A numerical method is presented for the computation of unsteady, three-dimensional potential flows in hydraulic pumps and turbines. The superelement method has been extended in order to eliminate slave degrees of freedom not only from the governing Laplace equation, but also from the Kutta conditions. The resulting superelement formulation is invariant under rotation. Therefore the geometrical symmetry of the flow channels in the rotor can be exploited. This makes the method especially suitable to performing fully coupled computations of the unsteady flow phenomena in both rotor and stator, the so-called rotor–stator interaction.The developed numerical method is used to simulate the unsteady flow in an industrial mixed-flow pump. Two types of simulation are considered: one in which unsteady wakes behind the trailing edges of the rotor blades are taken into account and one in which these are neglected. Results are given that show the importance of unsteady flow phenomena. However, the computed head-capacity curve is hardly influenced by whether or not unsteady wakes are taken into account. Copyright © 1999 John Wiley & Sons, Ltd.