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Cessation and reversals of large-scale structures in square Rayleigh–Bénard cells

  • Andrés Castillo-Castellanos (a1) (a2), Anne Sergent (a2) (a3), Bérengère Podvin (a2) and Maurice Rossi (a1)


We consider direct numerical simulations of turbulent Rayleigh–Bénard convection inside two-dimensional square cells. For Rayleigh numbers $Ra=10^{6}$ to $Ra=5\times 10^{8}$ and Prandtl numbers $Pr=3$ and $Pr=4.3$ , two types of flow regimes are observed intermittently: consecutive flow reversals (CR), and extended cessations (EC). For each regime, we combine proper orthogonal decomposition (POD) and statistical tools on long-term data to characterise the dynamics of large-scale structures. For the CR regime, centrosymmetric modes are dominant and display a coherent dynamics, while non-centrosymmetric modes fluctuate randomly. For the EC regime, all POD modes follow Poissonian statistics and a non-centrosymmetric mode is dominant. To explore further the differences between the CR and EC regimes, an analysis based on a cluster partition of the POD phase space is proposed. This data-driven approach confirms the successive mechanisms of the generic reversal cycle in CR as proposed in Castillo-Castellanos et al. (J. Fluid Mech., vol. 808, 2016, pp. 614–640). However, these mechanisms may take one of multiple paths in the POD phase space. Inside the EC regime, this approach reveals the presence of two types of coherent time sequences (weak reversals and actual cessations) and more rarely intense plume crossings. Finally, we analyse within a range of Rayleigh numbers up to turbulent flow, the relation between dynamical regimes and the POD energetic contents as well as the residence time in each cluster.


