Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-26T11:14:17.709Z Has data issue: false hasContentIssue false
  • English
  • Français

Turbulent Rayleigh–Bénard convection in a cylindrical container with aspect ratio Γ = 0.50 and Prandtl number Pr = 4.38

Published online by Cambridge University Press:  18 February 2011

STEPHAN WEISS  and
GUENTER AHLERS
STEPHAN WEISS
Affiliation:
Department of Physics, University of California, Santa Barbara, CA 93106, USA
GUENTER AHLERS*
Affiliation:
Department of Physics, University of California, Santa Barbara, CA 93106, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Measurements of the Nusselt number and properties of the large-scale circulation (LSC) are presented for turbulent Rayleigh–Bénard convection in water-filled cylindrical containers (Prandtl number Pr = 4.38) with aspect ratio Γ = 0.50. They cover the range 2 × 108 ≲ Ra ≲ 1 × 1011 of the Rayleigh number Ra. We confirm the occurrence of a double-roll state (DRS) of the LSC and focus on the statistics of the transitions between the DRS and a single-roll state (SRS). The fraction of the run time when the SRS existed varied continuously from about 0.12 near Ra = 2 × 108 to about 0.8 near Ra = 1011, while the fraction of the run time when the DRS could be detected changed from about 0.4 to about 0.06 over the same range of Ra. We determined separately the Nusselt number of the SRS and the DRS, and found the former to be larger than the latter by about 1.6% (0.9%) at Ra = 1010 (1011). We report a contribution to the dynamics of the SRS from a torsional oscillation similar to that observed for cylindrical samples with Γ = 1.00. Results for a number of statistical properties of the SRS are reported, and some are compared with the cases Γ = 0.50, Pr = 0.67 and Γ = 1.00, Pr = 4.38. We found that genuine cessations of the SRS were extremely rare and occurred only about 0.3 times per day, which is less frequent than for Γ = 1.00; however, the SRS was disrupted frequently by roll-state transitions and other less well-defined events. We show that the time derivative of the LSC plane orientation is a stochastic variable which, at constant LSC amplitude, is Gaussian distributed. Within the context of the LSC model of Brown & Ahlers (Phys. Fluids, vol. 20, 2008b, art. 075101), this demonstrates that the stochastic force due to the small-scale fluctuations that is driving the LSC dynamics has a Gaussian distribution.

JFM classification

Turbulent Flows: Turbulent convection
Type
Papers
Information
Journal of Fluid Mechanics , Volume 676 , 10 June 2011 , pp. 5 - 40
Copyright
Copyright © Cambridge University Press 2011

