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Rayleigh and Prandtl number scaling in the bulk of Rayleigh–Bénard turbulence

Phys. Fluids 17, 055107 (2005); doi:10.1063/1.1884165

Published 5 May 2005

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Enrico Calzavarini
Dipartimento di Fisica, Università di Ferrara, Via Paradiso 12, I-43100 Ferrara, Italy and INFN, Via Paradiso 12, I-43100 Ferrara, Italy

Detlef Lohse
Department of Applied Physics and J. M. Burgers Centre for Fluid Dynamics, University of Twente, 7500 AE Enschede, The Netherlands

Federico Toschi
IAC-CNR, Istituto per le Applicazioni del Calcolo and INFN, Via Paradiso 12, I-43100 Ferrara, Italy Viale del Policlinico 137, I-00161 Roma, Italy

Raffaele Tripiccione
Dipartimento di Fisica, Università di Ferrara, Via Paradiso 12, I-43100 Ferrara, Italy and INFN, Via Paradiso 12, I-43100 Ferrara, Italy
The Ra and Pr number scaling of the Nusselt number Nu, the Reynolds number Re, the temperature fluctuations, and the kinetic and thermal dissipation rates is studied for (numerical) homogeneous Rayleigh–Bénard turbulence, i.e., Rayleigh–Bénard turbulence with periodic boundary conditions in all directions and a volume forcing of the temperature field by a mean gradient. This system serves as model system for the bulk of Rayleigh–Bénard flow and therefore as model for the so-called "ultimate regime of thermal convection." With respect to the Ra dependence of Nu and Re we confirm our earlier results [D. Lohse and F. Toschi, "The ultimate state of thermal convection," Phys. Rev. Lett. 90, 034502 (2003)] which are consistent with the Kraichnan theory [R. H. Kraichnan, "Turbulent thermal convection at arbitrary Prandtl number," Phys. Fluids 5, 1374 (1962)] and the Grossmann–Lohse (GL) theory [S. Grossmann and D. Lohse, "Scaling in thermal convection: A unifying view," J. Fluid Mech. 407, 27 (2000); "Thermal convection for large Prandtl number," Phys. Rev. Lett. 86, 3316 (2001); "Prandtl and Rayleigh number dependence of the Reynolds number in turbulent thermal convection," Phys. Rev. E 66, 016305 (2002); "Fluctuations in turbulent Rayleigh–Bénard convection: The role of plumes," Phys. Fluids 16, 4462 (2004)], which both predict Nu~Ra1/2 and Re~Ra1/2. However the Pr dependence within these two theories is different. Here we show that the numerical data are consistent with the GL theory Nu~Pr1/2, Re~Pr–1/2. For the thermal and kinetic dissipation rates we find epsilontheta/(kappaDelta2L–2)~(Re  Pr)0.87 and epsilonu/(nu3L–4)~Re2.77, both near (but not fully consistent) the bulk dominated behavior, whereas the temperature fluctuations do not depend on Ra and Pr. Finally, the dynamics of the heat transport is studied and put into the context of a recent theoretical finding by Doering et al. ["Comment on ultimate state of thermal convection" (private communication)]. ©2005 American Institute of Physics
History: Received 9 October 2004; accepted 1 February 2005; published 5 May 2005
Permalink: http://link.aip.org/link/?PHFLE6/17/055107/1
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KEYWORDS and PACS

Keywords
PACS
  • 47.27.Eq
    Turbulence simulation and modeling
  • 47.27.Te
    Convection and heat transfer in fluid dynamics
  • 47.20.-k
    Hydrodynamic stability
  • 47.10.+g
    General theory of fluid dynamics
  • YEAR: 2005

