The Navier-Stokes equations are solved numerically for two-dimensional steady viscous laminar flows. The problems considered in this study are limited to internal flows; however, it is believed that the scheme can be readily extended to external flows also. The equations are solved on a grid made up of triangular shaped elements. The grid is generated based on the method of Delaunay triangulation. Boundary points are given as input. The computational domain is then discretized by adding new points such that they do not violate the Delaunay criterion for triangulation;A finite-volume approach is used to discretize the conservation law form of the compressible flow equations written in terms of primitive variables. The delta form of the equations is used and the equations are time marched using either an implicit Gauss-Seidel iterative procedure or a solver based on a conjugate gradient like method. Two forms of Gauss-Seidel relaxation are considered. One solver only retains the diagonal terms of the equations on the left hand side. The other solver retains the entire diagonal block on the left hand side and uses a lower-upper inversion to obtain a solution. A four color scheme is employed to vectorize the block Gauss-Seidel relaxation procedure. This increases the memory requirements minimally and decreases the computer time spent solving the resulting system of equations substantially. A factor of 7.6 speedup in the solver is typical for the viscous equations. A preconditioning matrix is added to the equations so that low Mach number flows can be solved economically. The structure of the added time term allows for steady state as well as unsteady solutions although only steady state applications will be considered in this study;Numerical results are obtained for inviscid flow over a bump in a channel at subsonic and transonic conditions for validation with structured solvers. Viscous results are computed for developing flow in a channel, a symmetric sudden expansion, periodic tandem cylinders in a cross-flow, and a four-port valve. Comparisons are made with available results obtained by other investigators.