A method is presented for solving the two-dimensional Navier-Stokes equations on a solution-adaptive grid of both structured and unstructured meshes. Flow near airfoil surfaces is modeled using an implicit finite difference algorithm on a structured O-type mesh. The flow equations in the blade passages are written in a cell-vertex finite volume formulation and are solved on an unstructured mesh using a Runge-Kutta explicit algorithm. Both the structured and unstructured grid also include solution dependent adaptation to allow resolution of flow features with a minimum of grid points. The structured mesh divides to locally add grid lines, while the unstructured mesh allows the addition or removal of individual cells. An overlapping interface region is used to conservatively communicate flow variable information between the two grids. The quasi-three-dimensional effects of streamtube contraction and radius change are included to allow calculation of modern turbomachine designs. A study is included to determine the effect on cacade parameters of inclusion of viscous terms in the solution of the flow equations in the unstructured domain. Quasi-three-dimensional computations of flow through a transonic compressor and turbine cascade are compared with experimental data.

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