A Three-Dimensional Analytical Model for Wind Turbine Wakes from near to Far Field: Incorporating Atmospheric Stability Effects

Abstract

In response to the critical demand for improved characterization of atmospheric stability effects in wind turbine wake prediction, this study proposes and systematically validates a new analytical wake model that incorporates atmospheric stability effects. In recent years, research on wake models with atmospheric stability effects has primarily followed two approaches: incorporating stability through high-fidelity numerical simulations or modifying classical analytical wake models. While the former offers clear mechanical insights, it incurs high computational costs, whereas the latter improves efficiency yet often suffers from near-wake prediction biases under stable stratification, lacks a unified framework covering the entire wake region, and relies heavily on case-specific calibration of key parameters. To overcome these limitations, this study introduces a stability-dependent turbulence expansion term with a square of a cosine function and the stability sign parameter, enabling the model to dynamically respond to varying atmospheric conditions and overcome the reliance of traditional models on neutral atmospheric assumptions. It achieves physically consistent descriptions of turbulence suppression under stable conditions and convective enhancement under unstable conditions. A newly developed far-field decay function effectively coordinates near-wake and far-wake evolution, maintaining computational efficiency while significantly improving prediction accuracy under complex stability conditions. The Present model has been validated against field measurements from the Scaled Wind Farm Technology (SWiFT) facility and the Alsvik wind farm, demonstrating superior performance in predicting wake velocity distributions on both vertical and horizontal planes. It also exhibits strong adaptability under neutral, stable, and unstable atmospheric conditions. This proposed framework provides a reliable tool for wind turbine layout optimization and power output forecasting under realistic atmospheric stability conditions.

1. Introduction

As a crucial component of clean and renewable energy, wind energy plays an increasingly important role in the global energy transition. With the continuous development of wind power, wind farm layout optimization and operational efficiency improvement have become key research priorities. Among the various challenges involved, the wake effect between wind turbines stands out as a core factor affecting overall wind farm performance. The wake leads to reduced incoming wind speed and increased turbulence intensity for downstream turbines, consequently causing power losses and structural fatigue damage. Therefore, accurate prediction of wind turbine wake characteristics is of significant scientific and engineering importance for the micro-siting, power prediction, and lifetime assessment of wind farms [1].

From the perspective of atmospheric structure, as noted by Cermak and Cochran [2], the atmosphere within approximately 1000 m above the ground surface typically constitutes the atmospheric boundary layer. The lowest 100 m of the boundary layer is defined as the atmospheric surface layer. Recently, to enhance wind energy capture efficiency and conserve land or sea resources, the wind power industry has been actively advancing the design and development of large-scale wind turbines [3]. Since 2019, when the hub height of offshore wind turbines surpassed 103 m and the rotor diameter reached 150 m [4], the industry has entered a phase of accelerated advancement. By the end of 2025, the world’s first 26 MW offshore wind turbine with a hub height of 185 m and a rotor diameter of 310 m will have been successfully connected to the grid in Shandong Province, China. This trend indicates that modern wind turbines now operate beyond the atmospheric surface layer, and their aerodynamic performance is increasingly dependent on physical processes throughout the entire atmospheric boundary layer. Therefore, as a key factor governing wind shear, turbulent mixing, and energy transfer within the boundary layer, atmospheric stability is of growing importance in wind turbine wake assessment.

Atmospheric stability is generally categorized into stable, neutral, and unstable conditions. Under unstable atmospheric conditions, the air that is warmer than the surrounding environment expands and rises, promoting the vertical development of turbulence to considerable heights. Under neutral atmospheric conditions, the air at the same temperature as its environment remains at its original height, resulting in only weak mixing [5]. Under stable atmospheric conditions, the cooler air compresses and sinks, suppressing vertical motion and weakening vertical mixing [6]. By directly influencing the wind speed profile and turbulence structure, atmospheric stability governs the recovery rate and spatial diffusion of wind turbine wakes [7,8,9,10,11,12,13,14]. Both field measurements and high-fidelity numerical simulations illustrate that the spatial distributions of wake velocity deficit and added turbulence intensity under different atmospheric stability conditions show significant differences. As demonstrated by Pérez et al. [15], the annual energy production (AEP) varies by more than 16% under different stability conditions. Specifically, under stable conditions, the wind turbine wake recovers slowly, resulting in a persistent velocity deficit over a long downstream distance, which leads to a lower AEP. In contrast, under unstable conditions, intense turbulent mixing facilitates faster wake recovery, yielding a higher AEP. Therefore, incorporating atmospheric stability into wake models is essential for improving wake prediction accuracy, enabling refined assessment of wind resources, and ensuring structural safety of turbines.

Traditional wake models, such as the classic Jensen (PARK) model and its derivatives, rely mostly on the assumption of neutral atmospheric conditions and depend heavily on empirical parameters. This limitation is typically exemplified by the variability of the wake decay coefficient in the Jensen model across different wind farms. Peña et al. [16] found that in the Sexbierum case, the wake decay coefficient of the Jensen model should be adjusted from the typical onshore value of 0.075 to a site-specific value of 0.038. In response, some researchers have focused on quantifying the wake decay coefficient. Cheng and Porté-Agel [17] proposed a physics-based wake model, drawing on scalar dispersion in turbulence. This model accounts for the influence of ambient turbulence intensity based on Taylor’s diffusion theory. Vahidi and Porté-Agel [18] developed a physics-based model by introducing a filtered turbulence scale to account for the combined effects of ambient and turbine-induced turbulence. Bastankhah and Porté-Agel [19] proposed a concise analytical wake model based on the conservation of mass and momentum and a Gaussian-shaped velocity deficit, which requires a wake growth rate parameter to determine the wake velocity. Building upon this, Niayifar and Porté-Agel [20] further enhanced the model by determining the wake growth rate from the local streamwise turbulence intensity. It should be noted that the key empirical constant was derived via linear regression of Large Eddy Simulation (LES) data, and its agreement with field measurements requires further validation. Ishihara and Qian [21] established a relationship linking the standard deviation of the Gaussian distribution to the turbine thrust coefficient and the incoming turbulence intensity, thereby overcoming the reliance on empirical parameters in wake models. Nevertheless, such modified models are proposed under neutral atmospheric conditions, and their applicability should be further verified.

Recently, some researchers have focused on incorporating atmospheric stability into wake models for a more comprehensive assessment of wind turbine wakes, with the main methodologies falling into two categories: methods based on numerical simulations and methods involving the modification of previous analytical wake models. In the realm of numerical simulations, Kale et al. [22,23] employed LES within the Weather Research and Forecasting Model (WRF) framework with Monin–Obukhov similarity theory to generate stable and unstable atmospheric boundary layers, using an actuator disk model for turbine simulation. They investigated the aerodynamic characteristics of full-scale wind turbines under different atmospheric conditions using a generalized actuator disk model. However, this approach is computationally expensive. Furthermore, under stable stratification, it exhibits significant deviations in wake velocity deficit and delayed wake recovery due to low turbulence intensity, high wind shear, and wind veer. Nygaard et al. [24] developed a simplified two-dimensional Reynolds-averaged Navier–Stokes (RANS) model based on the Boussinesq buoyancy approximation, which incorporates vertical energy transfer under different stability conditions by simplifying the governing equations into a parabolic form. This method qualitatively reproduces the influence of updrafts and downdrafts on the wake, but prioritizes mechanistic insight over quantitative accuracy.

