Linear stability of natural convection in spherical annuli

Natural convection in a Boussinesq fluid filling the narrow gap between two isothermal, concentric spheres at different temperatures depends strongly on radius ratio, Prandtl number, and Grashof number. When the inner sphere has a higher temperature than the outer sphere, and for fixed values of radius ratio and Prandtl number, experiments show the flow to be steady and axisymmetric for sufficiently small Grashof number and quasi-periodic and axisymmetric for Grashof numbers greater than a critical value. It is our hypothesis that the observed transition is a flow bifurcation. This hypothesis is examined by solving an appropriate eigenvalue problem. 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* 10⁴ to 2.63 * 10³ as Prandtl number Pγ increases from zero to 0.7, for radius ratios of 0.900 and 0.950. A transitional Prandtl number Pr_t exists such that for Pγ < Pr_t the bifurcation is time-periodic and axisymmetric. For Pγ > Pγ_tthe bifurcation is steady and non-axisymmetric with wavenumber two. A first approximation to the bifurcated flow is obtained using the critical eigenvectors. For Pγ < Pγ_tthe 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 as Pγ increases toward Pr_t. An important feature of the non-axisymmetric bifurcation for Pγ > Pr_t is a set of four cells located at each pole of the annulus in which the radial velocity 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.