Actuator Disk Theory—Steady and Unsteady Models

Abstract

In this paper, the classical work on steady actuator disk theory is recalled and the interpretation of the flow as the continuous shedding of vortex rings of constant strength that causes the slip stream is shown to be consistent with the classical model, yet offers a different approach that provides more flexibility for the extension to unsteady flow. The ring model is in agreement with the conservation law theorems which indicate that the axial inductance inside the streamtube, far downstream, is twice that at the rotor disk. The actuator disk relationship between the power and the axial inductance at the rotor plane has been used previously to construct the prescribed wake in the 3D vortex line model (VLM) of the author. The local pitch of the vortex sheet helices is based on the axial inductance along the streamtube, therefore, it is important to be able to calculate the axial inductance in all cases. In unsteady flow, the strength of the shed rings varies with power absorbed by the rotor and a new relationship between the two is derived to first-order, using the unsteady Bernoulli equation, which allows to calculate the inductance at the rotor and in the wake with the Biot-Savart law, even in cases where the Betz limit is exceeded. This more rigorous model can replace the semi empirical models such as dynamic stall and dynamic inflow used in the blade element method (BEM). The proposed new model has been incorporated in the VLM and applied to two sets of experiments, the Tjaereborg in-field tests and the NREL wind tunnel campaign, for which test data has been collected and serves for the comparison. The simulations show an overall good agreement with the experimental data. In some tests, the power coefficient exceeds the Betz limit, in which cases there are no solutions to the steady actuator disk theory, but the new model provides a solution for the inductance even in this case.