Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-19T15:49:22.275Z Has data issue: false hasContentIssue false
  • English
  • Français

Three-dimensional instability of axisymmetric buoyant convection in cylinders heated from below

Published online by Cambridge University Press:  26 April 2006

M. Wanschura ,
H. C. Kuhlmann  and
H. J. Rath
M. Wanschura
Affiliation:
Center of Applied Space Technology and Microgravity, ZARM - University of Bremen, Am Fallturm, 28359 Bremen, Germany
H. C. Kuhlmann
Affiliation:
Center of Applied Space Technology and Microgravity, ZARM - University of Bremen, Am Fallturm, 28359 Bremen, Germany
H. J. Rath
Affiliation:
Center of Applied Space Technology and Microgravity, ZARM - University of Bremen, Am Fallturm, 28359 Bremen, Germany
Get access Rights & Permissions [Opens in a new window]

Abstract

The stability of steady axisymmetric convection in cylinders heated from below and insulated laterally is investigated numerically using a mixed finite-difference/Chebyshev collocation method to solve the base flow and the linear stability equations. Linear stability boundaries are given for radius to height ratios γ from 0.9 to 1.56 and for Prandtl numbers Pr = 0.02 and Pr = 1. Depending on γ and Pr, the azimuthal wavenumber of the critical mode may be m = 1, 2, 3, or 4. The dependence of the critical Rayleigh number on the aspect ratio and the instability mechanisms are explained by analysing the energy transfer to the critical modes for selected cases. In addition to these results the onset of buoyant convection in liquid bridges with stress-free conditions on the cylindrical surface is considered. For insulating thermal boundary conditions, the onset of convection is never axisymmetric and the critical azimuthal wavenumber increases monotonically with γ. The critical Rayleigh number is less then 1708 for most aspect ratios.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 326 , 10 November 1996 , pp. 399 - 415
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

Bohm, J., Lüdge, A. & Schröder, W. 1994 Crystal growth by floating zone melting. In Handbook of Crystal Growth Vol. 2a : Bulk Crystal Growth (ed. D. T. J. Hurle) p. 213. Elsevier.
Buell, J. C. & Catton, I. 1983 The effect of wall conduction on the stability of a fluid in a right circular cylinder heated from below. Trans. ASME J. Heat Transfer 105, 255.Google Scholar
Busse, F. H. 1978 Non-linear properties of thermal convection. Rep. Prog. Phys. 41, 1929.Google Scholar
Charlson, G. S. & Sani, R. L. 1970 Thermoconvective instability in a bounded cylindrical fluid layer. Intl J. Heat Mass Transfer 13, 1479.Google Scholar
Charlson, G. S. & Sani, R. L. 1971 On thermoconvective instability in a bounded cylindrical fluid layer. Intl J. Heat Mass Transfer 14, 2157.Google Scholar
Charlson, G. S. & Sani, R. L. 1975 Finite amplitude axisymmetric thermoconvective flows in a bounded cylindrical layer of fluid. J. Fluid Mech. 71, 209.Google Scholar
Chen, Y. Y. 1992a Boundary conditions and linear analysis of finite-cell Rayleigh—Bénard convection. J. Fluid Mech. 241, 549.Google Scholar
Chen, Y. Y. 1992b Finite-size effects on linear stability of pure-fluid convection. Phys. Rev. A 45, 3727.Google Scholar
Crespo del Arco, E. & Bontoux, P. 1989 Numerical simulation and analysis of axisymmetric convection in a vertical cylinder: An effect of Prandtl number. Phys. Fluids A 1, 1348.Google Scholar
Hardin, G. R. & Sani, R. L. 1993 Buoyancy-driven instability in a vertical cylinder: Binary fluids with Soret effect. Part 2: Weakly non-linear solutions. Intl J. Numer. Meth. Fluids 17, 755.Google Scholar
Hardin, G. R., Sani, R. L., Henry, D. & Roux, B. 1990 Buoyancy-driven instability in a vertical cylinder: Binary fluids with Soret effect. Part 1: General theory and stationary stability results. Intl J. Numer. Meth. Fluids 10, 79.Google Scholar
Koschmieder, E. L. 1993 Bénard Cells and Taylor Vortices. Cambridge University Press.
Liang, S. F., Vidal, A. & Acrivos, A. 1969 Buoyancy-driven convection in cylindrical geometries. J. Fluid Mech. 36, 239.Google Scholar
Müller, G., Neumann, G. & Weber, W. 1984 Natural convection in vertical Bridgeman configurations. J. Cryst. Growth 70, 78.Google Scholar
Neumann, G. 1986 Berechnung der thermischen Auftriebskonvektion in Modellsystemen zur Kristallzüchtung. Dissertation, Technische Fakultät der Universität Erlangen-Nürnberg.
Neumann, G. 1990 Three-dimensional numerical simulation of buoyancy-driven convection in vertical cylinders heated from below. J. Fluid Mech. 214, 559.Google Scholar
Rosenblat, S. 1982 Thermal convection in a vertical circular cylinder. J. Fluid Mech. 122, 395.Google Scholar
Stork, K. & Müller, U. 1975 Convection in boxes: An experimental investigation in vertical cylinders and annuli. J. Fluid Mech. 71, 231.Google Scholar
Wagner, C., Friedrich, R. & Narayanan, R. 1994 Comments on the numerical investigation of Rayleigh and Marangoni convection in a vertical cylinder. Phys. Fluids 6, 1425.Google Scholar
Wanschura, M., Shevtsova, V. M., Kuhlmann, H. C. & Rath, H. J. 1995 Convective instability mechanisms in thermocapillary liquid bridges. Phys. Fluids 7, 912.Google Scholar