**1. Introduction**

Nowadays, the actuator line method (ALM) has been widely used in wind farm simulations due to its capability of wind turbine wakes simulation and its numerical stabilization and low computational cost. This method was developed by Sørensen and Shen [1] in 2002 to overcome the disadvantage of Blade Element Momentum theory (BEM), which cannot simulate the wake characteristics of wind turbines. By combining the computational fluid dynamics (CFD) method and blade element theory, the ALM method avoids the calculation of the boundary layer flow and thus greatly reduces the computational cost compared with resolved CFD approaches. The benefits of low computational cost have two aspects. Firstly, the mesh used in ALM simulation is more regular than the resolved approaches, which means that the orthogonality of the mesh is much better. Therefore, the Large Eddy Simulation (LES) turbulence model, which is more accurate but computationally expensive and sensitive to mesh quality compared with RANS-based turbulence models, can be easily applied to ALM simulations. Combining with the LES model, ALM can make good simulations of velocity field and turbulence field in the wake region [2,3] and it has advantages in wind turbine simulations when the inlet condition is complex, such as the atmospheric boundary layer condition [4,5]. Secondly, due to its low computational cost, ALM can be used in large-scale problems [4,6–8] and can be easily coupled with structural models [9,10]. Therefore, ALM is suitable for wind farm simulations and fluid-structure interaction simulations. In summary, ALM nowadays has become the most potential method in wind turbine simulations, especially in wind farm simulations and fluid-structure interaction simulations.

Regularization kernel was originally introduced to the ALM approach to avoid the numerical singularity [1]. During the ALM approach, the aerodynamic forces of wind turbine are calculated according to the blade element theory and the wind velocity field is calculated by solving the Navier–Stocks equations. Therefore, a regularization kernel must be employed to smoothly apply these aerodynamic forces to the Navier–Stocks equations and a uniform three-dimensional Gaussian function which is suggested by Sørensen and Shen [1] is widely used as the standard regularization kernel.

The regularization kernel also affects the conceptual shape of the wind turbine blade. When using the standard regularization kernel, the conceptual shape of a wind turbine blade will be like a cylinder [11], which is inconsistent with its real shape. Martínez-Tossas et al. [12] proposed a two-dimensional elliptical Gaussian function as the regularization kernel and its direction is based on the global coordinates. Churchfield et al. [13] developed an anisotropic Gaussian function as the regularization kernel whose direction is determined by the local coordinates of each blade element. By using these anisotropic kernels, the shaped of wind turbine blades can be better modeled, which will alleviate the need of tip correction and improve the simulation near the blade tip.

The gaussian width used in the regularization kernel was found to have great influence on rotor torque predictions, wake characteristics [2,14], and may cause new requirement for the mesh [15,16]. Troldborg states that the value of should be at least twice the local grid length to avoid numerical oscillation. Martínez-Tossas et al. [12] and Shives et al. [17] suggest that the value of should be a quarter of the local airfoil chord length. Shives also recommends limiting grid size to a quarter of . Churchfield et al. [13] states that should be around 0.035 times the rotor diameter when using the standard regularization kernel. Pankaj et al. [14] developed and tested a series of guidelines for choosing ALM parameters and the results showed that the appropriate should be determined by a function of blade aspect ratio, grid size and a empirical constant. As for the anisotropic kernel, Martínez-Tossas et al. [12] studied the influence of the chord length parameter <sup>c</sup> and the thickness parameter <sup>t</sup> for two dimensional flow and the result shows that <sup>c</sup> ≈ 0.4*c* and <sup>t</sup> < 0.2c are optimal. Churchfield et al. [13] studied the 3-dimensional wake characteristic of NREL (National Renewable Energy Laboratory) phase VI wind turbine with <sup>c</sup> = 0.85*c*, <sup>t</sup> = 0.85*t* and simulations using the anisotropic kernel are more consistent with the experiment than the results of the standard kernel.

However, there is still confusion about the optimal value of the gaussian width and the effect of each parameter used in anisotropic kernel on ALM simulation result is still unclear. The recommended value of the gaussian width from the studies above do not agree with each other. Due to the author's experience, these recommended values of do not always lead to reliable results. Although the anisotropic kernel was developed in Churchfield's study [13], the parameters used in anisotropic kernel were not systematically studied. Furthermore, different parameters of <sup>c</sup> = 0.4*c*, <sup>t</sup> = 0.2*c* were also used in this study for simulations of NREL 5MW wind turbine and the reason were not explained. Furthermore, the wake effect has great influence on the rotor torque of downstream wind turbines so the wake characteristics is significant in wind farm simulations and must be experimentally validated. In summary, the influence of the gaussian width used in regularization kernel and anisotropic kernel needs further study.

