Abstract
The Betz factor, i.e., the value 16∕27 for the power coefficient, is widely expected to give an upper limit for the performance of any wind turbine. In the present study, an analytical model of a vertical-axis wind turbine with straight vertical wings is developed. A goal of the work is to study if the one-dimensional Betz theory gives an upper limit of the performance of wind turbines when two-dimensional effects are included. The two-dimensional and time-dependent potential flow is solved by a conformal map of the wing sections to circles. The stagnation points are determined by the Kutta condition. The calculated power coefficient exceeds the Betz limit by a large factor. This is due to a completely different flow pattern compared to the one-dimensional Betz theory. In aerodynamic potential flow, the expanding flux tube of Betz is replaced by an asymptotic flow consisting of a superposition of homogeneous flow and a circulation around the wings. Moreover, the total torque on a turbine with three or more blades is found to be constant. The Betz theory does not take into account the two-dimensional flow effects and the velocity of the rotating wind turbine. By including these effects, far more optimistic theoretical results for the performance of vertical-axis wind turbines are obtained.
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