Investigation of Wake Expansion for Spanwise Arranged Turbines in the Offshore Wind Farm by Large Eddy Simulation

Abstract

The issue of wind turbine wake effects in the offshore environment has become increasingly important with the development of offshore wind farms. The problem of wake dispersion from turbines plays a crucial role in evaluating the wake velocity deficit and solving the optimization problem of wind farms. This study focuses on the wake expansion of spanwise arranged turbines using Large Eddy Simulation (LES). Firstly, numerical models are compared with the data from previous studies to validate their accuracy. Secondly, the study analyses wake structures for varying lateral spacings in spanwise turbine configurations using the actuator line model (ALM). Lastly, by comparing the predictions of wake expansion between existing models, a modified model considering added turbulence is proposed and then validated using LES data, significantly enhancing accuracy for predicting the wake width under different array spacings in the wind farm.

1. Introduction

Traditional energy sources such as oil and coal have been widely exploited and utilized globally for over a century. Their extraction and processing technologies have reached a relatively mature stage. Although global oil production has increased by over 50% since 1970, the growth rate of newly discovered oil reserves has significantly declined. It has dropped from approximately 9% in 2000 to less than 2% in 2022, indicating a gradual weakening of the economic viability for continued development [1]. However, clean energy, as an emerging and rapidly growing source of power, has seen remarkable expansion. It offers a sustainable pathway to address the environmental challenges posed by fossil fuels. According to the International Energy Agency (IEA) report, global renewable energy annual installed capacity has increased nearly eightfold over the past decade. In 2023, the additional installed capacity reached 507 GW, almost 50% higher than in 2022. The future development prospects are widely regarded as promising [2].

For higher power generation efficiency and better economic benefits, modern wind farms often adopt large-scale cluster layouts. As offshore wind power expands to deeper and farther offshore areas, the cost of wind turbines with traditional foundations, such as monopiles and jacket structures, will be higher compared to those with floating foundations. Floating wind turbines are expected to become the preferred choice for offshore wind farm development [3]. Zhang et al. [3] proposed a new configuration for floating wind turbine platforms that uses a single-row arrow-shaped layout of wind turbines. To maximize wind capture while minimizing space occupation, the spanwise distance between turbines in the direction perpendicular to the structure heading is 1.2 times the rotor diameter (D). The self-adaptive property of platforms was verified under different wind conditions. Lee et al. [4] designed a quadrilateral multi-turbine platform with four 3 MW wind turbines on top spaced less than 2D. Coupled hydrodynamic response analysis was performed on the platform integrating wave energy converters. Bashetty et al. [5] designed a pentagonal platform structure with five 8 MW turbines installed on columns connected to a semi-submersible platform. The upstream turbine spacing is 1.5D, and the inflow velocity of downstream turbines on the platform was analyzed by numerical simulations. The designs of the aforementioned floating platforms increase the operational efficiency of turbine platforms by arranging multiple turbines within a limited space.

Computational fluid dynamics (CFD) methods use numerical models to simulate the airflow around wind turbines and to analyze the dynamic changes in turbine wakes by solving fluid dynamics control equations. The three main numerical simulation techniques include the Reynolds-averaged Navier–Stokes (RANS) method, large eddy simulation (LES), and direct numerical simulation (DNS). Among them, the RANS method is widely used to solve various engineering problems due to its simple operation, low computational demand, and ability to meet the accuracy requirements of engineering applications. However, when using the RANS method for research in the early days, it was found that there were significant errors in simulating the velocity of wind turbine wakes. DNS provides higher accuracy, but its demand for computing resources is even more stringent, limiting its widespread application. In the field of wind power generation, the LES method has become the main numerical technology for studying wind turbine wakes due to its high accuracy and acceptable computational requirements in simulating wind field flow [6,7,8]. Porté-Agel et al. [9] successfully simulated key flow field variables such as mean velocity, turbulence intensity, and power spectra in wind turbine wakes using LES. Ishihara and Qian [10] analyzed the wake velocity distribution downstream of a single wind turbine using LES, indicating that the velocity deficit distribution exhibits a self-similar Gaussian curve. Liu et al. [11] used the actuator line model coupled with LES technology to study the wake expansion of a single wind turbine under different inflow wind speeds and turbulence conditions. The wake exhibits non-linear expansion under different operating conditions. Stevens et al. [12] investigated the power output of wind farms by LES, with different spanwise spacings between turbines from 3.5D to 8D. Reduced power output was observed with decreasing spanwise spacing. Bastankhah and Abkar [13] analyzed the variation of velocity deficit in the streamwise direction of four-rotor turbine wakes. A lower deficit was found compared to the wake of a single rotor turbine in the downstream range of 2D–8D. The wake velocity is increased with rotor spacing from 1D to 1.25D. Lower wake deficit and turbulence intensity were also observed in the LES study on four-rotor turbines by Ghaisas et al. [14], with rotor spacing from 1D to 2D. Through a LES study of two wind turbine arrays with 3D and 6D spacing in a wind farm, Liu et al. [2] found that the wake expansion rate on the outer side of the array is lower at small spacings.