Compared to methods based on numerical simulations, some researchers have devoted efforts to a more concise and practical methodology: directly incorporating atmospheric stability parameters into analytical wake models. Emeis [25] developed a top-down wind park model that integrates atmospheric stability effects on both momentum flux and momentum loss. Although it quantifies the macroscopic impact of atmospheric stability on the velocity deficit at hub height, it fails to capture the dynamics of individual turbine wakes and wind direction shifts. Abkar and Porté-Agel [26] characterize the differential growth of the wake in the lateral and vertical directions under different atmospheric conditions using a two-dimensional elliptical Gaussian distribution, in which wake growth rates were obtained by fitting LES results. Han et al. [27] established a simple logarithmic model linking wake expansion to turbulence, thereby indirectly incorporating atmospheric stability. While it efficiently captures key wake features, this model lacks an explicit stability parameter and performs poorly in the near-wake region under very low turbulence intensity. Cheng et al. [28] linked atmospheric stability to the lateral turbulence intensity and proposed a linear relationship between the lateral turbulence intensity and the wake expansion rate. However, as the authors noted, the model possesses inherent limitations in the near-wake region and under low incoming turbulence intensity. Xiao et al. [29] proposed the 3D-Stability-COUTI model, which incorporates the Obukhov length to predict wakes under different atmospheric stability conditions without empirical tuning. While accurate in general, its summation of two cosine functions results in an overemphasis of the bimodal velocity structure in the near-wake region under stable atmospheric conditions with low turbulence intensity and high wind shear.

Overall, the two aforementioned methodologies for incorporating atmospheric stability into wake models present distinct advantages and limitations. High-fidelity numerical methods like LES can directly resolve turbulent structures and wake expansion under different atmospheric stability conditions, offering clear physical mechanisms and broad applicability. However, their high computational cost limits their practicality for routine engineering applications. In contrast, modified analytical wake models include atmospheric stability parameters like the Monin–Obukhov length to approximate atmospheric stability effects to maintain computational efficiency. Nevertheless, their capacity to describe nonlinear responses under stable atmospheric stability conditions remains limited.

In summary, while progress has been made in establishing wake models considering atmospheric stability, several limitations still exist: (i) Traditional models exhibit significant deviations in predicting wake velocity deficit and wake recovery process under stable atmospheric conditions; (ii) most wake models lack a unified framework that is both adaptable to different atmospheric stability conditions and capable of sustaining high prediction accuracy throughout both the near-wake and far-wake regions; (iii) key parameters in analytical wake models considering atmospheric stability often require case-specific calibration. These limitations urgently call for the development of a new generation of wake prediction methodologies: First, an atmospheric stability parameterization scheme should be established based on atmospheric boundary layer theory to accurately describe velocity profiles and the wake recovery process. Second, it is necessary to construct a unified wake prediction framework covering the entire wake region. Ultimately, by establishing functional relationships between key wake model parameters and atmospheric stability, an analytical wake model balancing physical fidelity with engineering applicability can be proposed.

Therefore, the structure of this paper is organized as follows. Section 2 proposes a new wake model considering atmospheric stability, including fundamentals of atmospheric stability, previous wake models with atmospheric stability, the proposal of a new wake model with atmospheric stability, and a methodology for prediction accuracy assessment. Section 3 provides a performance assessment of the present wake model. In this section, field measurements from two cases—Vestas V27 wind turbine and Danwin 180 kW wind turbine—are adopted to assess wake model prediction performance. Finally, a summary and conclusions are presented in Section 4.

2. Proposal of a New Wake Model Considering Atmospheric Stability

2.1. Fundamentals of Atmospheric Stability

Figure 1 schematically illustrates the influence of atmospheric stability on wind profiles and wind turbine wakes. The figure presents vertical wind speed profiles and the downstream expansion of the wake under stable, neutral, and unstable conditions. Under stable stratification, strong wind shear and suppressed vertical mixing lead to a narrow wake with high velocity gradients. In contrast, unstable conditions enhance turbulent mixing, resulting in a fuller velocity profile and a wider, more diffused wake. The neutral case lies between these two regimes. These distinct flow characteristics highlight the necessity of incorporating atmospheric stability effects into wind turbine wake modeling.

The difference between the inflow wind speed profiles under different atmospheric conditions, as schematically shown in Figure 1, can be quantitatively described using the Monin–Obukhov Similarity Theory (MOST) [30]. Built upon the characteristic length scale proposed by Obukhov [31], MOST provides a universal framework for describing the turbulent structures within the atmospheric surface layer. This length scale is defined by the relative importance of buoyant production to mechanical shear production of turbulence kinetic energy. The core of the theory lies in applying dimensional analysis to simplify the complex turbulent transfer processes into functions of a single dimensionless parameter, ζ [32]. The governing equations are given in Equation (1) [33]:

$$\zeta ={\displaystyle \frac{z}{L}}={\displaystyle \frac{\kappa zg{\theta }_{*}}{\overline{\theta }{u}_{*}^2}}$$

(1)

In Equation (1), $\zeta =z/L$ is a stability parameter in which $z$ is the height above the ground surface and $L$ is the Obukhov length scale. $\kappa$ is the Karman constant. $g$ is gravitational acceleration. ${\theta }_{*}$ is a temperature scale and $\overline{\theta }$ is the mean potential temperature. $u_{*}$ is the friction velocity.

Based on the definition of the stability parameter, the logarithmic wind speed can be derived in Equation (2) [30],

$${\psi }_m(\zeta )={\displaystyle \frac{\kappa z}{u_{*}}}{\displaystyle \frac{\partial \overline{U}}{\partial z}}$$

(2)

where ${\psi }_m$ is the non-dimensional wind shear and $\overline{U}$ is the mean wind speed. To directly incorporate the Obukhov length L as a key stability parameter into the logarithmic wind speed profile, a piecewise function formulation was developed based on a series of field observations from the 1980s through dimensional analysis and empirical fitting [34,35,36]. The following form, given in Equation (3), has since become the widely adopted standard model,

$${\psi }_m=\left\{\begin{array}{c}-4.7(\zeta ), L>0, stable\\ 0, \left|L\right|\to \infty , neutral\\ 2ln[{\displaystyle \frac{1}{2}}(1+\varsigma )]+ln[{\displaystyle \frac{1}{2}}(1+{\varsigma }^2)]-2{tan}^{-1}(\varsigma )+{\displaystyle \frac{\pi }{2}}, \\ L<0, unstable\end{array}\right.$$

(3)

where $\varsigma ={\left[1-15\zeta \right]}^{1/4}$. As illustrated by Monin and Obukhov [30], the turbulent heat flux is directed downward under stable stratification, i.e., $L>0$; upward under unstable stratification, i.e., $L<0$; and is negligible under neutral stratification, i.e., $\left|L\right|\to \infty$. According to surface layer theory [37], Equation (4) can be derived for the surface layer over flat and homogeneous terrain,

$$U_0(z)={\displaystyle \frac{u_{*}}{\kappa }}[ln({\displaystyle \frac{z}{z_0}})-{\psi }_m(\zeta )]$$

(4)

where $U_0(z)$ is the wind speed at height $z$. Dividing $U_0(z)$ by the wind speed at the hub height, $U_{0,H}(z_H)$, eliminates $u_{*}$ and yields the following equation for the incoming flow wind speed profile that accounts for atmospheric stability, given in Equation (5),

$$U_0(z)=U_{0,H}{\displaystyle \frac{ln(\frac{z}{z_0})-{\psi }_m(\zeta )}{ln(\frac{z_H}{z_0})-{\psi }_m({\zeta }_H)}}$$

(5)

where ${\psi }_m({\zeta }_H)={\psi }_m(z_H/L)$. Similarly, the incoming flow turbulence intensity profile that accounts for atmospheric stability, given in Equation (6),

$$I_0(z)=I_{0,H}{\displaystyle \frac{ln(\frac{z_H}{z_0})-{\psi }_m({\zeta }_H)}{ln(\frac{z}{z_0})-{\psi }_m(\zeta )}}$$

(6)

where $I_{0,H}$ is the turbulence intensity at the hub height. Therefore, this derivation enables the calculation of the incoming flow wind speed and turbulence intensity profiles via Equations (5) and (6), using the hub-height wind speed and turbulence intensity obtained from nacelle-based sensors such as LiDAR or 3D scanning LiDAR.

