A Newton method with adaptive finite elements for solving phase-change problems with natural convection

We present a new numerical system using finite elements with mesh adaptivity for the simulation of solid–liquid phase change systems. In the liquid phase, the natural convection flow is simulated by solving the incompressible Navier–Stokes equations with Boussinesq approximation. A variable viscosity model allows the velocity to progressively vanish in the solid phase, through an intermediate mushy region. The phase change is modeled by introducing an implicit enthalpy source term in the heat equation. The final system of equations describing the liquid–solid system by a single domain approach is solved using a Newton iterative algorithm. The space discretization is based on a P2–P1 Taylor–Hood finite elements and mesh adaptivity by metric control is used to accurately track the solid–liquid interface or the density inversion interface for water flows.The numerical method is validated against classical benchmarks that progressively add strong non-linearities in the system of equations: natural convection of air, natural convection of water, melting of a phase-change material and water freezing. Very good agreement with experimental data is obtained for each test case, proving the capability of the method to deal with both melting and solidification problems with convection. The presented numerical method is easy to implement using FreeFem++ software using a syntax close to the mathematical formulation.