Skip to main content
×
Home

The unifying theory of scaling in thermal convection: the updated prefactors

  • Richard J. A. M. Stevens (a1) (a2), Erwin P. van der Poel (a2), Siegfried Grossmann (a3) and Detlef Lohse (a2)
Abstract
Abstract

The unifying theory of scaling in thermal convection (Grossmann & Lohse, J. Fluid. Mech., vol. 407, 2000, pp. 27–56; henceforth the GL theory) suggests that there are no pure power laws for the Nusselt and Reynolds numbers as function of the Rayleigh and Prandtl numbers in the experimentally accessible parameter regime. In Grossmann & Lohse (Phys. Rev. Lett., vol. 86, 2001, pp. 3316–3319) the dimensionless parameters of the theory were fitted to 155 experimental data points by Ahlers & Xu (Phys. Rev. Lett., vol. 86, 2001, pp. 3320–3323) in the regime $3\times 1{0}^{7} \leq \mathit{Ra}\leq 3\times 1{0}^{9} $ and $4\leq \mathit{Pr}\leq 34$ and Grossmann & Lohse (Phys. Rev. E, vol. 66, 2002, p. 016305) used the experimental data point from Qiu & Tong (Phys. Rev. E, vol. 64, 2001, p. 036304) and the fact that $\mathit{Nu}(\mathit{Ra}, \mathit{Pr})$ is independent of the parameter $a$ , which relates the dimensionless kinetic boundary thickness with the square root of the wind Reynolds number, to fix the Reynolds number dependence. Meanwhile the theory is, on the one hand, well-confirmed through various new experiments and numerical simulations; on the other hand, these new data points provide the basis for an updated fit in a much larger parameter space. Here we pick four well-established (and sufficiently distant) $\mathit{Nu}(\mathit{Ra}, \mathit{Pr})$ data points and show that the resulting $\mathit{Nu}(\mathit{Ra}, \mathit{Pr})$ function is in agreement with almost all established experimental and numerical data up to the ultimate regime of thermal convection, whose onset also follows from the theory. One extra $\mathit{Re}(\mathit{Ra}, \mathit{Pr})$ data point is used to fix $\mathit{Re}(\mathit{Ra}, \mathit{Pr})$ . As $\mathit{Re}$ can depend on the definition and the aspect ratio, the transformation properties of the GL equations are discussed in order to show how the GL coefficients can easily be adapted to new Reynolds number data while keeping $\mathit{Nu}(\mathit{Ra}, \mathit{Pr})$ unchanged.