Although LES demonstrates excellent simulation accuracy, it has high computational resource requirements. Especially when simulating complex scenarios or multiple turbine layouts, a large number of grids and smaller time steps are required, leading to increased computation time. Compared to numerical simulation methods, the use of analytical wake models presents significant advantages in wind turbine design and optimization of wind farms. These models can effectively improve the computational efficiency of wake velocity. Crespo et al. [15] provided a comprehensive overview and analysis of different analytical methods for modeling wind turbine wakes, which can be used as predictive and design tools for wind turbines and wind farms. Currently, widely used wake analysis models include the Jensen model [16] and the BP model [17]. These classical models have provided references for subsequent researchers, leading to the emergence of a series of improved models. For example, Tian et al. [18] introduced a trigonometric function description of lateral velocity distribution to improve wake simulation based on the Jensen model. Porté-Agel et al. [19] proposed a more sophisticated three-dimensional wake model based on the BP model. Liu et al. [20] optimized the traditional one-dimensional Jensen model and proposed a new approach for a full-field wake model, considering the influence of terrain factors on wake effects. By appropriately reflecting the spanwise distribution of wake velocity, the prediction accuracy of analytical wake models has been significantly improved. Therefore, accurately predicting the distribution scale of wake velocity, i.e., the wake width, is crucial for wake models. The accurate determination of this parameter plays a vital role in establishing both the scale and magnitude of the wake velocity deficit profile, thereby affecting power output prediction and layout optimization based on the data of velocity distribution. A comprehensive review of wake width models is provided in detail in the next section.

Previous research has largely been limited to conventional wind turbine configurations with large spacing, so further investigation is needed regarding the wake characteristics under novel configurations in floating offshore wind farms. With smaller wind turbine spacing (less than 2D), these configurations offer benefits in improved space utilization and reduced structural costs [21,22]. Furthermore, a properly limited spacing arrangement can even enhance wake recovery rate and power output by the acceleration effect [23,24]. In a wind farm, due to the arrangement of turbine arrays, the wakes between turbines inevitably interact with each other, affecting the wake distribution. However, current wake models primarily focus on the impact of wake interactions on velocity deficit while neglecting their effect on wake expansion. These models typically assume that the wake expansion of a single turbine is equivalent to that of a turbine within a wind farm. The earlier study [2] is limited to evaluating the wake width on one side at specific array spacings. The accuracy is probably compromised by asymmetric wake expansion within the array, and a systematic study on the effect of lateral spacing is absent. Therefore, further in-depth research is required on wake width models considering interaction effects under various lateral spacings.

LES directly simulates the large-scale vortex structures in fluids, providing higher spatial and temporal accuracy, making it suitable for complex flow fields. It can more accurately capture the transient characteristics and interactions of wake flows. The advantage of analytical wake models lies in their high computational efficiency and simplicity of implementation, but their development requires a foundation of appropriate empirical formulas and physical principles. Therefore, to address the aforementioned issues, this paper uses LES to simulate wind turbine wakes under different spacing arrangements and investigates the effects of turbine layout on wake expansion. The numerical model used is first introduced and validated. Then, by analyzing the LES results, the influence of varying arrangement spacing on wake expansion is discussed. The performance of several wake width models is analyzed through comparison with LES results. A modified model is proposed and validated using LES data, aiming to improve the accuracy of wake width predictions through the added turbulence, considering the effect of wake interactions.

The content of this paper is as follows: Section 2 introduces wake width models; Section 3 presents numerical analysis methods and validates the numerical method used in this paper; Section 4 discusses the numerical results of wake expansion of turbine clusters and the modification of wake width models; and Section 5 draws conclusions.

2. An Overview of the Wake Width Model

After passing through the blades of a wind turbine, the wind experiences a decrease in speed due to aerodynamic effects and energy loss. This reduction in speed creates a distinctive flow pattern behind the turbine, known as the wake. The wake is characterized by lower wind speeds and increased turbulence compared to the undisturbed upstream flow. The Jensen model, proposed by Jensen [16] in 1983, is a commonly used wake model based on the principles of mass conservation and empirical parameters. The Jensen model indicates that the wake effect is influenced by various parameters, including the thrust characteristics of the turbine, turbulent intensity in the environment, and the distance between the target location and the turbine generating the wake. It assumes a uniform distribution of wind and incompressible air within the region, leading to the following relationship:

$$\rho \pi r_x^{2}u=\rho \pi r_0^2{u}_1+\rho \pi (r_x^2-r_0^2)u_0,$$

(1)

where, r_x is the wake radius at position x downstream from the turbine; r₀ denotes the initial radius of the wake region; u₀ represents the inflow wind speed; u₁ is the velocity of the wind just after passing the turbine blades; u is the velocity of the wake at position x; and ρ represents reference air density.

The model assumes that the wake behind the turbine expands linearly, and it uses the wake expansion coefficient k to describe the rate of wake expansion. The relationship for k is as follows:

$$r_x=kx+r_0.$$

(2)

By rearranging terms, the expression for the Jensen model can be obtained as

$$u=u_0\left[1-{\displaystyle \frac{2a}{{(1+\frac{kx}{r_0})}^2}}\right],$$

(3)

where, a is the axial induction factor; x is the axial distance between the downstream calculation point and the rotor; and u denotes the wake velocity at that point. Typically, empirical values are used, with a value of 0.075 [25] for onshore turbines and 0.05 [25,26] for offshore turbines. If the surface roughness of the calculation area is known, it can be calculated using an empirical formula provided by Frandsen [27]:

$$k={\displaystyle \frac{1}{2\ln (\frac{z}{z_0})}},$$

(4)

where, z represents the height of the wind turbine’s hub; and z₀ denotes the surface roughness. This empirical formula considers the influence of surface roughness, further enriching the theoretical foundation of wake models, improving the accuracy of wake models, and making them more applicable to real-world scenarios. Although there have been many subsequent improvements in analytical models, the parameter for evaluating wake dispersion remains a key factor. Tian et al. [18] further refined the wake expansion coefficient by considering the impact of added turbulence within the wake, expressed by

$$k=k_F\cdot I/I_u,$$

(5)

$$I=0.4{\displaystyle \frac{C_T}{x/D}}+I_u,$$

(6)

