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Scaling of hard thermal turbulence in Rayleigh-Bénard convection

Published online by Cambridge University Press:  26 April 2006

Bernard Castaing ,
Gemunu Gunaratne ,
François Heslot ,
Leo Kadanoff ,
Albert Libchaber ,
Stefan Thomae ,
Xiao-Zhong Wu ,
Stéphane Zaleski  and
Gianluigi Zanetti
Bernard Castaing
Affiliation:
CNRS-CRTBT, 25 Avenue des Martyrs-P.B. 166X, 38042 Grenoble Cedex, France
Gemunu Gunaratne
Affiliation:
The Research Institutes, The University of Chicago, 5640 S. Ellis Avenue, Chicago, IL 60637, USA
François Heslot
Affiliation:
Collège de France, Matiere Condensée, Place Marcellin Berthelot, 75005 Paris, France
Leo Kadanoff
Affiliation:
The Research Institutes, The University of Chicago, 5640 S. Ellis Avenue, Chicago, IL 60637, USA
Albert Libchaber
Affiliation:
The Research Institutes, The University of Chicago, 5640 S. Ellis Avenue, Chicago, IL 60637, USA
Stefan Thomae
Affiliation:
Institut für Festkörperforschung der KFA, Postfach 1913, D-5170 Jülich, W. Germany
Xiao-Zhong Wu
Affiliation:
The Research Institutes, The University of Chicago, 5640 S. Ellis Avenue, Chicago, IL 60637, USA
Stéphane Zaleski
Affiliation:
Laboratoire de Physique Statistique, ENS, 24 rue Lhomond, 75231 Paris Cedex 05, France
Gianluigi Zanetti
Affiliation:
The Research Institutes, The University of Chicago, 5640 S. Ellis Avenue, Chicago, IL 60637, USA
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Abstract

An experimental study of Rayleigh-Bénard convection in helium gas at roughly 5 K is performed in a cell with aspect ratio 1. Data are analysed in a ‘hard turbulence’ region (4 × 107 < Ra < 6 × 1012) in which the Prandtl number remains between 0.65 and 1.5. The main observation is a simple scaling behaviour over this entire range of Ra. However the results are not the same as in previous theories. For example, a classical result gives the dimensionless heat flux, Nu, proportional to $Ra^{\frac{1}{3}}$ while experiment gives an index much closer to $\frac{2}{7}$. A new scaling theory is described. This new approach suggests scaling indices very close to the observed ones. The new approach is based upon the assumption that the boundary layer remains in existence even though its Rayleigh number is considerably greater than unity and is, in fact, diverging. A stability analysis of the boundary layer is performed which indicates that the boundary layer may be stabilized by the interaction of buoyancy driven effects and a fluctuating wind.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 204 , July 1989 , pp. 1 - 30
Copyright
© 1989 Cambridge University Press

