## 1 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 ADT and the RAST, whose origins are discussed hereafter, rely on the axial momentum balance in their formulation. The VRT is different from the ADT and the RAST in the sense that these two theories rely on the idealization that the wind speed at the wind turbine's (WT) rotor is the average of the wind speed in the wake and the free-stream speed of the wind upstream of the WT. The VRT considers the fact that the wind slows down as it flows past the WT, resulting in an induction or interference factor due to solidity \(a_{s}\), and that a further reduction in the wind speed occurs in the wake, resulting in an axial induction factor due to wake rotation \(a_{w}\). There is also a tangential or azimuthal component of the induction factor due to wake rotation \(a^{\prime}\), commonly called the azimuthal or angular induction factor. The RAST also makes use of the angular induction factor. Thus, per the VRT, the axial induction factor at the WT wake is \(\delta=a_{s}+a_{w}\) and not twice the induction factor at the WT rotor. It is worth noting that for \(a=0.5\), the freestream speed in the axial direction at the Trefftz plane is zero in the cases of the ADT and the RAST, whereas it is non-zero in the case of the VRT because \(a_{\text{s}}=0.5\) does not necessarily result in a \(\delta=1\). The VRT represents the wake of a horizontal-axis WT (HAWT) with a conical control volume (CV) and applies the Biot-Savart law [4] to the CV to determine \(a^{\prime}\) and \(a_{w}\). 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 [PERSON] 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. Upstream the HAWT, the air pressure is above atmospheric due to the slowing down of the airstream. As the airstream passes through the WT, it loses pressure [1]. Improvements on this theory were made by [PERSON] [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. [PERSON] [4] attempted to extend the ADT to unsteady fluid flow. In his treatise, he expatiated on both the steady and unsteady flow past a HAWT using a vortex tube as his CV. The vortex tube consisted of a continuous succession of vortex rings shed by the rotor and carried downstream by the flow. Suffice it to say, [PERSON]'s CV is quasi-cylindrical and does not seem to account for the expansion of the flow resulting from the slowing down of the air as it passes through the rotor. After rigorous mathematical manipulations, this researcher showed that at steady state, the induced velocity downstream the rotor at the Trefftz plane is twice the induced velocity at the rotor. In his bid to estimate the maximum power extracted by a HAWT, [PERSON] used the one-dimensional mass conservation and momentum conservation laws in conjunction with the ADT first developed by [PERSON] and [PERSON] for propellers [5]. Other schools of thought [1] hold that the ADT is the brainchild of [PERSON]. The [PERSON] model dwells on the premise that the power extracted by the WT is solely derived from the pressure drop across the rotor. This pressure drop results in a static thrust on the rotor [6].