An analytical method is presented,which enables the non-uniform velocity and pressure distributions at the impeller inlet of a pump to be accurately computed.The analyses are based on the potential flow theory and the geometrical similarity of the streamline distribution along the leading edge of the impeller blades.The method is thus called streamline similarity method (SSM).The obtained geometrical form of the flow distribution is then simply described by the geometrical variable G (s) and the first structural constant GI.As clearly demonstrated and also validated by experiments,both the flow velocity and the pressure distributions at the impeller inlet are usually highly non-uniform.This knowledge is indispensible for impeller blade designs to fulfill the shockless inlet flow condition.By introducing the second structural constant GⅡ,the paper also presents the simple and accurate computation of the shock loss,which occurs at the impeller inlet.The introduction of two structural constants contributes immensely to the enhancement of the computational accuracies.As further indicated,all computations presented in this paper can also be well applied to the non-uniform exit flow out of an impeller of the Francis turbine for accurately computing the related mean values.