The present study aimed to investigate two numerical solutions of the Navier-Stokes equations. For this purpose, the mentioned flow equations were written in two different formulations, namely (i) velocity-pressure and (ii) vorticity-stream function formulations. Solution algorithms and boundary conditions were presented for both formulations and the efficiency of each formulation was investigated by considering a two-dimensional low laminar flow around a square pile in a rectangular computational domain. Simulations under the same conditions were conducted to assess the difference between results generated by both formulations. Furthermore, the accuracy of the results was analyzed through a comparison of the results with the available reference data. In addition, computational efficiency of both formulations was investigated in term of computation time. The corresponding results indicated that both formulations are adequate to the case used in the present study. Moreover, performed simulations showed that solving the vorticity-stream function form of the flow equations is faster than solving the velocity-pressure form of those equations for simulating a two-dimensional laminar flow around a square pile.