Supersonic Turbines Convergent-Divergent Vaned Nozzles Losses and Deviation Models

Abstract This article continues a series of articles devoted to the development of design and analysis methodology for supersonic turbines to be incorporated into the turbomachine design system. The earlier published articles in this series are devoted to algorithms for the design and calculation of supersonic convergent-divergent nozzles of two types — drilled [GT2022-83387, GT2023-101760] and vaned [GT2024-125893]. Supersonic nozzles, as part of the turbine stage, are widely used in high-loaded steam and gas turbines, aircraft engines, jet propulsion units, rocket turbo pumps, and others. The scope of the already published article on convergent-divergent (CDV) supersonic nozzles was limited by the design algorithm and the method for calculating the flow coefficients of such nozzles. In the current article, the presentation of the methodology of CDV nozzle losses and flow deviation is continued. The developed methodology accounts for a broad range of operational modes, including design and off-design regimes. The ANSYS CFX was the main tool to study flow structure and extract the required information for calculation model development. The methodology includes effects of different geometrical and regime parameters reflecting conditions that CDV nozzles could meet in the practice of high-loaded supersonic turbines of different applications. Proposed nozzle losses and deviation models are based on a detailed study of supersonic flow physics into channels of different geometry and operation conditions. The proposed variant of the losses model includes two constituents — the design point losses model and the off-design correction factor. The design mode losses model is presented as a sum of profile losses and secondary (end walls) losses. Profile losses are a function of several parameters, varying in broad ranges such as — design pressure ratio, fluid-specific heat ratio, nozzle axis angle, Reynolds number and walls roughness, trailing edge shape (cylindrical vs. cut-ff) and thickness, distance from trailing edge (TE) and effect of shock waves from downstream located blades leading edge. The secondary losses are a function of the nozzle’s relative height. The effect of the off-design operation is accounted for by the correction factor, which is a complex function of nozzle design pressure ratio, isentropic exponent, and regime parameter. The physics of supersonic flow is illustrated by CFD results to support some assumptions used in loss and deviation model development. The results presented in this article, together with earlier published [GT2024-125893] have completed the development of design and computational models for CDV nozzles, which include discharge coefficients, losses, and deviation. The design algorithm and models were incorporated into the turbomachine design system and used to create a variant of the supersonic stage for verification purposes in CFD. The verification showed a good match in terms of losses in the nozzles in different modes of operation. The developed mathematical model of CDV nozzles is recommended for use in the design system of turbomachines for supersonic stages.