The governing equations considered in this study are the time dependent, three-dimensional Reynolds-averaged Navier-Stokes equations:
The viscous fluxes are simplified by incorporating the thin layer
assumption (Baldwin and Lomax, 1978). In the current study, viscous
terms are retained in the direction normal to the hub and shroud
surfaces, and in the direction normal to the blade surfaces. To extend
the equations of motion to turbulent flows, an eddy viscosity
formulation is used. The turbulent viscosity, 
, is calculated
using the two-layer Baldwin-Lomax (1978) algebraic turbulence
model.
The numerical algorithm used in the three-dimensional computational procedure consists of a time-marching, implicit, finite-difference scheme. The procedure is third-order spatially accurate and first-order temporally accurate. The inviscid fluxes are discretized according to the scheme developed by Roe (1981). The viscous fluxes are calculated using standard central differences. An alternating direction, approximate-factorization technique is used to compute the time rate changes in the primary variables. In addition, Newton sub-iterations are used at each global time step to increase stability and reduce linearization errors. For all cases investigated in this study, two Newton sub-iterations were performed at each time step. Further details on the numerical procedure can be found in Gundy-Burlet (1992).
The Navier-Stokes analysis uses O- and H-type zonal grids to discretize the rotor-stator flow field and facilitate relative motion of the rotor. The O-grids are body-fitted to the surfaces of the airfoils and generated using an elliptic equation solution procedure. They are used to properly resolve the viscous flow in the blade passages and to easily apply the algebraic turbulence model. Algebraically generated H-grids are used to discretize the rest of the passage in the vicinity of the airfoil.