Vortex Ring Theory—An Alternative to the Existing Actuator Disk and Rotating Annular Stream Tube Theories

James Agbormbai,Liang Li,Weidong Zhu

doi:10.3390/app11146576

Abstract

Currently, the actuator disk theory (ADT) and the rotating annular stream-tube theory (RAST), both of which predicate on the axial momentum and generalized momentum theories, among others, are commonly used in investigating the aerodynamic characteristics of horizontal axis wind turbines (HAWTs). These theories, which are based on a rotor with an infinite number of blades, typically do not properly capture the flow physics of wind blowing past the rotors of HAWTs. A vortex ring theory (VRT) that analyzes HAWTs based solely on the characteristics of fluids flowing past obstructions is proposed. The VRT is not predicated on the assertion that the induced velocity in the wake is twice the induced velocity at the rotor. On the contrary, it splits the axial induction factor in the wake into two components, namely, the induction or interference factor due to the solidity of the rotor and the induction factor due to the wake of the rotor aw; aw and its azimuthal counterpart are determined using the Biot–Savart law. The pressure differences across the rotor segments of a HAWT are derived from the Bernoulli equation for all the three theories. Blade segment/local areas based on the blade sectional geometry of the rotor are used in the case of the VRT to estimate the local forces. All the calculations in this study are based on the design parameters of the 5 MW National Renewable Energy Laboratory’s reference offshore wind turbine. Pressure differences are plotted as functions of local radii using the calculated axial and azimuthal induction factors for each theory. The local power coefficient is plotted as a function of the local tip-speed ratio, while the local thrust coefficient is plotted as a function of the local radii for all the three theories. There is piece-wise agreement between the VRT, the ADT, the RAST and numerical and experimental data available in the literature.

Highlights

This paper compares a vortex ring theory (VRT) that is not based on the general momentum theory to the existing actuator disk theory (ADT) [1] and the rotating annular stream-tube theory (RAST) [2,3]
The VRT considers the fact that the wind slows down as it flows past the wind turbine (WT), resulting in an induction or interference factor due to solidity as, and that a further reduction in the wind speed occurs in the wake, resulting in an axial induction factor due to wake rotation aw
There is no gainsaying that the VRT better captures the aerodynamics of horizontal axis wind turbines (HAWTs) than the RAST and the ADT

Summary

Introduction

This paper compares a vortex ring theory (VRT) that is not based on the general momentum theory to the existing actuator disk theory (ADT) [1] and the rotating annular stream-tube theory (RAST) [2,3]. The CV is a conical helix since the flow past a WT rotor is solenoidal and a potential flow This conical CV should not be misconstrued to be a diffuser, as is typical of shrouded WTs. The intent of this paper is to delve into the aerodynamics of commercial-scale unshrouded WTs. The axial momentum balance across the rotor of a HAWT forms the basis upon which Rankine formulated the ADT [1]. The axial momentum balance across the rotor of a HAWT forms the basis upon which Rankine formulated the ADT [1] He represented the rotor with a permeable disk of cross-sectional area A, which extracts energy from the air flowing past it by reducing its pressure. Improvements on this theory were made by Froude [5] who included the effects of the wake in his calculations and stated that the induced velocity at the wake of an actuator disk is twice the induced velocity at the disk

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Conclusion