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Aspect ratio dependence of heat transfer and large-scale flow in turbulent convection

Published online by Cambridge University Press:  12 May 2010

J. BAILON-CUBA ,
M. S. EMRAN  and
J. SCHUMACHER
J. BAILON-CUBA
Affiliation:
Institut für Thermo- und Fluiddynamik, Technische Universität Ilmenau, Postfach 100565, D-98684 Ilmenau, Germany
M. S. EMRAN*
Affiliation:
Institut für Thermo- und Fluiddynamik, Technische Universität Ilmenau, Postfach 100565, D-98684 Ilmenau, Germany
J. SCHUMACHER
Affiliation:
Institut für Thermo- und Fluiddynamik, Technische Universität Ilmenau, Postfach 100565, D-98684 Ilmenau, Germany
*
†Email address for correspondence: mohammad.emran@tu-ilmenau.de
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Abstract

The heat transport and corresponding changes in the large-scale circulation (LSC) in turbulent Rayleigh–Bénard convection are studied by means of three-dimensional direct numerical simulations as a function of the aspect ratio Γ of a closed cylindrical cell and the Rayleigh number Ra. The Prandtl number is Pr = 0.7 throughout the study. The aspect ratio Γ is varied between 0.5 and 12 for a Rayleigh number range between 107 and 109. The Nusselt number Nu is the dimensionless measure of the global turbulent heat transfer. For small and moderate aspect ratios, the global heat transfer law Nu = A × Raβ shows a power law dependence of both fit coefficients A and β on the aspect ratio. A minimum of Nu(Γ) is found at Γ ≈ 2.5 and Γ ≈ 2.25 for Ra = 107 and Ra = 108, respectively. This is the point where the LSC undergoes a transition from a single-roll to a double-roll pattern. With increasing aspect ratio, we detect complex multi-roll LSC configurations in the convection cell. For larger aspect ratios Γ ≳ 8, our data indicate that the heat transfer becomes independent of the aspect ratio of the cylindrical cell. The aspect ratio dependence of the turbulent heat transfer for small and moderate Γ is in line with a varying amount of energy contained in the LSC, as quantified by the Karhunen–Loève or proper orthogonal decomposition (POD) analysis of the turbulent convection field. The POD analysis is conducted here by the snapshot method for at least 100 independent realizations of the turbulent fields. The primary POD mode, which replicates the time-averaged LSC patterns, transports about 50% of the global heat for Γ ≥ 1. The snapshot analysis enables a systematic disentanglement of the contributions of POD modes to the global turbulent heat transfer. Although the smallest scale – the Kolmogorov scale ηK – and the largest scale – the cell height H – are widely separated in a turbulent flow field, the LSC patterns in fully turbulent fields exhibit strikingly similar texture to those in the weakly nonlinear regime right above the onset of convection. Pentagonal or hexagonal circulation cells are observed preferentially if the aspect ratio is sufficiently large (Γ ≳ 8).

Type
Papers
Information
Journal of Fluid Mechanics , Volume 655 , 25 July 2010 , pp. 152 - 173
Copyright
Copyright © Cambridge University Press 2010