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References

Anderson, A. C.: 1973 Low-noise ac bridge for resistance thermometry at low temperatures. Rev. Sci. Instrum. 44, 1475–1477.Google Scholar
Behringer, R. P.: 1985 Rayleigh-Bénard convection and turbulence in liquid helium. Rev. Mod. Phys. 57, 657–687.Google Scholar
Brunt, D.: 1927 The period of simple vertical oscillations in the atmosphere. Q. J. R. Met. Soc. 53, 30–32.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.: 1978a Non-linear properties of thermal convection. Rep. Prog. Phys. 41, 1929–1967.Google Scholar
Busse, F. H.: 1978b The optimum theory of turbulence. Adv. Appl. Mech. 18, 77–121.Google Scholar
Chan, S.-K.: 1971 Infinite Prandtl number turbulent convection. Stud. Appl. Math. 1, 13–49.Google Scholar
Chandrasekhar, 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
Datta, S. K.: 1965 Stability of spiral flow between concentric circular cylinders at low axial Reynolds number. J. Fluid Mech. 21, 635–640.Google Scholar
Deardorff, J. W. & Willis, G. E., 1965 The effect of two dimensionality upon the suppression of thermal turbulence. J. Fluid Mech. 23, 337–353.Google Scholar
Eckmann, J.-P.: 1981 Roads to turbulence in dissipative dynamical systems. Rev. Mod. Phys. 53, 643–654.Google Scholar
Fitzjarrald, D. E.: 1976 An experimental study of turbulent convection in air. J. Fluid Mech. 73, 693–719.Google Scholar
Foster, T. J. & Waller, S., 1985 Experiments on convection at very high Rayleigh numbers. Phys. Fluids 28, 455.Google Scholar
Gage, K. S. & Reid, W. H., 1968 The stability of thermally stratified plane Poiseuille flow. J. Fluid Mech. 33, 21–32.Google Scholar
Garron, A. M. & Goldstein, R. J., 1973 Velocity and heat transfer measurements in thermal convection. Phys. Fluids 16, 1818–1825.Google Scholar
Goldstein, R. J. & Chu, T. Y., 1971 Turbulent convection in a horizontal layer of air. Prog. Heat Mass Transfer 2, 55–75.Google Scholar
Goldstein, R. J. & Tokuda, S., 1980 Heat transfer by thermal convection at high Rayleigh numbers. Intl J. Heat Mass Transfer 23, 738–740.Google Scholar
Gough, D. O., Spiegel, E. A. & Toomre, J., 1975 Modal equations for cellular convection. J. Fluid Mech. 68, 695–719.Google Scholar
Herring, J. R.: 1963 Investigation of problems in thermal convection. J. Atmos. Sci. 20, 325–338.Google Scholar
Herring, J. R.: 1964 Investigation of problems in thermal convection: rigid boundaries. J. Atmos. Sci. 21, 277–290.Google Scholar
Heslot, F., Castaing, B. & Libchaber, A. 1987 Transition to turbulence in helium gas. Phys. Rev. A 36, 5870–5873.Google Scholar
Howard, L. N.: 1963 Heat transport by turbulent convection. J. Fluid Mech. 17, 405–432.Google Scholar
Howard, L. N.: 1966 Convection at high Rayleigh number. Applied Mechanics, Proc. of the 11th Congr. of Appl. Mech. Munich (Germany) (ed. H. Görtler), pp. 1109–1115. Springer.
Hughes, T. H. & Reid, W. H., 1967 The stability of spiral flow between rotating cylinders. Phil. Trans. R. Soc. Lond. A A263, 57–91.Google Scholar
Ingersoll, A. P.: 1966 Thermal convection with shear at high Rayleigh number. J. Fluid Mech. 25, 209–228.Google Scholar
Koschmieder, E. L.: 1974 Bénard convection. Adv. Chem. Phys. 26, 177–212.Google Scholar
Kraichnan, R. H.: 1962 Turbulent thermal convection at arbitrary Prandtl number. Phys. Fluids 5, 1374–1389.Google Scholar
Krishnamurti, R. & Howard, L. N., 1981 Large scale flow generation in turbulent convection. Proc. Natl Acad. Sci. 78, 1981–1985.Google Scholar
Krueger, E. R., Gross, A. & Di Prima, R. C. 1966 On the relative importance of Taylor-vortex and non-axisymmetric modes in flow between rotating cylinders. J. Fluid Mech. 24, 521–538.Google Scholar