2.2. Previous Wake Models with Atmospheric Stability

In this study, two previous wake models considering atmospheric stability, 3D-Stability-COUTI [29] and Cheng2019 [28], are adopted for comparison with the present wake model. A brief review of the two wake models is provided below to establish a foundation for the subsequent comparative analysis.

Xiao et al. [29] proposed an integrative three-dimensional cosine-shaped analytical model, named 3D-Stability-COUTI, for characterizing wake velocity and turbulence intensity under varying atmospheric stability conditions. Firstly, the essential input parameters for calculating the velocity deficit encompass the wind turbine specifications, atmospheric inflow conditions and terrain characteristics, as shown in

Table 1. Key input parameters for the present wake model, the 3D-Stability-COUTI model [29], and the Cheng2019 model [28].

Wake Model Considering Atmospheric Stability		Present Wake Model	3D-Stability-COUTI Model [29]	Cheng2019 Model [28]
Wind turbine	Rotor diameter, $D$	○	○	○
	Hub height, $z_H$	○	○	○
	Thrust coefficient curve, $C_t$	○	○	○
Atmospheric inflow	Monin–Obukhov length, $L$	○	○	○
	Wind speed at hub height, $U_{0,H}$	○	○	○
	Turbulence intensity at hub height, $I_{0,H}$	○	○	○
Terrain	Surface roughness length, $z_0$	○	○	○
	Angle of latitude, $\varphi$	×	×	○

Note: Filled circles (○) indicate presence; cross marks (×) indicate absence.

.

Then, according to Table 1, the wind speed, $U_{0,H}$, and turbulence intensity, $I_{0,H}$, at hub height, are essential input parameters for deriving the incoming wind velocity and turbulent intensity profiles via Equations (5) and (6), considering atmospheric stability.

Subsequently, the wake radius $r_w(x)$ at any downstream position is determined using a stability-corrected wake width model, which incorporates the Obukhov length and is represented by Equations (7) and (8),

$$r_w(x)=D{C}_t^{\kappa }{I}_{0,H}^{\frac{\kappa }{2}}{\left({\displaystyle \frac{x}{D}}\right)}^{\kappa +{\delta }_L}+{\displaystyle \frac{D}{2}}max(-0.2·ln({\displaystyle \frac{x}{D}})+0.2,0)$$

(7)

$${\delta }_L=-0.05{(z_H/L)}^{\kappa /2}$$

(8)

where $D$ is the rotor diameter. $\kappa$ is the Karman constant. $C_t$ is the thrust coefficient. $(x,y,z)$ is streamwise, spanwise, and vertical coordinates, respectively. In Equation (8), when $L<0$, the term ${(z_H/L)}^{\kappa /2}$ is evaluated using the real-valued odd root operation, i.e., ${(-a)}^{1/5}=-\sqrt[5]{a}$ for $a>0$, ensuring that ${\delta }_L$ becomes positive under unstable conditions and thereby enhances wake expansion.

The third step focuses on predicting the wake velocity at any downstream position through a two-stage process: a prediction step that calculates the 1D average wake velocity deficit $∆u^{*}(x)$, shown in Equation (9), and a correction step that redistributes $∆u^{*}(x)$ in 3D space using a dual-cosine shape function, shown in Equation (10), to obtain the detailed velocity deficit distribution $∆u(x,y,z)$, shown in Equation (11). Finally, the wake velocity at any position in the wake field $U_w(x,y,z)$ is provided by combining the incoming wind velocity with the calculated velocity deficit distribution, shown in Equation (12).

$$∆u^{*}(x)={\displaystyle \frac{(1-\sqrt{1-C_t})}{{(\frac{r_w(x)}{(-0.1·ln(x/D)+1.3)·(D/2)})}^2}}$$

(9)

$$\begin{gathered} ∆u_{+}=\left\{\begin{array}{c}A·\cos (K\times (r-\theta )),\theta -r_{\mathrm{w}}^{\prime }\le r<\theta +r_{\mathrm{w}}^{\prime }\\ 0,\mathrm{else}\end{array}\right. \\ ∆u_{-}=\left\{\begin{array}{c}A·\cos (K\times (r+\theta )),-\theta -r_{\mathrm{w}}^{\prime }\le r<-\theta +r_{\mathrm{w}}^{\prime }\\ 0,\mathrm{else}\end{array}\right. \end{gathered}$$

(10)

$$∆u(x,y,z)=∆u_{+}+∆u_{-}$$

(11)

$$U_w(x,y,z)=U_0(z)·(1-∆u(x,y,z))$$

(12)

where $A=\pi ·r_w(x)·∆u^{*}(x)/(4r_w^{\prime })$. $r_w^{\prime }$ is the half-width of the single-peak deficit profile, calculated by $r_w^{\prime }=r_w(x)-0.3D$. $\theta$ represents the radial distance from the wake centerline to the location of the maximum single-peak deficit, which is determined empirically as $\theta =0.3D$. $K$ is functionally dependent on $x$ and is computed as $K=\pi /(2r_w^{\prime })$. $r=\sqrt{{(y-y_C)}^2+{(z-z_H)}^2}$ is the radial distance from the wake center position $(x,y_C,z_H)$.

Cheng et al. [28] incorporated the Obukhov length into the parameterization of the lateral turbulence intensity to account for the effects of atmospheric stability. This wake model, referred to herein as the Cheng2019 model, requires the input parameters listed in Table 1. The incoming velocity and turbulence intensity profiles are calculated using Equations (5) and (6). Then, the lateral turbulence intensity I_v, a parameter critical to wake expansion, is determined using the recommendation of ESDU (Equation (13)), which incorporates Coriolis effects via boundary layer height estimation,

$$I_v=I_0[1-0.22{cos}^4({\displaystyle \frac{\pi z}{2h}})]$$

(13)

where $I_0$ is the incoming turbulence intensity. $h$ is the boundary layer height, which is estimated by,

$$h={\displaystyle \frac{u_{*}}{6f}}={\displaystyle \frac{\kappa U_0}{6f[ln(z/z_0)-{\psi }_m(\zeta )]}}$$

(14)

where $f=2\mathit{\Omega }sin(\varphi )$ is the Coriolis parameter, in which $\varphi$ is the angle of latitude and $\mathit{\Omega }=72.9\times {10}^{-6}$ rad/s is a constant presenting the angular rotation of the Earth. Based on Equation (13), the wake expansion rate k_w(x) is uniquely related to I_v through a linear empirical formula given in Equation (15),

$$k_w=0.223I_v+0.022$$

(15)

Consequently, the intercept $\epsilon$ in the wake width growth function is derived as a function of k_w(x), as presented in Equation (16),

$$\epsilon =-1.91k_w+0.34$$

(16)

Based on k_w(x) from Equation (15) and $\epsilon$ from Equation (16), the standard deviation $\mathsf{\sigma }$ of the Gaussian wake profile is then obtained by Equation (17),

$${\displaystyle \frac{\sigma }{D}}=k_w{\displaystyle \frac{x}{D}}+\epsilon$$

(17)

The velocity deficit across the wake is subsequently computed with the Gaussian-shaped distribution, formulated in Equation (18),

$$∆u(x,y,z)=U_H(1-\sqrt{1-{\displaystyle \frac{C_t}{8{(\sigma /D)}^2}}})\times \mathrm{e}\mathrm{x}\mathrm{p}(-{\displaystyle \frac{1}{2{(\sigma /D)}^2}}\times {({\displaystyle \frac{r}{D}})}^2)$$

(18)

Finally, the wake velocity at any position in the wake field is calculated by combining the above results via Equation (12).

