Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-16T04:14:14.906Z Has data issue: false hasContentIssue false
  • English
  • Français

Rigid bounds on heat transport by a fluid between slippery boundaries

Published online by Cambridge University Press:  13 July 2012

Jared P. Whitehead  and
Charles R. Doering
Jared P. Whitehead*
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043, USA Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
Charles R. Doering
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043, USA Department of Physics, University of Michigan, Ann Arbor, MI 48109-1120, USA Center for the Study of Complex Systems, University of Michigan, Ann Arbor, MI 48109-1107, USA
*
†Email address for correspondence: whitehead@lanl.gov
Get access Rights & Permissions [Opens in a new window]

Abstract

Rigorous bounds on heat transport are derived for thermal convection between stress-free horizontal plates. For three-dimensional Rayleigh–Bénard convection at infinite Prandtl number (), the Nusselt number () is bounded according to where is the standard Rayleigh number. For convection driven by a uniform steady internal heat source between isothermal boundaries, the spatially and temporally averaged (non-dimensional) temperature is bounded from below by in three dimensions at infinite and by in two dimensions at arbitrary , where is the heat Rayleigh number proportional to the injected flux.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 707 , 25 September 2012 , pp. 241 - 259
Copyright
Copyright © Cambridge University Press 2012

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

1. Ahlers, G. 2009 Turbulent convection. Physics 2 (74).CrossRefGoogle Scholar
2. Ahlers, G., Funfschilling, D. & Bodenschatz, E. 2009a Transitions in heat transport by turbulent convection at Rayleigh numbers up to . New J. Phys. 11, 123001.CrossRefGoogle Scholar
3. Ahlers, G., Grossmann, S. & Lohse, D. 2009b Heat transfer and large-scale dynamics in turbulent Rayleigh–Bénard convection. Rev. Mod. Phys. 81, 503–537.CrossRefGoogle Scholar
4. Busse, F. H. 1969 On Howard’s upper bound for heat transport by turbulent convection. J. Fluid Mech. 37, 457–477.CrossRefGoogle Scholar
5. Constantin, P. & Doering, C. R. 1999 Infinite Prandtl number convection. J. Stat. Phys. 94, 159–172.CrossRefGoogle Scholar
6. Coron, F. 1989 Derivation of slip boundary conditions for the Navier–Stokes system from the Boltzmann equation. J. Stat. Phys. 54 (3/4), 829–857.CrossRefGoogle Scholar
7. Doering, C. R. & Constantin, P. 1992 Energy dissipation in shear driven turbulence. Phys. Rev. Lett. 69 (11), 1648–1651.CrossRefGoogle ScholarPubMed
8. Doering, C. R. & Constantin, P. 1996 Variational bounds on energy dissipation in incompressible flows. Part 3. Convection. Phys. Rev. E. 53 (6), 5957–5981.CrossRefGoogle Scholar
9. Doering, C. R. & Constantin, P. 2001 On upper bounds for infinite Prandtl number convection with or without rotation. J. Math. Phys. 45 (2), 784–795.CrossRefGoogle Scholar
10. 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.CrossRefGoogle Scholar
11. Funfschilling, D., Bodenschatz, E. & Ahlers, G. 2009 Search for the ultimate state in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 103, 014503.CrossRefGoogle Scholar
12. Grossmann, S. & Lohse, D. 2000 Scaling in thermal convection: a unifying theory. J. Fluid Mech. 407, 27–56.CrossRefGoogle Scholar
13. Grossmann, S. & Lohse, D. 2001 Thermal convection for large Prandtl number. Phys. Rev. Lett. 86, 3316–3319.CrossRefGoogle Scholar
14. Grossmann, S. & Lohse, D. 2002 Prandtl and Rayleigh number dependence of the Reynolds number in turbulent thermal convection. Phys. Rev. E 66, 016305.CrossRefGoogle ScholarPubMed
15. Grossmann, S. & Lohse, D. 2011 Multiple scaling in the ultimate regime of thermal convection. Phys. Fluids 23, 045108.CrossRefGoogle Scholar
16. 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 (2), 024502.CrossRefGoogle Scholar
17. Howard, L. N. 1963 Heat transport by turbulent convection. J. Fluid Mech. 17, 405–432.CrossRefGoogle Scholar
18. 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
19. Johnston, H. & Doering, C. R. 2009 Comparison of turbulent thermal convection between conditions of constant temperature and constant flux. Phys. Rev. Lett. 102, 064501.CrossRefGoogle ScholarPubMed