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ISSN:
1070-6631 (print)   1089-7666 (online)
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REFERENCES (61)
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  1. S. Grossmann and D. Lohse, "Scaling in thermal convection: A unifying view," J. Fluid Mech. 407, 27 (2000).
  2. S. Grossmann and D. Lohse, "Thermal convection for large Prandtl number," Phys. Rev. Lett. 86, 3316 (2001).
  3. S. Grossmann and D. Lohse, "Prandtl and Rayleigh number dependence of the Reynolds number in turbulent thermal convection," Phys. Rev. E 66, 016305 (2002).
  4. S. Grossmann and D. Lohse, "Fluctuations in turbulent Rayleigh–Bénard convection: The role of plumes," Phys. Fluids 16, 4462 (2004).
  5. B. Castaing, G. Gunaratne, F. Heslot, L. Kadanoff, A. Libchaber, S. Thomae, X. Z. Wu, S. Zaleski, and G. Zanetti, "Scaling of hard thermal turbulence in Rayleigh–Bénard convection," J. Fluid Mech. 204, 1 (1989).
  6. X. Z. Wu, L. Kadanoff, A. Libchaber, and M. Sano, "Frequency power spectrum of temperature-fluctuation in free-convection," Phys. Rev. Lett. 64, 2140 (1990).
  7. X. Z. Wu and A. Libchaber, "Scaling relations in thermal turbulence: The aspect ratio dependence," Phys. Rev. A 45, 842 (1992).
  8. G. Zocchi, E. Moses, and A. Libchaber, "Coherent structures in turbulent convection: An experimental study," Physica A 166, 387 (1990).
  9. A. Belmonte, A. Tilgner, and A. Libchaber, "Boundary layer length scales in thermal turbulence," Phys. Rev. Lett. 70, 4067 (1993).
  10. L. P. Kadanoff, "Turbulent heat flow: Structures and scaling," Phys. Today 54(8), 34 (2001).
  11. X. Chavanne, F. Chilla, B. Castaing, B. Hebral, B. Chabaud, and J. Chaussy, "Observation of the ultimate regime in Rayleigh–Bénard convection," Phys. Rev. Lett. 79, 3648 (1997).
  12. X. Chavanne, F. Chilla, B. Chabaud, B. Castaing, and B. Hebral, "Turbulent Rayleigh–Bénard convection in gaseous and liquid He," Phys. Fluids 13, 1300 (2001).
  13. P. E. Roche, B. Castaing, B. Chabaud, and B. Hebral, "Observation of the 1/2 power law in Rayleigh–Bénard convection," Phys. Rev. E 63, 045303 (2001).
  14. P. E. Roche, B. Castaing, B. Chabaud, and B. Hebral, "Prandtl and Rayleigh numbers dependences in Rayleigh–Bénard convection," Europhys. Lett. 58, 693 (2002).
  15. E. D. Siggia, "High Rayleigh number convection," Annu. Rev. Fluid Mech. 26, 137 (1994).
  16. J. Niemela, L. Skrebek, K. R. Sreenivasan, and R. Donnelly, "Turbulent convection at very high Rayleigh numbers," Nature (London) 404, 837 (2000).
  17. J. Niemela, L. Skrbek, K. R. Sreenivasan, and R. J. Donnelly, "The wind in confined thermal turbulence," J. Fluid Mech. 449, 169 (2001).
  18. J. Niemela and K. R. Sreenivasan, "Confined turbulent convection," J. Fluid Mech. 481, 355 (2003).
  19. S. Grossmann and D. Lohse, "On geometry effects in Rayleigh–Bénard convection," J. Fluid Mech. 486, 105 (2003).
  20. X. Xu, K. M. S. Bajaj, and G. Ahlers, "Heat transport in turbulent Rayleigh–Bénard convection," Phys. Rev. Lett. 84, 4357 (2000).
  21. G. Ahlers and X. Xu, "Prandtl-number dependence of heat transport in turbulent Rayleigh–Bénard convection," Phys. Rev. Lett. 86, 3320 (2001).
  22. A. Nikolaenko and G. Ahlers, "Nusselt number measurements for turbulent Rayleigh–Bénard convection," Phys. Rev. Lett. 91, 084501 (2003).
  23. R. Verzicco and R. Camussi, "Prandtl number effects in convective turbulence," J. Fluid Mech. 383, 55 (1999).
  24. R. Verzicco, "Turbulent thermal convection in a closed domain: Viscous boundary layer and mean flow effects," Eur. Phys. J. B 35, 133 (2003).
  25. R. Verzicco and R. Camussi, "Numerical experiments on strongly turbulent thermal convection in a slender cylindrical cell," J. Fluid Mech. 477, 19 (2003).
  26. R. Camussi and R. Verzicco, "Convective turbulence in mercury: Scaling laws and spectra," Phys. Fluids 10, 516 (1998).
  27. X. D. Shang, X. L. Qiu, P. Tong, and K. Q. Xia, "Measured local heat transport in turbulent Rayleigh–Bénard convection," Phys. Rev. Lett. 90, 074501 (2003).
  28. Y. B. Du and P. Tong, "Enhanced heat transport in turbulent convection over a rough surface," Phys. Rev. Lett. 81, 987 (1998).
  29. Y. Shen, P. Tong, and K. Q. Xia, "Turbulent convection over rough surfaces," Phys. Rev. Lett. 76, 908 (1996).
  30. Y. B. Du and P. Tong, "Turbulent thermal convection in a cell with ordered rough boundaries," J. Fluid Mech. 407, 57 (2000).
