Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2025-01-02T18:41:05.013Z Has data issue: false hasContentIssue false
  • English
  • Français

Flow state transition induced by emergence of orbiting satellite eddies in two-dimensional turbulent Rayleigh–Bénard convection

Published online by Cambridge University Press:  16 October 2024

Zhen-Yuan Gao Open the ORCID record for Zhen-Yuan Gao [Opens in a new window] ,
Xin Tao Open the ORCID record for Xin Tao [Opens in a new window] ,
Shi-Di Huang Open the ORCID record for Shi-Di Huang [Opens in a new window] ,
Yun Bao  and
Yi-Chao Xie Open the ORCID record for Yi-Chao Xie [Opens in a new window]
Zhen-Yuan Gao
Affiliation:
Center for Complex Flows and Soft Matter Research and Department of Mechanics and Aerospace Engineering, Southern University of Science and Technology, Shenzhen 518055, PR China
Xin Tao
Affiliation:
State Key Laboratory for Strength and Vibration of Mechanical Structures and School of Aerospace, Xi'an Jiaotong University, Xi'an 710049, PR China
Shi-Di Huang*
Affiliation:
Center for Complex Flows and Soft Matter Research and Department of Mechanics and Aerospace Engineering, Southern University of Science and Technology, Shenzhen 518055, PR China Guangdong Provincial Key Laboratory of Turbulence Research and Applications, Southern University of Science and Technology, Shenzhen 518055, PR China
Yun Bao
Affiliation:
School of Aeronautics and Astronautics, Sun Yat-sen University, Shenzhen 518107, PR China
Yi-Chao Xie*
Affiliation:
State Key Laboratory for Strength and Vibration of Mechanical Structures and School of Aerospace, Xi'an Jiaotong University, Xi'an 710049, PR China
*
†Email addresses for correspondence: [email protected], [email protected]
†Email addresses for correspondence: [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We report a numerical investigation of a previously noticed but less explored flow state transition in two-dimensional turbulent Rayleigh–Bénard convection. The simulations are performed in a square domain over a Rayleigh number range of $10^7 \leq Ra \leq 2 \times 10^{11}$ and a Prandtl number range of $0.25 \leq Pr \leq 20$. The transition is characterized by the emergence of multiple satellite eddies with increasing $Ra$, which orbit around and interact with the main vortex roll in the system. Consequently, the main roll is squeezed to a smaller size compared with the domain and wanders around in the bulk region irregularly and extensively. This is in sharp contrast to the flow state before the transition, which is featured by a domain-sized circulatory roll with its vortex centre ‘condensed’ near the domain's centre. Detailed velocity field analysis reveals that there exists an abrupt increase in the energy fluctuations of the Fourier modes during the transition. Based on this phase-transition-like signal, the critical condition for the transition is found to follow a scaling relation as $Ra_t \sim Pr^{1.41}$ where $Ra_t$ is the critical Rayleigh number for the transition. This scaling relation is quantitatively explained by a phenomenological model grounded on the bistability behaviour (i.e. spontaneous and stochastic switching between the two flow states) observed at the edge of the transition. The model can also account for the effects of aspect ratio on the transition reported in the literature (van der Poel et al., Phys. Fluids, vol. 24, 2012).

JFM classification

Turbulent Flows: Turbulent convection
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 997 , 25 October 2024 , A54
Check for updates
Copyright
© The Author(s), 2024. 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.)

Footnotes

‡

Z.-Y.G. and X.T. contributed equally to this work.

