On the aerodynamic design of axial flow turbines by an implicit upwind axisymmetric solver

An inverse time-marching solution of Euler equations is performed in the meridional plane of a complete machine. The method predicts the flowfield and blade cambersurface geometry that correspond to a specified load distribution in the blade regions. The implicit approach is used to solve Euler equations with a specified mass flowrate at the outlet section. This boundary condition leads to typical turbine solutions of the inverse problem. The implicit upwind scheme is based on a time linearization of Osher's flux difference splitting operator. This operator employs a high-order Riemann solution that provides second-order space accuracy. For infinite-span turbine cascades working in the low-subsonic range, the new method has convergence times and residuals that are one order of magnitude lower than previous explicit methods. It keeps the same convergence properties up to the sonic outflow conditions, while explicit methods fail in the high-subsonic range. When endwalls are included in the computation to design finite-span cascades or complete turbine stages, the method still encounters some numerical instabilities that can be partly overcome using a sufficient mesh resolution in the spanwise direction.