We demonstrate that, if a globally smooth virtual circulation-preserving velocity exists, Kelvin’s and Helmholtz’s theorems can be extended to some non-ideal flows which are viscous, baroclinic or with non-conservative body forces. Then we track vortex surfaces frozen in the virtual velocity in the non-ideal flows, based on the evolution of a vortex-surface field (VSF). For a flow with a viscous-like diffusion term normal to the vorticity, we obtain an explicit virtual velocity to accurately track vortex surfaces in time. This modified flow is dissipative but prohibits reconnection of vortex lines. If a globally smooth virtual velocity does not exist, an approximate virtual velocity can still facilitate the tracking of vortex surfaces in non-ideal flows. In a magnetohydrodynamic Taylor–Green flow, we find that the conservation of vorticity flux is significantly improved in the VSF evolution convected by the approximate virtual velocity instead of the physical velocity, and the spurious vortex deformation induced by the Lorentz force is eliminated.

Abstract

I n s t i t u t e of Mechanics, Chinese A c a d e m y of Sciences, Beijing 100080, C h i n a ShengHeat transfer coefficients for horizontally immersed tubes have been studied in a model of ICFB (Internally Circulating Fluidized Bed). The characteristics in ICFB were found to be significantly different from those in bubbling bed. There is a flowing zone with high velocity in the heat exchange zone.The heat transfer coefficients strongly depend on the fluidized velocity in the flowing zone. The heat exchange process and suitable bed temperature can be controlled according to this feature. Based on the results of the experiments, a formulation for heat transfer has been developed.Keywords: internally circulation, fluidized b e d , i m m e r s e d tube, heat t r a n s f e r .