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Reynolds-number measurements for low-Prandtl-number turbulent convection of large-aspect-ratio samples

Published online by Cambridge University Press:  22 May 2013

James Hogg  and
Guenter Ahlers
James Hogg
Affiliation:
Department of Physics and ITST, University of California, Santa Barbara, CA 93106, USA
Guenter Ahlers*
Affiliation:
Department of Physics and ITST, University of California, Santa Barbara, CA 93106, USA
*
†Email address for correspondence: [email protected]
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Abstract

We present experimental results for the Reynolds number ${\mathit{Re}}_{U} $ based on the horizontal mean-flow velocity $U$ and for ${\mathit{Re}}_{V} $ based on the root-mean-square horizontal fluctuation velocity $V$ for turbulent Rayleigh–Bénard convection in a cylindrical sample of aspect ratio $\Gamma = 10. 9$ over the Prandtl number range $0. 18\leq \mathit{Pr}\leq 0. 88$. The results were derived from space–time cross-correlation functions of shadowgraph images, using the elliptic approximation of He & Zhang (Phys. Rev. E, vol. 73, 2006, 055303). The data cover the Rayleigh number range from $3\times 1{0}^{5} $ to $2\times 1{0}^{7} $. We find that ${\mathit{Re}}_{U} $ is nearly two orders of magnitude smaller than the values given by the Grossmann–Lohse (GL) model (Grossmann & Lohse, Phys. Rev. E, vol. 66, 2002, 016305) for $\Gamma = 1. 00$ and attribute this difference to averaging caused by lateral random diffusion of the large-scale circulation cells in large-$\Gamma $ samples. For the fluctuations we found ${\mathit{Re}}_{V} = {\tilde {R} }_{0} {\mathit{Pr}}^{\alpha } {\mathit{Ra}}^{\eta } $, with ${\tilde {R} }_{0} = 0. 31$, $\alpha = - 0. 53\pm 0. 11$ and $\eta = 0. 45\pm 0. 03$. That result agrees well with the GL model. The close agreement of the coefficient ${\tilde {R} }_{0} $ must be regarded as a coincidence because the GL model was for $\Gamma = 1. 00$ and for a mean-flow velocity $U$.

JFM classification

Convection: Convection Turbulent Flows: Turbulent Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 725 , 25 June 2013 , pp. 664 - 680
Copyright
©2013 Cambridge University Press 

