A method of solution is presented for the threedimensional, unsteady Navier - Stokes equations of an incompressible fluid. The method is based on the principle of artificial compressibility, extended to time-dependent solutions by the concept of dual time stepping. Higher order upwinding (Quick interpolation, and Roe-type splitting) is used for the spatial discretization. The properties of the method were tested by means of different validated flow problems. Present computations concern with the unsteady, and three-dimensional problem of the breakdown of an isolated vortex. The temporal evolution and the internal structure of the breakdown bubble is investigated and compared with experimental observations.