A nonlinear and semi-analytical actuator disk method for ducted wind turbines with hubs of general shape

Nowadays, several theoretical, numerical and experimental works have shown that a ducted wind turbine can exhibit a power coefficient significantly greater than the one of a bare turbine characterised by the same thrust coefficient. However, mostly because of the high cost and weight of the duct, this kind of device has never experienced a wide diffusion for large power and ground applications. Conversely, small-sized ducted wind turbines are recently gaining new attention since the abovementioned weight and cost concerns have a minor relative impact on the overall cost. However, small-sized wind turbines are often characterised by an appreciable value of the hub to tip radii ratio, especially when the hub has to accommodate the brake, the gear box and the generator. Under those circumstances the blockage and the associated acceleration induced by the hub on the rotor are important and cannot be neglected. Therefore, it becomes highly desirable disposing of efficient and accurate analysis methods which could properly handle a prescribed hub geometry. In this work the ducted actuator disk method of Bontempo and Manna1 is extended to deal with general hub shapes. The method returns the exact and implicit solution of the steady, axisymmetric, inviscid and incompressible flow through the rotor as given by the superposition of infinite ring vortices properly arranged inside the rotor wake and along the duct and hub surfaces. The proposed approach strongly couples the flows induced by the rotor, the duct and the hub. Moreover, it fully takes into account the wake rotation and divergence. To give an idea of the potential of the method, the contours of the three velocity components and the traces in the meridional plane of the streamlines are shown in the figure reported below. Finally, the method can also be successfully employed to analyse and verify the validity of the basic assumptions used in the momentum theory. Moreover, thanks to its reduced computational cost, it can be successfully applied in the very first stage of any design scheme based on the repeated-analysis concept.