References

Ahlers, G., Grossmann, S. & Lohse, D. 2009 Heat transfer and large scale dynamics in turbulent Rayleigh–Bénard convection. Rev. Mod. Phys. 81, 503–537.CrossRefGoogle Scholar
Alexakis, A. & Biferale, L. 2018 Cascades and transitions in turbulent flows. Phys. Rep. 767–769, 1–101.CrossRefGoogle Scholar
Bao, Y., Chen, J., Liu, B.-F., She, Z.-S., Zhang, J. & Zhou, Q. 2015 Enhanced heat transport in partitioned thermal convection. J. Fluid Mech. 784, R5.CrossRefGoogle Scholar
Bao, Y., Luo, J. & Ye, M. 2018 Parallel direct method of DNS for two-dimensional turbulent Rayleigh–Bénard convection. J. Mech. 34, 159–166.CrossRefGoogle Scholar
Benavides, S.J. & Alexakis, A. 2017 Critical transitions in thin layer turbulence. J. Fluid Mech. 822, 364–385.CrossRefGoogle Scholar
Bouchet, F. & Simonnet, E. 2009 Random changes of flow topology in two-dimensional and geophysical turbulence. Phys. Rev. Lett. 102, 094504.CrossRefGoogle ScholarPubMed
Brent, A.D., Voller, V.R. & Reid, K.J. 1988 Enthalpy-porosity technique for modeling convection-diffusion phase change: application to the melting of a pure metal. Numer. Heat Transfer 13, 297–318.CrossRefGoogle Scholar
Brown, E. & Ahlers, G. 2006 Rotations and cessations of the large-scale circulation in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 568, 351–386.CrossRefGoogle Scholar
Brown, E., Nikolaenko, A. & Ahlers, G. 2005 Reorientation of the large-scale circulation in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 95, 084503.CrossRefGoogle ScholarPubMed
Castillo-Castellanos, A., Sergent, A., Podvin, B. & Rossi, M. 2019 Cessation and reversals of large-scale structures in square Rayleigh–Bénard cells. J. Fluid Mech. 877, 922–954.CrossRefGoogle Scholar
Cattaneo, F., Emonet, T. & Weiss, N. 2003 On the interaction between convection and magnetic fields. Astrophys. J. 588, 1183–1198.CrossRefGoogle Scholar
Chandra, M. & Verma, M.K. 2011 Dynamics and symmetries of flow reversals in turbulent convection. Phys. Rev. E 83, 067303.CrossRefGoogle ScholarPubMed
Chen, X., Huang, S.-D., Xia, K.-Q. & Xi, H.-D. 2019 Emergence of substructures inside the large-scale circulation induces transition in flow reversals in turbulent thermal convection. J. Fluid Mech. 877, R1.CrossRefGoogle Scholar
Chen, X.-Y., Xie, Y.-C., Yang, J.-C. & Ni, M.-J. 2023 Strong coupling of flow structure and heat transport in liquid metal thermal convection. J. Fluid Mech. 975, A21.CrossRefGoogle Scholar
Chillà, F. & Schumacher, J. 2012 New perspectives in turbulent Rayleigh–Bénard convection. Eur. Phys. J. E 35, 1–25.CrossRefGoogle ScholarPubMed
Chong, K.L. & Xia, K.-Q. 2016 Exploring the severely confined regime in Rayleigh–Bénard convection. J. Fluid Mech. 805, R4.CrossRefGoogle Scholar
Ciliberto, S., Cioni, S. & Laroche, C. 1996 Large-scale flow properties of turbulent thermal convection. Phys. Rev. E 54, R5901.CrossRefGoogle ScholarPubMed
Cisneros, L.H., Kessler, J.O., Ganguly, S. & Goldstein, R.E. 2011 Dynamics of swimming bacteria: transition to directional order at high concentration. Phys. Rev. E 83, 061907.CrossRefGoogle ScholarPubMed
Cortet, P.-P., Chiffaudel, A., Daviaud, F. & Dubrulle, B. 2010 Experimental evidence of a phase transition in a closed turbulent flow. Phys. Rev. Lett. 105, 214501.CrossRefGoogle Scholar
Favier, B., Guervilly, C. & Knobloch, E. 2019 Subcritical turbulent condensate in rapidly rotating Rayleigh–Bénard convection. J. Fluid Mech. 864, R1.CrossRefGoogle Scholar
Funfschilling, D., Brown, E. & Ahlers, G. 2008 Torsional oscillations of the large-scale circulation in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 607, 119–139.CrossRefGoogle Scholar
