The results shows that with the increase of rotational speed, the high efficiency area of turbine gradually becomes wider, and there is little difference of the maximum efficiency at different rotational speeds. The head loss, pressure distribution, turbulence kinetic energy, and unsteady pressure pulsation were also clarified. To study the influence of rotational speed on the performance of hydraulic turbine in turbine mode (T-type turbine), the performance of turbine at different rotational speed was predicted through theoretical analysis, based on the Navier-Stokes equation and standard k-ε turbulence model, numerical simulation was used to study the performance of turbine at different rotational speed. The method presented in this paper can be well applied to create the extended Cordier diagram for the use of shape factors for pumps and compressors. Exclusively for Francis turbines, the impeller shape factor also combines the blade profile (angle) at the impeller inlet with design parameters. The Cordier diagram for the diameter number can be replaced by the new diagram for the turbine shape factors. For Pelton turbines, it is representative of the ratio between runner diameter and jet diameter for Francis and Kaplan turbines, it is representative of the ratio between the height of the wicket gate and the diameter of the impeller. This shape factor has a clear physical meaning in contrast to the diameter number in the conventional Cordier diagram. The part of extending the Cordier diagram is based on the application of design equations for hydraulic turbines and the introduction of the turbine shape factor. Both turbine type and impeller diameter can be more accurately determined. The Cordier lines (or bands) for Francis and Kaplan turbines have been recalculated based on the use of most available empirical relations without artificial smoothing of the boundaries between the different types of hydraulic turbines. The first part of this work completes the Cordier diagram by adding the effect of the number of injector nozzles in Pelton turbines. The Cordier diagram for hydraulic turbines has been accurately improved and extended.
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