Fixed Cartesian grid based numerical model for solidification process of semi-transparent materials I: Modelling and verification

An Eulerian–Lagrangian model was developed to numerically solve solidification process of semi-transparent, absorbing, emitting and anisotropically scattering materials. The model was based on the Fixed Cartesian Grid Front Tracking Algorithm combined with the Immersed Boundary Technique and for the first time was used for tracking highly curved, internal boundaries in participating medium with fluid flow taken into account. The convection in semitransparent melt was considered by applying the Fractional Step Method at a collocated grid. The Least Square Gradient Reconstruction Based Technique was used to find unknown temperature, velocities and pressure gradients at reshaped control volumes centres as well as to find unknown temperature, velocities and pressure values and gradients at control volume faces which were internally cut by the immersed boundary. The developed model enabled dealing accurately with different thermophysical and optical properties in the solid and the liquid phase of solidifying materials. The phenomena such as absorption, emission and diffuse or specular reflection of the thermal radiation at the opaque solid–liquid interface as well as transmission of the thermal radiation at the totally transparent solid–liquid interface were also introduced in proposed model. Several test cases, with fixed internal boundaries and with moving solid–liquid interface were solved to perform numerical verification of the model.