Grossmann, G. & Lohse, D. 2000 Scaling in thermal convection: a unifying theory. J. Fluid Mech. 407, 27–56.CrossRefGoogle Scholar
Grossmann, S. & Lohse, D. 2001 Thermal convection for large Prandtl numbers. Phys. Rev. Lett. 86, 3316.CrossRefGoogle ScholarPubMed
Hartmann, D.L., Moy, L.A. & Fu, Q. 2001 Tropical convection and the energy balance at the top of the atmosphere. J. Clim. 14, 4495–4511.2.0.CO;2>CrossRefGoogle Scholar
He, J.-C, Bao, Y. & Chen, X. 2023 Scaling transition of thermal dissipation in turbulent convection. Phys. Fluids 35, 015126.CrossRefGoogle Scholar
He, J.-C., Fang, M.-W., Gao, Z.-Y., Huang, S.-D. & Bao, Y. 2021 Effects of Prandtl number in two-dimentional turbulent convection. Chin. Phys. B 30, 094701.CrossRefGoogle Scholar
Huang, S.-D. & Xia, K.-Q. 2016 Effects of geometric confinement in quasi-2-D turbulent Rayleigh–Bénard convection. J. Fluid Mech. 794, 639–654.CrossRefGoogle Scholar
Huang, Y.-X. & Zhou, Q. 2013 Counter-gradient heat transport in two-dimensional turbulent Rayleigh–Bénard convection. J. Fluid Mech. 737, R3.CrossRefGoogle Scholar
Huisman, S.G., van der Veen, R.C.A., Sun, C. & Lohse, D. 2014 Multiple states in highly turbulent Taylor–Couette flow. Nat. Commun. 5, 3820.CrossRefGoogle ScholarPubMed
van Kan, A. & Alexakis, A. 2020 Critical transition in fast-rotating turbulence within highly elongated domains. J. Fluid Mech. 899, A33.CrossRefGoogle Scholar
van Kan, A., Nemoto, T. & Alexakis, A. 2019 Rare transitions to thin-layer turbulent condensates. J. Fluid Mech. 878, 356–369.CrossRefGoogle Scholar
Kraichnan, R.H. 1967 Inertial ranges in two-dimensional turbulence. Phys. Fluids 10, 1417–1423.CrossRefGoogle Scholar
Labarre, V., Fauve, S. & Chibbaro, S. 2023 Heat-flux fluctuations revealing regime transitions in Rayleigh–Bénard convection. Phys. Rev. Fluids 8, 053501.CrossRefGoogle Scholar
Landau, L.D. & Lifshitz, E.M. 1987 Fluid Mechanics. Pergamon.Google Scholar
Li, X.-M., He, J.-D., Tian, Y., Hao, P. & Huang, S.-D. 2021 Effects of Prandtl number in quasi-two-dimensional Rayleigh–Bénard convection. J. Fluid Mech. 915, A60.CrossRefGoogle Scholar
Li, Y.-Z., Chen, X., Xu, A. & Xi, H.-D. 2022 Counter-flow orbiting of the vortex centre in turbulent thermal convection. J. Fluid Mech. 935, A19.CrossRefGoogle Scholar
Lindborg, E. 2023 Reynolds-number scaling and convergence time scale in two-dimensional Rayleigh–Bénard convection. J. Fluid Mech. 973, A9.CrossRefGoogle Scholar
Linkmann, M., Hohmann, M. & Eckhardt, B. 2020 Non-universal transitions to two-dimensional turbulence. J. Fluid Mech. 892, A18.CrossRefGoogle Scholar
Lohse, D. & Toschi, F. 2003 Ultimate state of thermal convection. Phys. Rev. Lett. 90, 034502.CrossRefGoogle ScholarPubMed
Lohse, D. & Xia, K.-Q. 2010 Small-scale properties of turbulent Rayleigh–Bénard convection. Annu. Rev. Fluid Mech. 42, 335–364.CrossRefGoogle Scholar
Musacchio, S. & Boffetta, G. 2019 Condensate in quasi-two-dimensional turbulence. Phys. Rev. Fluids 4, 022602.CrossRefGoogle Scholar
Pandey, A. 2021 Thermal boundary layer structure in low-Prandtl-number turbulent convection. J. Fluid Mech. 910, A13.CrossRefGoogle Scholar
Pathria, R.K. & Beale, P.D. 2011 Statistical Mechanics. Elsevier.Google Scholar
Peng, Y., Liu, Z.Y. & Cheng, X. 2021 Imaging the emergence of bacterial turbulence: phase diagram and transition kinetics. Sci. Adv. 7, eabd1240.CrossRefGoogle ScholarPubMed
Podvin, B. & Sergent, A. 2015 A large-scale investigation of wind reversal in a square Rayleigh–Bénard cell. J. Fluid Mech. 766, 172–201.CrossRefGoogle Scholar
van der Poel, E.P., Ostilla-Mónico, R., Verzicco, R., Grossmann, S. & Lohse, D. 2015 Logarithmic mean temperature profiles and their connection to plume emissions in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 115, 154501.CrossRefGoogle ScholarPubMed
