A low-order panel method for the analysis/design of ducted wind turbines

The concept of enclosing a turbine in a duct is quite old and many numerical and experimental investigations have been carried out to clarify the operating principles of this kind of device. However, the fundamentals of ducted wind turbines are often studied using new formulations of the axial momentum theory, i.e. adopting the simplified framework of a uniformly-loaded actuator disk without wake rotation. In this simplified context, a new low-order panel method approach has been developed and verified. The method, which can also take into account the hub presence, employs a surface vorticity approach to model the duct, the hub, and the wake surfaces. The wake divergence is fully taken into account as well as the mutual interaction between the duct, the hub, and the wake. The sheet vortices are discretized using a low-order panel method. Specifically, the continuous sheet vortices are replaced by a set of ring vortices located in the pivotal points of the panels. A semi-infinite right circular cylinder vortex is also used to properly model the far wake. The shape and the density strength of the wake sheet are evaluated using two conditions. Firstly, the vortex sheet stability condition is enforced all along the wake, that is the static pressures just above and beneath the sheet are required to be equal. Secondly, the vortex sheet is constrained to align with the overall induced flow field, namely it is compelled to be a streamsurface. On the duct and hub surfaces the impermeability condition is imposed adapting the Martensen1 vortex method to axisymmetric bodies. The results of the method have been verified using an actuator disk approach based on classical CFD techniques. The agreement is excellent both for field and wall quantities. To give an idea of the potential of the method, the contours of the velocity components, as well as those of the static and total pressure coefficient, are shown in the figure reported below. Since, increasing the number of panels, the method recovers with the exact solution of the inviscid flow around a ducted and uniformly-loaded disk without wake rotation, its results are used to analyse and verify the outcomes of the momentum theory approaches. 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.