A two-dimensional transient fixed-grid enthalpy-based numerical method is developed to analyze the effects of turbulent transport during a binary alloy solidification process. Turbulence effects are introduced through standard k-epsilon equations, where coefficients are appropriately modified to account for phase-change. Microscopically-consistent estimates are made regarding temperature-solute coupling in a non-equilibrium solidification situation. The model is tested against laboratory experiments performed using an NH4Cl-H2O system in a rectangular cavity cooled and solidified from the top. Particular emphasis is laid on studying the interaction between Rayleigh-Benard type convection and directional solidification in the presence of turbulent transport. Numerical predictions are subsequently compared with experimental results regarding flow patterns, interface growth and evolution of the temperature field, and the agreement is found to be good.