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, 045303.CrossRefGoogle ScholarPubMed
van der Poel, E.P., Stevens, R.J.A.M. & Lohse, D. 2013 Comparison between two- and three-dimensional Rayleigh–Bénard convection. J. Fluid Mech. 736, 177–194.CrossRefGoogle Scholar
van der Poel, E.P., Stevens, R.J.A.M., Sugiyama, K. & Lohse, D. 2012 Flow states in two-dimensional Rayleigh–Bénard convection as a function of aspect-ratio and Rayleigh number. Phys. Fluids 24, 085104.CrossRefGoogle Scholar
Qiu, X.-L., Yao, S.H. & Tong, P. 2000 Large-scale coherent rotation and oscillation in turbulent thermal convection. Phys. Rev. E 61, R6075–R6078.CrossRefGoogle ScholarPubMed
Ren, L., Tao, X., Zhang, L., Ni, M.-J., Xia, K.-Q. & Xie, Y.-C. 2022 Flow states and heat transport in liquid metal convection. J. Fluid Mech. 951, R1.CrossRefGoogle Scholar
Samuel, R. & Verma, M.K. 2024 Bolgiano–Obukhov scaling in two-dimensional Rayleigh–Bénard convection at extreme Rayleigh numbers. Phys. Rev. Fluids 9, 023502.CrossRefGoogle Scholar
Schmalzl, J., Breuer, M. & Hansen, U. 2004 On the validity of two-dimensional numerical approaches to time-dependent thermal convection. Europhys. Lett. 67, 390.CrossRefGoogle Scholar
Seshasayanan, K. & Alexakis, A. 2018 Condensates in rotating turbulent flows. J. Fluid Mech. 841, 434–462.CrossRefGoogle Scholar
Seshasayanan, K., Benavides, S.J. & Alexakis, A. 2014 On the edge of an inverse cascade. Phys. Rev. E 90, 051003.CrossRefGoogle ScholarPubMed
Shishkina, O., Stevens, R.J.A.M., Grossmann, S. & Lohse, D. 2010 Boundary layer structure in turbulent thermal convection and its consequences for required numerical resolution. New J. Phys. 12, 075022.CrossRefGoogle Scholar
Stevens, R.J.A.M., van der Poel, E.P, Grossmann, S. & Lohse, D. 2013 The unifying theory of scaling in thermal convection: the updated prefactors. J. Fluid Mech. 730, 295–308.CrossRefGoogle Scholar
Sugiyama, K., Calzavarini, E., Grossmann, S. & Lohse, D. 2009 Flow organization in two-dimensional non-Oberbeck–Boussinesq Rayleigh–Bénard convection in water. J. Fluid Mech. 637, 105–135.CrossRefGoogle Scholar
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, 034503.CrossRefGoogle ScholarPubMed
Vogt, T., Horn, S., Grannan, A.M. & Aurnou, J.M. 2018 Jump rope vortex in liquid metal convection. Proc. Natl Acad. Sci. USA 115, 12674–12679.CrossRefGoogle ScholarPubMed
Wang, Q., Verzicco, R., Lohse, D. & Shishkina, O. 2020 Multiple states in turbulent large-aspect-ratio thermal convection: what determines the number of convection rolls. Phys. Rev. Lett. 125, 074501.CrossRefGoogle ScholarPubMed
Wang, Q., Xia, S.-N., Wang, B.-F., Sun, D.-J., Zhou, Q. & Wan, Z.-H. 2018 Flow reversals in two-dimensional thermal convection in tilted cells. J. Fluid Mech. 849, 355–372.CrossRefGoogle Scholar
Weeks, E.R., Tian, Y., Urbach, J.S., Ide, K., Swinney, H.L. & Ghil, M. 1997 Transitions between blocked and zonal flows in a rotating annulus with topography. Science 278, 1598–1601.CrossRefGoogle Scholar
Wei, P. 2021 The persistence of large-scale circulation in Rayleigh–Bénard convection. J. Fluid Mech. 924, A28.CrossRefGoogle Scholar
Weiss, S. & Ahlers, G. 2011 Turbulent Rayleigh–Bénard convection in a cylindrical container with aspect ratio $\varGamma =0.50$ and Prandtl number $Pr=4.38$. J. Fluid Mech. 676, 5–40.CrossRefGoogle Scholar
de Wit, X.M., Aguirre Guzmán, A.J., Clercx, H.J.H. & Kunnen, R.P.J. 2022 Discontinuous transitions towards vortex condensates in buoyancy-driven rotating turbulence. J. Fluid Mech. 936, A43.CrossRefGoogle Scholar
