2.1.1. Actuator Disk Model

The AD model neglects the geometry detail of individual wind turbine blades. It represents the rotating blades as a fixed 2D porous disk exerting a uniform thrust on the flow, which is the numerical reflection of the perforated disk model usually used in wind tunnel experiments. Neither the rotational effect nor the non-uniform force distribution are considered in the model employed in this work. The axial thrust force *fT* per unit area is uniformly distributed over the entire rotor disk surface *A* and is expressed with the thrust coefficient *C*<sup>T</sup> and the inflow velocity *V*ref:

$$f\_T = \frac{1}{2}\rho V\_{\text{ref}}^2 \mathbb{C}\_{\text{T}}.\tag{1}$$

where *ρ* is the density of air. The reference velocity *V*ref is defined to be equal to the freestream velocity in uniform inflow condition. In turbulent inflow simulations, the present work approximately calculates *V*ref by averaging the velocity on a disk of the rotor's size at one diameter in front of the real turbine. The trust coefficient *C*<sup>T</sup> remains to be determined according to the turbine operation state. In this work, *C*<sup>T</sup> is set to be equal to that of the AS simulations to ensure a fair comparison between the two models.

#### 2.1.2. Actuator Surface Model

The AS model represents the geometry of an individual wind turbine blade with a simplified two dimensional surfaces of zero thickness, which is formed by chords at different radial locations [11,12]. In the actuator model employed in this work, the aerodynamic forces on the surface vary with the radial position and are determined by the tabulated airfoil data in the same ways as the AL model as follows:

$$\mathbf{L} = \frac{1}{2} \rho \mathbf{C\_L} \mathbf{c} |V\_{\text{ref}}|^2 \mathbf{e\_L} \tag{2}$$

and

$$\mathbf{D} = \frac{1}{2} \rho \mathbf{C}\_{\mathrm{D}} \mathbf{c} |V\_{\mathrm{ref}}|^2 \mathbf{e}\_{\mathrm{D} \prime} \tag{3}$$

where **L** and **D** are the lift and drag force per unit length, *ρ* is the density of air, *c* is the chord length, *V*ref is the flow velocity relative to the rotating blade, **e**<sup>L</sup> and **e**<sup>L</sup> are unit directional vectors for lift and drag forces. *C*<sup>L</sup> and *C*<sup>D</sup> are the lift and the drag coefficients defined in 2D airfoil tables as a function of Reynolds number and the angle of attack. Corrections including the 3D stall delay model of Du and Selig [24] and the tip loss correction of Shen et al. [25,26] are applied.

After calculating **L** and **D**, the force **f** per unit area on the surface model is calculated by:

$$\mathbf{f} = (\mathbf{L} + \mathbf{D})/\boldsymbol{\varepsilon}.\tag{4}$$

The reacting forces exerting by the blade on the air are then distributed to the background Eulerian grid points with a smoothed discrete delta function [27].
