A fully explicit two-dimensional flow solver, based on a four-stage Runge-Kutta scheme, has been developed and used to predict two-dimensional viscous flow through turbomachinery cascades for which experimental data are available. The formulation is applied to the density-weighted time-averaged Navier-Stokes equations. Several features of the technique improve the ability of the code to predict high Reynolds number flows on highly stretched grids. These include a low Reynolds number compressible form of the \(k-\epsilon\) turbulence model, anisotropic scaling of artificial dissipation terms, and locally varying timestep evaluation based on hyperbolic and parabolic stability considerations. Comparisons between computation and experiment are presented for both a supersonic and a low-subsonic compressor cascade.