Hostname: page-component-76fb5796d-x4r87 Total loading time: 0 Render date: 2024-04-25T09:38:45.933Z Has data issue: false hasContentIssue false
  • English
  • Français

Bounds on Rayleigh–Bénard convection with imperfectly conducting plates

Published online by Cambridge University Press:  18 October 2010

RALF W. WITTENBERG
RALF W. WITTENBERG*
Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, BC V5A 1S6, Canada
*
†Email address for correspondence: ralf@sfu.ca
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate the influence of the thermal properties of the boundaries in turbulent Rayleigh–Bénard convection on analytical upper bounds on convective heat transport. We model imperfectly conducting bounding plates in two ways: using idealized mixed thermal boundary conditions (BCs) of constant Biot number η, continuously interpolating between the previously studied fixed temperature (η = 0) and fixed flux (η = ∞) cases; and by explicitly coupling the evolution equations in the fluid in the Boussinesq approximation through temperature and flux continuity to identical upper and lower conducting plates. In both cases, we systematically formulate a bounding principle and obtain explicit upper bounds on the Nusselt number Nu in terms of the usual Rayleigh number Ra measuring the average temperature drop across the fluid layer, using the ‘background method’ developed by Doering and Constantin. In the presence of plates, we find that the bounds depend on σ = d/λ, where d is the ratio of plate to fluid thickness and λ is the conductivity ratio, and that the bounding problem may be mapped onto that for Biot number η = σ. In particular, for each σ > 0, for sufficiently large Ra (depending on σ) we show that Nu ≤ c(σ) R1/3 ≤ CRa1/2, where C is a σ-independent constant, and where the control parameter R is a Rayleigh number defined in terms of the full temperature drop across the entire plate–fluid–plate system. In the Ra → ∞ limit, the usual fixed temperature assumption is a singular limit of the general bounding problem, while fixed flux conditions appear to be most relevant to the asymptotic Nu–Ra scaling even for highly conducting plates.

