A fixed grid method based on an enthalpy-porosity formulation for liquid/solid phase transition is extended to compute the time-dependent solutal convection in the melt during directional solidification of alloys. A finite volumes approximation is used for uniform and refined grids with a second order Euler scheme. The ability of the method to describe accurately the flow transitions and regimes is considered with respect to the results of the linear theory of stability and of available spectral accurate calculations.