Hostname: page-component-f554764f5-rj9fg Total loading time: 0 Render date: 2025-04-09T21:54:35.871Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-Boussinesq and penetrative convection in a cylindrical cell

Published online by Cambridge University Press:  20 April 2006

R. W. Walden  and
Guenter Ahlers
R. W. Walden
Affiliation:
Bell Laboratories, Murray Hill, NJ 07974
Guenter Ahlers
Affiliation:
Bell Laboratories, Murray Hill, NJ 07974 and Department of Physics, University of California, Santa Barbara, CA 93106
Get access Rights & Permissions [Opens in a new window]

Abstract

Measurements of Rayleigh numbers R and Nusselt numbers N have been made for a cylindrical cell having a radius to height ratio of 4·72 and containing liquid helium I. The upper surface of the cell was maintained at a constant temperature T2 near T0 = 2·178 K where the fluid density has a maximum. The heat input to the cell bottom was varied quasi-statically while the lower surface temperature T1 was recorded. The penetration parameter [Pscr ]≡(T1−T2)/(T1−T0) ranged between 0·1 and 3 for our experiments.

For [Pscr ] < 1, the initial slope of N(R) for R near the critical Rayleigh number Rc was independent of [Pscr ]. For ${\cal P}\;{\mathop {\raise-2pt\hbox{$>$}}\limits^{\raise-8pt\hbox{$\sim$}}}\;1$, two hysteresis loops of N(R) were observed. One of them occurred very near Rc and is interpreted in terms of the inverted bifurcation associated with the onset of cellular convection of non-Boussinesq systems. The other, for R > Rc, is believed to correspond to the transition from cellular to two-dimensional flow. Near Rc and with [Pscr ] > 2 we also observed multistability, with the states believed to correspond to different numbers of convective cells.

For large [Pscr ] the onset of time-dependent flow occurred much closer to Rc than is the case for Boussinesq systems.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 109 , August 1981 , pp. 89 - 114
Copyright
© 1981 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.)

