Three-phase numerical simulations of solidification with natural convection in a vertical cylindrical annulus

Abstract

We present numerical simulations of solidification in a vertical cylindrical annulus with temporal evolution of three interfaces, i.e., solid–liquid, solid–gas, and liquid–gas, and with the presence of natural convection. The numerical technique used is an axisymmetric front-tracking/finite difference method in which the interface separating two phases is represented by connected elements moving on a stationary grid. The governing Navier–Stokes and energy equations are solved in the entire domain with the no-slip and isothermal boundary conditions treated by interpolation techniques. A simple tri-junction condition is included due to the presence of three phases. Effects of various dimensionless parameters such as the Prandtl number Pr, the Stefan number St, the Rayleigh number Ra, the Weber number We, the dimensionless initial temperature of the liquid θ0, and the density ratio of the solid to liquid phases ρsl are investigated. The effect of the tri-junction in terms of the growth angle ϕgr is also considered. Numerical results show that the shape of the solidified phase is strongly affected by ρsl and ϕgr. Volume expansion (ρsl<1.0) produces a U-shaped top surface while shrinkage (ρsl>1.0) forms a cavity near the outer wall of the annulus. An increase in ϕgr results in an increase in its slope near the outer wall. Without volume change (ρsl=1.0), the top surface of the solidified phase becomes more curved with an increase in Pr or We. In contrast, varying St in the range of 0.01–1.0, Ra in the range of 103–106 or θ0 in the range of 1.0–2.0 has only a very minor effect on the top surface. Concerning the circulation flows induced by natural convection, the remarkable effects are yielded by variation of Ra and θ0: increasing Ra or θ0 results in an increase in the strength and the number of circulations. The circulations along with the interfacial tension force are the sources of the top front and solidification interface deformation. In addition, the effects of these parameters on the solidification rate are also investigated.