Volume 3, issue 1 | Copyright
Wind Energ. Sci., 3, 345-352, 2018
© Author(s) 2018. This work is distributed under
the Creative Commons Attribution 4.0 License.

Research articles 07 Jun 2018

Research articles | 07 Jun 2018

The second curvature correction for the straight segment approximation of periodic vortex wakes

David H. Wood David H. Wood
  • Department of Mechanical and Manufacturing Engineering, University of Calgary, Calgary T2N 1N4, AB, Canada

Abstract. 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.

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Short summary
The vortices in the wakes of wind turbines are often approximated by short, straight vortex segments, which cannot reproduce the curvature singularity in the induced velocity. They can also have a second error due to the periodicity: the vortices return to close proximity of the point at which the velocity is calculated. The second error is assessed by representing the far wake of a turbine as a row of vortex rings. The error is quantified and a simple correction is developed.
The vortices in the wakes of wind turbines are often approximated by short, straight vortex...