A 2D1/2 model for natural convection and solidification in a narrow enclosure

Efficient numerical models are derived for problems of natural convection and material solidification in a horizontal differentially heated slender cavity. These 2D1/2 models are obtained by averaging the equations of momentum, heat, and mass conservation along the transverse direction assuming both a constant temperature and a well defined velocity profile in this direction. Based on our former works, the transverse velocity profile is assumed to be either a Poiseuille profile (2Dp1/2 model), or Hartmann-type profiles featuring two boundary layers on the sides of a uniform bulk (2DH1/2 model). For this 2DH1/2 model, however, a parameter δ (giving the boundary layer thickness) has to be adjusted: optimal values have been found in a large range of the control parameters and expressed as a reliable fitted function of Gr. The ability of the model to reproduce 3D results in a 2D framework is investigated in a large range of the control parameters (Prandtl number Pr and Grashof number Gr); the validity domain of the model in this parameter space is also clarified and rigorously defined. A good precision is obtained for natural convection problems (intensity of the flow, temperature field) as well as for solid-liquid phase change problems (shape, position, and evolution of the front). A comparison with unpublished experimental data of solidification of pure tin is also conducted. The boundary conditions for the simulation are first defined after a post-treatment of the time-dependent experimental data in order for them to be representative of the experimental process despite a significant and time dependent thermal resistance between the walls of the crucible and the liquid. A very good agreement is observed between the 2DH1/2 model and the experimental measurements for this pure tin solidification experiment in the AFRODITE setup.