where, k_F is the expansion coefficient calculated by Equation (4); I is the turbulence intensity in the wake; I_u is the ambient streamwise turbulence intensity at the hub height; C_T represents the thrust coefficient; and D is the rotor diameter. Recognizing the lower growth rate of wake expansion in the very far wake region, Tian et al. [28] proposed a non-linear empirical model for calculating wake width, expressed by

$$r_x=2r_d{C}_T^{0.4}{I}_u^{0.2}{\left(x/D\right)}^{0.4},$$

(7)

where r_d denotes the rotor radius. Liu et al. [16] utilized numerical simulation results to derive a formula suitable for calculating the wake width in offshore wind farms. They conducted a validation and comparison of the formula. The fitted formula is as follows:

$${\displaystyle \frac{r_x}{D}}=1.86C_T^{0.29}{I}_v^{0.7}{\left({\displaystyle \frac{x}{D}}\right)}^{0.38}+0.42{\left(1-C_T\right)}^{-0.058},$$

(8)

where I_v is the lateral turbulence intensity.

In addition to Jensen-type models, the BP model [17] also assumes that the wake grows linearly with downstream distance. The formula for calculating wake width is as follows:

$$r_x=2\sigma ,$$

(9)

where σ presents the standard deviation of the Gaussian-shaped velocity deficit at position x downstream of the turbine, calculated as follows:

$${\displaystyle \frac{\sigma }{D}}=k^{\ast }{\displaystyle \frac{x}{D}}+\epsilon ,$$

(10)

where, k^* is the growth rate in the BP model; D is the rotor diameter; and ε represents the initial wake width, calculated by $0.2\sqrt{\beta }$ with $\beta =\left(1+\sqrt{1-C_T}\right)/\left(2\sqrt{1-C_T}\right)$. Through LES data, Niayifar and Porté-Agel [29] proposed a relationship between k^* and I_u, accounting for the added turbulence intensity, expressed by

$$k^{\ast }=0.3837I+0.003678,$$

(11)

$$I=\sqrt{I_{+}^2+I_u^2},$$

(12)

$$I_{+}=0.73a^{0.8325}{I}_u^{0.0325}{\left(x/D\right)}^{-0.32},$$

(13)

where, I₊ represents the added turbulence intensity; and a is the induction factor. Carbajo Fuertes et al. [30], through fitting full-scale field experiments, made corrections to the BP model, defined as

$$k^{\ast }=0.35I_u,$$

(14)

ε is denoted by

$$\epsilon =-0.6685I_u+0.34,$$

(15)

Cheng et al. [31] suggested that the wake expansion is more strongly affected by lateral turbulence and proposed a model expressed by

$$k^{\ast }=0.223I_v+0.022,$$

(16)

k^* is generally accepted that the wake growth rate k is twice the standard deviation growth rate k^* [32].

In a wind farm composed of turbine arrays, the interaction between wakes affects not only the velocity distribution but also the wake expansion. This influence persists even if the downstream turbines are not located within the velocity deficit region of the upstream turbines [2]. However, the above models do not take this effect into account. A high-precision numerical simulation method is employed to investigate this effect, which is introduced in the following section.

3. Numerical Model

3.1. Large Eddy Simulation (LES)

Large Eddy Simulation is an advanced numerical simulation method used in the field of computational fluid dynamics (CFD) to simulate complex flow phenomena. In LES, the flow field is decomposed into a combination of large-scale eddies and small-scale turbulence, where large-scale eddies are directly simulated while models are employed to parameterize the small-scale turbulence. In LES, the Navier–Stokes equations are decomposed into two parts: a filtered equation describing large-scale motion and a subgrid-scale model describing small-scale turbulence. The filtered continuity and momentum equations are given as follows:

$${\displaystyle \frac{\mathsf{\partial }{\overline{u}}_i}{\mathsf{\partial }{x}_i}}=0,$$

(17)

$${\displaystyle \frac{\mathsf{\partial }{\overline{u}}_i}{\mathsf{\partial }t}}+{\displaystyle \frac{\mathsf{\partial }({\overline{u}}_i{\overline{u}}_j)}{\mathsf{\partial }{x}_j}}=-{\displaystyle \frac{\mathsf{\partial }\widehat{p}}{\mathsf{\partial }{x}_i}}-{\displaystyle \frac{\mathsf{\partial }{\tau }_{ij}^D}{\mathsf{\partial }{x}_j}}-{\displaystyle \frac{1}{{\rho }_0}}{\displaystyle \frac{\mathsf{\partial }{p}_0(x,y)}{\mathsf{\partial }{x}_i}}+F_{ext},$$

(18)

where, t represents time; u represents wind speed; ρ₀ represents reference air density; τ_ij^D represents the deviatoric part of the fluid stress tensor; F_ext represents external forces in the flow field; and $\widehat{p}$ represents the modified pressure, with the expression as follows:

$$\widehat{p}={\displaystyle \frac{\overline{p}(x,y,z,t)}{{\rho }_0}}-{\displaystyle \frac{p_0(x,y)}{{\rho }_0}}+{\displaystyle \frac{{\tau }_{kk}}{3}}+gz,$$

(19)

where, $\overline{p}$ denotes the mean pressure; p₀ represents the static pressure; τ_kk denotes the trajectory of the stress tensor; and g represents the gravitational acceleration, typically taken as 9.81 m/s². The external forces in the flow field include forces applied by wind turbine blades, Coriolis force, and buoyancy, with the expression for F_ext as follows:

$$F_{ext}={\displaystyle \frac{1}{{\rho }_0}}{F}_i+g\left({\displaystyle \frac{\overline{\theta }-{\theta }_0}{{\theta }_0}}\right){\delta }_{i3}-{\epsilon }_{i3k}f{\overline{u}}_k,$$