Xi, H.-D., Lam, S. & Xia, K.-Q. 2004 From laminar plumes to organized flows: the onset of large-scale circulation in turbulent thermal convection. J. Fluid Mech. 503, 47–56.CrossRefGoogle Scholar
Xi, H.-D. & Xia, K.-Q. 2007 Cessations and reversals of the large-scale circulation in turbulent thermal convection. Phys. Rev. E 75, 066307.CrossRefGoogle ScholarPubMed
Xi, H.-D. & Xia, K.-Q. 2008 Flow mode transitions in turbulent thermal convection. Phys. Fluids 20, 055104.CrossRefGoogle Scholar
Xi, H.-D, Zhou, S.-Q., Zhou, Q., Chan, T.-S. & Xia, K.-Q. 2009 Origin of the temperature oscillation in turbulent thermal convection. Phys. Rev. Lett. 102, 044503.CrossRefGoogle ScholarPubMed
Xia, H., Byrne, D., Falkovich, G. & Shats, M. 2011 Upscale energy transfer in thick turbulent fluid layers. Nat. Phys. 7, 321–324.CrossRefGoogle Scholar
Xia, K.-Q. 2013 Current trends and future directions in turbulent thermal convection. Theor. Appl. Mech. Lett. 3, 052001.CrossRefGoogle Scholar
Xia, K.-Q., Huang, S.-D., Xie, Y.-C. & Zhang, L. 2023 Tuning heat transport via coherent structure manipulation: recent advances in thermal turbulence. Natl Sci. Rev. 10, nwad012.CrossRefGoogle ScholarPubMed
Xie, J.-H. & Huang, S.-D. 2022 Bolgiano–Obukhov scaling in two-dimensional isotropic convection. J. Fluid Mech. 942, A19.CrossRefGoogle Scholar
Xie, Y.-C., Ding, G.-Y. & Xia, K.-Q. 2018 Flow topology transition via global bifurcation in thermally driven turbulence. Phys. Rev. Lett. 120, 214501.CrossRefGoogle ScholarPubMed
Xie, Y.-C., Wei, P. & Xia, K.-Q. 2013 Dynamics of the large-scale circulation in high-Prandtl-number turbulent thermal convection. J. Fluid Mech. 717, 322–346.CrossRefGoogle Scholar
Xu, A., Chen, X. & Xi, H.-D. 2021 Tristable flow states and reversal of the large-scale circulation in two-dimensional circular convection cells. J. Fluid Mech. 910, A33.CrossRefGoogle Scholar
Yanagisawa, T., Yamagishi, Y., Hamano, Y., Tasaka, Y., Yoshida, M., Yano, K. & Takeda, Y. 2010 Structure of large-scale flows and their oscillation in the thermal convection of liquid gallium. Phys. Rev. E 82, 016320.CrossRefGoogle ScholarPubMed
Yang, Y., Chen, W., Verzicco, R. & Lohse, D. 2020 Multiple states and transport properties of double-diffusive convection turbulence. Proc. Natl Acad. Sci. USA 117, 14676–14681.CrossRefGoogle ScholarPubMed
Zhang, Y. & Zhou, Q. 2024 Low-Prandtl-number effects on global and local statistics in two-dimensional Rayleigh–Bénard convection. Phys. Fluids 36, 015107.CrossRefGoogle Scholar
Zhang, Y., Zhou, Q. & Sun, C. 2017 Statistics of kinetic and thermal energy dissipation rates in two-dimensional turbulent Rayleigh–Bénard convection. J. Fluid Mech. 814, 165–184.CrossRefGoogle Scholar
Zhou, Q., Stevens, R.J.A.M., Sugiyama, K., Grossmann, S., Lohse, D. & Xia, K.-Q. 2010 Prandtl–Blasius temperature and velocity boundary-layer profiles in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 664, 297–312.CrossRefGoogle Scholar
Zhou, Q., Xi, H.-D., Zhou, S.-Q., Sun, C. & Xia, K.-Q. 2009 Oscillations of the large-scale circulation in turbulent Rayleigh–Bénard convection: the sloshing mode and its relationship with the torsional mode. J. Fluid Mech. 630, 367–390.CrossRefGoogle Scholar
Zhu, X., Mathai, V., Stevens, R.J.A.M., Verzicco, R. & Lohse, D. 2018 Transition to the ultimate regime in two-dimensional Rayleigh–Bénard convection. Phys. Rev. Lett. 120, 144502.CrossRefGoogle Scholar
Zwirner, L., Tilgner, A. & Shishkina, O. 2020 Elliptical instability and multiple-roll flow modes of the large-scale circulation in confined turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 125, 054502.CrossRefGoogle ScholarPubMed
Supplementary material: File

