Highly accurate error estimate of the momentum theory as applied to wind turbines

The paper provides a verification of the well-known momentum theories by comparing their results with those of a semi-analytical method based on the exact solution of the flow through an actuator disk. In fact, many error sources are disseminated in these theories. Specifically, the axial momentum balance is diffusely applied in an approximate differential form, and several linearization are customarily introduced in the equations of the motion. In the present study, an analytical formulation of these two kinds of errors is provided and a set of data generated with controlled accuracy is used to quantify them both in terms of global (power coefficient) and local (axial velocity at the disk plane) quantities. The overall errors increase by increasing the thrust coefficient and decreasing the tip speed ratio. Although the errors can be generally considered small in terms of global performance coefficients, the differences between the two methods are significant when looking at local quantities. In particular, for the cases presented herein, the momentum theory is shown to overestimate the axial velocity at the tip and at the disk centre regions and to underestimate this velocity elsewhere. For low values of the tip speed ratio, an underestimation also occurs in the zone near to the disk centre.