(20)

where, F_i represents the force applied by the wind turbine; $\overline{\theta }$ denotes the potential temperature; θ₀ is the reference temperature; δ_i3 represents the Kronecker delta symbol; ε_i3k denotes the alternating unit tensor; and f is the Coriolis parameter defined as f = 2Ωsinϕ, where Ω is the Earth’s rotation rate and ϕ is the latitude. By combining the potential temperature equation, continuity equation, and momentum equation, we can obtain the potential temperature. The potential temperature equation is as follows:

$${\displaystyle \frac{\mathsf{\partial }\overline{\theta }}{\mathsf{\partial }t}}+{\displaystyle \frac{\mathsf{\partial }({\overline{u}}_j\overline{\theta })}{\mathsf{\partial }{x}_j}}={\displaystyle \frac{\mathsf{\partial }{q}_j}{\mathsf{\partial }{x}_j}},$$

(21)

where q_j represents the temperature flux.

3.2. Wind Turbine Modeling

In the numerical simulation of wind turbines, fully modeling the geometric model of the turbine would increase the difficulty of grid partitioning and consume computational resources. Therefore, when studying wind turbine wake issues, it is common practice not to directly model the turbine geometry but instead to simulate its effect by adding volume force source terms in the flow field. There are currently two commonly used wind turbine models: the actuator disk model (ADM) and the actuator line model (ALM). Previous studies have shown that coupling these two models with LES results in relatively small discrepancies between ALM-coupled LES simulations and experimental data in the near-field region, while in the far-field region, both model-coupled LES simulations agree well with experimental data [33].

The ALM was first proposed by Sorensen and Shen [34], utilizing volume forces at actuator points to simulate the influence of wind turbine blades on the flow field. The calculation of blade volume forces is based on blade element momentum theory, where at time t, the volume force at position (x, y, z) is given by

$$F_i(x,y,z,t)={\displaystyle \sum _{j=1}^{N}f(x_j,y_j,z_j,t)\frac{1}{{\epsilon }_i^3{\pi }^{3/2}}\exp \left[-{\left(\frac{d_j}{{\epsilon }_i}\right)}^2\right]},$$

(22)

where, (x_j, y_j, z_j) represents the coordinates of the j-th actuator point; d_j denotes the distance between the j-th actuator point and (x, y, z); and ε_i is a parameter used to adjust the regularization of the load distribution. The NREL 5 MW wind turbine [35] is used in the following simulations, with a rotor diameter of 126 m and a hub height of 90 m.

3.3. Numerical Setup

The solver used in this study is the SOWFA developed by the National Renewable Energy Laboratory (NREL) [36], with a version of SOWFA-2.4.x. The PISO (Pressure-Implicit with Splitting of Operators) algorithm [37] is employed during the solving process. The simulations in this study utilized the computing resources of the National Supercomputing Center (Tianhe-1A), with 140 cores allocated for each simulation case. In LES, the design of inflow boundary conditions is crucial as it directly impacts the accuracy and credibility of the simulation results. This paper utilizes a precursor simulation method to generate inflow boundary conditions. The numerical simulation in this paper is mainly divided into two stages: the first stage is the precursor simulation, and the second stage is the wind turbine simulation. The precursor simulation plays a crucial role in LES by simulating the turbulent flow field without incorporating the wind turbine model. Through appropriate grid partitioning, it effectively captures the detailed flow structures and vortex characteristics.

The computational domain in this paper is set as a rectangular prism with dimensions of 3000 m × 3000 m × 1000 m, as shown in Figure 1, and the initial grid size is 10 m × 10 m × 10 m. Regarding the choice of wind direction, Churchfield et al. [36] mentioned in their study that in typical numerical simulations, the wind direction is specified as 270°, aligning it with the positive x-direction of the computational domain. This alignment results in the formation of long streamwise turbulent structures that continue to circulate along the constant y-direction. In the simulation, the combination of periodicity along the lateral boundaries and flow inclined to the boundaries somewhat disrupts these structures. Therefore, the wind direction at a 90 m hub height is set to 240°, as shown in Figure 1, and the average wind speed is 8 m/s. The pre-simulation requires sufficient time to reach quasi-equilibrium conditions. In this study, the duration of the pre-simulation is set to 20,600 s, and the time step is 0.5 s. The first 20,000 s are allocated for generating turbulent flow fields, while the subsequent 600 s are utilized for saving relevant data to serve as inflow boundary conditions for the wind turbine simulation. To reduce computational expenses during the initial 20,000 s, a variable time step approach is implemented, with a maximum Courant number of 0.75. In numerical simulations, accurately simulating the atmospheric boundary layer (ABL) and turbulence flow is crucial for the analysis of wind turbine wakes. The realistic characteristics of the ABL determine the accuracy of the inflow conditions, while turbulence intensity directly affects the structure and evolution of the wake. By carefully capturing the velocity profiles, temperature gradients, and turbulence characteristics within the ABL, the simulation can more realistically reproduce the wake behavior behind wind turbines. Therefore, in the next stage for wind turbine simulations, the upstream inlet boundary condition uses the turbulent data saved from the inflow plane during the last 600 s of the pre-simulation as input, while the downstream outlet boundary condition is set with zero velocity and temperature gradients to allow wakes to exit. The boundary conditions for the other sides remain the same as in the pre-simulation. It takes some time for the wind turbine simulation to reach a steady state; therefore, data from the last 400 s of the simulation is used for calculating the required physical quantities in this study.