Gao et al. supplementary movie 1

Movie of the condensed flow state for Pr = 0.25, Ra = 2 × 107, depicting by the temperature field superimposed with the velocity vectors.
Download Gao et al. supplementary movie 1(File)
File 2.7 MB
Supplementary material: File

Gao et al. supplementary movie 2

Movie of the condensed flow state for Pr = 0.7, Ra = 1 × 108, depicting by the temperature field superimposed with the velocity vectors.
Download Gao et al. supplementary movie 2(File)
File 2.6 MB
Supplementary material: File

Gao et al. supplementary movie 3

Movie of the condensed flow state for Pr = 4.3, Ra = 1 × 109, depicting by the temperature field superimposed with the velocity vectors.
Download Gao et al. supplementary movie 3(File)
File 3 MB
Supplementary material: File

Gao et al. supplementary movie 4

Movie of the condensed flow state for Pr = 20, Ra = 1 × 1010, depicting by the temperature field superimposed with the velocity vectors.
Download Gao et al. supplementary movie 4(File)
File 4 MB
Supplementary material: File

Gao et al. supplementary movie 5

Movie of the uncondensed flow state for Pr = 0.25, Ra = 8 × 108, depicting by the temperature field superimposed with the velocity vectors.
Download Gao et al. supplementary movie 5(File)
File 7.9 MB
Supplementary material: File

Gao et al. supplementary movie 6

Movie of the uncondensed flow state for Pr = 0.7, Ra = 5 × 109, depicting by the temperature field superimposed with the velocity vectors.
Download Gao et al. supplementary movie 6(File)
File 7.8 MB
Supplementary material: File

Gao et al. supplementary movie 7

Movie of the uncondensed flow state for Pr = 4.3, Ra = 1 × 1011, depicting by the temperature field superimposed with the velocity vectors.
Download Gao et al. supplementary movie 7(File)
File 7 MB
Supplementary material: File

Gao et al. supplementary movie 8

Movie of the uncondensed flow state for Pr = 20, Ra = 2 × 1011, depicting by the temperature field superimposed with the velocity vectors.
Download Gao et al. supplementary movie 8(File)
File 8.3 MB
Supplementary material: File

Gao et al. supplementary movie 9

Movie shows the extraction of the vortex center of the largest roll for Pr = 4.3, Ra = 1 × 1011. By applying a threshold ω < −8 to the original vorticity distribution (left), the strong clockwise-rotating vortices, including the largest roll, have been identified (middle), sorted and color-labeled (right). The star indicates the identified vortex center (centroid) of the largest roll.
Download Gao et al. supplementary movie 9(File)
File 9 MB
Supplementary material: File

Gao et al. supplementary movie 10

Movie of a spontaneous switch from the condensed to uncondensed state around t ≈ 16050 for Pr = 4.3, Ra = 7.5 × 109, reflecting the bi-stability behaviour the system exhibits.
Download Gao et al. supplementary movie 10(File)
File 8.6 MB