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Linear stability and bifurcation of natural convection flows in narrow gap, concentric spherical annulus enclosures
Abstract
Natural convection in a fluid filling the narrow gap between two isothermal, concentric spheres at different temperatures is strongly dependent on radius ratio, Prandtl number, and Grashof number. The gravitational acceleration vector is not everywhere parallel to the temperature gradient, and so the base flow is non-quiescent. Hence this problem is different from the spherical analog of the Rayleigh-Benard problem. For fixed values of radius ratio and Prandtl number, the flow is steady and axisymmetric for sufficiently small Grashof number, or quasi-periodic and axisymmetric for Grashof numbers greater than a critical value. The hypothesis that the transition is a flow bifurcation is tested by solving an appropriate eigenvalue problem for infinitesimal disturbances to the base flow in a Boussinesq fluid. The numerical solution of the eigenvalue problem involves the use of poloidal and toroidal potentials; and a new spectral method, called the modified tau method, which eliminates spurious eigenvalues. The critical Grashof number, critical eigenvalues, and corresponding eigenvectors are obtained as functions of the radius ratio, Prandtl number, and longitudinal wavenumber. Critical Grashof numbers range from 1.18 $\times$ 10$\sp4$ to 2.63 $\times$ 10$\sp3$ as Prandtl number Pr increases from zero to 0.7, for radius ratios of 0.900 and 0.950. A transitional Prandtl number $Pr\sb{t}$ exists such that for $Pr < Pr\sb{t}$ the bifurcation is time-periodic and axisymmetric. For $Pr > Pr\sb{t}$ the bifurcation is steady and non-axisymmetric with wavenumber two. This transition is due to a change in the identity of the critical eigenvalue. The first approximation to the bifurcated flow is obtained using the critical eigenvectors. For $Pr < Pr\sb{t}$ the bifurcation sets in as a cluster of relatively strong cells with alternating directions of rotation. The cells remain fixed in location, but pulsate with time. The cluster moves toward the top of the annulus at Pr increases toward $Pr\sb{t}$. An important feature of the non-axisymmetric bifurcation for $Pr > Pr\sb{t}$ is a set of four cells located at each pole of the annulus. The radial alternates direction in moving from any one cell to an adjacent one. For fixed radius ratio, the average Nusselt number at criticality varies only slightly with Prandtl number. Rayleigh numbers compare favorably with extrapolations of experimental data.
Subject Area
Mechanical engineering
Recommended Citation
Gardner, David R, "Linear stability and bifurcation of natural convection flows in narrow gap, concentric spherical annulus enclosures" (1988). ETD collection for University of Nebraska-Lincoln. AAI8818620.
https://digitalcommons.unl.edu/dissertations/AAI8818620