2.3. Hypotheses, Boundary Conditions, and Accuracy Assessment Method of the Proposed Analytical Wake Model

To enhance the applicability of the wake model under different atmospheric stability conditions, an improved analytical wake model is proposed in this study, termed the Present model. Prior to introducing the governing equations, the fundamental hypotheses and boundary conditions of the model are clarified to establish a clear physical framework and provide a direct rationale for the turbulence modeling approach.

• Model hypotheses

  (1)

  Steady-state, incompressible flow: Wake evolution is modeled under a quasi-steady assumption, considering time-averaged flow fields. This approach is typical for engineering-scale wake predictions.

  (2)

  Far-wake dominance by turbulent diffusion: In the far-wake region ($x/D=5$), wake recovery is primarily governed by turbulent mixing between the wake and the ambient flow, rather than by deterministic vortex structures from the near-wake.

  (3)

  Empirical closure for stability effects: The complex, nonlinear effects of atmospheric stability on wake turbulence are parameterized through a stability sign parameter S and a stability-corrected wake expansion term, ${\delta }_L$. This approach provides a simplified yet physically interpretable closure scheme.

• Boundary conditions

The complete set of boundary conditions for the wake flow problem is specified as follows:

(1)

Inflow boundary condition: Downstream of the rotor plane (at x = 0 in Figure 1), the inflow is prescribed by the vertical profiles of wind speed, U₀(z), and turbulence intensity, I₀(z), given by Equations (5) and (6). These profiles incorporate the influence of atmospheric stability via the Obukhov length, L.

(2)

Lateral and vertical boundary conditions: At distances sufficiently far from the wake centerline in the lateral (y) and vertical (z) directions, the wake flow approaches the free-stream conditions, characterized by U → U₀(z) and I → I₀(z).

(3)

Ground boundary condition: A standard no-slip condition at the ground is implicitly considered within the logarithmic profile of the incoming flow, U₀(z) and I₀(z), which accounts for the surface roughness length, z₀.

3.

Turbulence modeling rational

Regarding turbulence modeling, the Present model does not solve the Reynolds-averaged Navier–Stokes equations explicitly. Instead, the critical effects of ambient turbulence and its modulation by atmospheric stability are incorporated parametrically through two key mechanisms integrated into the governing equations: (i) the stability-corrected wake growth model (via the term ${\delta }_L$ in the wake width equation), which intrinsically represents the turbulent diffusion process adjusted by stability; and (ii) the cosine-squared velocity deficit shape function, which provides a simplified analytical closure for the turbulent shear stress effects in the wake, modulated by the parameter S.

4.

Accuracy assessment method

To quantitatively assess the prediction accuracy of the wake models under various atmospheric stability conditions, the hit rate metric, denoted as $q$, is adopted in this study. This metric, employed by Chen and Ishihara [38,39], provides a normalized measure of the agreement between model predictions and experimental observations. The hit rate is defined in Equation (19),

$$q={\displaystyle \frac{1}{N}}\sum _{i=1}^N{n}_i, with n_i=\left\{\begin{array}{c}1, \left|{\displaystyle \frac{M_i-P_i}{P_i}}\right|\le 15\% or 20\%\\ 0, else\end{array}\right.$$

(19)

where $M_i$ represents the ith observed value from field measurements, and P_i denotes the corresponding ith predicted value obtained from the wake models. The variable $N$ indicates the total number of data points included in the comparison.

The hit rate $q$ ranges from 0 to 1, where $q=1$ indicates perfect agreement between predictions and measurements, and values approaching 0 signify increasing deviation. This metric is particularly suitable for evaluating wake model performance because it incorporates relative error in a normalized form, giving equal importance to errors across different velocity regimes—unlike absolute error measures, which can be dominated by high-speed data points. The application of this metric allows for a consistent and interpretable comparison of model performance across the different atmospheric stability regimes investigated in this work.

2.4. Proposal of New Wake Model with Atmospheric Stability

Specifically, the equations for calculating wake width in the Present model remain consistent with those in the 3D-Stability-COUTI (Equations (7) and (8)). However, the range of values for the velocity deficit has been modified from $(\theta -r_w^{\prime })\le r<(\theta +r_w^{\prime })$ and $(-\theta -r_w^{\prime })\le r<(-\theta +r_w^{\prime })$ in 3D-Stability-COUTI to $-r_w\le r<r_w$. Concurrently, the velocity deficit formula has been revised from the sum of two cosine functions in 3D-Stability-COUTI to the square of a cosine function, as shown in Equation (20),

$$∆u(x,y,z)=\left\{\begin{array}{c}A·[B+C(1-S)]·\eta ({\displaystyle \frac{x}{D}})·{\displaystyle \frac{x}{D}}·{cos}^2(E·r), \\ if-r_w\le r\le r_w\\ 0, otherwise\end{array}\right.;$$

(20)

where $A=\pi ·r_w(x)·∆u^{*}(x)/(4r_w^{\prime })$. $\eta (x/D)$ is the far-field decay function. The key parameters $B$, $C$, and $E$ are optimized via regression against and determined by sensitivity analysis based on field measurement data from Refs. [29,40,41,42]. The results of sensitively analysis, as presented in Figure 2, indicate that $B=0.3$, $C=0.15$, and $E=0.06$ are the optimal constants for the proposed analytical wake model.

$S$ is the stability sign parameter, defined as Equations (21) and (22),

$$\eta ({\displaystyle \frac{x}{D}})=\left\{\begin{array}{c}1, if {\displaystyle \frac{x}{D}}\le 5\\ exp[-F({\displaystyle \frac{x}{D}}-5)], if {\displaystyle \frac{x}{D}}>5\end{array}\right.$$

(21)

$$S=\left\{\begin{array}{c}1, L>0, stable\\ -1, L<0 and \left|L\right|\to \infty , unstable and neutral\end{array}\right.$$

(22)

The selection of $x/D=5$ as the demarcation between the near- and far-wake regions is physically justified by the evolution of tip vortex dynamics and its impact on the wake structure. The assignment of values to the parameter $S$ quantifies the influence of atmospheric stability on wake behavior. Under stable conditions ($S=1$), the velocity deficit amplitude is governed by the baseline term, reflecting suppressed wake dissipation. Under unstable to neutral conditions ($S=-1$), the amplitude increases significantly, capturing enhanced turbulent mixing and faster wake recovery. This parameterization thus provides a physically consistent representation of how thermal stratification affects wake development.

