Potential Flow Calculations of Axisymmetric Ducted Wind Turbines

An incompressible potential-flow vortex method has been constructed to analyze the flow field of a ducted wind turbine following that outlined by Lewis (1991). Attention is paid to balancing the momentum change in the flow to the total longitudinal forces acting on the duct-turbine combination: the pressure force on the actuator disk plus the pressure forces acting on the duct, which typically includes a negative component of drag due to high leading-edge suction. These forces are shown to balance the momentum changes in the flow, resulting in a model for power output from a ducted wind turbine over a wide range of pressure changes across the actuator disk. The results are compared to the Betz actuator disk model and it is shown that the maximum power output from a ducted turbine occurs at a lower value of pressure drop/momentum extraction than that for a bare turbine. In this method, vortex panels are used to define the shape of the duct and to account for the flow changes due to the pressure drop across the turbine disk. It may seem surprising that potential flow can be used to describe the flow of a ducted wind turbine, but under the assumption of uniform changes in static and stagnation pressure across the turbine disk, as is done in the Betz model, both regions of the flow have a potential function. The velocity field, which is irrotational, is unchanged across the actuator disk. The two regions of flow are defined by the stagnation pressure change across the actuator disk and by the cylindrical vortex sheet that originates from the trailing edge of the duct as a result of the pressure changes across the disk. In experiments in which the actuator disk is replaced by a uniform screen, the change in stagnation pressure is not likely to be constant across the actuator disk due to the non-uniform velocity at the disk entry. This calls into question the use of screens to represent an actuator disk across which the change in stagnation pressure is assumed to be constant. However, these errors are of the same order as the application of actuator disk theory itself to a problem as complex as a wind turbine with a finite or an infinite number of blades. In any case this potential flow model should be considered to be a descendent of the Betz model and other similar analysis of ducted propellers and should give useful insights into the operation of ducted wind turbines. The advantages of such a simplified model are that basic questions can be investigated without the complexity of a full CFD simulation: basic questions such as the momentum balance in the flow; the forces acting on the duct; the role of the Kutta condition in comparison to its usual form for an airfoil in a uniform potential