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References

Ahlers, G. 2009 Turbulent convection. Physics 2, 74–1–7.CrossRefGoogle Scholar
Ahlers, G., Grossmann, S. & Lohse, D. 2009 Heat transfer and large-scale dynamics in turbulent Rayleigh–Bénard convection. Rev. Mod. Phys. 81, 503–538.Google Scholar
Ahlers, G. & Xu, X. 2001 Prandtl-number dependence of heat transport in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 86, 3320–3323.Google Scholar
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, 152–173.CrossRefGoogle Scholar
Bodenschatz, E., Pesch, W. & Ahlers, G. 2000 Recent developments in Rayleigh–Bénard convection. Annu. Rev. Fluid Mech. 32, 709–778.Google Scholar
Bosbach, J., Weiss, S. & Ahlers, G. 2012 Plume fragmentation by bulk interactions in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 108, 054501.Google Scholar
Boushehri, A., Bzowski, J., Kestin, J. & Mason, E. A. 1987 Equilibrium and transport properties of eleven polyatomic gases at low density. J. Phys. Chem. Ref. Data 16 (3), 445–466.CrossRefGoogle Scholar
Braker, W. & Mossman, A. L. 1974 The Matheson Unabridged Gas Data Book: A Compilation of Physical and Thermodynamic Properties of Gases. Matheson Gas Products.Google Scholar
Brown, E. & Ahlers, G. 2008 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.Google Scholar
Brown, E., Funfschilling, D. & Ahlers, G. 2007 Anomalous Reynolds-number scaling in turbulent Rayleigh–Bénard convection. J. Stat. Mech. 2007, P10005.Google Scholar
Brown, E., Nikolaenko, A. & Ahlers, G. 2005 Reorientation of the large-scale circulation in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 95, 084503.Google Scholar
Chavanne, X., Chilla, F., Chabaud, B., Castaing, B. & Hebral, B. 2001 Turbulent Rayleigh–Bénard convection in gaseous and liquid He. Phys. Fluids 13, 1300–1320.Google Scholar
Cross, M. C. & Hohenberg, P. C. 1993 Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851–1112.Google Scholar
de Bruyn, J. R., Bodenschatz, E., Morris, S. W., Trainoff, S., Hu, Y., Cannell, D. S. & Ahlers, G. 1996 Apparatus for the study of Rayleigh–Bénard convection in gases under pressure. Rev. Sci. Instrum. 67, 2043.Google 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.Google Scholar
Funfschilling, 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.Google Scholar
Gracki, J. A., Flynn, G. P. & Ross, J. 1969 Viscosity of nitrogen, helium, hydrogen, and argon from $- 100$ to 25 C up to 150–250 atm. J. Chem. Phys. 51 (9), 3856–3863.Google Scholar
Grossmann, S. & Lohse, D. 1993 Characteristic scales in Rayleigh–Bénard turbulence. Phys. Lett. A 173, 58–62.Google Scholar
Grossmann, S. & Lohse, D. 2002 Prandtl and Rayleigh number dependence of the Reynolds number in turbulent thermal convection. Phys. Rev. E 66, 016305.CrossRefGoogle ScholarPubMed
Hartlep, T., Tilgner, A. & Busse, F. H. 2005 Transition to turbulent convection in a fluid layer heated from below at moderate aspect ratio. J. Fluid Mech. 544, 309–322.CrossRefGoogle Scholar
He, G.-W. & Zhang, J.-B. 2006 Elliptic model for space–time correlations in turbulent shear flows. Phys. Rev. E 73, 055303.Google Scholar
He, X., Ching, E. & Tong, P. 2011 Locally averaged thermal dissipation rate in turbulent thermal convection: a decomposition into contributions from different temperature gradient components. Phys. Fluids 23, 025106-1–13.Google Scholar
He, X., Funfschilling, D., Nobach, H., Bodenschatz, E. & Ahlers, G. 2012 Transition to the ultimate state of turbulent Rayleigh–Beńard convection. Phys. Rev. Lett. 108, 024502.CrossRefGoogle Scholar
He, X., He, G. & Tong, P. 2010 Small-scale turbulent fluctuations beyond Taylor’s frozen-flow hypothesis. Phys. Rev. E 81, 065303.Google Scholar
He, X. & Tong, P. 2011 Kraichnan’s random sweeping hypothesis in homogeneous turbulent convection. Phys. Rev. E 83, 037302.Google Scholar
Hilsenrath, J., Hoge, H. J., Beckett, C. W., Masi, J. F., Benedict, W. S., Nuttall, R. L., Fano, L., Touloukian, Y. S. & Woodley, H. W. 1960 Tables of Thermodynamic and Transport Properties of Air, Carbon Dioxide, Carbon Monoxide, Hydrogen, Nitrogen, Oxygen, and Steam. Pergamon.Google Scholar
Hoogland, J., van den Berg, H. & Trappeniers, N. 1985 Measurements of the viscosity of sulfur hexafluoride up to 100 bar by a capillary-flow viscometer. Physica A 134, 169–192.Google Scholar
Howard, R. & LaBonte, B. 1980 A search for large-scale convection cells in the solar atmosphere. Astrophys. J. 239, 738–745.Google Scholar
Kestin, J. & Imaishi, N. 1985 Int. J. Thermophys. 6, 107.Google Scholar
Krishnamurti, R. 1970 On the transition to turbulent convection. Part 2. The transition to time-dependent flow. J. Fluid Mech. 42, 309–320.CrossRefGoogle Scholar
Lam, S., Shang, X. D., Zhou, S. Q. & Xia, K.-Q. 2002 Prandtl-number dependence of the viscous boundary layer and the Reynolds-number in Rayleigh–Bénard convection. Phys. Rev. E 65, 066306.Google Scholar
Lappa, M. 2010 Thermal Convection: Patterns, Evolution and Stability. John Wiley & Sons.Google Scholar
Lemmon, E., Huber, M. & McLinden, M. 2007 NIST standard reference database 23: reference fluid thermodynamic and transport properties-REFPROP, version 8.0. National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg.Google Scholar
Liu, L. & Ahlers, G. 1997 Rayleigh–Bénard convection in binary gas mixtures: thermophysical properties and the onset of convection. Phys. Rev. E 55, 6950.CrossRefGoogle Scholar
Lohse, D. & Xia, K.-Q. 2010 Small-scale properties of turbulent Rayleigh–Bénard convection. Annu. Rev. Fluid Mech. 42, 335–364.Google Scholar
Niemela, J. J., Skrbek, L., Sreenivasan, K. R. & Donnelly, R. 2000 Turbulent convection at very high Rayleigh numbers. Nature 404, 837–840.Google 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.Google Scholar
Oda, A., Uematsu, M. & Watanabe, K. 1983 An equation of state for sulfur hexafluoride in the range of temperatures 222 to 500 K up to 50 MPa. Bull. JSME 26 (219), 1590–1596.Google Scholar
Qiu, X. L. & Tong, P. 2001 Large scale velocity structures in turbulent thermal convection. Phys. Rev. E 64, 036304.Google Scholar
Rahmstorf, S. 2000 The thermohaline ocean circulation: a system with dangerous thresholds?. Climate Change 46, 247–256.Google Scholar
Rasenat, S., Hartung, G., Winkler, B. & Rehberg, I. 1989 Exp. Fluids 7, 412–420.CrossRefGoogle Scholar
Sugiyama, K., Calzavarini, E., Grossmann, S. & Lohse, D. 2007 Non-Oberbeck–Boussinesq effects in two-dimensional Rayleigh–Bénard convection in glycerol. Europhys. Lett. 80, 34002.Google Scholar
Trainoff, S. P. & Cannell, D. S. 2002 Physical optics treatment of the shadowgraph. Phys. Fluids 14, 1340–1363.Google Scholar
Uribe, F., Mason, E. & Kestin, J. 1990 Thermal conductivity of nine polyatomic gases at low density. J. Phys. Chem. Ref. Data 19 (5), 1123–1136.Google Scholar
Verzicco, R. & Camussi, R. 1999 Prandtl number effects in convective turbulence. J. Fluid Mech. 383, 55–73.Google Scholar
Weiss, S. & Ahlers, G. 2011 Turbulent Rayleigh–Bénard convection in a cylindrical container with aspect ratio $\Gamma = 0. 50$ and Prandtl number $\mathit{Pr}= 4. 38$ . J. Fluid Mech. 676, 5–40.Google Scholar
Xi, H.-D. & Xia, K.-Q. 2008 Flow mode transition in turbulent thermal convection. Phys. Fluids 20, 055104.Google Scholar
Xia, K.-Q. 2007 Two clocks for a single engine in turbulent convection. J. Stat. Mech. N11001.Google Scholar
Xu, X., Bajaj, K. M. S. & Ahlers, G. 2000 Heat transport in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 84, 4357–4360.Google Scholar
Zhao, X. & He, G.-W. 2009 Space–time correlations of fluctuating velocities in turbulent shear flows. Phys. Rev. E 79, 046316.Google Scholar
Zhong, J.-Q., Funfschilling, D. & Ahlers, G. 2009 Enhanced heat transport by turbulent two-phase Rayleigh–Bénard convection. Phys. Rev. Lett. 102, 124501.Google Scholar
Zhou, Q., Li, C.-M., Lu, Z.-M. & Liu, Y.-L. 2011 Experimental investigation of longitudinal space–time correlations of the velocity field in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 683, 94–111.CrossRefGoogle Scholar

