Three-dimensional sharp and conservative VOF method for the simulation of binary solidification

We present a 3D adaptive octree based numerical method for the simulation of binary solidification, with the temperature and the concentration fields being strongly coupled at the solidification front. The volume of fluid approach (VOF) is used to advance the front, which is geometrically reconstructed as a sharp interface separating the liquid and the solid domains. On basis of this, the embedded boundary method (EBM), together with the finite volume method, is employed to discretize the temperature and concentration fields at two sides of the interface in a sharp manner. The major novelty of the present study is displayed in several aspects: It is the first attempt to develop fully sharp schemes for the simulation of binary solidification in the VOF framework, while previous VOF-type methods consider the jump conditions at the interface as source terms; second the geometrically reconstructed interface and the EBM enable us to capture those discontinuities accurately without any artificial smearing, and particularly, we will show how to construct a second-order scheme for the flux jump condition of the temperature field across the interface; finally compared to other sharp schemes, we will show that the present method has natural advantages to adapt with the fluid flow solvers, because the hybrid VOF-EBM method guarantees good performance in solving the complex solid-fluid coupling problems based on the finite volume framework. Owing to all these advantages, we will show that regardless of pure substance or binary solution, all the numerical results agree well with the benchmark results. In particular, we apply the numerical method to model the solidification of a Ni-Cu alloy, which is thought to be very challenging owing to the high Lewis number. Besides, coupling to the fluid flow solver, we demonstrate that this method can predict the solidification/melting of an ice cylinder/sphere accurately in a salted solution under forced convection, by which the fluid, temperature and concentration equations are all solved sharply.