In this study, a simulation of a series of wind turbines is conducted initially for model validation. For details on model validation, please refer to the next section. Subsequently, simulations are performed for different lateral spacing configurations of parallel wind turbines. As research on dual-rotor floating wind turbines becomes increasingly widespread [24], the extreme case of relatively small lateral spacing is first considered, with the lateral spacing set to 1.05D. Figure 1a illustrates a schematic of the computational domain for parallel wind turbines with a lateral spacing of 1.05D. The origin is located at the intersection of the coordinate axes in the bottom-left corner of the figure. The short black lines represent the wind turbines, and rectangles denote areas of grid refinement. Solid rectangles represent primary refinement zones, measuring 14D in length, 6.05D in width, and 405 m in height, with a 1D spacing between the outer and inner rectangles. Dashed rectangles represent secondary refinement zones, measuring 12D in length, 4.05D in width, and 279 m in height, with wind turbine axes spaced at 1.05D apart. Subsequently, simulations are conducted starting from 1.5D and incrementing every 0.5D up to 4.5D. When setting up secondary refinement zones, this paper follows the principle of maintaining a distance of 1D from the edge of the rotor to the boundary of the secondary refinement zone, 5D upstream from the boundary of the secondary refinement zone, and 7D downstream from the boundary of the secondary refinement zone. When the lateral spacing of wind turbines is greater than 3D, the refinement zones undergo changes. Figure 1b depicts a schematic of parallel wind turbines with a lateral spacing of 3.5D, where the number of secondary refinement zones increases to two. This adjustment is made because, as the lateral spacing of wind turbines increases, setting a single secondary refinement zone would result in excessively large zones and waste computational resources. The solid rectangle measures 14D in length, 8.5D in width, and 405 m in height. The secondary refinement rectangles for wind turbines 0 and 1 (depicted as dashed rectangles in Figure 1b) are both 12D long, 3D wide, and 279 m high. These two secondary refinement zones are symmetrically distributed, with a spacing of 3.5D between the hub axes of the wind turbines. Furthermore, this is also an operating condition for a single wind turbine, as depicted in Figure 1c, to serve as a reference for studying the wake expansion under different lateral spacing conditions in Section 4. Detailed grid schematics are shown in Figure 1d,e.

Table 1. Numerical settings.

Parameters	Value
Computational domain (m)	3000 × 3000 × 3000
∆x (m)	2.5
∆y (m)	2.5
∆z (m)	2.5
Simulation time for precursor (s)	20,600
Simulation time step of wind turbine (s)	0.02
Number of cores	140

summarizes the numerical settings, where the dimensions of the refined grid in the X, Y, and Z directions in the refinement area are denoted as ∆x, ∆y, and ∆z, respectively.

Table 2. Case settings.

Case No.	N	LD (D)
1	1	-
2	2	1.05
3	2	1.50
4	2	2.00
5	2	2.50
6	2	3.00
7	2	3.50
8	2	4.00
9	2	4.50

presents the operating conditions. In the table, N denotes the number of wind turbines, and L_D denotes the lateral spacing between two columns of the wind turbines. The surface roughness is 0.001 m, representing the typical offshore conditions [2].

3.4. Model Validation

In numerical simulation research, grid sensitivity analysis and model validation are crucial steps. Churchfield et al. [36] showed that a grid refinement of approximately 50 cells across the rotor is sufficient to resolve wake structures, including tip vortices. A grid sensitivity analysis conducted in our earlier work [2] of the NREL 5 MW turbine revealed that the simulation of velocity deficit was independent of grid resolution within the range of 1.25 m to 5 m. Consequently, as in the earlier work, a resolution of 2.5 m is employed for grid refinement around the turbines and their wakes. Research by Troldborg [38] and Martinez et al. [39] on the regularization parameter ε_i consistently showed that excessively small coefficients result in numerical instabilities, whereas excessively large coefficients make it difficult to clearly present tip vortex structures. They also found that an ε_i equal to twice the grid length provides a suitable balance between these two aspects. Thus, ε_i is set to 5 in this work. In this section, the results obtained from the LES coupled with ALM are compared and validated by the data presented in the study of Churchfield et al. [36].

After the pre-simulation is completed, the wind turbine simulation is conducted, which involves adding the wind turbine model based on the recorded data of the pre-simulation. To meet the requirements of the ALM and improve calculation accuracy, the grid around the wind turbine needs to be refined. Figure 2 depicts a top view of the computational domain. The solid lines inside indicate single grid refinement, with a grid size of 5 m × 5 m × 5 m after refinement, while the dashed lines inside represent double grid refinement, with a grid size of 2.5 m × 2.5 m × 2.5 m. The outer solid rectangle has a length of 19D and a width of 5D, while the inner dashed rectangle has a length of 17D and a width of 3D. The two turbines are arranged in series within the refined region, with a radial distance of 7D.

Figure 3 shows the time-averaged streamwise velocity deficit profiles of wind turbine wakes. Figure 3a,b compare the horizontal velocity deficit profiles at the hub height of Turbines 1 and 2, respectively. Figure 3c,d compare the vertical velocity deficit profiles along the hub axis of Turbines 1 and 2, respectively. The continuous profiles are spaced 1D apart, with sections at 0D, 1D, 2D, 3D, 4D, and 5D downstream of the turbine. Each interval length in the profiles represents a velocity deficit ratio of 0.6. The black lines represent the numerical simulation data from this study, while the red lines represent the data from the study by Churchfield et al. [36].

By comparing the simulation results in this paper with the data from Churchfield et al. [36], it is found that the normalized velocity deficit at hub height in the horizontal and vertical planes of this paper closely matches the data in Churchfield et al. [36]. The average relative error is 6.28%.

Through a thorough analysis of the model validation process, the primary source of the average relative error probably lies in the use of an improved ALM. This model is based on the ALM used in the study by Churchfield et al. [36]. It incorporates aerodynamic simulations of the wind turbine nacelle and tower to capture complex aerodynamic phenomena that primarily affect the velocity distribution in the near wake region. As a result, there may be some deviation between the simulation results in this paper and the data from Churchfield et al. [36].

