A staggered finite volume upwind algorithm for solving the compressible Navier-Stokes equations and the \(k-\omega\) turbulence model equations has been developed for computing cascade flows. Roe's upwind scheme is used to discretize the convective terms of the Navier-Stokes equations, and a third-order upwind scheme is used for the convective terms of the \(k-\omega\) equations. All of the diffusion terms are discretized with the central-difference method. By the use of a combination of cell-centered and cell-vertex schemes, the method maintains a small stencil for all of the diffusion terms and makes the Navier-Stokes equations and \(k-\omega\) equations strongly coupled. The algorithm was tested for a flat-plate flow, a supersonic wedge cascade flow, and a low-pressure turbine cascade flow at design and off-design conditions.