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Antonia, R. A. 1981 Conditional sampling in turbulence measurement. Annu. Rev. Fluid Mech. 13 (1), 131156.10.1146/annurev.fl.13.010181.001023
Arthur, D. & Vassilvitskii, S. 2007 k-means++: The advantages of careful seeding. In Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 10271035. Society for Industrial and Applied Mathematics.
Bai, K., Ji, D. & Brown, E. 2016 Ability of a low-dimensional model to predict geometry-dependent dynamics of large-scale coherent structures in turbulence. Phys. Rev. E 93 (2), 023117.
Bailon-Cuba, J., Emran, M. S. & Schumacher, J. 2010 Aspect ratio dependence of heat transfer and large-scale flow in turbulent convection. J. Fluid Mech. 655, 152173.10.1017/S0022112010000820
Bell, J. B., Colella, P. & Glaz, H. M. 1989 A second-order projection method for the incompressible Navier–Stokes equations. J. Comput. Phys. 283, 257283.10.1016/0021-9991(89)90151-4
Brown, E. & Ahlers, G. 2006 Rotations and cessations of the large-scale circulation in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 568, 351386.10.1017/S0022112006002540
Brown, E., Nikolaenko, A. & Ahlers, G. 2005 Reorientation of the large-scale circulation in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 95, 084503.10.1103/PhysRevLett.95.084503
Castillo-Castellanos, A., Sergent, A. & Rossi, M. 2016 Reversal cycle in square Rayleigh–Bénard cells in turbulent regime. J. Fluid Mech. 808, 614640.10.1017/jfm.2016.647
Chandra, M. & Verma, M. K. 2011 Dynamics and symmetries of flow reversals in turbulent convection. Phys. Rev. E 83 (6), 067303.
Chandra, M. & Verma, M. K. 2013 Flow reversals in turbulent convection via vortex reconnections. Phys. Rev. Lett. 110 (11), 114503.10.1103/PhysRevLett.110.114503
Das, A., Ghosal, U. & Kumar, K. 2000 Asymmetric squares as standing waves in Rayleigh–Bénard convection. Phys. Rev. E 62 (3), R3051.
Faranda, D., Podvin, B. & Sergent, A. 2019 On reversals in 2D turbulent Rayleigh–Bénard convection: insights from embedding theory and comparison with proper orthogonal decomposition analysis. Chaos 29, 033110.10.1063/1.5081031
Fauve, S., Herault, J., Michel, G. & Pétrélis, F. 2017 Instabilities on a turbulent background. J. Stat. Mech. 6, 064001.
Foroozani, N., Niemela, J. J., Armenio, V. & Sreenivasan, K. R. 2017 Reorientations of the large-scale flow in turbulent convection in a cube. Phys. Rev. E 95 (3), 033107.
Giannakis, D., Kolchinskaya, A., Krasnov, D. & Schumacher, J. 2018 Koopman analysis of the long-term evolution in a turbulent convection cell. J. Fluid Mech. 847, 735767.10.1017/jfm.2018.297
Grossmann, S. & Lohse, D. 2003 On geometry effects in Rayleigh–Bénard convection. J. Fluid Mech. 486, 105114.10.1017/S0022112003004270
Holmes, P., Lumley, J. L., Berkooz, G. & Rowley, C. W. 2012 Turbulence, Coherent Structures, Dynamical Systems and Symmetry, 2nd edn. Cambridge University Press.10.1017/CBO9780511919701
Horn, S. & Schmid, P. J. 2017 Prograde, retrograde, and oscillatory modes in rotating Rayleigh–Bénard convection. J. Fluid Mech. 831, 182211.10.1017/jfm.2017.631
Hughes, G. O., Gayen, B. & Griffiths, R. W. 2013 Available potential energy in Rayleigh–Bénard convection. J. Fluid Mech. 729, R3.10.1017/jfm.2013.353
Hussain, A. K. M. F. 1986 Coherent structures and turbulence. J. Fluid Mech. 173, 303356.10.1017/S0022112086001192
Jain, A. K. 2010 Data clustering: 50 years beyond k-means. Pattern Recognition Lett. 31 (8), 651666.10.1016/j.patrec.2009.09.011
Kadanoff, L. P. 2001 Turbulent heat flow: structures and scaling. Phys. Today 54 (8), 3439.10.1063/1.1404847
Kaiser, E., Noack, B. R., Cordier, L., Spohn, A., Segond, M., Abel, M., Daviller, G., Östh, J., Krajnovi, S. & Niven, R. K. 2014 Cluster-based reduced-order modelling of a mixing layer. J. Fluid Mech. 754, 365414.10.1017/jfm.2014.355
Lhuillier, F., Hulot, G. & Gallet, Y. 2013 Statistical properties of reversals and chrons in numerical dynamos and implications for the geodynamo. Phys. Earth Planet. Inter. 220, 1936.10.1016/j.pepi.2013.04.005
Lumley, J. L. 1967 The structure of inhomogeneous turbulent flows. In Atmospheric Turbulence and Radio Wave Propagation (ed. Yaglom, A. M. & Tatarski, V. I.), pp. 166178. Nauka.
McFadden, P. L. & Merrill, R. T. 1986 Geodynamo energy source constraints from palaeomagnetic data. Phys. Earth Planet. Inter. 43 (1), 2233.10.1016/0031-9201(86)90118-4
Merrill, R., McElhinny, M. & McFadden, P. 1998 Chapter five reversals of the earth’s magnetic field. In The Magnetic Field of the Earth: Paleomagnetism, the Core, and the Deep Mantle, International Geophysics, vol. 63, pp. 163215. Academic Press.
