Hostname: page-component-745bb68f8f-d8cs5 Total loading time: 0 Render date: 2025-02-11T12:13:22.890Z Has data issue: false hasContentIssue false
  • English
  • Français

Rayleigh number scaling in numerical convection

Published online by Cambridge University Press:  26 April 2006

Robert M. Kerr
Robert M. Kerr
Affiliation:
Geophysical Turbulence Program, National Center for Atmospheric Research, PO Box 3000, Boulder, CO 80307-3000, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Using direct simulations of the incompressible Navier-Stokes equations with rigid upper and lower boundaries at fixed temperature and periodic sidewalls, scaling with respect to Rayleigh number is determined. At large aspect ratio (6:6:1) on meshes up to 288 × 288 × 96, a single scaling regime consistent with the properties of ‘hard’ convective turbulence is found for Pr = 0.7 between Ra = 5 × 104 and Ra = 2 × 107. The properties of this regime include Nu ∼ RaβT with βT = 0.28 ≈ 2/7, exponential temperature distributions in the centre of the cell, and velocity and temperature scales consistent with experimental measurements. Two velocity boundary-layer thicknesses are identified, one outside the thermal boundary layer that scales as Ra−1/7 and the other within it that scales as Ra−3/7. Large-scale shears are not observed; instead, strong local boundary-layer shears are observed in regions between incoming plumes and an outgoing network of buoyant sheets. At the highest Rayleigh number, there is a decade where the energy spectra are close to k−5/3 and temperature variance spectra are noticeably less steep. It is argued that taken together this is good evidence for ‘hard’ turbulence, even if individually each of these properties might have alternative explanations.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 310 , 10 March 1996 , pp. 139 - 179
Copyright
© 1996 Cambridge University Press