During the operation process of a wind turbine, under the influence of turbulent fluctuations, tip vortices undergo random displacements that amplify with downstream distance (Yen et al. [43]). The wake region is commonly divided into the near-wake, intermediate-wake, and far-wake regions: in the near-wake region (<2D), large-scale vortices within the shear layer lead to a bimodal distribution of turbulence intensity. As the wake develops further into the intermediate wake (3D to 5D), the shear layer continues to expand toward the wake centerline until the far-wake regions [44,45,46]. To accurately capture this transition in the velocity deficit decay, the far-field decay function is designed to remain unity (i.e., inactive) in the near-wake region ($x/D\le 5$), thereby preserving the near-wake structure, and initiates an exponential decay beyond $x/D=5$. This formulation effectively represents the enhanced mixing with the free-stream flow in the fully developed far-wake region, where the wake recovery is dominated by turbulent diffusion rather than organized vortex structures.

Regarding the stability sign parameter, a value of 1 is assigned under stable atmospheric conditions to represent the suppression of vertical mixing and lateral wake expansion. This is achieved by modulating the exponential term in the wake expansion model, effectively capturing the physics of turbulent diffusion. Conversely, under unstable and neutral atmospheric conditions, the stability sign parameter is set to −1 to enhance the wake expansion term. This models the accelerated wake recovery from intensified turbulent mixing, allowing for wake width to dynamically respond to atmospheric stability, rather than relying solely on the empirical linear growth assumed under neutral conditions.

Furthermore, unlike the 3D-Stability-COUTI model, which constructs a bimodal velocity deficit profile through the superposition of two cosine functions, the Present model simplifies the lateral distribution of the velocity deficit by adopting a modified cosine-squared function. This modification shifts the model’s focus from characterizing the bimodal structure to a streamlined, universal description. While retaining the critical flow characteristics in the near-wake region, this approach significantly reduces mathematical complexity. By integrating this simplified velocity deficit distribution with the accurate incoming profiles and the wake width model with the stability sign parameter, the Present model achieves effective wake predictions across the entire wake region under different atmospheric stability conditions, without introducing additional empirical parameters. Consequently, it demonstrates a marked improvement in overall prediction accuracy.

To quantitatively compare the core formulations of the Present model against the 3D-Stability-COUTI and Cheng2019 models,

Table 2. Comparison of analytical formulas for wake models considering atmospheric stability.

Wake Model Considering Atmospheric Stability			Proposed Wake Model	3D-Stability-COUTI Model [29]	Wake Model Based on Monin–Obukhov Similarity Theory [28]
Inflow	Incoming flow	$\mathrm{Incoming} \mathrm{wind} \mathrm{velocity}, U_0(z)$	$U_0(z)=U_H·{\displaystyle \frac{ln(z/z_0)-{\psi }_m(z/L)}{ln(z_H/z_0)-{\psi }_m(z_H/L)}}$ $\mathrm{where} {\psi }_m=\left\{\begin{array}{c}-4.7({\displaystyle \frac{z}{L}}), L>0, stable\\ 0, \left|L\right|\to \infty , neutral\\ 2ln[{\displaystyle \frac{1}{2}}(1+\varsigma )]+ln[{\displaystyle \frac{1}{2}}(1+{\varsigma }^2)]-2{tan}^{-1}(\varsigma )+{\displaystyle \frac{\pi }{2}}, \\ L<0, unstable\end{array}\right.$$\mathrm{and} \mathsf{\varsigma }={[1-15(z/L)]}^{1/4}$		$U_0(z)={\displaystyle \frac{u_{*}}{\kappa }}[ln({\displaystyle \frac{z}{z_0}})-{\psi }_m({\displaystyle \frac{z}{L}})]$ $\mathrm{where} u_{*}={\displaystyle \frac{\kappa U_H}{ln(\frac{z_H}{z_0})-{\psi }_m(\frac{z_H}{L})}}$$; \mathsf{\varsigma }={[1-15(z/L)]}^{1/4}$
		$\mathrm{Incoming} \mathrm{turbulence} \mathrm{intensity}, I_0(z)$	$I_0(z)=I_H·{\displaystyle \frac{ln(z/z_0)-{\psi }_m(z/L)}{ln(z_H/z_0)-{\psi }_m(z_H/L)}}$		$I_0(z_H)=\gamma {\displaystyle \frac{2.5u_{*}}{U_H}}$ $\mathrm{where} \mathsf{\gamma }=0.8$$\mathrm{for} {\displaystyle \frac{2.5u_{*}}{U_H}}<0.12$$\mathrm{and} \gamma =0.$$7 \mathrm{for} {\displaystyle \frac{2.5u_{*}}{U_H}}>0.12$
Wake model	Wake geometry	$\mathrm{Width}, r_w(x)$	$r_w(x)=D{C}_t^{\kappa }{I}_{0,H}^{\frac{\kappa }{2}}{({\displaystyle \frac{x}{D}})}^{\kappa +{\delta }_L}+{\displaystyle \frac{D}{2}}max(-0.2·ln({\displaystyle \frac{x}{D}})+0.2,0)$ $\mathrm{where} {\delta }_L=-0.05{(\frac{z_H}{L})}^{{\displaystyle \frac{\kappa }{2}}}$		$r_w(x)=\sigma =k_w·x+\epsilon ·D$ $\mathrm{where} k_w=0.223I_v+0.022$$; \mathsf{\epsilon }=-1.91k_w+0.34$$; I_v=I_0(1-0.22{cos}^4({\displaystyle \frac{\pi z}{2h}}));$$h={\displaystyle \frac{u_{*}}{12\Omega sin(\varphi )}}$;
	Wake velocity	$\mathrm{Wake} \mathrm{velocity}, U_w(x,y,z)$	$U_w(x,y,z)=U_0(z)·(1-∆u(x,y,z))$ $\mathrm{where} \mathrm{velocity} \mathrm{deficit}, ∆u(x,y,z)=$ $\left\{\begin{array}{c}A·[0.3+0.15(1-S)]·\eta ({\displaystyle \frac{x}{D}})·{\displaystyle \frac{x}{D}}·{cos}^2(0.06r), \\ if-r_w\le r\le r_w\\ 0, otherwise\end{array}\right.$ $A={\displaystyle \frac{\pi }{2}}·{\displaystyle \frac{r_w(x)}{2r_w^{’}}}·∆u^{*}(x)$$; ∆u^{*}(x)={\displaystyle \frac{(1-\sqrt{1-C_t})}{{(\frac{r_w(x)}{(-0.1·ln(x/D)+1.3)·(D/2)})}^2}};$$\mathrm{S}=\left\{\begin{array}{c}1, L>0\\ -1, L<0 and \left|\mathrm{L}\right|\to \infty \end{array}\right.;$$\eta ({\displaystyle \frac{x}{D}})=\left\{\begin{array}{c}1, if {\displaystyle \frac{x}{D}}\le 5\\ exp[-0.15({\displaystyle \frac{x}{D}}-5)], if {\displaystyle \frac{x}{D}}>5\end{array}\right.$	$U_w(x,y,z)=U_0(z)·(1-∆u(x,y,z))$ $\mathrm{where} \mathrm{velocity} \mathrm{deficit}, ∆u(x,y,z)=∆u_{+}+∆u_{-}$ $∆u_{+}=\left\{\begin{array}{c}A·\cos (K\times (r-\theta )),\theta -r_{\mathrm{w}}^{\mathrm{’}}\le r<\theta +r_{\mathrm{w}}^{\mathrm{’}}\\ 0,\mathrm{else}\end{array}\right.$ $∆u_{-}=\left\{\begin{array}{c}A·\cos (K\times (r+\theta )),-\theta -r_{\mathrm{w}}^{\mathrm{’}}\le r<-\theta +r_{\mathrm{w}}^{\mathrm{’}}\\ 0,\mathrm{else}\end{array}\right.$ $\mathrm{and} \theta =0.6r_d$$r_w^{’}=r_w(x)-0.6r_d;$$K={\displaystyle \frac{\pi }{2r_w^{’}}}$$A={\displaystyle \frac{\pi }{2}}·{\displaystyle \frac{r_w(x)}{2r_w^{’}}}·∆u^{*}(x)$$∆u^{*}(x)={\displaystyle \frac{(1-\sqrt{1-C_t})}{{(\frac{r_w(x)}{(-0.1·ln(x/D)+1.3)·(D/2)})}^2}}$	$U_w(x,y,z)=U_0(z)·(1-∆u(x,y,z))$ $\mathrm{where} \mathrm{velocity} \mathrm{deficit}, ∆u(x,y,z)=U_H(1-\sqrt{1-{\displaystyle \frac{C_t}{8{(\sigma /D)}^2}}})\times \mathrm{e}\mathrm{x}\mathrm{p}(-{\displaystyle \frac{1}{2{(\sigma /D)}^2}}\times {({\displaystyle \frac{r}{D}})}^2)$$\mathrm{and} {\displaystyle \frac{\sigma }{D}}=k_w{\displaystyle \frac{x}{D}}+\epsilon$$\beta ={\displaystyle \frac{1}{2}}{\displaystyle \frac{1+\sqrt{1-C_t}}{\sqrt{1-C_t}}}$;

