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Stability of vertical natural convection boundary layers: some numerical solutions

Published online by Cambridge University Press:  29 March 2006

C. A. Hieber  and
B. Gebhart
C. A. Hieber
Affiliation:
Department of Thermal Engineering, Cornell University Clarkson College of Technology, Potsdam, New York.
B. Gebhart
Affiliation:
Department of Thermal Engineering, Cornell University
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Abstract

Linear stability theory is applied to the natural convection boundary layer arising from a vertical plate dissipating a uniform heat flux. By using a numerical procedure which is much simpler than those previously employed on this problem, computer solutions are obtained for a much larger range of the Grashof number (G). For a Prandtl number (σ) of 0·733, it is found that, as G → ∞: the effect of temperature coupling vanishes more rapidly than that of viscosity; the upper branch of the neutral curve is oscillatory but does approach a finite non-zero inviscid asymptote. For moderate and large values of σ, a loop appears in the neutral stability curve as a result of the merging of two unstable modes. As σ → ∞, the mode associated with the uncoupled (i.e. Orr–Sommerfeld) problem rapidly becomes less unstable than that arising from the temperature coupling, with the stability characteristics being independent of the thermal capacity of the plate. For small values of σ, only one unstable mode is found to exist with the coupling effect being negligible in the case of large thermal capacity plates but markedly destabilizing when the thermal capacity is small.

By obtaining numerical results out to G ≈ 1010 for the cases σ = 0·733 and 6·7, it becomes possible to attempt to directly relate the theory to the actual observance of turbulent transition. Based upon comparison with available experimental data, empirical correlations are obtained between the linear stability theory and the régimes in which: (i) the boundary layer is first noticeably oscillatory; (ii) the mean (temporal) flow quantities first deviate significantly from those of laminar flow.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 48 , Issue 4 , 27 August 1971 , pp. 625 - 646
Copyright
© 1971 Cambridge University Press

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