A multi-sweep relaxation procedure is considered in the following study for two-dimensional (2-D) and three-dimensional (3-D) internal flows with interaction using reduced (pressure-elliptic) Navier-Stokes (RNS) equations. The velocities and pressures are fully coupled in the algorithm, and the pressure gradient in the streamwise momentum equation is appropriately treated to simulate the elliptic interaction. A Poisson type of equation is not required. The continuity equation is directly satisfied everywhere in the flow domain. Incompressible flow solutions are obtained for a 2-D channel with a backward facing step and a 3-D duct with square cross section. The solution for these test cases demonstrates the capability of the RNS algorithm to solve internal flows with strong interaction encountered in turbomachinery passages.