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Sidewall effects in Rayleigh–Bénard convection

Published online by Cambridge University Press:  17 February 2014

Richard J. A. M. Stevens ,
Detlef Lohse  and
Roberto Verzicco
Richard J. A. M. Stevens
Affiliation:
Department of Science and Technology and J.M. Burgers Center for Fluid Dynamics, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands Department of Mechanical Engineering, Johns Hopkins University, Baltimore, MD 21218, USA
Detlef Lohse
Affiliation:
Department of Science and Technology and J.M. Burgers Center for Fluid Dynamics, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands
Roberto Verzicco*
Affiliation:
Department of Science and Technology and J.M. Burgers Center for Fluid Dynamics, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands Department of Industrial Engineering, Università di Roma ‘Tor Vergata’, Via del Politecnico 1, 00133, Roma
*
†Email address for correspondence: verzicco@uniroma2.it
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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.

JFM classification

Convection: Convection
Type
Papers
Information
Journal of Fluid Mechanics , Volume 741 , 25 February 2014 , pp. 1 - 27
Copyright
© 2014 Cambridge University Press 

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References

Ahlers, G. 2000 Effect of sidewall conductance on heat-transport measurements for turbulent Rayleigh–Bénard convection. Phys. Rev. E 63, 015303.Google ScholarPubMed
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.Google Scholar
Ahlers, G., Bodenschatz, E., Funfschilling, D. & Hogg, J. 2009a Turbulent Rayleigh–Bénard convection for a Prandtl number of 0.67. J. Fluid Mech. 641, 157–167.Google Scholar
Ahlers, G., Funfschilling, D. & Bodenschatz, E. 2009b Transitions in heat transport by turbulent convection at Rayleigh numbers up to $10^{15}$. New J. Phys. 11, 123001.Google Scholar
Ahlers, G., Funfschilling, D. & Bodenschatz, E. 2011 Addendum to transitions in heat transport by turbulent convection at Rayleigh numbers up to $10^{15}$. New J. Phys. 13, 049401.Google Scholar
Ahlers, G., Grossmann, S. & Lohse, D. 2009c Heat transfer and large-scale dynamics in turbulent Rayleigh–Bénard convection. Rev. Mod. Phys. 81, 503–537.Google Scholar
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.Google 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
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.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
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.Google Scholar
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, 35–60.Google Scholar
Fleischer, A. S. & Goldstein, R. J. 2002 High-Rayleigh-number convection of pressurized gases in a horizontal enclosure. J. Fluid Mech. 469, 1–12.Google Scholar
Funfschilling, D., Bodenschatz, E. & Ahlers, G. 2009 Search for the ‘ultimate state’ in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 103, 014503.Google Scholar
Grossmann, S. & Lohse, D. 2000 Scaling in thermal convection: A unifying view. J. Fluid Mech. 407, 27–56.Google Scholar
Grossmann, S. & Lohse, D. 2001 Thermal convection for large Prandtl number. Phys. Rev. Lett. 86, 3316–3319.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.Google Scholar
Grossmann, S. & Lohse, D. 2004 Fluctuations in turbulent Rayleigh–Bénard convection: The role of plumes. Phys. Fluids 16, 4462–4472.Google Scholar
Grossmann, S. & Lohse, D. 2011 Multiple scaling in the ultimate regime of thermal convection. Phys. Fluids 23 (4), 045108.Google Scholar
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.Google Scholar
Hébert, F., Hufschmid, R., Scheel, J. & Ahlers, G. 2010 Onset of Rayleigh–Bénard convection in cylindrical containers. Phys. Rev. E 81, 046318.Google Scholar
Johnston, H. & Doering, C. R. 2009 Comparison of turbulent thermal convection between conditions of constant temperature and constant flux. Phys. Rev. Lett. 102, 064501.Google Scholar
Kraichnan, R. H. 1962 Turbulent thermal convection at arbritrary Prandtl number. Phys. Fluids 5, 1374–1389.Google Scholar
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, 422–442.Google Scholar
Niemela, J., Skrbek, L., Sreenivasan, K. R. & Donnelly, R. 2000 Turbulent convection at very high Rayleigh numbers. Nature 404, 837–840.Google Scholar
Niemela, J., Skrbek, L., Sreenivasan, K. R. & Donnelly, R. J. 2001 The wind in confined thermal turbulence. J. Fluid Mech. 449, 169–178.Google Scholar
Niemela, J. & Sreenivasan, K. R. 2003 Confined turbulent convection. J. Fluid Mech. 481, 355–384.Google Scholar
Niemela, J. & Sreenivasan, K. R. 2006 Turbulent convection at high Rayleigh numbers and aspect ratio 4. J. Fluid Mech. 557, 411–422.Google Scholar
Niemela, J. J. & Sreenivasan, K. R. 2010 Does confined turbulent convection ever attain the ‘asymptotic scaling’ with $1/2$ power?. New J. Phys. 12, 115002.Google Scholar
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).Google Scholar
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.Google Scholar
Roche, P. E., Castaing, B., Chabaud, B. & Hebral, B. 2002 Prandtl and Rayleigh numbers dependences in Rayleigh–Bénard convection. Europhys. Lett. 58, 693–698.Google Scholar
Roche, P. E., Castaing, B., Chabaud, B., Hebral, B. & Sommeria, J. 2001b Side wall effects in Rayleigh–Bénard experiments. Eur. Phys. J. B 24, 405–408.Google Scholar
Roche, P.-E., Gauthier, F., Kaiser, R. & Salort, J. 2010 On the triggering of the ultimate regime of convection. New J. Phys. 12, 085014.Google Scholar
Scheel, J. D., Kim, E. & White, K. R. 2012 Thermal and viscous boundary layers in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 711, 281–305.Google Scholar
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.Google Scholar
Shishkina, O. & Thess, A. 2009 Mean temperature profiles in turbulent Rayleigh–Bénard convection of water. J. Fluid Mech. 633, 449–460.CrossRefGoogle Scholar
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, 31–43.Google Scholar
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, 295–308.Google Scholar
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, 495–507.Google Scholar
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, 165–174.Google Scholar
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.Google Scholar
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.Google Scholar
Urban, P., Musilová, V. & Skrbek, L. 2011 Efficiency of heat transfer in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 107, 014302.Google Scholar
Verzicco, R. 2002 Sidewall finite conductivity effects in confined turbulent thermal convection. J. Fluid Mech. 473, 201–210.Google Scholar
Verzicco, R. & Camussi, R. 1997 Transitional regimes of low-Prandtl thermal convection in a cylindrical cell. Phys. Fluids 9, 1287–1295.Google Scholar
Verzicco, R. & Orlandi, P. 1996 A finite-difference scheme for three-dimensional incompressible flow in cylindrical coordinates. J. Comput. Phys. 123, 402–413.CrossRefGoogle Scholar
Weiss, S. & Ahlers, G. 2011 Turbulent Rayleigh–Bénard convection in a cylindrical container with aspect ratio $\Gamma =0.50$ and Prandtl number $Pr = 4.38$. J. Fluid Mech. 676, 5–40.Google Scholar
Xi, H. D. & Xia, K. Q. 2008 Flow mode transitions in turbulent thermal convection. Phys. Fluids 20, 055104.Google Scholar