In this paper, an implicit fractional step method for the solution of the two-dimensional, time dependent, incompressible Navier-Stokes equations is presented. The current method was developed for use on an unstructured grid made up of triangles. The basic principles of this method are that the evaluation of the time evolution is split into intermediate steps and that for the spatial discretization of the flow equations, a finite volume discretization for an unstructured triangular mesh is used. The current methodology, which has been tested extensively for steady problems, is now applied for the numerical simulation of unsteady flows. Several cases were tested, such as plane or axisymmetric channels, a square cavity and an axisymmetric stenosis.