All theories now in use for the design of the bladings of Kaplan turbines and pumps are based on the assumptions that the fluid moves on concentric cylinders and that no trailing vorticity is shed off the blading. Approximate corrections are here found arising from the fact that the flow does not, in practice, satisfy these assumptions. These corrections are the more important the smaller the ratio of hub diameter to tip diameter. An approximation to them is found from investigation of the axially symmetrical mean flow in the rotor, which is obtained when the velocity components at any point are replaced by their mean values with respect to the angular co-ordinate. This approximation is the more accurate the smaller the pitch-chord ratio. It is found that the three-dimensional effects in the flow are due to the transverse component of the vorticity (“ring vorticity”). Formulas for the velocity components induced by it are given. Two distinct problems arise in connection with the design of a Kaplan blading, viz., (a) to find the shape of the blade and the pressure distribution for a prescribed load distribution, under the optimum working condition, and (b) to find the performance and pressure distribution for a prescribed blade shape, after a change of pitch. The theory applies also to multistage axial-flow turbomachines where the mean velocity is not too high for knowledge of the incompressible flow to be of interest.

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