg/mmm/course.pdf

[7] Samarskii A. A. and Vabishchevich P. N., Numerical Heat Transfer, Librokom, Moscow, 2009

[8] Logg A., Mardal K.-A., Wells G., Automated Solution of Differential Equations by the Finite Element Method, The FEniCS book, 84, Springer Science Business Media, Berlin, 2012 | MR | Zbl

[9] Otto F., Seis C., “Rayleigh–Bénard convection: improved bounds on the nusselt number”, J. Math. Phys., 52:8 (2011), 083702 | DOI | MR | Zbl

[10] Sakievich P. J., Peet Y. T., Adrian R. J, “Large-scale thermal motions of turbulent Rayleigh–Bénard convection in a wide aspect-ratio cylindrical domain”, Intern. J. Heat Fluid Flow, 61 (2016), 183–196 | DOI

[11] Vynnycky M., Masuda Y., “Rayleigh–Bénard convection at high rayleigh number and infinite Prandtl number: Asymptotics and numerics”, Phys. Fluids, 25:11 (2013), 113602 | DOI | Zbl

[12] Vabishchevich P. N. (ed.), Computational Technologies. Profesional Level, URSS, 2014

[13] Alnaes M., Blechta J., Hake J., Johansson A., Kehlet B., Logg A., Richardson C., Ring J., Rognes M. E., and Wells G. N, “The FEniCS Project Version 1.5”, Arch. Numer. Software, 3 (2015), 9–23

[14] Lir J. T. and Lin T. F., “Visualization of roll patterns in Rayleigh–Bénard convection of air in a rectangular shallow cavity”, Intern. J. Heat Mass Transfer, 44:15 (2001), 2889–2902 | DOI | Zbl