Hydrodynamic stability of Rayleigh-Bénard convection with constant heat flux boundary condition
Authors:
H. Park and L. Sirovich
Journal:
Quart. Appl. Math. 49 (1991), 313-332
MSC:
Primary 76E15; Secondary 76M25, 76R99
DOI:
https://doi.org/10.1090/qam/1106395
MathSciNet review:
MR1106395
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Abstract: We study the onset of thermal instability with the heat flux prescribed on the fluid boundaries. Assuming Boussinesq fluid, the Landau equation, which describes the evolution of the amplitude of the convection cells, is derived using the small amplitude expansion technique. For the case of a three-dimensional rectangular box with aspect ratio (8, 4, 1), the incipient convection cell is a two-dimensional one at $pr = 0.72$, which is confirmed by the numerical solution of the three-dimensional Boussinesq equation with a Chebyshev-Fourier pseudospectral code. The secondary bifurcation gives rise to an oscillatory two-dimensional roll for the same Prandtl number at $R = 2.0{R_c}$ and the motion becomes three dimensional at $R = 2.8{R_c}$.
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H. Jeffreys, The stability of a layer of fluid heated below, Philos. Mag. (7) 2, 833–44 (1926)
L. Kleiser and U. Schuman, Spectral Simulations of the Laminar-Turbulent Transition Process in Plane Poiseuille Flow, Spectral Methods for Partial Differential Equations, editors: R. G. Voigt, D. Gottlieb, and M. Y. Hussaini, SIAM, 141–163 (1984)
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C. Normand, Y. Pomeau, and M. G. Velarde, Rev. Modern Phys. 49, 581 (1977)
N. Riahi and F. H. Busse, Nonlinear convection in a layer with nearly insulating boundaries, J. Fluid Mech. 96, 243–256 (1980)
A. Schlüter, D. Lortz, and F. Busse, On the stability of steady finite amplitude convection, J. Fluid Mech. 23, 129–44 (1965)
L. Sirovich, H. Tarman, and M. Maxey, Analysis of turbulent thermal convection, Sixth Symposium of Turbulent Shear Flow, Toulouse, 1987
L. Sirovich and H. Park, Turbulent thermal convection in a finite domain: Part I: Theory, Phys. Fluids A, 2 (9), 1649–1658 (1990)
E. M. Sparrow, R. J. Goldstein, and V. K. Jonsson, Thermal instability in a horizontal fluid layer: effect of boundary conditions and non-linear temperature profile, J. Fluid Mech. 18, 513–28 (1964)
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Article copyright:
© Copyright 1991
American Mathematical Society