A Front-Tracking Method for Three-Phase Computations of Solidification with Volume Change

Abstract

We use a front-tracking method to simulate solidi cation with volume change of a droplet on a xed cooling plate. The problem includes temporal evolution of three interfaces, i.e., solid-liquid, solid-air, and liquid-air, that are explicitly tracked under the assumption of axisymmetry. The solid-liquid interface is propagated with a normal velocity that is calculated from the normal temperature gradient across the front and the latent heat. The liquid-air front is advected by the velocity interpolated from nearest bulk uid ow velocities. Accordingly, the evolution of the solid-air front is simply the temporal imprint of the triple point at which simple and straightforward conditions are imposed. The governing Navier-Stokes equations are solved for the whole domain, setting the velocities in the solid phase to zero and with the non-slip condition on the solid-liquid interface. Computational results are compared with exact solutions for twodimensional Stefan problems and with corresponding experimental results, and show good agreement.

Study, Modelling and Implementation of the Level Set Method Used in Micromachining Processes

“…2. 9. It can be observed that the large oscillations disappears quickly while the parts with smoother curvatures remain longer.…”

“…For instance, an implementation of the geometric approach, also known as front tracking or marker particle method, has been developed recently for fluid interfaces in compressible flows [8], as well as an algorithm to simulate solidification with volume change of a droplet on a fixed cooling plate [9]. The VOF method is still used for many 10 (a) (b) applications too, such as the simulation of wall to liquid heat transfer [10] or the simulation of electrohydrodynamic two-phase flows [11].…”

“…A simple example is given by the Burgers' equation: 9) which describes the motion of a compressible fluid in one dimension. The solution of this equation can develop discontinuities, known as shocks, even for arbitrarily smooth initial data.…”

“…In contrast to complex MEMS structures, such as experiments (a), (b), (c), (g) and (h), a much smaller grid could be used without sacrificing accuracy in the case of simpler topologies with no sensitive details, such as experiments (d)-(g). 9.33gum 2.46gum…”

“…4 Find a proper initial surface with algorithm 10. 5 Build the SDF φ of the initial surface by applying the propagating algorithm 9 and keep these values (φ 0 ) for reinitialization process. 6 Calculate the time steps for convection model ∆t c , energy model ∆t e , and reinitialization process ∆t, according to (3.21), (3.16), and (3.31) respectively.…”