In this study, a new method is developed to measure the three-dimensional velocity field more efficiently and less expensively. Borrowing the idea of a frozen rotor, which is widely used in CFD simulations, the wake characteristics are reconstructed from the simultaneously gathered data of hot-wire anemometer and encoder. This measurement approach not only rebuilds the velocity distribution in a plane (along with or perpendicular to the main flow) but also reconstructs the whole wake region of a wind turbine. In this study, the influence of Gaussian width used in ALM with

anisotropic regularization kernel is studied. Validated by the experimental results of power and wake characteristics, the relationship among the parameters of the anisotropic regularization kernel, physical scale of the blade, and mesh grid size are determined. This relationship will be used in further studies of the coupled aeroelastic wake behavior of a wind turbine based on ALM.

#### **2. Method**

## *2.1. Actuator Line Method (ALM)*

The actuator line method is realized using OpenFOAM which is an open-source computational fluid software. The Large Eddy Simulation (LES) turbulence model is employed in this study because of its accuracy in wake simulations. The equations are shown as Equations (1) and (2).

$$\frac{\partial \overline{u\_i}}{\partial \mathbf{x\_i}} = 0 \tag{1}$$

$$
\rho \frac{\partial \overline{u}\_i}{\partial t} + \rho \frac{\partial \left(\overline{u}\_i \,\, \overline{u}\_j\right)}{\partial \mathbf{x}\_j} = -\frac{\partial p}{\partial \mathbf{x}\_i} + \frac{\partial}{\partial \mathbf{x}\_j} \left[\mu \left(\frac{\partial \overline{u}\_i}{\partial \mathbf{x}\_j} + \frac{\partial \overline{u}\_j}{\partial \mathbf{x}\_i}\right)\right] + \frac{\partial \tau^s\_{ij}}{\partial \mathbf{x}\_j} + f \tag{2}
$$

where *u* is the filtered velocity vector field, *p* is the scalar field of pressure, μ is a scalar represent the kinematic viscosity, τ*<sup>s</sup> ij* = −ρ *uiuj* − *ui uj* is called the subgrid-scale (SGS) Reynolds stress [18], and the standard Smagorinsky SGS model is employed in this study.

*f* is the source term which represents the wind turbine blade forces in ALM. When considering the wind turbine blade as a series of blade elements, the force along the blade can be calculated according to Equation (3). Here *Cl* and *Cd* are the lift and drag coefficient, respectively. ρ is the density of air, *v* is the inlet velocity of the blade element, *c* is the chord length, and *L* is the length of the each blade element.

$$F\_{elemmt} = (F\_{l\prime}F\_d) = \left(\frac{1}{2}\rho v^2 c \mathcal{C}\_l L, \frac{1}{2}\rho v^2 c \mathcal{C}\_d L\right) \tag{3}$$

The forces calculated by Equation (3) are point forces and a regularization kernel must be employed to avoid a numerical singularity, as shown Equation (4). Traditionally, a uniform three-dimensional Gaussian function is employed as the standard regularization kernel in the actuator line method. is Gaussian width which adjusts the strength of this regularization kernel.

$$f = \sum F\_{\text{element}} \otimes \eta\_{\text{c}} \tag{4}$$

$$\eta\_{\mathfrak{c}} = \frac{1}{\mathfrak{e}^3 \pi^{3/2}} \mathbf{e}^{-(\frac{\mathfrak{c}}{\mathfrak{e}})^2} \tag{5}$$

However, this uniform function will lead an imprecise approximation of the shape of the wind turbine blade. Although the chord length and the twist angle of the blade elements vary a lot from the root to the tip, the shape of the blade in actuator line model will be like a cylinder because of this uniform function. Furthermore, the uniform smooth function will cause the blade element force to be over concentrated along the chord direction, but more scattered along the thickness direction of the blade element at the meantime. Recently, an anisotropic regularization kernel as shown in Equation (6) was developed to overcome the disadvantages of the standard one.

$$\eta\_{\mathfrak{c}} = \frac{1}{\mathfrak{e}\_{\mathfrak{c}} \mathfrak{e}\_{\mathfrak{t}} \mathfrak{e}\_{\mathfrak{l}} \pi^{3/2}} \mathrm{e}^{-\left(\frac{r\_{\mathfrak{c}}}{a\mathfrak{c}}\right)^{2} - \left(\frac{r\_{\mathfrak{l}}}{a\_{\mathfrak{l}}}\right)^{2} - \left(\frac{r\_{\mathfrak{l}}}{a\_{\mathfrak{l}}}\right)^{2}} \tag{6}$$

Here *rc*, *rt*, *rl* are the distances between the grid center and the force point in local coordinates of each blade element and *<sup>c</sup>*, *<sup>t</sup>*, *<sup>l</sup>* are the corresponding Gaussian widths. Figure 1 shows the influence range and the strength distribution of these two kernels, the blue curve represents the standard kernel and the red one represents the anisotropic kernel

**Figure 1.** A schematic diagram of the influence range and strength distribution of two types of regularization kernel, the blue curve represents the standard kernel and the red one represents the anisotropic kernel.
