@Article{cmes.2009.049.217,
AUTHOR = {Gerardo Anguiano-Orozco, Rubén Avila},
TITLE = {Vortex Ring Formation within a Spherical Container with Natural Convection},
JOURNAL = {Computer Modeling in Engineering \& Sciences},
VOLUME = {49},
YEAR = {2009},
NUMBER = {3},
PAGES = {217--254},
URL = {http://www.techscience.com/CMES/v49n3/25386},
ISSN = {1526-1506},
ABSTRACT = {A numerical investigation of the transient, three dimensional, laminar natural convection of a fluid confined in a spherical container is carried out. Initially the fluid is quiescent with a uniform temperature *T*_{i} equal to the temperature of the wall of the container. At time *t*=0, the temperature of the wall is suddenly lowered to a uniform temperature *T*_{w}=0. The natural convection, that conducts to a vortex ring formation within the sphere, is driven by a terrestrial gravity force (laboratory gravity) and by the step change in the temperature of the wall. A scaling analysis of a simplified transient, two dimensional model, formulated in the cylindrical coordinate system, provides a qualitative description of the flow in the spherical enclosure, from start up (including the three stages of the transient process: boundary layer development, stratification and cooling-down) to the time at which the system reaches the new thermal equilibrium condition (uniform temperature *T*_{w}) without motion. The governing three dimensional Navier-Stokes equations for an incompressible fluid, formulated in the Cartesian coordinate system, have been numerically solved by using the *h/p* spectral element method. The Rayleigh number is in the range: 1 ×10^{3} ≤ *Ra* ≤ 1.5 ×10^{5}. The average Nusselt number *Nu*^{¯¯¯} as a function of time is evaluated at the wall of the container. The results provided by the spectral element method are in agreement with the scaling analysis results for low *Ra* numbers, *Ra* ≤ 1 ×10^{4}. As the *Ra* number is increased, in the range: 1 ×10^{5} ≤ *Ra* ≤ 1.5 ×10^{5}, the flow becomes unstable and oscillatory in the stratification stage. The temperature, vorticity and pressure fields for the three stages of the transient process are presented.},
DOI = {10.3970/cmes.2009.049.217}
}