Skip to main content Accessibility help
×
×
Home

Instability of steady natural convection in a vertical fluid layer

  • R. F. Bergholz (a1)
Abstract

The instability of steady natural convection of a stably stratified fluid between vertical surfaces maintained at different temperatures is analysed. The linear stability theory is employed to obtain the critical Grashof and Rayleigh numbers, for widely varying levels of the stable background stratification, for Prandtl numbers ranging from 0·73 to 1000 and for the limiting case of infinite Prandtl number. The energetics of the critical disturbance modes also are investigated. The numerical results show that, if the value of the Prandtl number is in the low to moderate range, there is a transition from stationary to travelling-wave instability if the stratification exceeds a certain magnitude. However, if the Prandtl number is large, the transition, with increasing stratification, is from travelling-wave to stationary instability. The theoretical predictions are in excellent agreement with the experimental observations of Elder (1965) and of Vest & Arpaci (1969), for stationary instability, and in fair to good agreement with the experimental results of Hart (1971), for travelling-wave instability.

Copyright
References
Hide All
Batchelor, G. K. 1954 Heat transfer by free convection across a closed cavity between vertical boundaries at different temperatures. Quart. Appl. Math. 12, 209233.
Bergholz, R. F. 1976 Ph.D. thesis, Dept. of Applied Mechanics and Engineering Science, University of Michigan.
Birikh, R. V. 1966 On small perturbations of a plane parallel flow with a cubic velocity profile. Prikl. Mat. Mekh. 30, 356361.
Birikh, R. V., Gershuni, G. Z., Zhukhovitskii, E. M. & Rudakov, R. N. 1969 Stability of the steady convective motion of a fluid with a longitudinal temperature gradient. Prikl. Mat. Mekh. 33, 958968.
Birikh, R. V., Gershuni, G. Z., Zhukhovitskii, E. M. & Rudakov, R. N. 1972 On oscillatory instability of plane-parallel convective motion in a vertical channel. Prikl. Mat. Mekh. 36, 745748.
Eckert, E. R. G. & Carlson, W. O. 1961 Natural convection in an air layer enclosed between two vertical plates with different temperatures. Int. J. Heat Mass Transfer 2, 106120.
Elder, J. W. 1965 Laminar free convection in a vertical slot. J. Fluid Mech. 23, 7798.
Elder, J. W. 1966 Numerical experiments with free convection in a vertical slot. J. Fluid Mech. 24, 823.
Gershuni, G. Z. 1953 Stability of plane convective motion of a liquid. Zh. Tekn. Fiz. 23, 18381844.
Gill, A. E. 1966 The boundary layer regime for convection in a rectangular cavity. J. Fluid Mech. 26, 515536.
Gill, A. E. & Davey, A. 1969 Instabilities of a buoyancy-driven system. J. Fluid Mech. 35, 775798.
Gill, A. E. & Kirkham, C. C. 1970 A note on the stability of convection in a vertical slot. J. Fluid Mech. 42, 125127.
Gotoh, K. & Ikeda, N. 1972 Asymptotic solution of the instability problem of channel flows with antisymmetric velocity profile. J. Phys. Soc. Japan 32, 845850.
Gotoh, K. & Mizushima, J. 1973 The stability of convection between two parallel vertical walls. J. Phys. Soc. Japan 34, 14081413.
Gotoh, K. & Satoh, M. 1966 The stability of a natural convection between two parallel vertical planes. J. Phys. Soc. Japan 21, 542548.
Hart, J. E. 1971 Stability of the flow in a differentially heated inclined box. J. Fluid Mech. 47, 547576.
Kerczek, C. Von 1973 Ph.D. thesis, Johns Hopkins University.
Korpela, S. A., GÖZÜM, D. & Baxi, C. B. 1973 On the stability of the conduction regime of natural convection in a vertical slot. Int. J. Heat Mass Transfer 16, 16831690.
Macgregor, R. K. & Emery, A. F. 1969 Free convection through vertical plane layers–moderate and high Prandtl number fluids. J. Heat Transfer 91, 391403.
Mizushima, J. & Gotoh, K. 1976 The stability of natural convection in a vertical fluid layer. J. Fluid Mech. 73, 6575.
Moler, C. B. & Stewart, G. W. 1973 An algorithm for generalized eigenvalue problems. SIAM J. Numer. Anal. 10, 241256.
Mordchelles-Regnier, G. & Kaplan, C. 1963 Visualization of natural convection on a plane wall and in a vertical gap by differential interferometry. Transitional and turbulent regimes. Heat Transfer Fluid Mech. Inst., pp. 94111.
Nachtsheim, P. R. 1963 Stability of free-convection boundary-layer flows. N.A.S.A. Tech. Note D-2089.
Oshima, Y. 1971 Experimental studies of free convection in a rectangular cavity. J. Phys. Soc. Japan 30, 872.
Rudakov, R. N. 1966 On small perturbations of convective motion between vertical parallel planes. Prikl. Mat. Mekh. 30, 362368.
Rudakov, R. N. 1967 Spectrum of perturbations and stability of convective motion between vertical planes. Prikl. Mat. Mekh. 31, 376383.
Vahl Davis, G. De 1968 Laminar natural convection in an enclosed rectangular cavity. Int. J. Heat Mass Transfer 11, 16751693.
Vahl Davis, G. De & Mallinson, G. D. 1975 A note on natural convection in a vertical slot. J. Fluid Mech. 72, 8793.
Vest, C. M. & Arpaci, V. S. 1969 Stability of natural convection in a vertical slot. J. Fluid Mech. 36, 115.
Wilkes, J. O. & Churchill, S. W. 1966 The finite-difference computation of natural convection in a rectangular enclosure. A.I.Ch.E. J. 12, 161166.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed