Computational investigation is conducted during time-dependent behaviors induced by solutal convection in the melt for directional solidification of a binary alloy. Effects of solutal Rayleigh number on flow, heat, and mass transfer are studied for upward Bridgman configuration restricted to fluid phase. The results compare well with spectral ones exhibiting various hydrodynamic solutal transitions (symmetric, asymmetric, time-dependent). The finite-volumes method is tested for various uniform and refined meshes near the walls to obtain spectral accuracy. Calculations are extended to configurations including melt/solid interactions by the use of a homogeneous formulation and a linearized equilibrium phase diagram.