The periodic, helical vortex wakes of wind turbines, propellers, and helicopters are often approximated using straight vortex segments which cannot reproduce the binormal velocity associated with the local curvature. This leads to the need for the first curvature correction, which is well known and understood. It is less well known that under some circumstances, the binormal velocity determined from straight segments needs a second correction when the periodicity returns the vortex to the proximity of the point at which the velocity is required. This paper analyzes the second correction by modelling the helical far wake of a wind turbine as an infinite row of equispaced vortex rings of constant radius and circulation. The ring spacing is proportional to the helix pitch. The second correction is required at small vortex pitch, which is typical of the operating conditions of large modern turbines. Then the velocity induced by the periodic wake can greatly exceed the local curvature contribution. The second correction is quadratic in the inverse of the number of segments per ring and linear in the inverse spacing. An approximate expression is developed for the second correction and shown to reduce the errors by an order of magnitude.

It is common for computational models of the wakes of helicopters,
propellers, and wind turbines to use straight vortex segments whose position
is iterated until they follow the local flow and the vortex is force free.
Solving the Biot–Savart integral gives the induced velocity used in the
iteration. Figure

Schematic of two turns of constant radius helical vortices with

A well-known difficulty of the straight segment approximation is that it does
not reproduce the binormal velocity due to the curvature of the vortex line
(e.g.

Curvature in the wakes of rotors is often associated with vortex periodicity,
the “return” of a vortex to the proximity of the control point, which can
cause a significant contribution to the binormal velocity.

Vortex ring representation of the helical wake in Fig. 1 and the corresponding vortex segment approximation.

The paper is organized as follows. The next section introduces the vortex ring model of the wake. In the following section, the induced velocity for the periodic component of the wake over a range of vortex spacings is found in terms of its Biot–Savart integral. Section 4 describes the calculation of the induced velocity for the straight segment approximation, determines the second curvature correction, and tests its accuracy. The final section contains the conclusions.

For a point with the same radius as a single vortex ring and distance

The test case used here to investigate the second correction models the
far wake of a wind turbine as an infinite row of equispaced vortex rings of
constant spacing,

The ring vortex wake is consistent with the “Joukowsky” model of the wake,
used by

The integrands for the two influence coefficients for a control
point on the vortex ring and

Testing corrections for the straight segment approximation requires an
accurate evaluation of the series in Eq. (

The very limited experimental information on the velocity of the vortices in
wind turbine wakes are in general agreement with this argument.

A closed form sum for

Values of the influence coefficient for varying

Each of the rings not containing the control point was approximated by an
even number,

Relative error of the straight segment approximation for the
conditions in Table 1 (

Angular contribution of straight segments to the influence
coefficient for

Figure

Variation in relative error with

One of these complexities is that the control point may not align with the
junction of segments on (in this case) adjacent rings. The effect of this can
be investigated by using non-zero

Variation in the straight segment approximation to

The second curvature error was shown in the last section to be caused largely
by aligned segments on the rings either side of the control point. For
increasing

Terms in Eq. (

The terms in Eq. (

The widely used straight segment approximation for approximating
the curved and periodic vortex wakes of wind turbines, propellers, and
helicopters can have two errors associated with the wake curvature. The first
is the well-known error in reproducing the locally induced binormal velocity.
This is usually accommodated by a cut-off in the Biot–Savart determination of
the vortex velocity using Eqs. (

By modelling the far wake of a wind turbine as an infinite row of equispaced vortex rings, two important results were obtained. First, it was shown that the velocity associated with the second error dominates at the small spacings typical of modern wind turbine operation. The available experimental evidence on wake structure is consistent with this finding. Then it is shown that the second error is quadratic in the number of segments per revolution and inversely proportional to the spacing of the rings, which is proportional to the pitch of a more realistic, but more difficult, helical wake. The model to investigate the second correction is artificial in that a single, infinite row of vortex rings of constant spacing, radius, and circulation is not applicable to the near wake. Nevertheless the model demonstrated the general importance of the rings adjacent to the control point at which the velocity is being calculated. These adjacent rings contribute over 80 % of the correction that is needed because the straight segment approximation does not correctly determine the contribution to the induced velocity from the closest parts of the adjacent rings, called the aligned segments.

It was also shown that the best behaviour possible for the second error is cubic in the product of the number of segments per revolution and the vortex spacing. It is likely, however, that larger numbers of vortex segments would be needed to achieve this error than are used in practice. This result was generalized to develop a second correction that improves the computed induced velocity by nearly 1 order of magnitude.

The MATLAB codes used in this study are available from the author.

The author declares that he has no conflict of interest.

This work is part of a research project on wind turbine aerodynamics funded by the NSERC Discovery Grants Program. Edited by: Alessandro Bianchini Reviewed by: Joseph Saverin and Wang Xiaodong