summarizes the key mathematical representations governing the velocity deficit, wake width, and their stability-dependent modifications for each model.

It should be noted that the present model is designed to predict the time-averaged (steady-state) wake velocity field, under the core assumption that both the inflow conditions and the wake structure remain statistically stationary. Consequently, the model does not resolve high-frequency transient phenomena such as wake meandering, periodic vortex shedding, or instantaneous wake oscillations. Given this characteristic, the model is primarily suited for applications that rely on the mean wake deficit, including annual energy production estimation under representative atmospheric conditions, macro-layout optimization, and long-term assessment of wake interference. It should be explicitly noted that the model is not intended for analyzing short-term dynamic loads, instantaneous wake interactions, or studies requiring resolution of high-frequency wake fluctuations. Users are advised to select the appropriate modeling approach based on the spatiotemporal scales relevant to their specific engineering problem.

3. Performance Assessment of the Present Wake Model

3.1. Case 1: Vestas V27 Wind Turbine

For the first validation case of the Present model, field measurements of wind turbine wakes acquired under different atmospheric stability conditions at the Scaled Wind Farm Technology (SWiFT) facility in Lubbock, TX, USA are employed [46]. The site is characterized by flat terrain and homogeneous surface cover, effectively minimizing flow distortions induced by complex topography. Wake measurements were conducted using a rear-facing, nacelle-mounted SpinnerLidar operating in a specific scanning pattern. Each full scan, completed in approximately 2 s, provides line-of-sight velocity estimates at 984 points distributed over a spherical surface. The field measurement data of these points were subsequently linearly interpolated onto a regular grid in the lateral ($y$) and vertical ($z$) directions with a resolution finer than 1 m. In terms of the measurement setup, the LiDAR scanned the wake from 2D (D is the rotor diameter) to 5D downstream of the Vestas V27 wind turbine for both neutral and stable conditions, with a spacing of 1D between consecutive planes. Under unstable conditions, measurements were focused exclusively on the 3D downstream. Further details regarding the field measurement setup are available in Refs. [41,42]. A summary of the input parameters for Case 1 under neutral, unstable, and stable atmospheric conditions is provided in

Table 3. Summary of input parameters for Case 1 under neutral, unstable, and stable atmospheric conditions [41,42,46].

Atmospheric Stability	Stable	Unstable	Neutral
Rotor diameter, D (m)	27	27	27
Hub height, zH (m)	32.1	32.1	32.1
Wind turbine location, Φ (°)	33.60795	33.60795	33.60795
Surface roughness height, z0 (m)	0.0275	0.0275	0.0275
Thrust coefficient, Ct	0.83	0.81	0.70
Wind speed at hub height, UH	4.8	6.7	8.7
Turbulence intensity at hub height, IH	0.034	0.126	0.107
Obukhov length, L (m)	8.69	−112.36	2500

.

Figure 3 shows the normalized wind speed profiles along the vertical centerline ($y=0$) for Case 1 under stable, unstable, and neutral atmospheric conditions, while Figure 4 illustrates the corresponding profiles on the horizontal plane ($z=z_H$) for the same case and stability conditions. The normalized wind speed is defined as the ratio of the local wind speed to that at the hub height, i.e., u ratio = U(_z)/U_H. Based on the field measurements, the vertical profiles of the wind turbine wake exhibit obvious asymmetry, which physically results from the complex interaction between turbulent mixing in the wake and the surface effects in the near-ground region. Likewise, the horizontal profiles also display asymmetry due to the rotational motion of the blades.

The three wake models accounting for atmospheric stability perform differently in capturing this asymmetric characteristic. Under stable atmospheric conditions (Figure 3a and Figure 4a), the 3D-Stability-COUTI model underestimates the velocity deficit at the hub height and produces a bimodal structure that is inconsistent with field measurements. This phenomenon is closely related to the formula of wake velocity deficit, which is defined as the sum of two cosine functions. This mathematical structure tends to overestimate the bimodal characteristics in the wake region. According to Refs. [44,45,47], the bimodal structure in the wake is typically observed within a downstream region extending to approximately 2D. On the other hand, the Cheng2019 model exhibits a systematic underestimation of the velocity deficit across all planes. This deviation may stem from the fact that its parameterization scheme for longitudinal turbulence intensity does not adequately respond to stable atmospheric conditions. Under unstable atmospheric conditions (Figure 3b and Figure 4b), the field measurement data are only valid on a single plane, where all three models demonstrate high prediction accuracy. Under neutral atmospheric conditions (Figure 3c and Figure 4c), both the 3D-Stability-COUTI and Cheng2019 models consistently underestimate the velocity deficit across all the planes. In contrast, the Present model agrees well with the field measurements at x = 2D and 3D downstream but tends to slightly overestimate the velocity deficit at $x=4\mathrm{D}$ and 5D.

The validation metrics of the Present, the 3D-Stability-COUTI, and Cheng2019 models on both vertical and horizontal planes under stable, unstable, and neutral atmospheric conditions are presented in Figure 5. The corresponding hit rates are summarized in

Table 4. Hit rates of predicted wind speed against field measurements of Case 1 by Present model, 3D-Stability-COUTI model, and Cheng2019 model (measurement data from [41,42,46]).