Molenaar, D., Clercx, H. J. H. & Van Heijst, G. J. F. 2004 Angular momentum of forced 2D turbulence in a square no-slip domain. Physica D 196, 329340.10.1016/j.physd.2004.06.001
Ni, R., Huang, S.-D. & Xia, K.-Q. 2015 Reversals of the large-scale circulation in quasi-2D Rayleigh–Bénard convection. J. Fluid Mech. 778, R5.10.1017/jfm.2015.433
Niemela, J. J., Skrbek, L., Sreenivasan, K. R. & Donnelly, R. J. 2001 The wind in confined thermal convection. J. Fluid Mech. 449, 169178.10.1017/S0022112001006310
Okabe, A., Boots, B., Sugihara, K. & Chiu, S. N. 2009 Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, vol. 501. John Wiley & Sons.
Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V. et al. 2011 Scikit-learn: Machine learning in Python. J. Mach. Learn. Res. 12, 28252830.
Petschel, K., Wilczek, M., Breuer, M., Friedrich, R. & Hansen, U. 2011 Statistical analysis of global wind dynamics in vigorous Rayleigh–Bénard convection. Phys. Rev. E 84 (2), 026309.
Podvin, B. & Sergent, A. 2015 A large-scale investigation of wind reversal in a square Rayleigh–Bénard cell. J. Fluid Mech. 766, 172201.10.1017/jfm.2015.15
Podvin, B. & Sergent, A. 2017 Precursor for wind reversal in a square Rayleigh–Bénard cell. Phys. Rev. E 95 (1), 013112.
van der Poel, E. P., Stevens, R. J. A. M. & Lohse, D. 2011 Connecting flow structures and heat flux in turbulent Rayleigh–Bénard convection. Phys. Rev. E 84 (4), 045303.
Popinet, S. 2003 Gerris: a tree-based adaptive solver for the incompressible euler equations in complex geometries. J. Comput. Phys. 190 (2), 572600.10.1016/S0021-9991(03)00298-5
Popinet, S. 2009 An accurate adaptive solver for surface-tension-driven interfacial flows. J. Comput. Phys. 228 (16), 58385866.10.1016/
Popinet, S. 2015 A quadtree-adaptive multigrid solver for the Serre–Green–Naghdi equations. J. Comput. Phys. 302, 336358.10.1016/
Schmid, P. J. 2010 Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 528.10.1017/S0022112010001217
Shraiman, B. I. & Siggia, E. D. 1990 Heat transport in high-Rayleigh-number convection. Phys. Rev. A 42 (6), 36503653.10.1103/PhysRevA.42.3650
Sirovich, L. 1987 Turbulence and the dynamics of coherent structures part I: coherent structures. Q. Appl. Maths 45 (3), 561571.10.1090/qam/910462
Sreenivasan, K. R., Bershadskii, A. & Niemela, J. J. 2002 Mean wind and its reversal in thermal convection. Phys. Rev. E 65 (5), 056306.
Sugiyama, K., Ni, R., Stevens, R. J. A. M., Chan, T. S., Zhou, S.-Q., Xi, H.-D., Sun, C., Grossmann, S., Xia, K.-Q. & Lohse, D. 2010 Flow reversals in thermally driven turbulence. Phys. Rev. Lett. 105 (3), 034503.10.1103/PhysRevLett.105.034503
Valet, J.-P., Fournier, A., Courtillot, V. & Herrero-Bervera, E. 2012 Dynamical similarity of geomagnetic field reversals. Nature 490 (7418), 8993.10.1038/nature11491
Van Heijst, G. J. F., Clercx, H. J. H. & Molenaar, D. 2006 The effects of solid boundaries on confined two-dimensional turbulence. J. Fluid Mech. 554, 411431.10.1017/S002211200600886X
Vasilev, A. Y. & Frick, P. G. 2011 Reversals of large-scale circulation in turbulent convection in rectangular cavities. JETP Lett. 93 (6), 330334.10.1134/S0021364011060117
Wicht, J., Stellmach, S. & Harder, H. 2009 Numerical Models of the Geodynamo: From Fundamental Cartesian Models to 3D Simulations of Field Reversals. pp. 107158. Springer.
Winters, K. B., Lombard, P. N., Riley, J. J. & D’Asaro, E. A. 1995 Available potential energy and mixing in density-stratified fluids. J. Fluid Mech. 289, 115128.10.1017/S002211209500125X
Xi, H.-D. & Xia, K.-Q. 2007 Cessations and reversals of the large-scale circulation in turbulent thermal convection. Phys. Rev. E 75 (6), 066307.
Xi, H.-D. & Xia, K.-Q. 2008a Azimuthal motion, reorientation, cessation, and reversal of the large-scale circulation in turbulent thermal convection: a comparative study in aspect ratio one and one-half geometries. Phys. Rev. E 78 (3), 036326.
Xi, H.-D. & Xia, K.-Q. 2008b Flow mode transitions in turbulent thermal convection. Phys. Fluids 20 (5), 055104.10.1063/1.2920444
Xi, H.-D., Zhou, Q. & Xia, K.-Q. 2006 Azimuthal motion of the mean wind in turbulent thermal convection. Phys. Rev. E 73 (5), 056312.
Zhang, X. & Zikanov, O. 2015 Two-dimensional turbulent convection in a toroidal duct of a liquid metal blanket of a fusion reactor. J. Fluid Mech. 779, 3652.10.1017/jfm.2015.421
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Cessation and reversals of large-scale structures in square Rayleigh–Bénard cells

  • Andrés Castillo-Castellanos (a1) (a2), Anne Sergent (a2) (a3), Bérengère Podvin (a2) and Maurice Rossi (a1)


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