A finite difference method for phase change problems.
Charn-Jung Kim · Deep Blue (University of Michigan) · 1991
A new finite-difference method based on coordinate transformations is presented to treat phase change of pure substances, and diffusion-controlled solidification of binary alloys. The numerical method is formulated to be consistent with the well-established solution methodology used for fixed-boundary problems. Some novel features of the proposed method are: enhanced capability of inclusion of density difference between phases; unified treatment of multiple moving-boundaries; and conservation-based discretization of pseudo-velocities created by the immobilization of moving boundaries. A general transformed equation is employed which preserves conservative forms and thus reflects the conservation principles in a moving, curvilinear control volume in the physical coordinate. Consequently, the discontinuities in the physical quantities at the moving interface are simply expressed as the continuity of the interfacial fluxes in the transformed coordinate. The unknown interface positions and/or the unknown interface temperature are determined from the conservation of the interfacial fluxes. Pseudo-velocities are discretized according to the geometrical relation associated with moving control-volumes in the physical coordinate so that the pseudo-velocity fields independently satisfy the mass continuity. This special treatment eliminates parasitical mass sources and allows the interfaces to move in both coordinate directions. The method is first tested against some diffusion-controlled phase-change of pure substances for which analytical solutions are available. These include the classical Stefan problem and the growth of a spherical bubble in a superheated liquid. Numerical results agree to exceptional accuracies with the analytical solutions. Next, the method is extended to treat two-dimensional geometries and the fluid motion. As a final example, diffusion-controlled solidification of binary alloys is considered. A general temperature-position-correction equation is developed and solved simultaneously for fast convergence. In order to encourage other potential application, the numerical method is presented in a general form to accommodate various types of moving boundaries including those between two immiscible fluids, between fluids and solids, as well as phase-change interfaces. Further extensions to other important problems appear promising owing to the generality of the method.