IntroductionT HE purpose of this study is to compare quantitatively several numerical solution techniques for solving twodimensional viscous flow problems, including separated flows, at moderate to high Reynolds numbers. These solution techniques have various degrees of implicitness, and they are compared on the basis of convergence, CPU time, storage requirements, and solution accuracy. The stream-function (\l/) vorticity (co) approach is used to formulate the partially parabolized Navier-Stokes (PPNS) equations that describe the two-dimensional incompressible flow. The PPNS equations are quite similar to the Navier-Stokes (NS) equations; streamwise diffusion is the only physical process neglected, and terms representing this diffusion are dropped from the NS equations. The governing equations arewhere the term otu t represents an artificial time-dependent term, and the velocity u = \l/ y and v --\l/ x . At the solid wall, the no-slip boundary condition requires u = 0 and v = 0. For external flows co = 0 and u = U(x) at the upper boundary. For internal flows the boundary conditions at the centerline of, e.g., a channel, are co = 0 and \l/ = \l/(x Q9 y c ). At the inflow boundary \l/ and co are prescribed, whereas at the outflow boundary the PPNS equations are reduced to the boundarylayer equations by neglecting the i/^ term.