A Physics-Informed Neural Network for the Design of Supersonic Turbine Stator Blades

The recovery of low- and medium-temperature waste heat using Organic Rankine Cycles (ORCs) is increasingly important for improving the efficiency and sustainability of industrial and energy systems. In compact ORC turboexpanders, high specific power output and large pressure ratios often require single- or two-stage turbines operating in transonic or supersonic regimes. Under these conditions, stator blade design is complicated by strong compressible-flow effects and, for organic working fluids, by real-gas thermodynamic behaviour. Conventional supersonic stator design methods, such as the method of characteristics, are mainly applicable to the diverging supersonic portion of the blade passage, while the converging region is typically defined using empirical or heuristic prescriptions. This paper presents a physics-informed neural-network-based inverse design method for supersonic turbine stator blades. The proposed framework generates the complete inter-blade passage, including both the converging and diverging regions, starting from a prescribed mean-line geometry and Mach number distribution. The velocity field is obtained by solving the governing equations of steady, inviscid, adiabatic, irrotational compressible flow within a PINN formulation. A hard boundary-condition strategy is used to impose the specified mean-line velocity distribution exactly, while real-fluid thermodynamic effects are incorporated through lookup tables for the speed of sound and density. The blade contours are then reconstructed from stream-function isolines predicted from the computed velocity field. The method is demonstrated for two working fluids: air, treated as a perfect gas, and toluene undergoing transcritical expansion. The resulting blade passages are first validated using inviscid CFD simulations, which show close agreement between the prescribed and computed mean-line Mach number distributions. Turbulent CFD simulations of the final blade cascades confirm smooth acceleration through the inter-blade passage, with no strong internal shocks and only weak fishtail shocks downstream of the trailing edge. For both fluids, the post-expansion ratio is approximately unity and the exit flow angle remains close to the prescribed blade metal angle, indicating well-matched supersonic stator designs. The results demonstrate that the proposed PINN-based inverse design method provides a systematic and physically consistent approach for generating supersonic stator blade profiles for both ideal-gas and real-gas turbine applications.