Hodograph method for turbine nozzle guide vane profiling

As an outgrowth of earlier investigations [1, 3, 2], a numerical method for design of turbine nozzle blade cascade by solving a singular boundary-value problem in the hodograph plane is presented. Since the relatively height of a nozzle channel is, as a rule, a small parameter, a two-dimensional approximation is valid. A first-stage nozzle blade cascade used for transformation of the incoming subsonic flow into supersonic one is profiled so that sonic lines are straight, the flow is uniform at the outlet of the cascade, and the flow velocity increases monotonically along the blade wall. The consequence of this fact is that boundary layer is unseparated and the ideal gas model used together with boundary layer theory is adequate. The supersonic part of the blade contour may be chosen so that it includes corner point on the straight sonic line. It results in essential reducing of the length of the nozzle channel what is rather efficient from the viewpoint of technical application. The turbine blade contour is found by solving a singular Dirichlet problem for Chaplygin equation on Riemann surface of logarithmic type in subsonic half-plane, where this equation is of degenerated elliptic type. Existence of unique solution is proved for considered problem [4]. For physical realizability of solution obtained in the hodograph plane it is needed to satisfy an additional condition that determines, in fact, the position of the singular point in the hodograph plain. The numerical method (which is a modification of the Ryaben'kii's method of difference potentials [5]) is used on the simple sheet of the Riemann surface. A number of sought for nozzle vane cascades is calculated.