Abstract
The velocity field induced by a right vortex cylinder is derived in details in this chapter. The results of this vortex model provides the basis of many of the analyses presented in this book. The vector potential and velocity field are expressed using the Biot–Savart law. The Biot–Savart law in terms of solid angle is also used. Results for finite, infinite and semi-infinite cylinders are provided. The case of longitudinal and tangential vorticity along the cylinder surface are considered. Cylinders of circular and arbitrary cross sections are investigated. The flow in the full domain or at key locations is presented. Different illustrations of the flow are provided. The derivations steps are provided in details in this chapter. Cylinders of tangential vorticity with arbitrary cross sections are considered in a first section. Different results are derived. It is shown in particular that the velocity induced by an infinite vortex cylinder is constant and equal to the vortex intensity inside the cylinder and equal to zero outside. It is shown that the velocity induced by a finite cylinder of tangential vorticity is linked to the velocity induced by source surfaces using the Neumann-to-Dirichlet map. The vector potential and velocity field induced by a vortex cylinder of circular cross section and tangential vorticity is derived next. Similar developments are used for the cylinder of longitudinal vorticity. A Matlab source code is provided to compute the velocity field induced by a semi-infinite cylinder of tangential vorticity. This code can directly be used to study the induction zone in front of a rotor. Most of the results presented in this section were published in the article titled “Cylindrical vortex wake model: right cylinder” (Branland and Gaunaa, Wind Energy, 524(1), 1–15, 2014, [2]). Results from this chapter are used: in Chaps. 17 and 18 to model a wind turbine (or rotor) in uniform inflow, in Chap. 24 to study the induction zone in front of a wind turbine, in Chap. 26 to model a wind turbine/rotor under unsteady situations.
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Notes
- 1.
In his article, the value \(V_C\) has been mistyped and should read \(V_C=1/(2r)\).
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Branlard, E. (2017). Flow Induced by a Right Vortex Cylinder. In: Wind Turbine Aerodynamics and Vorticity-Based Methods. Research Topics in Wind Energy, vol 7. Springer, Cham. https://doi.org/10.1007/978-3-319-55164-7_36
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DOI: https://doi.org/10.1007/978-3-319-55164-7_36
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