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

Adrian, R. J., Ferreira, R. T. D. S. & Boberg, T. 1986 Turbulent thermal convection in wide horizontal fluid layers. Exps. Fluids 4, 121–141.Google Scholar
Balachandar, S. & Sirovich, L. 1991 Probability distribution functions in turbulent convection. Phys. Fluids A 3, 919–927.Google Scholar
Batchelor, G. K. 1969 Computation of the energy spectrum in homogeneous two-dimensional turbulence. Phys. Fluids Suppl. II 12, 233–239.Google Scholar
Bolgiano, R. 1959 Turbulent spectra in a stably stratified atmosphere. J. Geophys. Res. 64, 2226–2229.Google Scholar
Busse, F. H. & Whitehead, J. A. 1974 Oscillatory and collective instabilities in large Prandtl number convection. J. Fluid Mech. 66, 67–79.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.Google Scholar
Champagne, F. H., Friehe, C. A., Larue, J. C. & Wyngaard, J. C. 1977 Flux measurements, flux estimation techniques, and fine-scale turbulence measurements in the unstable surface layer over land. J. Atmos. Sci. 34, 515–530.Google Scholar
Chandrasekar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Clarendon
Chu, T. Y. & Goldstein, R. J. 1973 Turbulent convection in a horizontal layer of water. J. Fluid Mech. 60, 141–159.Google Scholar
Curry, J. C., Herring, J. R., Loncaric, J. & Orszag, S. A. 1984 Order and disorder in two- and three-dimensional Bénard convection. J. Fluid Mech. 147, 1–38.Google Scholar
Deardorff, J. W. 1970 Convective velocity and temperature scales for the unstable planetary boundary layer and for Rayleigh-Bénard convection. J. Atmos. Sci. 27, 1211–1213.Google Scholar
Deardorff, J. W. & Willis, G. E. 1967 Investigation of turbulent thermal convection between horizontal plates. J. Fluid Mech. 28, 675–704.Google Scholar
Deardorff, J. W. & Willis, G. E. 1985 Further results from a laboratory model of the convective planetary boundary layer. Boundary-Layer Met. 32, 205–236.Google Scholar
Deluca, E. E., Whrne, J., Rosner, R. & Cattaneo, F. 1990 Numerical simulations of soft and hard turbulence: preliminary results for two-dimensional convection. Phys. Rev. Lett. 64, 2370–2373.Google Scholar
Domaradzki, J. A. & Metcalfe, R. W. 1988 Direct numerical simulations of the effects of shear on turbulent Rayleigh-Bénard convection. J. Fluid Mech. 193, 499–531.Google Scholar
Eidson, T. M., Hussaini, M. Y. & Zang, T. A. 1986 Simulation of the turbulent Rayleigh-Bénard problem using a spectral/finite difference technique. ICASE Rep. 86–6. Also in Direct and Large Eddy Simulation of Turbulence (ed. U. Schumann & R. Friedrich), pp. 188–209. Vieweg, 1986.
Fiedler, B. H. & Khairoutdinov, M. 1994 Cell broadening in three-dimensional thermal convection between poorly conducting boundaries: large eddy simulations. Beitr. Phys. Atmos. 67, 235–241.Google Scholar
Groutzbach, G. 1982 Direct numerical simulation of laminar and turbulent Bénard convection. J. Fluid Mech. 119, 27–53.Google Scholar
Groutzbach, G. 1983 Spatial resolution requirement for direct numerical simulation of Rayleigh-Bénard convection. J. Comput. Phys. 49, 241–264.Google Scholar
Herring, J. R., 1966 Some analytic results in the theory of thermal convection. J. Atmos. Sci. 23, 672–677.Google Scholar
Herring, J. R. & Kerr, R. M. 1982 Comparison of direct numerical simulations with predictions of two-point closures for isotropic turbulence convecting a passive scalar. J. Fluid Mech. 118, 205–219.Google Scholar
Heslot, F., Castaing, B. & Libchaber, A. 1987 Transition to turbulence in helium gas. Phys. Rev. A 36, 5870–5873.Google Scholar
Kaimal, J. C., Wyngaard, J. C., Haugen, D. A., Cotea, O., Izumi, Y., Caughey, S. J. & Readings, C. J. 1976 Turbulence structure in the convective boundary layer. J. Atmos. Sci. 33, 2152–2169.Google Scholar
Kerr, R. M. 1985 Higher-order derivative correlations and the alignment of small-scale structures in isotropic numerical turbulence. J. Fluid Mech. 153, 31–58.Google Scholar
Kerr, R. M. 1991 Velocity, scalar and transfer spectra in numerical turbulence. J. Fluid Mech. 211, 309–332.Google Scholar
Kerr, R. M., Herring, J. R. & Brandenburg, A. 1995 Large-scale structure in Rayleigh-Bénard convection with impenetrable side-walls. Chaos, Solitons and Fractals 5, 2047–2053.Google Scholar
Kim, J., Moin, P. & Moser, R. 1987 Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, 133–166.Google Scholar