Hit Rate	Atmospheric Stability	Plane	Present Model	3D-Stability-COUTI [29]	Cheng2019 [28]
$q(\left|{\displaystyle \frac{M_i-P_i}{P_i}}\right|\le 15\%)$	stable	XOZ	0.94	0.83	0.73
		XOY	0.73	0.59	0.65
	unstable	XOZ	0.97	0.97	0.97
		XOY	1.00	1.00	1.00
	neutral	XOZ	0.98	0.99	0.87
		XOY	0.96	0.95	0.87
$q(\left|{\displaystyle \frac{M_i-P_i}{P_i}}\right|\le 20\%)$	stable	XOZ	0.99	0.91	0.82
		XOY	0.87	0.69	0.74
	unstable	XOZ	1.00	1.00	1.00
		XOY	1.00	1.00	1.00
	neutral	XOZ	1.00	1.00	1.00
		XOY	1.00	1.00	1.00

. The prediction accuracy of the three models exhibits notable differences across different atmospheric stability conditions. The Present model exhibits the highest consistency with field measurements across most of the planes, demonstrating superior prediction accuracy in wake velocity. And it shows particularly robust performance in capturing both wake velocity and development under stable and unstable atmospheric conditions. The 3D-Stability-COUTI model demonstrates moderate prediction accuracy, though its performance varies with atmospheric stability and shows a relatively obvious systematic deviation under stable conditions. It should be noted that under neutral conditions on the XOZ plane, the hit rate of the 3D-Stability-COUTI model (0.99) is slightly higher than that of the present model (0.98). This minor discrepancy may be attributed to the ability of the 3D-Stability-COUTI model to capture subtle bimodal features present in the data under perfectly neutral and homogeneous ideal conditions. In contrast, the universal cosine-squared velocity deficit profile adopted by the Present model may introduce slight smoothing errors under such specific conditions. The Cheng2019 model presents higher prediction accuracy under unstable and neutral conditions, but tends to overestimate wake velocity under stable conditions, indicating its relatively limited adaptability to different atmospheric stability conditions. Overall, all models exhibit good consistency under neutral conditions, while the Present model achieves the most outstanding prediction accuracy under stable conditions.

Table 5. Comparison of fitting curve statistics for Case 1 by Present model, 3D-Stability-COUTI model, and Cheng2019 model in Figure 5 (measurement data from [41,42,46]).

Atmospheric Stability	Plane	Statistics of Fitting Line	Present Model	3D-Stabiliy-COUTI [29]	Cheng2019 [28]
Stable	XOZ	Slope	1.11	1.14	1.08
		R-Square	0.96	0.90	0.86
	XOY	Slope	0.82	0.86	0.62
		R-Square	0.76	0.59	0.69
Unstable	XOZ	Slope	0.78	0.58	0.83
		R-Square	0.96	0.88	0.94
	XOY	Slope	0.88	0.71	0.91
		R-Square	0.99	0.97	0.98
Neutral	XOZ	Slope	1.13	0.90	0.89
		R-Square	0.84	0.82	0.76
	XOY	Slope	1.02	0.78	0.78
		R-Square	0.81	0.84	0.88

presents a comparison of key statistics (slope and R²) for the fitting line of three wake models shown in Figure 5. Overall, the Present model demonstrates the best performance in terms of both prediction accuracy and stability, with slope values closest to the ideal value of 1 and the highest R² under most conditions. This indicates that it has the smallest systematic bias and provides the most robust and reliable predictions. The 3D-Stability-COUTI model shows moderate performance but exhibits noticeable variability; specifically, its R² is significantly lower on the XOY plane under stable conditions, and it tends to systematically exhibit an overestimation in the low-speed region and an underestimation in the high-speed region. In contrast, the Cheng2019 model is the most sensitive to atmospheric stability conditions, displaying clear systematic underestimation and data scatter under stable conditions, although its prediction performance improves under unstable conditions. In summary, among the three models, the Present model maintains balanced and superior prediction performance under all atmospheric stability conditions.

Figure 6 and Figure 7 show the color maps of normalized wind speed on the horizontal plane at hub height for Case 1 by the Present, 3D-Stability-COUTI, and Cheng2019 models under stable and unstable atmospheric conditions. It can be seen that the wake width under stable conditions is significantly narrower than that under unstable conditions—a trend that is successfully captured by all three wake models. However, the Cheng2019 model shows limited distinction between the wake widths under stable and unstable conditions, indicating that it has limited ability to differentiate between stable and unstable conditions. It is noteworthy that the Cheng2019 model fails to produce valid numerical solutions for $x\in (0,1.259D)$ under stable conditions and $x\in (0,1.259D)$ under unstable conditions, which confirms mathematical singularity at the near-wake region. On the other hand, from the color maps of the 3D-Stability-COUTI model, two distinct recirculation zones in the wake under both stable and unstable conditions can be observed, consistent with the bimodal wind profiles in Figure 4. The recirculation zones are notably smaller under unstable conditions, which explains why the bimodal structure in Figure 4b is less pronounced than in Figure 4a.

Based on the color maps of normalized wind speed on the vertical profiles in Figure 8 and Figure 9, all three models could capture a narrower wake under stable conditions and a wider wake with faster recovery under unstable conditions. The Present model and the 3D-Stability-COUTI model both produce continuous and physically consistent wake velocity fields under both stable and unstable conditions, clearly reflecting stability-dependent wake expansion. However, the 3D-Stability-COUTI model predicts recirculation zones extending beyond 2D downstream, which deviates from high-fidelity observations. In contrast, the Cheng2019 model fails in the near-wake region due to mathematical singularities. Overall, the present model demonstrates superior prediction accuracy and stability response compared to the other two models.

Atmospheric stability systematically governs wake morphology and recovery by modulating turbulent mixing intensity. The comparison between Figure 6a and Figure 7a reveals that under stable conditions ($L>0$), suppressed vertical mixing leads to a contracted wake width and delayed velocity deficit recovery, resulting in a persistent downstream influence. In contrast, unstable conditions ($L<0$) enhance convective mixing, promoting lateral wake expansion and accelerating momentum exchange with the free stream, thereby facilitating rapid recovery, as can be seen in Figure 8a and Figure 9a. Neutral conditions ($\left|L\right|\to \infty$) represent an intermediate baseline state. The present model successfully parameterizes this physical mechanism through the stability sign parameter S and the corrected expansion term ${\delta }_L$, enabling a unified description and accurate prediction of wake dynamics across varying stability regimes.

3.2. Case 2: Danwin 180 kW Wind Turbine

This section employs field measurements of wind turbine wakes acquired under different atmospheric stability conditions at the Alsvik wind farm, Gotland, Sweden [48], for the second validation case of the Present model. The site is characterized by a flat coastal strip, which is covered with short grass and some bushes. Wake measurements were conducted using two measurement towers, M1 and M2, equipped with sensors developed by the Department of Meteorology in Uppsala with a sampling frequency of 1 Hz. In terms of the measurement setup, the measurement data with winds from sea (200–290°) were selected. Under these conditions, M2 was positioned downstream of various turbines, enabling wake measurements at 4.2D, 6.1D, and 9.6D, while M1 provided the undisturbed incoming wind speed and turbulence intensity. Further details regarding the field measurement setup are available in Refs. [29,42]. A summary of the input parameters for Case 2 under neutral, unstable, and stable atmospheric conditions is provided in

Table 6. Summary of input parameters for Case 2 under neutral, unstable, and stable atmospheric conditions [29,42,48].

Atmospheric Stability	Stable	Unstable
Rotor diameter, D (m)	23	23
Hub height, zH (m)	31	31
Wind turbine location, Φ (°)	57.47467	57.47467
Surface roughness height, z0 (m)	0.0005	0.0005
Thrust coefficient, Ct	0.82	0.82
Wind speed at hub height, UH	8.0	8.0
Turbulence intensity at hub height, IH	0.085	0.085
Obukhov length, L(m)	35	100

.

