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Linear stability and bifurcation of natural convection within arbitrary-gap spherical annuli
Abstract
Natural convection between spherical shells maintained at distinct temperatures is dependent on the annular radius ratio, fluid Prandtl number, and the Grashof number. For a given radius ratio and Prandtl number, the flow changes from a steady, well defined flow for sufficiently small Grashof numbers to a quasi-periodic or secondary steady flow for Grashof numbers exceeding a critical value. This problem has been addressed only for narrow-gap annuli and only for a small range of Prandtl numbers. In those cases it was shown that the flow becomes unstable at the critical Grashof number and bifurcates to another stable flow. A linear stability analysis is performed for steady, axisymmetric natural convection of an arbitrary Prandtl number fluid within arbitrary-gap, concentric spherical annuli to three-dimensional, time-dependent disturbances. The fluid is Boussinesq, and a uniform gravity field acts vertically parallel to the poles. The basic motion is modeled by an asymptotic series solution in powers of ${\cal R}$, the square root of the Grashof number. The linearized disturbance equations are solved using a Chebyshev/Legendre tau spectral method. Two methods for determining the eigenvalues of the resulting algebraic eigenvalue problem are implemented. One, a full eigenvalue method, simply solves for all eigenvalues and sorts them. The second method, an iterative Arnoldi subspace method, reduces the full algebraic eigenvalue problem to a smaller one, thus reducing memory requirements. Additionally, a spectrum transformation is implemented which gives emphasis to the leading eigenvalues, which are the eigenvalues of interest. It is determined that for fixed Prandtl number and radius ratio there is a critical Grashof number, ${\cal R}\sb{c}$, at which the flow becomes unstable and bifurcates to another flow. Neutral stability results are obtained for radius ratios ranging from 20% to 92.5% and $Pr$ ranging from 0.1 to 100, mapping ${\cal R}\sb{c}$ as a function of $Pr$. It is determined that for fixed annular radius ratio, there exists a transition Prandtl number, $Pr\sb{t}$, below which the bifurcated flows are time periodic (a Hopf bifurcation) and above which the bifurcated flows are steady. For $PrPr\sb{t}$ the critical disturbances are generally three-dimensional and furthermore ${\cal R}\sb{c}$ is inversely proportional to the square root of $Pr$.
Subject Area
Mechanical engineering
Recommended Citation
TeBeest, Kevin G, "Linear stability and bifurcation of natural convection within arbitrary-gap spherical annuli" (1992). ETD collection for University of Nebraska-Lincoln. AAI9308199.
https://digitalcommons.unl.edu/dissertations/AAI9308199