A Fixed-Grid Front-Tracking Algorithm for Solidification Problems. Part I - Method and Validation

This paper presents a method for solving moving boundary problems associated with phase-change. This method is an explicit interface-tracking scheme that involves the reconstruction and advection of the moving interface on a fixed grid. Three distinct steps are undertaken to handle the movement of the interface: advection and reconstruction (tracking); calculation of normal velocities; and the solution of the governing equations for different phases. Details of each step and its implementation is provided. The transient heat diffusion equation in two space dimensions is the governing equation for energy transport. In order to validate the approach, results obtained from the front tracking scheme are compared with exact analytical solutions for melting problems in Cartesian and cylindrical coordinates. It is shown that the numerical and analytical results are in excellent agreement. Finally, problems involving the multi-dimensional solidification of a pure aluminum ingot subject to bi-directional heat extraction are presented. The results indicate that the algorithm was able to accurately track the front under these aggressive conditions which caused large curvature. In Part II, the front-tracking approach described herein is applied to directional solidification problems influenced by melt convection. § Submitted for publication in Numerical Heat Transfer, August 2002 ¶ Research Assistant † Professor, person to whom correspondence should be addressed ‡ Postdoctoral Research Associate 2 INTRODUCTION Moving liquid-vapor or solid-liquid interfaces encountered in phase change and materials processing problems may in general be highly distorted. The shape and position of such an interface is not known a priori, but instead must be obtained as part of the problem solution. The problem in such cases consists not only of a careful solution of the governing equations in the two phases but also of an accurate tracking of the interface in response to the thermal, compositional, or flow fields computed at each time step. The interface between phases has been variously treated in the literature with interface-capturing and interface-tracking methods [1]. In the interface-capturing category, an attempt is made to resolve the details of the structure of the interface. The interface is modeled only to the extent that the actual physical discontinuity is known to be someplace near the middle of the (temperature) gradient; no specific modeling is needed for the interface, except perhaps for finer meshing in its vicinity. The enthalpy method is an example of this one-domain method. The different phases are considered together in the solution, and the properties are either changed discretely at the interface or continuously over a range near the interface. The interface itself is generally not tracked. Methods of interface tracking, as distinct from capturing, may be divided into front-tracking and volume-tracking approaches. Interface-tracking studies in the literature were reviewed by Chen et al. [2] with reference to the simulation of bubble rise and distortion. In front tracking [3] the interface is identified by an ordered set of marker points located on the interface, and is represented by the distance between the points and some reference surface. A line connecting the marker points, usually a piecewise polynomial, represents the front. An irregular grid is used in the vicinity of the interface. In an extension of this method Unverdi and