The three-dimensional turbine geometry is based on that of experiments performed by Dring, et al.[11, 12] and Joslyn and Dring at United Technologies Research Center (UTRC). The experimental geometry is a - stage turbine with a 27 inch midspan radius, 6 inch span and airfoil aspect ratios of approximately unity. The turbine hub and casing are at constant radii. The axial gap between the first stator and rotor is approximately 50% of the first stage average axial chord, while the gap between the rotor and the second stator is approximately 67% of their average chord. The rotor tip clearance is approximately 1% of span. A large experimental database exists for this turbine, including time-averaged pressures at several spanwise locations on each airfoil, traverse data behind each airfoil, and surface flow visualizations.
The experimental configuration has 22 airfoils in the first stator row and 28 airfoils in each of the rotor and second stator rows for a total of 78 airfoils. A three-dimensional computation of the flow through the complete turbine configuration would be prohibitively expensive. To reduce the cost of the computation, the number of stators in the first row was increased to 28 and the size of the stators was reduced by a factor of 22/28 to maintain the same blockage. The flow is then assumed to be periodic from passage to passage, thereby allowing a reduction to a single blade or vane in each of the rows. This approximation has been used by Rai and Madavan et al. and was shown to not affect the time-averaged pressures in the first stage of this turbine, even with a much smaller axial gap of 15%.
The grids used to describe the hub and airfoil surfaces of the turbine are shown in figure 1 . Only every fourth point in the grid is shown for clarity. An inner grid and an outer grid is used to compute the flow around each of the airfoils. The inner "O" grids are clustered near the airfoil surface, hub and casing to resolve the viscous effects in those regions. The outer sheared Cartesian grids are clustered near the hub and casing to capture the viscous effects generated there. In addition to the two grids per airfoil, one inlet and one exit grid is used to compute the flow in the upstream and downstream regions respectively. Eight grids are thus used to describe the 3 airfoil system.
A hole which contains the entire blade or vane has been constructed in each of the outer grids. These "hole" points are not used in the calculation, since the inner grids are used to compute the flow near the airfoils. The inner grids overlap the outer grids to promote stability of the integration scheme. The outer grids also overlap each other along the slip boundary to increase stability in the hub slip region. The outer grids are allowed to slip past one another to simulate the relative motions between blades and vanes. The rotor blades are embedded within the inner "O" grid, so there are points between the tip of the rotor and the casing to capture the tip leakage flow without adding a special tip grid. The points within the rotor are treated similar to the "hole" points, since they are not used in the calculation.
The inner stator grid dimensions are 214 points in the wrap-around direction, 26 points in the surface-normal direction and 51 points in the radial direction. The inner rotor grid dimensions are 214 in the wrap-around direction, 45 points in the surface-normal direction and 51 points in the radial direction. The additional points in the rotor grid exist to discretize the rotor tip region. The outer grid axial-direction dimension varies, but averages 123 points in the axial direction, 81 points in the circumferential direction and 51 points radially. The inlet and outlet grids have the dimensions 34 by 81 by 51. The total number of grid points used for the 8 grid system was approximately 2.7 million grid points.