4. Results and Discussion

4.1. Analysis of Wake Expansion

This study primarily considers the influence of lateral spacing on the wake expansion, selecting data from Cases 1–9 to analyze the wake expansion. Figure 4 shows the time-averaged streamwise wake velocity contours at hub height on the horizontal plane obtained by ALM-coupled LES for parallelled wind turbines with lateral spacings of 1.05D, 1.5D, 2D, 2.5D, 3D, 3.5D, 4D, and 4.5D. Compared with the earlier study [2], the inflow field is more uniform in this study. As the flow direction is not parallel to periodic boundaries, the streamwise turbulent structures within the flow field are no longer constrained to fixed spanwise locations. The angle between the flow and the periodic boundaries disrupts the streamwise turbulent structures, resulting in a more random and realistic distribution pattern, which is consistent with the findings of Churchfield et al. [36]. From Figure 4, it can be seen that when the lateral spacing between adjacent wind turbines is 2D, the inner wake region between the two rows of turbines begins to show significant overlap. When the lateral spacing is 1.05D, the interaction between the turbines becomes extremely strong, and the inner wake boundaries of the two rows completely overlap, making it difficult to observe clear wake boundaries.

The wake on the left side of Turbine 0 (viewed from downstream to upstream) shows a more severe velocity deficit compared to the cases with a lateral spacing greater than 1.05D, with the area of significant velocity deficit, indicated by the dark blue region in the figure, extending about 6D downstream of the turbine. Similarly, the wake region on the right side of Turbine 1 (viewed from downstream to upstream) also shows an extension of the area with severe velocity deficits compared to the cases with larger spacing. The recovery of the wake velocity is significantly slower than in other cases. The lateral distribution of time-averaged wake velocity is shown in Figure 5. The coordinate origin is at the center of the rotor of Turbine 0. The velocity is normalized by u₀. The scale bar for normalized velocity is shown below the figure. The results show that the wake velocity at 8D to 10D downstream of Turbine 0, for Case 1, experiences a 20% to 26% loss of inflow velocity. In Case 2, this deficit further increases by an additional 2%. Considering the cubic relationship between power output and inflow velocity, this implies that wind turbines downstream of Turbine 0 could face power losses of up to 53% to 63%, with an additional 3% to 5% power loss compared to a single wind turbine. This deficit is attributable to wake superposition. For Case 2, the wakes of Turbine 0 and Turbine 1 overlap, leading to an increased velocity deficit downstream of Turbine 0. However, in Cases 3 and 4, a decrease in velocity deficit was observed in the overlap region at 8D to 10D downstream of Turbine 0, which implies a faster wake recovery rate. This finding is consistent with the results of Zhang et al. [24]. This is attributed to an acceleration effect generated by the close spacing of the two wind turbines [13], which cannot be adequately captured by simple wake superposition models (e.g., linear sum and square sum models).

While the wake regions experience velocity deficits, the turbulence intensity significantly increases. Figure 6 shows the lateral distribution of streamwise turbulence intensity at the hub height on the horizontal plane. The coordinate origin is at the location of Turbine 0. The scale bar for turbulent intensity is shown below the figure. The results demonstrate that variations in wind turbine spacing affect the downstream turbulence distribution. In Case 2, with extremely close spacing, the turbulent intensity in the wake overlap region exhibits a reduction compared to Case 1 (the single wind turbine scenario), up to 31% at the center position. Conversely, Cases 3 and 4 show a significant increase in turbulent intensity in the overlap region, reaching up to 35% and 29% at the center position, respectively. When the wakes do not directly overlap (Cases 5 to 9), the near-field turbulent intensity distribution is similar to Case 1. However, as downstream distance increases and the wakes diffuse, the turbulent intensity distribution gradually diverges. For Cases 5 to 9, the turbulent intensity at the far-field blade tip location exhibits varying degrees of enhancement, while a reduction is observed at the blade root. This effect diminishes with increasing lateral spacing.

The wake boundary is defined as the point where the time-averaged flow velocity within the wake region returns to 99% of the free-stream velocity [40]. The wake width is the lateral distance between the left and right boundaries of the wake. When the lateral spacing range is from 1.05D to 2D, the inner wake boundaries of Turbines 0 and 1 cross and merge. Therefore, the right boundary of Turbine 0 and the left boundary of Turbine 1 are chosen to calculate the wake width every 0.5D from 2D to 10D behind the turbines. When the lateral spacing is between 2.5D and 4.5D, the wakes of two turbines no longer overlap. So, the left and right boundaries of Wind Turbine 0 are selected to calculate the wake width every 0.5D from 2D to 10D behind the turbines. As shown in Figure 4, the left and right boundaries exhibit a distinct asymmetry during the wake evolution process, especially when the spacing between turbines is relatively small. The reason for this phenomenon lies in the rotation of the wind turbine rotor and the interaction between turbine wakes. In this study, considering the asymmetry of wakes, the definition of wake growth rate is the rate of increase in wake width. Figure 7 shows the linear fitting of wake growth rates for Cases 1–9. The vertical axis represents the wake width, and the horizontal axis represents the distance from the wind turbine. The gray area shows the error band for the wake width data, estimated while considering the statistical uncertainty in the time-averaged wake velocity. The bounds of the error band are determined by the wake width based on the mean value and standard deviation of wake velocity. The error band reflects the uncertainty of the wake boundary, attributable to velocity field fluctuations generated by turbulent diffusion and mixing as the wake propagates downstream. A reduced velocity deficit in the downstream wake region leads to a lower velocity gradient, thereby increasing the sensitivity of the wake boundary definition and contributing to greater uncertainty. For linear fitting, Case 1 represents the case of a single wind turbine with an intercept of $D\sqrt{\beta }$ [2]. For Cases 2 to 4, the intercept is the distance between the axes of the two rows of wind turbines plus $D\sqrt{\beta }$. For Cases 5 to 9, $D\sqrt{\beta }$ is chosen as the intercept.