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

Ahlers, G. 2009 Turbulent convection. Physics 2, 74.CrossRefGoogle Scholar
Ahlers, G., Bodenschatz, E., Funfschilling, D. & Hogg, J. 2009 a Turbulent Rayleigh–Bénard convection for a Prandtl number of 0.67. J. Fluid Mech. 641, 157–167.CrossRefGoogle Scholar
Ahlers, G., Brown, E., Fontenele Araujo, F., Funfschilling, D., Grossmann, S. & Lohse, D. 2006 a Non-Oberbeck–Boussinesq effects in strongly turbulent Rayleigh–Bénard convection. J. Fluid Mech. 569, 409–445.CrossRefGoogle Scholar
Ahlers, G., Brown, E. & Nikolaenko, A. 2006 b The search for slow transients, and the effect of imperfect vertical alignment, in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 557, 347–367.CrossRefGoogle Scholar
Ahlers, G., Calzavarini, E., Fontenele Araujo, F., Funfschilling, D., Grossmann, S., Lohse, D. & Sugiyama, K. 2008 Non-Oberbeck–Boussinesq effects in turbulent thermal convection in ethane close to the critical point. Phys. Rev. E 77, 046302 (1–16).CrossRefGoogle Scholar
Ahlers, G., Cannell, D. S., Berge, L. I. & Sakurai, S. 1994 Thermal conductivity of the nematic liquid crystal 4-n-pentyl-4'-cyanobiphenyl. Phys. Rev. E 49, 545–553.CrossRefGoogle ScholarPubMed
Ahlers, G., Grossmann, S. & Lohse, D. 2002 Hochpräzision im Kochtopf: Neues zur turbulenten Konvektion. Phys. J. 1 (2), 31–37.Google Scholar
Ahlers, G., Grossmann, S. & Lohse, D. 2009 b Heat transfer and large scale dynamics in turbulent Rayleigh–Bénard convection. Rev. Mod. Phys. 81, 503–538.CrossRefGoogle Scholar
Boussinesq, J. 1903 Theorie Analytique de la Chaleur, vol. 2. Gauthier-Villars.Google Scholar
Brown, E. & Ahlers, G. 2006 a Effect of the Earth's Coriolis force on turbulent Rayleigh–Bénard convection in the laboratory. Phys. Fluids 18, 125108 (1–15).CrossRefGoogle Scholar
Brown, E. & Ahlers, G. 2006 b Rotations and cessations of the large-scale circulation in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 568, 351–386.CrossRefGoogle Scholar
Brown, E. & Ahlers, G. 2007 a Large-scale circulation model of turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 98, 134501 (1–4).CrossRefGoogle ScholarPubMed
Brown, E. & Ahlers, G. 2007 b Temperature gradients, and search for non-Boussinesq effects, in the interior of turbulent Rayleigh–Bénard convection. Europhys. Lett. 80, 14001 (1–6).CrossRefGoogle Scholar
Brown, E. & Ahlers, G. 2008 a Azimuthal asymmetries of the large-scale circulation in turbulent Rayleigh–Bénard convection. Phys. Fluids 20, 105105 (1–15).CrossRefGoogle Scholar
Brown, E. & Ahlers, G. 2008 b A model of diffusion in a potential well for the dynamics of the large-scale circulation in turbulent Rayleigh–Bénard convection. Phys. Fluids 20, 075101 (1–16).CrossRefGoogle Scholar
Brown, E. & Ahlers, G. 2009 The origin of oscillations of the large-scale circulation of turbulent Rayleigh–Bénard convection. J. Fluid Mech. 638, 383–400.CrossRefGoogle Scholar
Brown, E., Nikolaenko, A. & Ahlers, G. 2005 a Reorientation of the large-scale circulation in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 95, 084503 (1–4).CrossRefGoogle ScholarPubMed
Brown, E., Nikolaenko, A., Funfschilling, D. & Ahlers, G. 2005 b Heat transport by turbulent Rayleigh–Bénard convection: effect of finite top- and bottom-plate conductivity. Phys. Fluids 17, 075108 (1–10).CrossRefGoogle Scholar
Busse, F. H. 1994 Convection-driven zonal flows and vortices in the major planets. Chaos 4, 123–134.CrossRefGoogle ScholarPubMed
Cardin, P. & Olson, P. 1994 Chaotic thermal convection in a rapidly rotating spherical shell: consequences for flow in the outer core. Phys. Earth Planet. Inter. 82, 235–259.CrossRefGoogle Scholar
Cattaneo, F., Emonet, T. & Weiss, N. 2003 On the interaction between convection and magnetic fields. Astrophys. J. 588, 1183–1198.CrossRefGoogle Scholar
Chillà, F., Rastello, M., Chaumat, S. & Castaing, B. 2004 Long relaxation times and tilt sensitivity in Rayleigh–Bénard turbulence. Eur. Phys. J. B 40, 223–227.CrossRefGoogle Scholar