Article purchase

Temporarily unavailable

References

Ahlers, G. 1974 Low-temperature studies of the Rayleigh—Bénard instability and turbulence. Phys. Rev. Lett. 33, 1185–1188.Google Scholar
Ahlers, G. 1975 The Rayleigh—Bénard instability at helium temperatures. In Fluctuations, Instabilities, and Phase Transitions (ed. T. Riste), pp. 181–193. Plenum.
Ahlers, G. 1980 Effect of departures from the Oberbeck—Boussinesq approximation on the heat transport of horizontal convecting fluid layers. J. Fluid Mech. 98, 137–148.Google Scholar
Ahlers, G. & Behringer, R. P. 1978 Evolution of turbulence from the Rayleigh—Bénard instability. Phys. Rev. Lett. 40, 712–716.Google Scholar
Ahlers, G., Cross, M. C., Hohenberg, P. C. & Safran, S. 1981 The amplitude equation near the convective threshold: application to time-dependent heating experiments. J. Fluid Mech. (to appear).Google Scholar
Ahlers, G. & Walden, R. W. 1980 Turbulence near onset of convection. Phys. Rev. Lett. 44, 445–448.Google Scholar
Azouni, M. A. & Normand, C. 1981 Thermoconvective instability of water between 0 and 4C. J. Fluid Mech. (submitted).
Behringer, R. P. & Ahlers, G. 1977 Heat transport and critical slowing down near the Rayleigh—Bénard instability in cylindrical containers. Phys. Lett. A 62, 329–331.Google Scholar
Behringer, R. P. & Ahlers, G. 1982 Heat transport and the time evolution of steady flow near the Rayleigh—Bénard instability. (To be published.)
Boger, D. V. & Westwater, J. W. 1967 Effect of buoyancy on the melting and freezing process. Trans. A.S.M.E. C, J. Heat Transfer 89, 81–89.Google Scholar
Brown, S. N. & Stewartson, K. 1978 On finite amplitude Bénard convection in cylindrical containers. Proc. Roy. Soc. A 360, 455–469.Google Scholar
Busse, F. H. 1967 The stability of finite amplitude cellular convection and its relation to an extremum principle. J. Fluid Mech. 30, 625–649.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Clarendon.
Charlson, G. S. & Sani, R. L. 1970 Thermoconvective instability in a bounded cylindrical fluid layer. Int. J. Heat Mass Transfer, 13, 1479–1496.Google Scholar
Davis, S. H. & Segel, L. A. 1968 Effects of surface curvature and property variations on cellular convection. Phys. Fluids 11, 470–476.Google Scholar
Deardorff, J. W., Willis, G. E. & Lilly, D. K. 1969 Laboratory investigation of non-steady penetrative convection. J. Fluid Mech. 35, 7–31.Google Scholar
Dubois, M., Berge, P. & Wesfreid, J. 1978 Non-Boussinesq convective structures in water near 4C. J. Phys. 39, 1253–1257.Google Scholar
Farhadieh, R. & Tankin, R. S. 1974 Interferometric study of two-dimensional Bénard convection cells. J. Fluid Mech. 66, 739–752.Google Scholar
Furumoto, A. & Rooth, C. 1961 Observations on convection in water cooled from below. Geophysical Fluid Dynamics 3, 14 pp. Woods Hole Oceanographic Inst.Google Scholar
Hoard, C., Robertson, C. & Acrivos, A. 1970 Experiments on the cellular structure in Bénard convection. Int. J. Heat Mass Transfer 13, 849–856.Google Scholar
Koschmieder, E. L. 1966 On convection on a uniformly heated plate. Beitr. Phys. Atmos. 39, 1–11.Google Scholar
Koschmieder, E. L. 1969 On the wavelength of convective motions. J. Fluid Mech. 35, 527–530.Google Scholar
Koschmieder, E. L. 1974 Bénard convection. Adv. Chem. Phys. 26, 177–212.Google Scholar
Koschmieder, E. L. & Pallas, S. G. 1974 Heat transfer through a shallow horizontal convecting fluid layer. Int. J. Heat Mass Transfer 17, 991–1002.Google Scholar
Krishnamurti, R. 1968 Finite amplitude convection with changing mean temperature. Part 1. Theory. Part 2. An experimental test of the theory. J. Fluid Mech. 33, 445–455; 457–463.Google Scholar
Legros, J. C., Longree, D. & Thomaes, G. 1974 Bénard problem in water near 4C. Physica 72, 410–414.Google Scholar
Martin, P. C. 1975 The onset of turbulence: a review of recent developments in theory and experiment. Proc. Int. Conf. on Statistical Physics, Budapest (ed. L. Pál & P. Szépfalusy). North-Holland.
Merker, G. P., Waas, P. & Grigubb, U. 1979 Onset of convection in a horizontal water layer with maximum density effects. Int. J. Heat Mass Transfer 22, 505–515.Google Scholar
Moore, D. R. & Weiss, N. O. 1973 Nonlinear penetrative convection. J. Fluid Mech. 61, 553–581.Google Scholar
Musman, S. 1968 Penetrative convection. J. Fluid Mech. 31, 343–360.Google Scholar
Myrup, L., Gross, D., Hoo, L. S. & Goddard, W. 1970 Upside down convection. Weather 25, 150–157.Google Scholar
Normand, C., Pomeau, Y. & Velarde, M. G. 1977 Convective instability: a physicist's approach. Rev. Mod. Phys. 49, 581–624.Google Scholar
Palm, E. 1960 On the tendency towards hexagonal cells in steady convection. J. Fluid Mech. 8, 183–192.Google Scholar
Palm, E., Ellingsen, T. & Gjevic, B. 1967 On the occurrence of cellular motion in Bénard convection. J. Fluid Mech. 30, 651–661.Google Scholar
Richter, F. M. 1978 Experiments on the stability of convection rolls in fluids whose viscosity depends on temperature. J. Fluid Mech. 89, 553–560.Google Scholar
Rossby, H. T. 1969 Bénard convection with and without rotation. J. Fluid Mech. 36, 309–335.Google Scholar
Schlüter, A., Lortz, D. & Busse, F. 1965 On the stability of finite amplitude convection. J. Fluid Mech. 23, 129–144.Google Scholar
Segel, L. A. 1965 The non-linear interaction of a finite number of disturbances to a layer of fluid heated from below. J. Fluid Mech. 21, 359–384.Google Scholar
Segel, L. A. & Stuart, J. T. 1962 On the question of the preferred mode in cellular thermal convection. J. Fluid Mech. 13, 289–306.Google Scholar
Silverston, P. L. 1958 Wärmedurchgang in Waagerechten Flüssigkeitsschichten. Forsch. Ing.-Wes. 24, 29–32; 59–69.Google Scholar
Somerscales, E. & Dougherty, T. S. 1970 Observed flow patterns at the initiation of convection in a horizontal liquid layer heated from below. J. Fluid Mech. 42, 755–768.Google Scholar
Spiegel, E. A. 1971 Convection in stars. I. Basic Boussinesq convection. Ann. Rev. Astron. Astrophys. 9, 323–352.Google Scholar
Sun, Z.-S., Chi, T. & Yen, Y.-C. 1969 Thermal instability of a horizontal layer of liquid with maximum density. A.I.Ch.E. J. 15, 910–915.Google Scholar
Tankin, R. S. & Farhadieh, R. 1971 Effects of thermal convection currents on formation of ice. Int. J. Heat Mass Transfer 14, 953–961.Google Scholar
Tippelskirch, H. V. 1956 Beitr. Phys. Atmos. 29, 218–233.
Townsend, A. A. 1966 Internal waves produced by a convective layer. J. Fluid Mech. 24, 307–319.Google Scholar
Veronis, G. 1963 Penetrative convection. Astrophysical J. 137, 641–663.Google Scholar
Walden, R. W. & Ahlers, G. 1979 Bull. Am. Phys. Soc. 24, 1135.
Whitehead, J. A. 1975 A survey of hydrodynamic instabilities. In Fluctuations, Instabilities, and Phase Transitions (ed. T. Riste), pp. 153–180. Plenum.
Yen, Y.-C. 1968 Onset of convection in a layer of water formed by melting ice from below. Phys. Fluids 11, 1263–1270.Google Scholar
Yen, Y.-C. & Galea, F. 1969 Onset of convection in a water layer formed continuously by melting ice. Phys. Fluids 12, 509–516.Google Scholar