The vaneless contra-rotating turbine (VCRT) has the potential to improve the thrust-to-weight ratio for future high performance propulsion systems. Without the inter-stage vane, the variation of the high pressure turbine (HPT) exit swirl, under off-design conditions, can introduce more significant effects on the operation of the low pressure turbine (LPT). This study performed both analytical and numerical work investigating the matching relationship between the two stages of one and one-half stage (1+1/2) VCRTs. The significant difference of 1+1/2 VCRTs from conventional turbines, under choking geometry-fixed conditions, was found the flow capacity at the station between HPT and LPT is not invariant. The matching relationship between HPT and LPT is indeed how the flow capacity at the station between the two stages varies for different matching conditions. Influenced by the variation of the flow capacity, both the total pressure ratio of HPT and the corrected mass flow rate at the inlet of LPT are confined by the matching relationship. The corrected rotational speeds of the two stages define different matching conditions. A correlation, defining the matching conditions that LPT can work with constant incidence, was also derived. With aid of the correlation, the matching conditions can be controlled assuring LPT work with stable high efficiency. In the end, an application of incorporating the matching relationship into the investigation of engine performances is demonstrated, in which, a turbojet performance cycle analysis was performed. This paper has established a framework guiding how to obtain the matching relationship between the two stages of 1+1/2 VCRTs and incorporate it into the investigation of engine performances.

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