JFM classification

Turbulent Flows: Turbulent convection Mathematical Foundations: Variational methods
Type
Papers
Information
Journal of Fluid Mechanics , Volume 665 , 25 December 2010 , pp. 158 - 198
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Ahlers, G., Funfschilling, D. & Bodenschatz, E. 2009 a Transitions in heat transport by turbulent convection at Rayleigh numbers up to 1015. New J. Phys. 11, 123001.CrossRefGoogle Scholar
Ahlers, G., Grossmann, S. & Lohse, D. 2009 b Heat transfer and large scale dynamics in turbulent Rayleigh-Bénard convection. Rev. Mod. Phys. 81, 503–537.Google Scholar
Amati, G., Koal, K., Massaioli, F., Sreenivasan, K. R. & Verzicco, R. 2005 Turbulent thermal convection at high Rayleigh numbers for a Boussinesq fluid of constant Prandtl number. Phys. Fluids 17, 121701.CrossRefGoogle Scholar
Balmforth, N. J., Ghadge, S. A., Kettapun, A. & Mandre, S. D. 2006 Bounds on double-diffusive convection. J. Fluid Mech. 569, 29–50.Google Scholar
Brown, E., Nikolaenko, A., Funfschilling, D. & Ahlers, G. 2005 Heat transport in turbulent Rayleigh-Bénard convection: effect of finite top- and bottom-plate conductivities. Phys. Fluids 17, 075108.Google Scholar
Busse, F. H. 1969 On Howard's upper bound for heat transport by turbulent convection. J. Fluid Mech. 37, 457–477.Google Scholar
Busse, F. H. & Riahi, N. 1980 Nonlinear convection in a layer with nearly insulating boundaries. J. Fluid Mech. 96, 243–256.CrossRefGoogle Scholar
Chapman, C. J., Childress, S. & Proctor, M. R. E. 1980 Long wavelength thermal convection between non-conducting boundaries. Earth Planet. Sci. Lett. 51, 362–369.Google Scholar
Chapman, C. J. & Proctor, M. R. E. 1980 Nonlinear Rayleigh-Bénard convection between poorly conducting boundaries. J. Fluid Mech. 101 (4), 759–782.CrossRefGoogle Scholar
Chaumat, S., Castaing, B. & Chillà, F. 2002 Rayleigh-Bénard cells: influence of the plates' properties. In Advances in Turbulence IX, Proceedings of the Ninth European Turbulence Conference (ed. Castro, I. P., Hancock, P. E. & Thomas, T. G.), pp. 159–162. CIMNE.Google Scholar
Chavanne, X., Chillà, F., Chabaud, B., Castaing, B. & Hébral, B. 2001 Turbulent Rayleigh-Bénard convection in gaseous and liquid He. Phys. Fluids 13, 1300–1320.CrossRefGoogle Scholar
Chillà, F., Rastello, M., Chaumat, S. & Castaing, B. 2004 Ultimate regime in Rayleigh-Bénard convection: the role of plates. Phys. Fluids 16, 2452–2456.CrossRefGoogle Scholar
Constantin, P. & Doering, C. R. 1996 Heat transfer in convective turbulence. Nonlinearity 9, 1049–1060.CrossRefGoogle Scholar
Cross, M. & Hohenberg, P. 1993 Pattern formation outside of equilibrium. Rev. Mod. Phys. 65 (3), 851–1112.Google Scholar
Doering, C. R. & Constantin, P. 1992 Energy dissipation in shear driven turbulence. Phys. Rev. Lett. 69 (11), 1648–1651.Google Scholar
Doering, C. R. & Constantin, P. 1996 Variational bounds on energy dissipation in incompressible flows. Part III. Convection. Phys. Rev. E 53 (6), 5957–5981.Google Scholar
Doering, C. R., Otto, F. & Reznikoff, M. G. 2006 Bounds on vertical heat transport for infinite-Prandtl-number Rayleigh-Bénard convection. J. Fluid Mech. 560, 229–241.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
Gertsberg, V. L. & Sivashinsky, G. I. 1981 Large cells in nonlinear Rayleigh-Bénard convection. Prog. Theor. Phys. 66, 1219–1229.CrossRefGoogle Scholar
Glazier, J. A., Segawa, T., Naert, A. & Sano, M. 1999 Evidence against ‘ultrahard’ thermal turbulence at very high Rayleigh numbers. Nature 398, 307–310.Google Scholar
Grigné, C., Labrosse, S. & Tackley, P. J. 2007 a Convection under a lid of finite conductivity: heat flux scaling and application to continents. J. Geophys. Res. 112, B08402.Google Scholar
Grigné, C., Labrosse, S. & Tackley, P. J. 2007 b Convection under a lid of finite conductivity in wide aspect ratio models: effect of continents on the wavelength of mantle flow. J. Geophys. Res. 112, B08403.Google Scholar
Grossmann, S. & Lohse, D. 2000 Scaling in thermal convection: a unifying theory. J. Fluid Mech. 407, 27–56.CrossRefGoogle Scholar
Guillou, L. & Jaupart, C. 1995 On the effects of continents on mantle convection. J. Geophys. Res. 100, 24217–24238.CrossRefGoogle Scholar
Heslot, F., Castaing, B. & Libchaber, A. 1987 Transitions to turbulence in helium gas. Phys. Rev. A 36, 5870–5873.CrossRefGoogle ScholarPubMed
Holmedal, B., Tveitereid, M. & Palm, E. 2005 Planform selection in Rayleigh-Bénard convection between finite slabs. J. Fluid Mech. 537, 255–270.CrossRefGoogle Scholar
Hopf, E. 1941 Ein allgemeiner Endlichkeitssatz der Hydrodynamik. Math. Ann. 117, 764–775.CrossRefGoogle Scholar
