Skip to main content
×
Home

Sidewall effects in Rayleigh–Bénard convection

  • Richard J. A. M. Stevens (a1) (a2), Detlef Lohse (a1) and Roberto Verzicco (a1) (a3)
Abstract
Abstract

We investigate the influence of the temperature boundary conditions at the sidewall on the heat transport in Rayleigh–Bénard (RB) convection using direct numerical simulations. For relatively low Rayleigh numbers $Ra$ the heat transport is higher when the sidewall is isothermal, kept at a temperature $T_c+\Delta /2$ (where $\Delta $ is the temperature difference between the horizontal plates and $T_c$ the temperature of the cold plate), than when the sidewall is adiabatic. The reason is that in the former case part of the heat current avoids the thermal resistance of the fluid layer by escaping through the sidewall that acts as a short-circuit. For higher $Ra$ the bulk becomes more isothermal and this reduces the heat current through the sidewall. Therefore the heat flux in a cell with an isothermal sidewall converges to the value obtained with an adiabatic sidewall for high enough $Ra$ (${\simeq }10^{10}$). However, when the sidewall temperature deviates from $T_c+\Delta /2$ the heat transport at the bottom and top plates is different from the value obtained using an adiabatic sidewall. In this case the difference does not decrease with increasing $Ra$ thus indicating that the ambient temperature of the experimental apparatus can influence the heat transfer. A similar behaviour is observed when only a very small sidewall region close to the horizontal plates is kept isothermal, while the rest of the sidewall is adiabatic. The reason is that in the region closest to the horizontal plates the temperature difference between the fluid and the sidewall is highest. This suggests that one should be careful with the placement of thermal shields outside the fluid sample to minimize spurious heat currents. When the physical sidewall properties (thickness, thermal conductivity and heat capacity) are considered the problem becomes one of conjugate heat transfer and different behaviours are possible depending on the sidewall properties and the temperature boundary condition on the ‘dry’ side. The problem becomes even more complicated when the sidewall is shielded with additional insulation or temperature-controlled surfaces; some particular examples are illustrated and discussed. It has been observed that the sidewall temperature dynamics not only affects the heat transfer but can also trigger a different mean flow state or change the temperature fluctuations in the flow and this could explain some of the observed differences between similar but not fully identical experiments.

Copyright
Corresponding author
Email address for correspondence: verzicco@uniroma2.it
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. 2012 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., Funfschilling D. & Bodenschatz E. 2009b Transitions in heat transport by turbulent convection at Rayleigh numbers up to inline-graphic$10^{15}$. New J. Phys. 11, 123001.
Ahlers G., Funfschilling D. & Bodenschatz E. 2011 Addendum to transitions in heat transport by turbulent convection at Rayleigh numbers up to inline-graphic$10^{15}$. New J. Phys. 13, 049401.
Ahlers G., Grossmann S. & Lohse D. 2009c Heat transfer and large-scale dynamics in turbulent Rayleigh–Bénard convection. Rev. Mod. Phys. 81, 503537.
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.
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.), Barcelona: International Center for Numerical Methods in Engineering, CIMNE.
Chavanne X., Chilla F., Chabaud B., Castaing B. & Hebral B. 2001 Turbulent Rayleigh–Bénard convection in gaseous and liquid he. Phys. Fluids 13, 13001320.
Chillà F., Rastello M., Chaumat S. & Castaing B. 2004 Long relaxation times and tilt sensitivity in Rayleigh–Bénard turbulence. Eur. Phys. J. B 40, 223227.
Fadlun E. A., Verzicco R., Orlandi P. & Mohd-Yusof J. 2000 Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulations. J. Comput. Phys. 161, 3560.
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., Bodenschatz E. & Ahlers G. 2009 Search for the ‘ultimate state’ in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 103, 014503.
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 (4), 045108.
He X., Funfschilling D., Nobach H., Bodenschatz E. & Ahlers G. 2012 Transition to the ultimate state of turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 108, 024502.
Hébert F., Hufschmid R., Scheel J. & Ahlers G. 2010 Onset of Rayleigh–Bénard convection in cylindrical containers. Phys. Rev. E 81, 046318.
Johnston H. & Doering C. R. 2009 Comparison of turbulent thermal convection between conditions of constant temperature and constant flux. Phys. Rev. Lett. 102, 064501.
Kraichnan R. H. 1962 Turbulent thermal convection at arbritrary Prandtl number. Phys. Fluids 5, 13741389.
Kunnen R. P. J., Stevens R. J. A. M., Overkamp J., Sun C., van Heijst G. J. F. & Clercx H. J. H. 2011 The role of Stewartson and Ekman layers in turbulent rotating Rayleigh–Bénard convection. J. Fluid Mech. 688, 422442.
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.
Niemela J. J. & Sreenivasan K. R. 2010 Does confined turbulent convection ever attain the ‘asymptotic scaling’ with inline-graphic$1/2$ power?. New J. Phys. 12, 115002.
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(R).
Roche P. E., Castaing B., Chabaud B. & Hebral B. 2001a Observation of the 1/2 power law in Rayleigh–Bénard convection. Phys. Rev. E 63, 045303.
Roche P. E., Castaing B., Chabaud B. & Hebral B. 2002 Prandtl and Rayleigh numbers dependences in Rayleigh–Bénard convection. Europhys. Lett. 58, 693698.
Roche P. E., Castaing B., Chabaud B., Hebral B. & Sommeria J. 2001b 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.
Scheel J. D., Kim E. & White K. R. 2012 Thermal and viscous boundary layers in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 711, 281305.
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.
Stevens R. J. A. M., Lohse D. & Verzicco R. 2011 Prandtl number dependence of heat transport in high Rayleigh number thermal convection. J. Fluid Mech. 688, 3143.
Stevens R. J. A. M., van der Poel E. P. & Lohse D. 2013 The unifying theory of scaling in thermal convection: The updated prefactors. J. Fluid Mech. 730, 295308.
Stevens R. J. A. M., Verzicco R. & Lohse D. 2010 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. 2005a Heat transport by turbulent Rayleigh–Bénard convection in 1m diameter cylindrical cells of widely varying aspect ratio. J. Fluid Mech. 542, 165174.
Sun C., Xi H. D. & Xia K. Q. 2005b 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.
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.
Verzicco R. 2002 Sidewall finite conductivity effects in confined turbulent thermal convection. J. Fluid Mech. 473, 201210.
Verzicco R. & Camussi R. 1997 Transitional regimes of low-Prandtl thermal convection in a cylindrical cell. Phys. Fluids 9, 12871295.
Verzicco R. & Orlandi P. 1996 A finite-difference scheme for three-dimensional incompressible flow in cylindrical coordinates. J. Comput. Phys. 123, 402413.
Weiss S. & Ahlers G. 2011 Turbulent Rayleigh–Bénard convection in a cylindrical container with aspect ratio inline-graphic$\Gamma =0.50$ and Prandtl number inline-graphic$Pr = 4.38$. J. Fluid Mech. 676, 540.
Xi H. D. & Xia K. Q. 2008 Flow mode transitions in turbulent thermal convection. Phys. Fluids 20, 055104.
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: 92 *
Loading metrics...

Abstract views

Total abstract views: 253 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 25th November 2017. This data will be updated every 24 hours.