Libchaber, A. & Maurer, J., 1982 A Rayleigh Bénard experiment: Helium in a small box. Nonlinear Phenomena at Phase Transitions and Instabilities, Proceedings NATO ASI, Geilo Mar 29-Apr 9, 1981 (ed. T. Riste), pp. 259–286. Plenum.
Long, R. R.: 1975 Some properties of turbulent convection with shear. Geophys. Fluid Dyn. 6, 337–350.Google Scholar
Long, R. R.: 1976 Relation between Nusselt number and Rayleigh number in turbulent thermal convection. J. Fluid Mech. 73, 445–451.Google Scholar
Lorenz, E. N.: 1963 Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141.Google Scholar
McCarthy, P. D.: 1973 Thermophysical properties of He from 2 to 1500 K with pressure to 1000 Atm. J. Phys. Chem. Ref. Data 2, 923–1024.Google Scholar
Malkus, W. V. R.: 1954a Discrete transitions in turbulent convection. Proc. R. Soc. Lond. A 225, 185–195.Google Scholar
Malkus, W. V. R.: 1954b The heat transport and spectrum of thermal turbulence. Proc. R. Soc. Lond. A A225, 196–212.Google Scholar
Malkus, W. V. R.: 1963 Outline of a theory of turbulent convection. In Theory and Fundamental Research in Heat Transfer. Pergamon.
Malkus, W. V. R. & Veronis, G. 1958 Finite amplitude cellular convection. J. Fluid Mech. 4, 225–260.Google Scholar
Monin, A. S. & Obukhov, A. M., 1953 Dimensionless characteristics of turbulence in the atmospheric surface layer. Dokl. Akad. Nauk USSR 93, 257–260.Google Scholar
Monin, A. S. & Obukhov, A. M., 1954 Basic turbulent mixing laws in the atmospheric surface layer. Trudy Geofiz. Inst. Akad. Nauk USSR no. 24 151, 163–187.Google Scholar
Monin, A. S. & Yaglom, A. M., 1971 Statistical Fluid Mechanics, vol. 1. The MIT Press.
Pellew, A. & Southwell, R. V., 1940 On maintained convective motion in a fluid heated from below. Proc. R. Soc. Lond. A 176, 312–343.Google Scholar
Prandtl, L.: 1932 Meteorologische Anwendungen der Strömungslehre. Beitr. Z. 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
Priestley, C. H. B.: 1959 Turbulent Transfer in the Lower Atmosphere. The University of Chicago Press.
Reid, W. H. & Harris, D. L., 1958 Some further results on the Bénard problem. Phys. Fluids 1, 102–110.Google Scholar
Roberts, P. H.: 1966 On non-linear Bénard convection. Non-Equilibrium Thermodynamics, Variational Techniques and Stability. Proc. Symp. University of Chicago, May 17–19, 1965 (ed. R. J. Donnelly, R. Herman & I. Prigogine), pp. 125–127. The University of Chicago Press.
Spiegel, E. A.: 1962 On the Malkus theory of turbulence. Mécanique de la turbulence, Colloque Internationaux de CNRS a Marseille, pp. 181–201. Éditions CNRS.
Spiegel, E. & Zaleski, S., 1984 Shear induced instability in reaction diffusion systems. Phys. Lett. 106A, 335–338.Google Scholar
Tanaka, H. & Miyata, H., 1980 Turbulent natural convection in a horizontal water layer heated from below. Intl J. Heat Mass Transfer 23, 1273–1281.Google Scholar
Threlfall, D. C.: 1975 Free convection in low-temperature gaseous helium. J. Fluid Mech. 67, 17–28.Google Scholar
Toomre, J., Gough, D. O. & Spiegel, E. A., 1977 Numerical solutions of single mode convection equations. J. Fluid Mech. 79, 1–31.Google Scholar
Toomre, J., Gough, D. O. & Spiegel, E. A., 1982 Time dependent solutions of multimode convection equations. J. Fluid Mech. 125, 99–122.Google Scholar
Townsend, A. A.: 1959 Temperature fluctuations over an heated surface. Fluid Mech. 5, 209–241.Google Scholar
Turner, J. S.: 1969 Buoyant plumes and thermals. Ann. Rev. Fluid Mech. 1, 29–44.Google Scholar
Väisälä 1925 Soc. Fennica, Commun. Phys. Maths 2, 38.