Howard, L. N. 1963 Heat transport by turbulent convection. J. Fluid Mech. 17, 405–432.Google Scholar
Hunt, J. C. R., Vrieling, A. J., Nieuwstadt, F. T. M. & Fernando, H. J. S. 2003 The influence of the thermal diffusivity of the lower boundary on eddy motion in convection. J. Fluid Mech. 491, 183–205.Google Scholar
Hurle, D. T. J., Jakeman, E. & Pike, E. R. 1967 On the solution of the Benard problem with boundaries of finite conductivity. Proc. R. Soc. Lond. A 296 (1447), 469–475.Google Scholar
Ierley, G. R., Kerswell, R. R. & Plasting, S. C. 2006 Infinite-Prandtl-number convection. Part 2. A singular limit of upper bound theory. J. Fluid Mech. 560, 159–227.CrossRefGoogle Scholar
Jenkins, D. R. & Proctor, M. R. E. 1984 The transition from roll to square-cell solutions in Rayleigh–Bénard convection. J. Fluid Mech. 139, 461–471.Google 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.Google Scholar
Kadanoff, L. P. 2001 Turbulent heat flow: structures and scaling. Phys. Today pp. 34–39.CrossRefGoogle Scholar
Kerswell, R. R. 1997 Variational bounds on shear-driven turbulence and turbulent Boussinesq convection. Physica D 100, 355–376.Google Scholar
Kerswell, R. R. 2001 New results in the variational approach to turbulent Boussinesq convection. Phys. Fluids 13 (1), 192–209.CrossRefGoogle Scholar
Kraichnan, R. H. 1962 Turbulent thermal convection at arbitrary Prandtl number. Phys. Fluids 5, 1374–1389.CrossRefGoogle Scholar
Lenardic, A. & Moresi, L. 2003 Thermal convection below a conducting lid of variable extent: heat flow scaling and two-dimensional, infinite Prandtl number numerical simulations. Phys. Fluids 15, 455–466.CrossRefGoogle Scholar
Malkus, M. V. R. 1954 The heat transport and spectrum of thermal turbulence. Proc. R. Soc. Lond. A 225, 196–212.Google Scholar
Nicodemus, R., Grossmann, S. & Holthaus, M. 1997 Improved variational principle for bounds on energy dissipation in turbulent shear flow. Physica D 101, 178–190.CrossRefGoogle Scholar
Niemela, J. J. & Sreenivasan, K. R. 2006 a Turbulent convection at high Rayleigh numbers and aspect ratio 4. J. Fluid Mech. 557, 411–422.Google Scholar
Niemela, J. J. & Sreenivasan, K. R. 2006 b The use of cryogenic helium for classical turbulence: promises and hurdles. J. Low Temp. Phys. 143, 163–212.Google Scholar
Normand, C., Pomeau, Y. & Velarde, M. G. 1977 Convective instability: a physicist's approach. Rev. Mod. Phys. 49, 581–624.CrossRefGoogle Scholar
Otero, J., Dontcheva, L. A., Johnston, H., Worthing, R. A., Kurganov, A., Petrova, G. & Doering, C. R. 2004 High-Rayleigh-number convection in a fluid-saturated porous layer. J. Fluid Mech. 500, 263–281.Google Scholar
Otero, J., Wittenberg, R. W., Worthing, R. A. & Doering, C. R. 2002 Bounds on Rayleigh–Bénard convection with an imposed heat flux. J. Fluid Mech. 473, 191–199.CrossRefGoogle Scholar
Plasting, S. C. & Kerswell, R. R. 2003 Improved upper bound on the energy dissipation rate in plane Couette flow: the full solution to Busse's problem and the Constantin-Doering-Hopf problem with one-dimensional background field. J. Fluid Mech. 477, 363–379.Google Scholar
Procaccia, I. & Sreenivasan, K. R. 2008 The state of the art in hydrodynamic turbulence: past successes and future challenges. Physica D 237, 2167–2183.CrossRefGoogle Scholar
Proctor, M. R. E. 1981 Planform selection by finite-amplitude thermal convection between poorly conducting slabs. J. Fluid Mech. 113, 469–485.Google Scholar
Roche, P.-E., Gauthier, F., Chabaud, B. & Hébral, B. 2005 Ultimate regime of convection: robustness to poor thermal reservoirs. Phys. Fluids 17, 115107.Google Scholar
Siggers, J. H., Kerswell, R. R. & Balmforth, N. J. 2004 Bounds on horizontal convection. J. Fluid Mech. 517, 55–70.CrossRefGoogle Scholar
Sparrow, E. M., Goldstein, R. J. & Jonsson, V. K. 1964 Thermal instability in a horizontal fluid layer: effect of boundary conditions and non-linear temperature profile. J. Fluid Mech. 18, 513–528.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
Verzicco, R. 2004 Effects of nonperfect thermal sources in turbulent thermal convection. Phys. Fluids 16, 1965–1979.Google Scholar
Verzicco, R. & Sreenivasan, K. R. 2008 A comparison of turbulent thermal convection between conditions of constant temperature and constant heat flux. J. Fluid Mech. 595, 203–219.CrossRefGoogle Scholar
Westerburg, M. & Busse, F. H. 2001 Finite-amplitude convection in the presence of finitely conducting boundaries. J. Fluid Mech. 432, 351–367.Google Scholar
Wittenberg, R. W. & Gao, J. 2010 Conservative bounds on Rayleigh-Bénard convection with mixed thermal boundary conditions. Eur. Phys. J. B 76, 565–580.CrossRefGoogle Scholar