Vortex interaction with a translating sphere in a stratified temperature field

Abstract

The three-dimensional interaction of an initially cylindrical vortex tube with a solid sphere in a stratified temperature field has been investigated by solving the Navier-Stokes and energy equations. Particular attention is given to the coupled effect on the sphere Nusselt number by the vortex advection and temperature stratification. Comparison with the sphere Nusselt number influenced by a vortex in a uniform temperature field [Masoudi and Sirignano, Int. J. Heat Mass Transfer, 1997, 40(15), 3663–3673] shows that stratification has a profound effect on the sphere Nusselt number, qualitatively and quantitatively. Transient Nusselt number patterns are entirely different from those in a uniform temperature field; temperature stratification influences the vortex impact by nearly three fold. The sphere Nusselt number can be represented as the sum of Nusselt numbers in a uniform free-stream temperature at the average value and in a stratified stream with the given temperature variation and average temperature of zero. Therefore, in contrast to the commonly studied uniform free-stream case, stratification introduces an explicit dependence of the sphere Nusselt number on the temperature quantities. A correlation quantifying the coupled effect of the vortex-temperature stratification on the sphere heating, signifying a self-similar pattern in this unsteady problem has been produced. This correlation is also shown to be approximately valid for a liquid sphere. When the product of the vortex circulation and the gas-field temperature gradient is positive, the sphere time-averaged Nusselt number increases monotonically with an increase in the vortex circulation and with an increase in the vortex initial distance from the flow symmetry axis; when such product is negative, the Nusselt number increases with a decrease in vortex circulation or the vortex distance. Beyond a certain range of the vortex initial distance, the time-averaged Nusselt number reaches an asymptotic value. Based on these findings, it can be shown that in spray combustion systems coupled vortex-temperature stratification could have significant effect on droplets convective heating and their eventual evaporation