Table 3. The linear fitting results of wake growth rates and relative errors for Cases 1 to 9.

Case No.	k	Relative Error (%)
1	0.07367	2.30
2	0.05691	2.32
3	0.09494	2.07
4	0.06309	0.93
5	0.07367	2.73
6	0.07629	2.59
7	0.07684	3.29
8	0.07257	3.31
9	0.07717	3.77

presents the linear fitting data for Cases 1 to 9, with an average relative error below 2.6%. Overall, the fitting results are satisfactory.

Figure 8 illustrates the variation of wake growth rate for each case. The blue line represents the wake growth rate for a single wind turbine. It can be found from the figure that when the lateral spacing is from 1.05D to 2D, the wake growth rate shows a significant difference compared to that of a single wind turbine, and there is no clear pattern observed with the variation of lateral spacing. As shown in Figure 4, when the lateral spacing is between 1.05D and 2D, the wakes of the two wind turbines overlap. The interaction in the wake regions is very strong, causing the wakes of the two turbines to merge into one, with no clear boundary between them. Therefore, a different method for calculating wake widths has to be used here compared to other cases. This may be the reason for the irregularity in the wake growth rate. When the lateral spacing exceeds 2D, the wake growth rate is similar to that of a single wind turbine. However, the wake expansion exhibits more pronounced non-linear behavior at downstream distances of 4D–8D. The results indicate that although the wakes of the two turbines do not directly overlap, they still influence each other. The disturbance caused by turbine wakes on the surrounding flow is not confined to the velocity deficit region but extends over a larger area. Additionally, an acceleration effect can be observed in the region between the two turbine wakes, and the closer the turbines are, the more pronounced this effect becomes. This effect has also been found in early studies with closely arranged rotors [24]. It is explained that the presence of rotors induces the blockage effect, resulting in accelerated surrounding flow [23]. As shown in Figure 4b–i, compared to a single wind turbine (Figure 4a), the presence of Wind Turbine 1 results in a decrease in the upstream inflow velocity within a 4D range. These disturbances affect the expansion of wakes.

4.2. Comparative Analysis of Wake Width Models

This section mainly presents the comparison of different wake width models with six cases. The parameters of each case are summarized in

Table 4. Parameters of six cases.

Case No.	CT	Iu	Iv
1	0.8144	0.0605	0.0446
5	0.818
6	0.8176
7	0.8173
8	0.817
9	0.8167

. Figure 9 shows the simulation results of a single wind turbine and lateral spacings of 2.5D, 3.5D, and 4.5D, along with a comparison of seven wake width models. The grey square scatter points in the figure represent the numerical simulation results of this paper. To better compare the numerical simulation results in this paper with the calculated results from seven different wake width models, the average relative errors between the two are calculated and are shown in

Table 5. The average relative error of wake width in the horizontal direction.

Wake Width Models	Average Relative Error (%)
	Case 1	Case 5	Case 6	Case 7	Case 8	Case 9	Avg.
Frandsen [27]	14.82	11.98	12.90	12.81	11.92	13.13	12.93
Tian et al. [18]	14.28	16.00	14.77	14.89	16.01	14.44	15.07
Niayifar and Porté-Agel [29]	28.83	36.96	35.50	35.56	37.00	35.04	34.82
Carbajo Fuertes et al. [30]	3.45	2.4	2.91	2.92	3.16	3.54	3.06
Cheng et al. [31]	5.51	5.55	6.05	5.79	7.04	6.95	6.15
Tian et al. [28]	14.94	21.06	19.78	19.84	21.11	19.39	19.35
Liu et al. [11]	3.4	4.34	4.61	5.37	4.09	3.69	4.25

.

From the comparison results, it can be observed from Figure 9 that the model proposed by Frandsen [27] underestimates the wake width in all cases. In contrast, the models proposed by Tian et al. [18], Niayifar and Porté-Agel [29], and Tian et al. [28] overestimate the wake width. Despite the fact that the models of Tian et al. [18] and Niayifar and Porté-Agel consider the effect of added turbulence, this also causes them to overestimate the wake growth rate. In comparison to other models, the prediction accuracy of Carbajo Fuertes et al. [30], Cheng et al. [31], and Liu et al. [11] is significantly higher. However, the model of Carbajo Fuertes et al. does not capture the non-linear characteristics of wake expansion, especially under varying lateral spacing conditions. The model of Cheng et al. overestimates the wake width in the far wake region beyond 7D. The model by Liu et al. slightly underestimates the wake width in the far wake and does not precisely reflect the non-linear wake behavior for different lateral spacing configurations.

4.3. Modification of Wake Width Model

In the comparison between the numerical simulation results for different lateral spacing conditions from the previous section and the calculated results of seven wake width models, it is observed that the model by Liu et al. partially captures non-linear wake behavior. However, wake interactions reduce its accuracy under different lateral spacing conditions. Therefore, in this section, modifications are made to the model, which considers the impacts of different lateral spacings on the added turbulence. The fitting results from Cases 5–9 are expressed as follows:

$${\displaystyle \frac{r_x}{D}}=1.86C_T^{0.29}{I}^{0.7}{\left({\displaystyle \frac{x}{D}}\right)}^{0.38}+0.563{\displaystyle \frac{r_{x0}}{D}},$$

(23)

$$I=\sqrt{I_{+}^2+I_v^2},$$

(24)

$$I_{+}=0.0661C_T^{0.0924}{I}_v^{-0.034}{\left(L_D/x_n\right)}^{-0.02},$$

(25)

where, r_x0 is the initial wake width calculated by Equation (8); and x_n is the length of the near wake defined by Vermeulen [41], details available in [29].

