Abstract
A complete analytical formulation of the vortex approach for the rotor with an ideal load distribution under Betz optimal condition needs some additional assumption about a correct choice of the helical pitch for vortex sheets in the rotor wake. An examination of the three evident assumptions (the pitch is independent from velocities induced by the wake; the pitch depends on the induced velocities in the far wake; the pitch depends on the induced velocities in the rotor plane) was considered by a comparison with the main restriction of the actuator disk theory – the Betz-Joukowsky limit. In the present investigation an analytical solution for the limit case of an infinite number of blades was used to re-examine the choice of the wake pitch.
Highlights
For planar wings of a finite span with a planar vortex wake in a uniform flow it’s well-known [1] that an elliptic distribution of the load along the lifting line corresponds to the lowest drag of the trailing vortex (Fig. 1a)
An examination of the three evident assumptions was considered by a comparison with the main restriction of the actuator disk theory - the Betz-Joukowsky limit
For a correct comparison it is necessary to replace an abstract pitch to a real parameter of the operating regimes Eq (1) for wind turbines which for each theory according to Eq (8) transform to the form: O0
Summary
For planar wings of a finite span with a planar vortex wake in a uniform flow it’s well-known [1] that an elliptic distribution of the load along the lifting line corresponds to the lowest drag of the trailing vortex (Fig. 1a). Betz [2] generalized this result for rotor and formulated an analogical condition for the optimum of a rotating propeller: the distribution of circulation along the lifting line replacing the blade should be such that the free vortex sheet trailing from it has an exact helical shape and moves uniformly along the rotor axis in the direction of the main flow (Fig. 1b). Those semiinfinite helical vortex sheets are usually replaced by an associating vortex system, which extends on both sides to infinity and moves in axial direction in equilibrium (Fig. 2a) In this case the circulation changes from the elliptic distribution along the wing to asymmetrical function along the blade and the task to find it was a challenge which Betz could not solve. The efficiency of the algorithm was confirmed in [8, 9] by good correlation with tabulated data which was calculated by direct simulation of the Goldstein function [11]
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