Abstract: The generation of prismatic unstructured grid for complex³ D geometries and the application of a new high resolution algorithm for the unstructured grids have been developed in consideration of the engineering background.The unstructured grids are generated on one given S₁ surface (i.e.a surface of constant x³ in a three dimensional curved coordinate ( x¹,x²,x³),while the grids on the other S₁ surfaces are constructed by a space mapping.The final grids obtained in this way become unstructured on S₁ surfaces and structured in x³ direction.Therefore they are composed of pentahedrons.In the generation of the grid on the S₁ surface,the triangle elements of 0 type grid are created near the walls.By improving Bowyer Watson method,a scheme of high efficiency is developed to produce unstructured grids in the region far from the walls.The new algorithm consists of a high resolution finite volume scheme for the 3 D Navier Stokes turbulent flow equations.Spatial discretization uses a finite volume.High order Taylor series expansion is employed to march the solution in time.Spacial volume and time step weighting of the residual are employed to produce a stable scheme.A second order smoothing operator and the minmod limiter are developed for shock capturing.The operator retains its order with unstructured grids.Some typical numerical examples, including a turbine stator blade row with film cooling,demonstrate the ability of the algorithm to resolve local flow features.