Kraichnan, R. H. 1967 Inertial ranges in two-dimensional turbulence. Phys. Fluids 10, 1417–1423.Google Scholar
Krishnamurti, R. 1995 Low frequency oscillations in turbulence convection: laboratory experiments. Fluid Dyn. Res. (in press).Google Scholar
Krishnamurti, R. & Howard, L. N. 1981 Large-scale flow generation in turbulent convection. Proc. Nat. Acad. Sci. 78, 1981–1985.Google Scholar
Lenschow, D. H., Wyngaard, J. C. & Pennell, W. T. 1980 Mean-field and second-moment budgets in a baroclinic, convective boundary layer. J. Atmos. Sci. 37, 1313–1326.Google Scholar
Mason, P. J. 1989 Large eddy simulation of a convective atmospheric boundary layer. J. Atmos. Sci. 46, 1492–1516.Google Scholar
Mclaughlin, J. B. & Orszag, S. A. 1982 Transition from periodic to chaotic thermal convection. J. Fluid Mech. 122, 123–142.Google Scholar
Meatais, O. & Lesieur, M. 1992 Spectral large-eddy simulation of isotropic and stably stratified turbulence. J. Fluid Mech. 239, 157–194.Google Scholar
Moeng, C. H. & Rotunno, R. 1990 Vertical-velocity skewness in the buoyancy-driven boundary layer. J. Atmos. Sci. 47, 1149–1162.Google Scholar
Moeng, C. H. & Wyngaard, J. C. 1988 Spectral analysis of large-eddy simulations of the convective boundary layer. J. Atmos. Sci. 45, 3573–3587.Google Scholar
Moser, R. D., Moin, P. & Leonard, A. 1983 A spectral numerical method for the Navier-Stokes equations with application to Taylor-Couette flow. J. Comput. Phys. 52, 524–544.Google Scholar
Prandtl, L. 1932 Meteorologische Anwendungen der Strömungslehre. Beitr. Phys. Atmos. 19, 188–202.Google Scholar
Priestley, C. H. B. 1954 Convection from a large horizontal surface. Austral. J. Phys. 7, 176–201.Google Scholar
Sano, M., Wu, X. Z. & Libchaber, A. 1989 Turbulence in helium-gas free convection. Phys. Rev. A 40, 6421–6430.Google Scholar
Schmidt, H. & Schumann, U. 1989 Conherent structure of the convective boundary layer derived from large-eddy simulations. J. Fluid Mech. 200, 511–562.Google Scholar
She, Z. S. 1989 On the scaling laws of thermal turbulent convection. Phys. Fluids A 1, 911–913.Google Scholar
Shraiman, B. I. & Siggia, E. D. 1990 Heat transport in high-Rayleigh-number convection. Phys. Rev. A 42, 3650–3653.Google Scholar
Sirovich, L., Balachandar, S. & Maxey, M. R. 1989 Simulations of turbulent thermal convection. Phys. Fluids A 1, 1911–1914.Google Scholar
Spalart, P. R. 1988 Direct simulation of a turbulent boundary layer up to Rθ = 1410. J. Fluid Mech. 187, 61–98.Google Scholar
Spalart, P. R., Moser, R. D. & Rogers, M. M. 1991 Spectral methods for the Navier-Stokes equations with one infinite and two periodic directions. J. Comput. Phys. 96, 297–324.Google Scholar
Sreenivasan, K. R. 1991 On local isotropy of passive scalars in turbulent shear flows. Proc. R. Soc. Lond. A 434, 165–182.Google Scholar
Thual, O. 1992 Zero Prandtl number convection J. Fluid Mech. 240, 229–258.Google Scholar
Tong, P. & Shen, Y. 1992 Relative velocity fluctuations in turbulent Rayleigh-Bénard convection. Phys. Rev. Lett. 69, 2066–2069.Google Scholar
Vincent, A. & Meneguzzi, M. 1991 The spatial structure and statistical properties of homogeneous turbulence. J. Fluid Mech. 225, 1–20.Google Scholar
Werne, J. 1993 Structure of hard-turbulent convection in two dimensions: Numerical evidence. Phys. Rev. E 48, 1020–1035.Google Scholar
Werne, J., Deluca, E. E., Rosner, R. & Cattaneo, F. 1991 Development of hard-turbulent convection in two dimensions: Numerical evidence. Phys. Rev. Lett. 67, 3519–3522.Google Scholar
Willis, G. E. & Deardorff, J. W. 1974 A laboratory model of diffusion into the convective planetary boundary layer. Q. J. R. Met. Soc. 102, 427–445.Google Scholar
Wu, X.-Z., Kadanoff, L., Libchaber, A. & Sano, M. 1990 Frequency power spectrum of temperature fluctuations in free convection. Phys. Rev Lett. 64, 2140–2143.Google Scholar
Wu, X.-Z. & Libchaber, A. 1992 Scaling relations in thermal turbulence: The aspect-ratio dependence. Phys. Rev. A 45, 842–845.Google Scholar
Wyngaard, J. C. & Weil, J. C. 1991 Transport asymmetry in skewed turbulence. Phys. Fluids A 3, 155–162.Google Scholar
Yakhot, V. 1992 The 4/5-Kolmogorov law for statistically stationary turbulence. Application to high Rayleigh number Bénard convection. Phys. Rev. Lett. 69, 769–771.Google Scholar
Zocchi, G., Moses, E. & Libchaber, A. 1990 Coherent structures in turbulent convection, an experimental study. Physica 166A, 387–407.Google Scholar