Abstract
A simple vortex model of a rotor with infinite number of blades and constant circulation is presented in this chapter. The model consists of a root vortex, a right semi-infinite vortex cylinder of tangential and longitudinal vorticity, and a bound vortex disk (actuator disk) of radial vorticity. This model was briefly presented in Sect. 5.2.3 The model forms the basis for the results derived in several chapters of this book, namely: Chaps. 18, 21, 25 and 26. Most of the analytical derivations regarding the vortex cylinder are given in Chap. 36. The reading of Chap. 36 and in particular in Sect. 36.2.2 is highly advised for an insight on the velocity field induced by a semi-infinite vortex cylinder of tangential vorticity and the flow properties at the rotor disk. The content of the current chapter is based on results presented in an article by the author, titled “Cylindrical vortex wake model: right cylinder” (Heyson and Katzoff, Langley Field, 1956, [2]). The chapter starts with a literature review on the cylindrical model. The key results of the model are described in the next section. It is shown that the model is consistent with the axial momentum theory under the assumption of large tip-speed ratio. It is also shown that a superposition of concentric systems under the assumption of infinite tip-speed ratio leads to the annuli-independence assumption that is used in Blade element theory and stream-tube analyses. The vortex cylinder model can be used to determine the velocity field analytically in the induction zone in front of a wind turbine or any rotor. This is applied in particular in this chapter to compute the velocity deficit upstream of a wind turbine on the rotor axis. A more thorough application to the induction zone of yawed and straight rotors is presented in Chap. 24.
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References
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Branlard, E. (2017). Cylindrical Vortex Model of a Rotor of Finite or Infinite Tip-Speed Ratios. 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_17
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