## 1 Introduction
The ALM is implemented by discretizing each rotor blade into span-wise segments. The blade element method (BEM) is used in conjunction with the flow field data returned by a Reynolds Average Navier-Stokes (RANS) solver to estimate the two-dimensional aerodynamic forces acting at the quarter chord of each span-wise segment. The force on each segment of radius \(r\) and length \(\delta r\) is the vector sum of the lift and drag. The ADT, the BEMT and three-dimensional blade-resolved methods are frequently used in the simulation of the flow past HAWT rotors. The ADT and the BEM allow for fast simulations but do not accurately simulate flow features, such as unsteady wake development [18]. The BEMT neither accurately predicts the axial velocity of flow past a HAWT rotor, especially around the inner part of the rotor where wake rotation accelerates the flow and the outer part where it decelerates the flow [19]. It is worth noting that the ABL is quite stable under nocturnal conditions, so much so that a considerable shear develops with an increase in wind speed as the height above the ground increases. Shear in the wind entering large HAWTs with mega-watt capacities is a frequent operational condition. Simulating such inflow conditions using the BEM, as is typically done using most common aero-elastic codes, is fraught with uncertainty. This uncertainty is the upshot of the lack of a well-documented procedure of the BEM model for sheared inflow [20]. Yet another method for simulating the flow past HAWTs is the actuator sector method (ASM). The ASM distributes the forces applied to the fluid azimuthally to maintain a continuous flow solution for increased time steps compared to the ALM. It is a hybrid actuator model that preserves the features of the ALM and reduces the computational expense associated with small time steps, which are typical of the latter. The ASM is the intermediary between the ALM and the ADT [21]. In this paper, a comparison is made between the ADT and the RAST derived by others [2; 3] and the VRT. To carry out the comparison, the design parameters of the United States Department of Energy's National Renewable Energy laboratory's (NREL) 5 MW reference offshore WT (OWT) are used. The WT to be investigated in this work is an upwind, three-bladed, 5 MW NREL reference HAWT. The OWT has a swept radius of 63 m, a hub height of 90 m, a hub diameter of 3 m, a cut-in rotational speed of 6.9 RPM and a rated rotational speed of 12.1 RPM. Its cut-in wind speed at 90 m hub height is 3 m/s, its rated wind speed at hub height is 11.4 m/s and its cut off wind speed is 25 m/s [22]. This paper is structured as follows: Section 2 is a critique of the generalized momentum theory that forms the basis for the ADT and the RAST. Section 3 and its subsections consist of the derivation of the basic equations of fluid dynamics from which the aerodynamic equations are developed. Section 4 outlines the steps followed in coding the necessary equations into MATLAB and the subsequent plotting of curves. Section 4 consists of the parameters of the 5 MW reference NREL OWT. Section 5 consists of the results and discussion of the results of the analysis based on the three theories; subsection below Section 5 is a comparison between the VRT results and existing numerical and experimental results. Section 6 is the conclusion of the paper.