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References

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
Amati, G., Koal, K., Massaioli, F., Sreenivasan, K. R. & Verzicco, R. 2005 Turbulent thermal convection at high Rayleigh numbers for Boussinesq fluid of constant Prandtl number. Phys. Fluids 17, 121701.CrossRefGoogle Scholar
Bodenschatz, E., Pesch, W. & Ahlers, G. 2000 Recent developments in Rayleigh–Bénard convection. Annu. Rev. Fluid Mech. 32, 709–778.CrossRefGoogle Scholar
Brown, E. & Ahlers, G. 2008 Azimuthal asymmetries of the large-scale circulation in turbulent Rayleigh–Bénard convection. Phys. Fluids 20, 105105.CrossRefGoogle Scholar
Bukai, M., Eidelman, A., Elperin, T., Kleeorin, N., Rogachevskii, I. & Sapir-Katiraie, I. 2009 Effect of large-scale coherent structures on turbulent convection. Phys. Rev. E 79, 066302.CrossRefGoogle ScholarPubMed
Busse, F. H. 2003 The sequence-of-bifurcations approach towards understanding turbulent fluid flow. Surv. Geophys. 24, 269–288.CrossRefGoogle Scholar
Busse, F. H. & Whitehead, J. A. 1971 Instabilities of convection rolls in a high Prandtl number fluid. J. Fluid Mech. 47, 305–320.CrossRefGoogle Scholar
Castaing, B., Gunaratne, G., Heslot, F., Kadanoff, L., Libchaber, A., Thomae, S., Wu, X.-Z., Zaleski, S. & Zanetti, G. 1989 Scaling of hard thermal turbulence in Rayleigh–Bénard convection. J. Fluid Mech. 204, 1–30.CrossRefGoogle Scholar
Charlson, G. S. & Sani, R. L. 1970 Thermoconvective instability in a bounded cylindrical fluid layer. Intl J. Heat Mass Transfer 13, 1479–1496.CrossRefGoogle Scholar
Charlson, G. S. & Sani, R. L. 1971 On the thermoconvective instability in a bounded cylindrical fluid layer. Intl J. Heat Mass Transfer 14, 2157–2160.CrossRefGoogle Scholar
Ching, E. S. C. & Tam, W. S. 2006 Aspect-ratio dependence of heat transport by turbulent Rayleigh–Bénard convection. J. Turbul. 7, N 72.CrossRefGoogle Scholar
Clever, R. M. & Busse, F. H. 1989 Three-dimensional knot convection in a layer heated from below. J. Fluid Mech. 198, 345–363.CrossRefGoogle Scholar
Croquette, V. 1989 Convective pattern dynamics at low Prandtl number. Part II. Contemp. Phys. 30, 153–171.CrossRefGoogle Scholar
Emran, M. S. & Schumacher, J. 2008 Fine-scale statistics of temperature and its derivatives in convective turbulence. J. Fluid Mech. 611, 13–34.CrossRefGoogle Scholar
Fitzjarrald, D. E. 1976 An experimental study of turbulent convection in air. J. Fluid Mech. 73, 693–719.CrossRefGoogle Scholar
Funfschilling, D., Brown, E., Nikolaenko, A. & Ahlers, G. 2005 Heat transport by turbulent Rayleigh–Bénard convection in cylindrical samples with aspect ratio one and larger. J. Fluid Mech. 536, 145–154.CrossRefGoogle Scholar
Gao, H., Metcalfe, G., Jung, T. & Behringer, R. P. 1987 Heat-flow experiments in liquid 4He with variable cylindrical geometry. J. Fluid Mech. 174, 209–231.CrossRefGoogle Scholar
Grossmann, S. & Lohse, D. 2000 Scaling in thermal convection: a unifying theory. J. Fluid Mech. 407, 27–56.CrossRefGoogle Scholar
Grossmann, S. & Lohse, D. 2003 On geometry effects in Rayleigh–Bénard convection. J. Fluid Mech. 486, 105–114.CrossRefGoogle Scholar
Grötzbach, G. 1983 Spatial resolution requirements for direct numerical simulation of the Rayleigh–Bénard convection. J. Comput. Phys. 49, 241–269.CrossRefGoogle Scholar
von Hardenberg, J., Parodi, A., Passoni, G., Provenzale, A. & Spiegel, E. A. 2008 Large-scale patterns in Rayleigh–Bénard convection. Phys. Lett. A 372, 2223–2229.CrossRefGoogle Scholar
Hartlep, T., Tilgner, A. & Busse, F. H. 2003 Large scale stuctures in Rayleigh–Bénard convection at high Rayleigh numbers. Phys. Rev. Lett. 91, 064501.CrossRefGoogle Scholar
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
Johnston, H. & Doering, C. R. 2009 A comparison of turbulent thermal convection between conditions of constant temperature and constant flux. Phys. Rev. Lett. 102, 064501.CrossRefGoogle ScholarPubMed
Kerr, R. M. 1996 Rayleigh number scaling in numerical convection. J. Fluid Mech. 310, 139–179.CrossRefGoogle Scholar