Hogg Supplementary Material

Movie 1 corresponding to Fig. 1a of the main document, running at approximately half of real-time speed. 11 mm square in the center of the cell, with Pr = 0.18, Ra = 1x10^{6}, 11 pixels/mm

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Video 1.2 MB

Hogg Supplementary Material

Movie 2 corresponding to Fig. 1b of the main document, running at approximately real-time speed. 94 mm square, with Pr=0.88, Ra=2x10^{7}, 5 pixels/mm.

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Video 5.1 MB

Hogg Supplementary Material

Movie 3 corresponding to Fig. 7a of the main document, running at approximately real-time speed. It shows processed shadowgraph images corresponding to Fig. 1b for Pr = 0.88 and Ra = 2x10^{7}. Only the dark (relatively warm) structures near the bottom of the sample remain. A 94 mm square was used with a resolution of 5 pixels/mm.

Download Hogg Supplementary Material(Video)
Video 5.2 MB

Hogg Supplementary Material

Movie 4 corresponding to Fig. 7b of the main document, running at approximately real-time speed. It shows processed shadowgraph images corresponding to Fig. 1b for Pr = 0.88 and Ra = 2x10^{7}. Only the light (relatively cold) structures near the top of the sample remain. A 94 mm square was used with a resolution of 5 pixels/mm.

Download Hogg Supplementary Material(Video)
Video 5 MB