Figure 10 shows the comparison between the numerical simulation results for different lateral spacing conditions and the predictions of the modified model. The horizontal axis represents the LES results, and the vertical axis represents the predictions of the modified model. The relative errors of the modified model are shown in

Table 6. The average relative error of modified wake width in the horizontal direction.

Case No.	Average Relative Error (%)
5	1.57
6	1.42
7	2.92
8	2.30
9	2.35
Avg.	1.91

. The comparison reveals that, compared to the average relative errors before modification, the average relative errors after considering the added turbulence are significantly reduced. This indicates that the modified wake width model aligns better with the numerical simulation data after the modification.

4.4. Validation of Modified Model

LES data from Liu et al. [2] is used to validate the modified wake width model. The hub height and rotor diameter of the wind turbine are 90 m and 126 m, respectively. The surface roughness covers typical offshore and rough sea conditions. The parameters of each case are summarized in

Table 7. Cases for validation.

Case No.	z0 (m)	LD (D)	CT	Iu	Iv
1	0.001	6	0.9342	0.0551	0.0397
2	0.016	6	0.9579	0.0768	0.0499
3	0.001	3	0.8819	0.0513	0.0414
4	0.016	3	0.9327	0.0768	0.0499

. Figure 11 shows the comparison between the LES results for different cases and the predictions of different models. The average relative errors between the two are listed in

Table 8. The average relative error of wake width models in the horizontal direction.

Wake Width Models	Average Relative Error (%)
	Case 1	Case 2	Case 3	Case 4	Avg.
Frandsen [27]	16.4	11.27	13.96	8.3	12.48
Tian et al. [18]	20.56	23.98	24.62	27.17	24.08
Niayifar and Porté-Agel [29]	37.4	41.78	34.07	42.48	38.93
Carbajo Fuertes et al. [30]	5.75	3.82	3.68	0.74	3.5
Cheng et al. [31]	8.92	8.5	5.88	5.45	7.19
Tian et al. [28]	15.92	22.69	14.99	25.43	19.76
Liu et al. [11]	2.03	7.31	2.63	8.83	5.2
Present	1.46	3.11	1.99	4.5	2.77

. The results demonstrate that the modification improves the predicting accuracy with an average relative error of 2.77%, and the modified model has the best performance under different inflow conditions and different lateral spacings. Considering the fitted cases discussed in the previous section, the comparison results highlight the broad applicability of the modified model across both typical offshore and rough sea conditions, as well as for lateral spacing configurations ranging from 2.5D to 6D.

5. Conclusions

This paper utilizes LES techniques to analyze the wake characteristics of spanwise wind turbines by coupling the ALM under varying lateral spacing. The velocity distribution, turbulence intensity, and boundary characteristics of wind turbine wakes at various lateral spacings are investigated through comparison with a single wind turbine wake. Additionally, the predictions of wake width from seven different wake width models are compared. Finally, a modified wake width model is introduced, incorporating the effects of added turbulence, and its performance is evaluated using LES data. Major findings are summarized as

(1)

The inflow setting that is not parallel to the flow field boundaries significantly improved the quality of the simulated flow field structure. This setting can enhance the reasonability and accuracy of the simulation.

(2)

At lateral spacings of 1.05D, 1.5D, and 2D between two columns of wind turbines, the wake interaction between turbines is significant due to the small lateral spacing. This leads to the merging of wake regions downstream of the wind turbines. Additionally, the presence of an additional wind turbine decreases the velocity of the upstream flow and accelerates the velocity of the surrounding flow of rotors.

(3)

Compared to a single wind turbine, a 1.05D spacing leads to a larger velocity deficit and a reduced turbulence intensity increase due to wake interaction. Conversely, faster wake recovery is observed at 1.5D and 2D. From 2.5D to 4.5D, turbulence intensity at the downstream blade tip location increases, but the extent of this change decreases with increasing spacing.

(4)

When the lateral spacing is between 1.05D and 2D, the wakes of two rows of wind turbines overlap, resulting in an irregular wake growth rate. When the lateral spacing is greater than 2D, the wake growth rate is similar to a single wind turbine. Nevertheless, the non-linear characteristics of wakes become more apparent at downstream distances of 4D–8D.

(5)

While the models by Carbajo Fuertes et al. [30], Cheng et al. [31], and Liu et al. [11] have better performance in the predictions of wake width, they cannot sufficiently reflect the non-linear behavior resulting from variations in spanwise spacing. By fitting LES results, the modified model considering added turbulence closely approximates the data from LES with minimum average relative errors across all cases. These results indicate that the modified wake width model can more accurately describe wake expansion in wind turbine arrays.

By improved understanding of the wake velocity profile scale, this work provides a crucial foundation for predicting wake velocity distribution and a valuable reference for the design and optimization of floating wind farms within a limited space. The wake expansion and interaction are more complicated between wind turbines with small lateral spacings. A pair of wind turbines serves as a fundamental building block on floating wind turbine platforms. Building upon a progressively deeper understanding of the wake characteristics of this basic unit, the wake characteristics within large floating wind farms, which are comprised of numerous such units, require further investigation. In addition, while the applicability of the model is validated using varying atmospheric turbulence intensities and surface roughness, the covered conditions are limited, and thus, its applicability warrants further examination. Changes in atmospheric stability due to thermal stratification were not considered in this study, which needs further research in the future.