A novel finite-volume formulation is proposed for unsteady solutions on complex geometries. A computer code based on a cell-centered finite-volume method is developed to solve both two-dimensional (2-D) and three-dimensional (3-D) Navier-Stokes equations for incompressible laminar flow on unstructured grids. A collocated (i.e., nonstaggered) arrangement of variables is used. The convective terms have provision for a variable upwinding factor, and the diffusion fluxes are computed in a novel and natural way. The pressure-velocity decoupling is avoided by momentum interpolation. The method is shown to have nearly second-order accuracy even on nonorthogonal grids. Some Navier-Stokes solutions, both 2-D and 3-D, are presented to verify the method with standard benchmark solutions. The comparison of present results with those in the literature is good. A computational study of 2-D laminar flow and heat transfer past a triangular cylinder in free stream is presented for the range 10 ≤ Re ≤ 200. Copyright © Taylor & Francis Group, LLC.