Vortex Ring Formation within a Spherical Container with Natural Convection

摘要

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.