Copyright
Corresponding author
Email addresses for correspondence: r.j.a.m.stevens@utwente.nl, d.lohse@utwente.nl
References
Hide All
Ahlers G. 2000 Effect of sidewall conductance on heat-transport measurements for turbulent Rayleigh–Bénard convection. Phys. Rev. E 63, 015303.
Ahlers G., Bodenschatz E., Funfschilling D., Grossmann S., He X., Lohse D., Stevens R. J. A. M. & Verzicco R. 2012a Logarithmic temperature profiles in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 109, 114501.
Ahlers G., Bodenschatz E., Funfschilling D. & Hogg J. 2009a Turbulent Rayleigh–Bénard convection for a Prandtl number of 0.67. J. Fluid Mech. 641, 157167.
Ahlers G., Grossmann S. & Lohse D. 2009b Heat transfer and large-scale dynamics in turbulent Rayleigh–Bénard convection. Rev. Mod. Phys. 81, 503537.
Ahlers G., He X., Funfschilling D. & Bodenschatz E. 2012b Heat transport by turbulent Rayleigh–Bénard convection for inline-graphic $\mathit{Pr}= 0. 8$ and inline-graphic $3\times 1{0}^{12} \leq \mathit{Ra}\leq 1{0}^{15} $ : aspect ratio inline-graphic $\Gamma = 0. 50$ . New J. Phys. 14, 063030.
Ahlers G. & Xu X. 2001 Prandtl-number dependence of heat transport in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 86, 33203323.
Breuer M., Wessling S., Schmalzl J. & Hansen U. 2004 Effect of inertia in Rayleigh–Bénard convection. Phys. Rev. E 69, 026302.
Brown E., Funfschilling D., Nikolaenko A. & Ahlers G. 2005 Heat transport by turbulent Rayleigh–Bénard convection: effect of finite top- and bottom conductivity. Phys. Fluids 17, 075108.
Burnishev Y., Segre E. & Steinberg V. 2010 Strong symmetrical non-Oberbeck–Boussinesq turbulent convection and the role of compressibility. Phys. Fluids 22, 035108.
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, 130.
Chaumat S., Castaing B. & Chilla F. 2002 Rayleigh–Bénard cells: influence of plate properties. In Advances in Turbulence IX (ed. Castro I. P., Hancock P. E. & Thomas T. G.). International Center for Numerical Methods in Engineering, CIMNE.
Chavanne X., Chilla F., Castaing B., Hebral B., Chabaud B. & Chaussy J. 1997 Observation of the ultimate regime in Rayleigh–Bénard convection. Phys. Rev. Lett. 79, 36483651.
Chavanne X., Chilla F., Chabaud B., Castaing B. & Hebral B. 2001 Turbulent Rayleigh–Bénard convection in gaseous and liquid He. Phys. Fluids 13, 13001320.
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, 111140.
Emran M. S. & Schumacher J. 2012 Conditional statistics of thermal dissipation rate in turbulent Rayleigh–Bénard convection. Eur. Phys. J. E 108, 3542.
Fleischer A. S. & Goldstein R. J. 2002 High-Rayleigh-number convection of pressurized gases in a horizontal enclosure. J. Fluid Mech. 469, 112.
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, 145154.
Glazier J. A., Segawa T., Naert A. & Sano M. 1999 Evidence against ultrahard thermal turbulence at very high Rayleigh numbers. Nature 398, 307310.
Grossmann S. & Lohse D. 2000 Scaling in thermal convection: a unifying view. J. Fluid. Mech. 407, 2756.
Grossmann S. & Lohse D. 2001 Thermal convection for large Prandtl number. Phys. Rev. Lett. 86, 33163319.
Grossmann S. & Lohse D. 2002 Prandtl and Rayleigh number dependence of the Reynolds number in turbulent thermal convection. Phys. Rev. E 66, 016305.
Grossmann S. & Lohse D. 2004 Fluctuations in turbulent Rayleigh–Bénard convection: the role of plumes. Phys. Fluids 16, 44624472.
Grossmann S. & Lohse D. 2011 Multiple scaling in the ultimate regime of thermal convection. Phys. Fluids 23, 045108.
He X., Funfschilling D., Bodenschatz E. & Ahlers G. 2012a Heat transport by turbulent Rayleigh–Bénard convection for inline-graphic $\Pr = 0. 8$ and inline-graphic $4\times 1{0}^{11} \leq \mathit{Ra}\leq 2\times 1{0}^{14} $ for aspect ratio inline-graphic $\Gamma = 1. 00$ . New J. Phys. 14, 103012.
He X., Funfschilling D., Nobach H., Bodenschatz E. & Ahlers G. 2012b Transition to the ultimate state of turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 108, 024502.
He X., He G. & Tong P. 2010 Small-scale turbulent fluctuations beyond Taylor’s frozen-flow hypothesis. Phys. Rev. E 81, 065303.
He X. & Tong P. 2011 Kraichnan’s random sweeping hypothesis in homogeneous turbulent convection. Phys. Rev. E 83, 037302.
He G. W. & Zhang J. B. 2006 Elliptic model for space–time correlations in turbulent shear flows. Phys. Rev. E 73, 055303.
Horanyi S., Krebs L. & Müller U. 1999 Turbulent Rayleigh–Bénard convection in low Prandtl number fluids. Intl J. Heat Mass Transfer 42, 39834003.
