A two-grid adaptive volume of fluid approach for dendritic solidification

A two-grid approach to simulating dendritic growth in the solidification of pure metals is presented. The approach is based on using a uniform Cartesian grid to solve the energy equation and an adaptive Cartesian grid of higher-resolution to solve the interface evolution, providing a more accurate representation of the interface. The energy equation is solved using a diffuse-interface method, and the advection equation of a discretized solid fraction function is solved using an unsplit VOF (volume of fluid) method, with PLIC (piecewise linear interface calculation) reconstruction of the interface. The thermal gradients at both sides of the interface, which are needed to obtain the front velocity, are calculated with the aid of the distance function to the reconstructed interface, which is obtained using an efficient method described in detail in this work. The influence of the grid resolution used to solve the advection equation on the accuracy of the method is analyzed. For the dendritic growth cases considered in this work, particularly, it was found that using a grid resolution for the advection equation two times higher than that used for the energy equation considerably improves the results, while keeping the CPU time consumed at an acceptable level. To underline the importance of an accurate estimation of the interface curvature, the results obtained using three techniques commonly used in VOF methods of different degrees of accuracy (a convolved VOF technique, a distance function technique and an improved height function technique, which provides second-order accuracy) are compared. The proposed methodology is assessed by comparing the numerical results with analytical solutions and with results obtained by different authors for the formation of complex dendritic structures in two and three dimensions.