This paper describes the development of a new through-flow method for the analysis of axial multistage turbines with cooling by air from compressor bleed. The method is based on a stream function approach and a finite element solution procedure. It includes a high-fidelity loss and deviation model with improved correlations. A radial distribution model of losses and a new spanwise mixing model are applied to simulate 3D flow effects. The calibration of the models is performed by calculation of a number of test cases with different configurations, with the aim of achieving high accuracy and optimum robustness for each of the test cases considered. Various types of cooling air injection were encompassed: film cooling, trailing edge injection, and disk/endwall coolant flow. There are two effects of air cooling: (i) increase in mass flow downstream of the injection surface and (ii) reduction of the gas total temperature connected with total pressure losses. For both of these effects, the appropriate 2D models were developed and applied. The code was applied to flow analysis and performance prediction of a newly developed industrial gas turbine. Comparison of the predicted results and measured test data for a number of parameters showed good agreement. The results of the validation confirmed that this method based on calibrated correlations can be considered a reliable tool for flow analysis and parameter variation during the design phase.