A numerical method to solve the three-dimensional Navier-Stokes equations for the flow in transonic turbine stages with tip gap leakage is presented. Viscous flow in a transonic turbine stage has been simulated. The high pressure difference at the rotor blade tip results in a supersonic jet. The relative motion of the casing wall is oriented against the tip leakage flow and tends to reduce it. Very large velocity gradients in the tip region pose a challenge for the numerical simulation. Computational results are compared with experimental data obtained in operation. Measurements include data for the tip leakage jet. The numerical method is based on a conservative finite volume cell–vertex scheme in cylindrical coordinates with central difference approximation and Runge–Kutta time stepping. Convergence is accelerated by use of a multigrid method and implicit residual smoothing with variable coefficients. The Baldwin–Lomax turbulence model is used for closure. The boundary condition treatment at inlet and outlet as well as the coupling of stator and rotor flow is achieved by use of non–reflective boundary conditions. The tip region is discretized by an additional grid within a multi-block approach.

The flow inside a diffuser which serves as outlet of the last turbine stage is strongly influenced by the tip leakage jet of the last blade row. For the computation of diffuser flow a quasi 3D Euler solver with a simple model for the approximation of viscous effects was developed. A body force which is calculated as a function of velocity gradients normal to the streamwise direction is included in the Euler equations. Computational cost for calculations using this method remains low, permitting its use as a design tool for turbine diffusers. Comparison of numerical results with experimental data for an annular axial radial diffuser at the outlet of a steam turbine model is presented. Flowfields for different tip clearances show a strong dependence of the pressure recovery on the tip leakage jet. Numerical results agree well with measurements.