  31. X. L. Qiu and P. Tong, "Onset of coherent oscillations in turbulent Rayleigh–Bénard convection," Phys. Rev. Lett. 87, 094501 (2001).
  32. S. Cioni, S. Ciliberto, and J. Sommeria, "Strongly turbulent Rayleigh–Bénard convection in mercury: Comparison with results at moderate Prandtl number," J. Fluid Mech. 335, 111 (1997).
  33. S. Ciliberto and C. Laroche, "Random roughness of boundary increases the turbulent convection scaling exponent," Phys. Rev. Lett. 82, 3998 (1999).
  34. K.-Q. Xia and S.-L. Lui, "Turbulent thermal convection with an obstructed sidewall," Phys. Rev. Lett. 79, 5006 (1997).
  35. K.-Q. Xia, S. Lam, and S. Q. Zhou, "Heat-flux measurement in high-Prandtl-number turbulent Rayleigh–Bénard convection," Phys. Rev. Lett. 88, 064501 (2002).
  36. S. Lam, X. D. Shang, S. Q. Zhou, and K. Q. Xia, "Prandtl-number dependence of the viscous boundary layer and the Reynolds-number in Rayleigh–Bénard convection," Phys. Rev. E 65, 066306 (2002).
  37. H. D. Xi, S. Lam, and K. Q. Xia, "From laminar plumes to organized flows: The onset of large-scale circulation in turbulent thermal convection," J. Fluid Mech. 503, 47 (2004).
  38. E. S. C. Ching, "Heat flux and shear rate in turbulent convection," Phys. Rev. E 55, 1189 (1997).
  39. E. S. C. Ching and K. F. Lo, "Heat transport by fluid flows with prescribed velocity fields," Phys. Rev. E 64, 046302 (2001).
  40. E. S. C. Ching and K. M. Pang, "Dependence of heat transport on the strength and shear rate of circulating flows," Eur. Phys. J. B 27, 559 (2002).
  41. T. Takeshita, T. Segawa, J. A. Glazier, and M. Sano, "Thermal turbulence in mercury," Phys. Rev. Lett. 76, 1465 (1996).
  42. A. Naert, T. Segawa, and M. Sano, "High-Reynolds-number thermal turbulence in mercury," Phys. Rev. E 56, R1302 (1997).
  43. T. Segawa, A. Naert, and M. Sano, "Matched boundary layers in turbulent Rayleigh–Bénard convection of mercury," Phys. Rev. E 57, 557 (1998).
  44. J. Sommeria, "The elusive ultimate state of thermal convection," Nature (London) 398, 294 (1999).
  45. J. A. Glazier, T. Segawa, A. Naert, and M. Sano, "Evidence against ultrahard thermal turbulence at very high Rayleigh numbers," Nature (London) 398, 307 (1999).
  46. C. Doering and P. Constantin, "Variational bounds on energy dissipation in incompressible flows: III. Convection," Phys. Rev. E 53, 5957 (1996).
  47. Z. A. Daya and R. E. Ecke, "Does turbulent convection feel the shape of the container?" Phys. Rev. Lett. 87, 184501 (2001).
  48. Z. A. Daya and R. E. Ecke, "Prandtl number dependence of interior temperature and velocity fluctuations in turbulent convection," Phys. Rev. E 66, 045301 (2002).
  49. S. Kenjeres and K. Hanjalic, "Numerical insight into flow structure in ultraturbulent thermal convection," Phys. Rev. E 66, 036307 (2002).
  50. M. Breuer, S. Wessling, J. Schmalzl, and U. Hansen, "Effect of inertia in Rayleigh–Bénard convection," Phys. Rev. E 69, 026302 (2004).
  51. R. H. Kraichnan, "Turbulent thermal convection at arbitrary Prandtl number," Phys. Fluids 5, 1374 (1962).
  52. E. A. Spiegel, "Convection in stars, I: Basic Boussinesq convection," Annu. Rev. Astron. Astrophys. 9, 323 (1971).
  53. D. Lohse and F. Toschi, "The ultimate state of thermal convection," Phys. Rev. Lett. 90, 034502 (2003).
  54. L. Biferale, E. Calzavarini, F. Toschi, and R. Tripiccione, "Universality of anisotropic fluctuations from numerical simulations of turbulent flows," Europhys. Lett. 64, 461 (2003).
  55. V. Borue and S. Orszag, "Turbulent convection driven by a constant temperature gradient," J. Sci. Comput. 12, 305 (1997).
  56. C.R. Doering, J.D. Gibbon, and A. Tanabe, "Comment on ultimate state of thermal convection," Phys. Rev. Lett. (private communication).
  57. E. Calzavarini, F. Toschi, and R. Tripiccione, "Evidences of Bolgiano–Obhukhov scaling in three-dimensional Rayleigh–Bénard convection," Phys. Rev. E 66, 016304 (2002).
  58. R. Tripiccione, "APEmille," Parallel Comput. 25, 1297 (1999).
  59. A. Bartoloni et al., "Status of APEmille," Nucl. Phys. B 106, 1043 (2002).
  60. S. Aumaitre and S. Fauve, "Statistical properties of fluctuations of the heat transfer in turbulent convection," Europhys. Lett. 62, 822 (2003).
  61. In our previous paper (Ref. 53) the overall magnitude of Nu was affected by a normalization error, hence all points of Fig. 1 of that paper should be multiplied by a factor 240 (corresponding to the grid size of our simulation). This of course does not affect the scaling exponent given there.

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