Cioni, S., Ciliberto, S. & Sommeria, J. 1997 Strongly turbulent Rayleigh–Bénard convection in mercury: comparison with results at moderate Prandtl number. J. Fluid Mech. 335, 111–140.CrossRefGoogle Scholar
van Doorn, E., Dhruva, B., Sreenivasan, K. R. & Cassella, V. 2000 Statistics of wind direction and its increments. Phys. Fluids 12, 1529–1534.CrossRefGoogle Scholar
Funfschiling, D., Brown, E., Nikolaenko, A. & Ahlers, G. 2005 Heat transport by turbulent Rayleigh–Bénard convection in cylindrical cells with aspect ratio one and larger. J. Fluid Mech. 536, 145–154.CrossRefGoogle Scholar
Funfschilling, D. & Ahlers, G. 2004 Plume motion and large-scale circulation in a cylindrical Rayleigh–Bénard cell. Phys. Rev. Lett. 92, 194502 (1–4).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
Glatzmaier, G., Coe, R., Hongre, L. & Roberts, P. 1999 The role of the Earth's mantle in controlling the frequency of geomagnetic reversals. Nature (London) 401, 885–890.CrossRefGoogle Scholar
Grossmann, S. & Lohse, D. 2000 Scaling in thermal convection: a unifying view. J. Fluid. Mech. 407, 27–56.CrossRefGoogle Scholar
Grossmann, S. & Lohse, D. 2001 Thermal convection for large Prandtl number. Phys. Rev. Lett. 86, 3316–3319.CrossRefGoogle Scholar
Hartmann, D. L., Moy, L. A. & Fu, Q. 2001 Tropical convection and the energy balance at the top of the atmosphere. J. Climate 14, 4495–4511.2.0.CO;2>CrossRefGoogle Scholar
Kadanoff, L. P. 2001 Turbulent heat flow: structures and scaling. Phys. Today 54 (8), 34–39.CrossRefGoogle Scholar
Lohse, D. & Xia, K.-Q. 2010 Small-scale properties of turbulent Rayleigh–Bénard convection. Annu. Rev. Fluid Mech. 42, 335–364.CrossRefGoogle Scholar
Marshall, J. & Schott, F. 1999 Open-ocean convection: observations, theory, and models. Rev. Geophys. 37, 1–64.CrossRefGoogle Scholar
Nikolaenko, A., Brown, E., Funfschilling, D. & Ahlers, G. 2005 Heat transport by turbulent Rayleigh–Bénard convection in cylindrical cells with aspect ratio one and less. J. Fluid Mech. 523, 251–260.CrossRefGoogle Scholar
Oberbeck, A. 1879 über die Wärmeleitung der Flüssigkeiten bei Berücksichtigung der Strömungen infolge von Temperaturdifferenzen. Ann. Phys. Chem. 7, 271–292.CrossRefGoogle Scholar
Rahmstorf, S. 2000 The thermohaline ocean circulation: a system with dangerous thresholds? Clim. Change 46, 247–256.CrossRefGoogle Scholar
Roche, P. E., Castaing, B., Chabaud, B. & Hebral, B. 2004 Heat transfer in turbulent Rayleigh–Bénard convection below the ultimate regime. J. Low. Temp. Phys. 134, 1011–1042.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., Rui, N., Stevens, R., Chan, T., 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 (1–4).CrossRefGoogle ScholarPubMed
Sun, C., Ren, L.-Y., Song, H. & Xia, K.-Q. 2005 a Heat transport by turbulent Rayleigh–Bénard convection in cylindrical cells of widely varying aspect ratios. J. Fluid Mech. 542, 165–174.CrossRefGoogle Scholar
Sun, C., Xi, H. D. & Xia, K. Q. 2005 b Azimuthal symmetry, flow dynamics, and heat transport in turbulent thermal convection in a cylinder with an aspect ratio of 0.5. Phys. Rev. Lett. 95, 074502 (1–4).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 (1–5).CrossRefGoogle ScholarPubMed
Xi, H.-D. & Xia, K.-Q. 2008 a Azimuthal motion, reorientation, cessation, and reversals of the large-scale circulation in turbulent thermal convection: a comparative study in aspect ratio one and one-half geometries. Phys. Rev. E 78, 036326 (1–11).CrossRefGoogle ScholarPubMed
Xi, H.-D. & Xia, K.-Q. 2008 b Flow mode transition in turbulent thermal convection. Phys. Fluids 20, 055104 (1–15).CrossRefGoogle Scholar
Xi, H. D., Zhou, Q. & Xia, K. Q. 2006 Azimuthal motion of the mean wind in turbulent thermal convection. Phys. Rev. E 73, 056312 (1–13).CrossRefGoogle ScholarPubMed
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 (1–4).CrossRefGoogle ScholarPubMed
Zhong, J.-Q. & Ahlers, G. 2010 Heat transport and the large-scale circulation in rotating turbulent Rayleigh–Bénard convection. J. Fluid Mech. 665, 300–333.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