Kerr R. & Herring J. R. 2000 Prandtl number dependence of Nusselt number in direct numerical simulations. J. Fluid Mech. 419, 325344.
Lakkaraju R., Stevens R. J. A. M., Verzicco R., Grossmann S., Prosperetti A., Sun C. & Lohse D. 2012 Spatial distribution of heat flux and fluctuations in turbulent Rayleigh–Bénard convection. Phys. Rev. E 86, 056315.
Landau L. D. & Lifshitz E. M. 1987 Fluid Mechanics. Pergamon.
Lohse D. & Xia K. Q. 2010 Small-scale properties of turbulent Rayleigh–Bénard convection. Annu. Rev. Fluid Mech. 42, 335364.
Niemela J., Skrbek L., Sreenivasan K. R. & Donnelly R. 2000 Turbulent convection at very high Rayleigh numbers. Nature 404, 837840.
Niemela J., Skrbek L., Sreenivasan K. R. & Donnelly R. J. 2001 The wind in confined thermal turbulence. J. Fluid Mech. 449, 169178.
Niemela J. & Sreenivasan K. R. 2003 Confined turbulent convection. J. Fluid Mech. 481, 355384.
Niemela J. & Sreenivasan K. R. 2006 Turbulent convection at high Rayleigh numbers and aspect ratio 4. J. Fluid Mech. 557, 411422.
Petschel K., Stellmach S., Wilczek M., Lülff J. & Hansen U. 2013 Dissipation layers in Rayleigh–Bénard convection: a unifying view. Phys. Rev. Lett. 110, 114502.
Qiu X. L. & Tong P. 2001 Large scale velocity structures in turbulent thermal convection. Phys. Rev. E 64, 036304.
Roche P. E., Castaing B., Chabaud B., Hebral B. & Sommeria J. 2001 Side wall effects in Rayleigh–Bénard experiments. Eur. Phys. J. B 24, 405408.
Roche P.-E., Gauthier F., Kaiser R. & Salort J. 2010 On the triggering of the ultimate regime of convection. New J. Phys. 12, 085014.
Rossby H. T. 1969 A study of Bénard convection with and without rotation. J. Fluid Mech. 36, 309335.
Schlichting H. 1979 Boundary Layer Theory, 7th edn. McGraw-Hill.
Shishkina O., Stevens R. J. A. M., Grossmann S. & Lohse D. 2010 Boundary layer structure in turbulent thermal convection and its consequences for the required numerical resolution. New J. Phys. 12, 075022.
Shishkina O. & Thess A. 2009 Mean temperature profiles in turbulent Rayleigh–Bénard convection of water. J. Fluid Mech. 633, 449460.
Siggia E. D. 1994 High Rayleigh number convection. Annu. Rev. Fluid Mech. 26, 137168.
Stevens R. J. A. M., Clercx H. J. H. & Lohse D. 2010a Boundary layers in rotating weakly turbulent Rayleigh–Bénard convection. Phys. Fluids 22, 085103.
Stevens R. J. A. M., Clercx H. J. H. & Lohse D. 2010b Optimal Prandtl number for heat transfer in rotating Rayleigh–Bénard convection. New J. Phys. 12, 075005.
Stevens R. J. A. M., Lohse D. & Verzicco R. 2011a Prandtl number dependence of heat transport in high Rayleigh number thermal convection. J. Fluid. Mech. 688, 3143.
Stevens R. J. A. M., Overkamp J., Lohse D. & Clercx H. J. H. 2011b Effect of aspect-ratio on vortex distribution and heat transfer in rotating Rayleigh–Bénard convection. Phys. Rev. E 84, 056313.
Stevens R. J. A. M., Verzicco R. & Lohse D. 2010c Radial boundary layer structure and Nusselt number in Rayleigh–Bénard convection. J. Fluid. Mech. 643, 495507.
Sun C., Ren L.-Y., Song H. & Xia K.-Q. 2005 Heat transport by turbulent Rayleigh–Bénard convection in 1m diameter cylindrical cells of widely varying aspect ratio. J. Fluid Mech. 542, 165174.
Sun C. & Xia K.-Q. 2005 Scaling of the Reynolds number in turbulent thermal convection. Phys. Rev. E 72, 067302.
Urban P., Hanzelka P., Kralik T., Musilova V., Srnka A. & Skrbek L. 2012 Effect of boundary layers asymmetry on heat transfer efficiency in turbulent Rayleigh–Bénard convection at very high Rayleigh numbers. Phys. Rev. Lett. 109, 154301.
Urban P., Musilová V. & Skrbek L. 2011 Efficiency of heat transfer in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 107, 014302.
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. (Submitted).
Verzicco R. 2002 Sidewall finite conductivity effects in confined turbulent thermal convection. J. Fluid Mech. 473, 201210.
Verzicco R. & Camussi R. 1999 Prandtl number effects in convective turbulence. J. Fluid Mech. 383, 5573.
Xia K.-Q., Lam S. & Zhou S. Q. 2002 Heat-flux measurement in high-Prandtl-number turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 88, 064501.
Zhao X. & He G.-W. 2009 Space–time correlations of fluctuating velocities in turbulent shear flows. Phys. Rev. E 79, 046316.
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, 94111.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 165 *
Loading metrics...

Abstract views

Total abstract views: 310 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 23rd November 2017. This data will be updated every 24 hours.