20. Julien, K., Legg, S., McWilliams, J. & Werne, J. 1995 Hard turbulence in rotating Rayleigh–Bénard convection. Phys. Rev. E 53 (6), R5557–R5560.CrossRefGoogle Scholar
21. Kraichnan, R. H. 1962 Turbulent thermal convection at arbitrary Prandtl number. Phys. Fluids 5, 1374–1389.CrossRefGoogle Scholar
22. Lu, L., Doering, C. R. & Busse, F. H. 2004 Bounds on convection driven by internal heating. J. Math. Phys. 45 (7), 2967–2986.CrossRefGoogle Scholar
23. Malkus, W. V. R. 1954 The heat transport and spectrum of thermal turbulence. Proc. R. Soc. Lond. A 225, 196–212.Google Scholar
24. 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
25. Otero, J. 2002 Bounds for the heat transport in turbulent convection. PhD thesis, University of Michigan, MI, Department of Mathematics.Google Scholar
26. 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
27. Otto, F. & Seis, C. 2011 Rayleigh–Bénard convection: improved bounds on the Nusselt number. J. Math. Phys. 52, 083702.CrossRefGoogle Scholar
28. Plasting, S. C. 2004 Turbulence has its limits: a priori estimates of transport properties in turbulent fluid flows. PhD thesis, University of Bristol.Google Scholar
29. Plasting, S. C. & Ierley, G. R. 2005 Estimates of heat transport in infinite Prandtl number convection. Part 1. Conservative bounds. J. Fluid Mech. 542, 343–363.CrossRefGoogle Scholar
30. Plasting, S. C. & Kerswell, R. R. 2003 Improved upper bound on the energy dissipation rate in plane Couette flow: the full solutions to Busse’s problem and the Constantin–Doering–Hopf problem with one-dimensional background fields. J. Fluid Mech. 477, 363–379.CrossRefGoogle Scholar
31. Rayleigh, Lord 1916 On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side. Phil. Mag. J. Sci. 32 (192), 529–546.CrossRefGoogle Scholar
32. Roberts, P. H. 1967 Convection in horizontal layers with internal heat generation. Theory. J. Fluid Mech. 30, 33–49.CrossRefGoogle Scholar
33. Roche, P.-E., Gauthier, F., Kaiser, R. & Salort, J. 2010 On the triggering of the ultimate regime of convection. New. J. Phys. 12, 085014.CrossRefGoogle Scholar
34. Sotin, C. & Labrosse, S. 1999 Three-dimensional thermal convection in an iso-viscous, infinite Prandtl number fluid heated from within and from below: applications to the transfer of heat through planetary mantles. Phys. Earth Planet. Inter. 112, 171–190.CrossRefGoogle Scholar
35. Sparrow, E. M., Goldstein, R. J. & Jonsson, V. K. 1963 Thermal instability in a horizontal fluid layer: effect of boundary conditions and nonlinear temperature profiles. J. Fluid Mech. 18, 513–528.CrossRefGoogle Scholar
36. Spiegel, E. A. 1971 Convection in stars. Part 1. Basic Boussinesq convection. Annu. Rev. Astron. Astrophys. 9, 323–352.CrossRefGoogle Scholar
37. Stevens, R. J. A. M., Lohse, D. & Verzicco, R. 2011 Prandtl and Rayleigh number dependence of heat transport in high Rayleigh number thermal convection. J. Fluid Mech. 688, 31–43.CrossRefGoogle Scholar
38. 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.CrossRefGoogle Scholar
39. Urban, P., Musilová, V. & Skrbek, L. 2011 Efficiency of heat transfer in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 107, 014302.CrossRefGoogle ScholarPubMed
40. Vitanov, N. K. & Busse, F. H. 1997 Bounds on the heat transport in a horizontal fluid layer with stress-free boundaries. Z. Angew. Math. Phys. 48, 310–324.Google Scholar
41. Wang, X. 2004 Infinite Prandtl number limit of Rayleigh–Bénard convection. Commun. Pure Appl. Math. 57, 1265–1285.CrossRefGoogle Scholar
42. Whitehead, J. P. & Doering, C. R. 2011a Internal heating driven convection at infinite Prandtl number. J. Math. Phys. 52, 093101.CrossRefGoogle Scholar
43. Whitehead, J. P. & Doering, C. R. 2011b The ultimate regime of two-dimensional Rayleigh–Bénard convection with stress-free boundaries. Phys. Rev. Lett. 106, 244501.CrossRefGoogle Scholar
44. Wittenberg, R. 2010 Bounds on Rayleigh–Bénard convection with imperfectly conducting plates. J. Fluid Mech. 665, 158–198.CrossRefGoogle Scholar
45. Yan, X. D. 2004 On limits to convective heat transport at infinite Prandtl number with or without rotation. J. Math. Phys. 45, 2718–2743.CrossRefGoogle Scholar
46. Zhu, Y. & Granick, S. 2002 Limits of the hydrodynamic no-slip boundary condition. Phys. Rev. Lett. 88 (10), 106102.CrossRefGoogle ScholarPubMed