Koschmieder, E. 1969 On the wavelength of convective motions. J. Fluid Mech. 35, 527–530.CrossRefGoogle Scholar
La Porta, A., Voth, G. A., Crawford, A. M., Alexander, J. & Bodenschatz, E. 2001 Fluid particle accelerations in fully developed turbulence. Nature 409, 1017–1019.CrossRefGoogle ScholarPubMed
Lathrop, D. P., Fineberg, J. & Swinney, H. L. 1992 Turbulent flow between concentric rotating cylinders at large Reynolds numbers. Phys. Rev. Lett. 68, 1515–1518.CrossRefGoogle Scholar
Niemela, J. J., Skrbek, L., Sreenivasan, K. R. & Donelly, R. J. 2000 Turbulent convection at very high Rayleigh numbers. Nature 404, 837–840.CrossRefGoogle ScholarPubMed
Niemela, J. J. & Sreenivasan, K. R. 2006 Turbulent convection at high Rayleigh numbers and aspect ratio 4. J. Fluid Mech. 557, 411–422.CrossRefGoogle Scholar
Oresta, P., Stringano, G. & Verzicco, R. 2007 Transitional regimes and rotation effects in Rayleigh–Bénard convection in a slender cylindrical cell. Eur. J. Mech. B/Fluids 26, 1–14.CrossRefGoogle Scholar
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
du Puits, R., Resagk, C. & Thess, A. 2007 Breakdown of wind in turbulent thermal convection. Phys. Rev. E 75, 016302.CrossRefGoogle ScholarPubMed
van Reeuwijk, M., Jonker, H. J. J. & Hanjalić, K. 2008 Wind and boundary layers in Rayleigh–Bénard convection. I. Analysis and modelling. Phys. Rev. E 77, 036311.CrossRefGoogle Scholar
Shishkina, O. & Wagner, C. 2006 Analysis of thermal dissipation rates in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 546, 51–60.CrossRefGoogle Scholar
Shishkina, O. & Wagner, C. 2008 Analysis of sheet-like thermal plumes in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 599, 383–404.CrossRefGoogle Scholar
Siggia, E. D. 1994 High Rayleigh number convection. Annu. Rev. Fluid Mech. 26, 137–168.CrossRefGoogle Scholar
Sirovich, L. & Park, H. 1990 Turbulent thermal convection in a finite domain. Part I. Theory. Phys. Fluids A 2, 1649–1668.CrossRefGoogle Scholar
Smith, T. R., Moehlis, J. & Holmes, P. 2005 Low-dimensional modelling of turbulence using Proper Orthogonal Decomposition: a tutorial. Nonlinear Dyn. 41, 275–307.CrossRefGoogle Scholar
Stein, R. F. & Nordlund, A 2006 Solar small-scale magnetoconvection. Astrophys. J. 642, 1246–1255.CrossRefGoogle Scholar
Sun, C., Ren, L.-Y., Song, H. & Xia, K.-Q. 2005 Heat transport by turbulent Rayleigh–Bénard convection in 1 m diameter cylindrical cells of widely varying aspect ratio. J. Fluid Mech. 542, 165–174.CrossRefGoogle Scholar
Strogatz, S. H. 1994 Nonlinear Dynamics and Chaos. Westview Press.Google Scholar
Verzicco, R. & Camussi, R. 2003 Numerical experiments on strongly turbulent thermal convection in a slender cylindrical cell. J. Fluid Mech. 477, 19–49.CrossRefGoogle Scholar
Verzicco, R. & Orlandi, P. 1996 A finite-difference scheme for three-dimensional incompressible flows in cylindrical coordinates. J. Comput. Phys. 123, 402–414.CrossRefGoogle Scholar
Verzicco, R. & Sreenivasan, K. R. 2007 A comparison of turbulent thermal convection between conditions of constant temperature and constant heat flux. J. Fluid Mech. 595, 203–219.CrossRefGoogle Scholar
Wu, X.-Z. & Libchaber, A. 1992 Scaling relation in thermal turbulence: the aspect-ratio dependence. Phys. Rev. A 45, 842–845.CrossRefGoogle ScholarPubMed
Zerihun Desta, T., Van Brecht, A., Quanten, S., Van Buggenhout, S., Meyers, J., Baelmans, M. & Berckmans, D. 2005 Modelling and control of heat transfer phenomena inside a ventilated air space. Energy Build. 37, 777–786.CrossRefGoogle Scholar
Xi, H.-D. & Xia, K.-Q. 2008 a 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, 036326.CrossRefGoogle ScholarPubMed
Xi, H.-D. & Xia, K.-Q. 2008 b Flow mode transitions in turbulent thermal convection. Phys. Fluids 20, 055104.CrossRefGoogle Scholar
Zhou, Q., Sun, C. & Xia, K.-Q. 2007 Morphological evolution of thermal plumes in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 98, 074501.CrossRefGoogle ScholarPubMed