@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 <i>T<sub>i</sub></i> equal to the temperature of the wall of the container. At time <i>t</i>=0, the temperature of the wall is suddenly lowered to a uniform temperature <i>T<sub>w</sub></i>=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 <i>T<sub>w</sub></i>) 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 <i>h/p</i> spectral element method. The Rayleigh number is in the range: 1 ×10<sup>3</sup> ≤ <i>Ra</i> ≤ 1.5 ×10<sup>5</sup>. The average Nusselt number <i>Nu<sup style="margin-left:-15px">¯¯¯</sup></i> 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 <i>Ra</i> numbers, <i>Ra</i> ≤ 1 ×10<sup>4</sup>. As the <i>Ra</i> number is increased, in the range: 1 ×10<sup>5</sup> ≤ <i>Ra</i> ≤ 1.5 ×10<sup>5</sup>, 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}
}