## 1 Introduction
Alternatively stated, a fluid particle passing through the WT rotor loses part of its kinetic energy (KE), viz, the flow through a WT slows down gradually from an upstream free-stream speed of \(U_{0}\) to an average value far downstream in the wake \(U_{\text{zb}}\). The static pressure increases from its upstream value \(P_{\infty}\) to a value \(P_{r}{}^{+}\) just in front of the rotor and then drops to \(P_{r}{}^{-}\) behind the rotor. The difference in static pressure between the rotor inlet and its outlet results in an axial force exerted on the rotor. Recovery of the pressure lost by the wind as it flows past the WT occurs gradually in the wake until the initial free-stream pressure \(P_{\infty}\) is attained. It is worth noting that \(P_{r}{}^{+}-P_{\infty}>P_{\infty}-P_{r}{}^{-}\)[7]. The flow past a WT rotor is decelerated. Ideally, this deceleration of flow can result in a maximum power extraction of 59.26% of the initial KE per unit time of the wind by a WT rotor as established from the ADT, which is based on the conservation laws of flow as discussed in the foregoing paragraph. As stated earlier, the ADT replaces the WT rotor with a permeable disk with a distributed load, yielding the same overall thrust as the actual rotor. Despite this far-reaching abstraction, the ADT is the basis of WT aerodynamics. The maximum efficiency achievable by a HAWT is attributed to [PERSON] by most German and British scholars, while Russians attribute it to [PERSON] [8]. In their bid to achieve good analytical results, [PERSON] et al. [9] developed an analytical method predicated on the conservation of mass and linear momentum in the context of the ADT and the assumption that the velocity deficit in the wake follows a double Gaussian distribution. Existing versions of the VRT, typically called ring vortex theories, predicate on the fact that the axial induction coefficient at the wake is \(2a\) or in other words the wind speed at the wake of a WT is \(U_{w}=U_{0}(1-2a)\). A case in point is the ring vortex actuator disk model by [PERSON] et al. [10]. [PERSON] et al. [10] proposed a ring-vortex actuator disk theory (RVADT) based on a uniformly loaded disk with no wake rotation, predicatedon the axial and generalized momentum theories, respectively. They used their theory to numerically investigate the effects of the hub of HAWTs on the flow past a WT rotor. Per these researchers, the hub blockage effects are more pronounced in small HAWTs whose hub radius to rotor radius ratios range between 25 and 30% than in large HAWTs, which typically have hub radius to rotor radius ratios ranging between 4 and 5%. These researchers assert that hub blockage effects on large HAWTs are lower than those of smaller HAWTs such that neglecting these effects has little impact on the solution for large HAWTs than for smaller HAWTs. The RVADT models the flow induced by the rotor disk and hub with ring sheet vortices, which are discretized through a classical panel method by utilizing a homogeneous Dirichlet boundary condition for the velocity just beneath the hub vortex sheet, while a free force condition is imposed along the entire boundary [10]. These actuator disk-based ring vortex theories have a drawback in the sense that when \(a=0.5\), the axial component of the wind speed at the wake is zero, something that causes the air particles to stagnate at the Trefftz plane and undergo circular motion. This is not what occurs in real life. The proposed VRT does not result in a zero axial wind speed at the wake when the interference factor is 0.5 because the induction factor due to wake rotation \(a_{w}\) is not necessarily 0.5. Other models have been used by numerical analysts to study WT wakes. Wake modeling approaches include empirical correlations, the ADT and full three-dimensional computational fluid dynamics (CFD) simulations. Analytical methods, such as [PERSON]'s blade element momentum theory (BEMT), have been extended to computational WT models, such as the vorticity-velocity model (VVM) [9].