Figure 10 shows the normalized wind speed profiles along the vertical centerline ($y=0$) for Case 2 under stable, unstable, and neutral atmospheric conditions, while Figure 11 illustrates the corresponding profiles on the horizontal plane ($z=z_H$) for the same case and stability conditions. The three wake models exhibit some deviations under stable and unstable atmospheric conditions. Under stable conditions (Figure 10a and Figure 11a), all models underestimate the velocity deficit at $x/D=4.2$, with the Present model exhibiting a relatively smaller underestimation. The 3D-Stability-COUTI model not only exhibits a more pronounced underestimation but also displays a bimodal velocity profile that deviates from the field measurement distribution. Similarly, the Cheng2019 model demonstrates underestimation. Under unstable conditions (Figure 10b and Figure 11b), the Present model switches to overestimating the near-wake velocity deficit, while the 3D-Stability-COUTI model shows improved predictive accuracy, and the Cheng2019 model remains relatively consistent in its performance.

The Danwin 180 kW turbine at the Alsvik site operates under relatively low turbulence intensity (I_H = 0.085), while the wind speed (U_H = 8.0 m/s) and thrust coefficient (C_t = 0.82) remain moderately high. This combination results in a strong initial wake but relatively slow turbulent diffusion. Under such flow conditions, the near-wake velocity deficit profile tends to exhibit a more concentrated and sharper peak near the wake centerline. The cosine-squared velocity deficit function adopted in the Present model (Equation (20)), with its predefined peak shape, may overestimate the velocity deficit magnitude in the vicinity of the centerline under low turbulence unstable conditions, leading to the overprediction of the velocity deficit at $x/D=4.2$.

This discrepancy highlights an opportunity to refine the near-wake velocity deficit formulation under unstable conditions with low ambient turbulence. One possible direction is to introduce a turbulence-intensity-modulated shape function that can adapt the deficit profile between a cosine-squared and a mildly bimodal form based on local turbulence intensities. Such a modification would allow the model to better capture near-wake physics while retaining its simplicity and stability-sensitivity in the far-wake.

The validation metrics of the Present, the 3D-Stability-COUTI, and Cheng2019 models for Case 2 on both vertical and horizontal planes under stable, unstable, and neutral atmospheric conditions are presented in Figure 12. The corresponding hit rates are summarized in

Table 7. Hit rates of predicted wind speed against field measurements of Case 2 by Present model, 3D-Stability-COUTI model, and Cheng2019 model (measurement data from [29,42,48]).

Hit Rate	Atmospheric Stability	Plane	Present Model	3D-Stability-COUTI [29]	Cheng2019 [28]
$q(\left|{\displaystyle \frac{M_i-P_i}{P_i}}\right|\le 15\%)$	stable	XOZ	0.93	0.97	0.89
		XOY	0.83	1.00	0.99
	unstable	XOZ	0.90	1.00	1.00
		XOY	1.00	1.00	1.00
$q(\left|{\displaystyle \frac{M_i-P_i}{P_i}}\right|\le 20\%)$	stable	XOZ	1.00	1.00	0.93
		XOY	0.93	1.00	1.00
	unstable	XOZ	1.00	1.00	1.00
		XOY	1.00	1.00	1.00

. The three models demonstrate reliable prediction accuracy under stable and unstable atmospheric conditions, each exhibiting distinct characteristics. Under stable conditions, the Present model shows good prediction accuracy in the vertical profile (XOZ plane), while the 3D-Stability-COUTI model maintains consistent performance across both profiles. Under stable atmospheric conditions, the hit rate of the 3D-Stability-COUTI model on the XOY plane is higher than that of the Present model. This is primarily due to the fact that the Present model employs a more concentrated modeling approach for the wake core region under stable, low-turbulence conditions. Since the actual wake profile is typically more uniform, the Present model tends to overestimate the velocity deficit near the centerline, which compromises its local prediction accuracy and hit rate on the horizontal plane. Similarly, under unstable conditions, the Present model remains highly sensitive to the peak shape of the wake center, leading to an overestimation of the velocity deficit on the vertical plane (XOZ) as well, thereby resulting in a lower hit rate compared to the 3D-Stability-COUTI model. In contrast, the Cheng2019 model shows some discrepancy in the vertical profile compared to the other models. Under unstable conditions, all models achieve relatively good agreement on the horizontal profile (XOY plane). For Case 2, with a relative error threshold of 15%, the hit rates of all three wake models exceed 80% in both vertical and horizontal profiles, indicating that all the models exhibit high prediction accuracy.

Table 8. Comparison of fitting curve statistics for Case 1 by Present model, 3D-Stability-COUTI model [29], and Cheng2019 model [28] in Figure 12 (measurement data from [29,42,48]).

Atmospheric Stability	Plane	Statistics of Fitting Line	Present Model	3D-Stabiliy-COUTI [29]	Cheng2019 [28]
Stable	XOZ	Slope	1.01	1.00	0.98
		R-Square	0.89	0.91	0.84
	XOY	Slope	1.21	0.80	0.72
		R-Square	0.80	0.82	0.87
Unstable	XOZ	Slope	1.25	0.67	0.87
		R-Square	0.82	0.79	0.77
	XOY	Slope	1.01	0.76	0.89
		R-Square	0.91	0.91	0.94

presents a comparison of key statistics (slope and R²) for the fitting line of three wake models shown in Figure 12. Specifically, the slope of the linear fitting line of the Present model is closest to the ideal value of 1.0 under most conditions, visually demonstrating its overall superior predictive accuracy. In contrast, the 3D-Stability-COUTI model shows the most significant deviation in fitted slope under unstable atmospheric conditions, with a slope of only 0.76 on the XOY plane and 0.67 on the XOZ plane. Additionally, the coefficient of determination R² for both planes is the lowest among all models. Although this model achieves a slightly higher hit rate than the Present model on the XOY plane under unstable conditions, its overall fitting statistics remain slightly inferior. The Cheng2019 model performs intermediately between the two aforementioned models under unstable conditions: while it exhibits a relatively high R² value (0.94) on the XOY plane, its fitted slope is still lower than that of the Present model, reflecting a certain degree of systematic underestimation. Under stable conditions, the performance of this model declines significantly, particularly on the XOY plane, where, despite having the highest R² (0.87), it is accompanied by the lowest fitted slope, further highlighting its limitations in prediction accuracy and systematic bias control under stable operating conditions. In summary, among the three models, the Present model maintains balanced and superior prediction performance under all atmospheric stability conditions.

The color maps of the normalized wind speed on the horizontal plane at hub height for Case 2 under different atmospheric stability conditions are displayed in Figure 13 and Figure 14. The present model produces a symmetric and continuous velocity deficit distribution under stable conditions, characterized by a well-defined wake boundary; under unstable conditions, it shows enhanced lateral wake expansion, reflecting its response to convective mixing. The 3D-Stability-COUTI model exhibits a unique bimodal structure in the near-wake region under stable conditions, while under unstable conditions, the wake becomes more uniform while maintaining a relatively compact low-speed core. In contrast, the Cheng2019 model yields a homogeneous fan-shaped velocity field under both stable and unstable conditions. This insensitivity to atmospheric stability is reflected in the only minor variations in wake width of the color maps under the two atmospheric stability conditions.

Figure 15 and Figure 16 show the normalized wind speed on the vertical plane at $y=0$ for Case 2 by the Present, the 3D-Stability-COUTI and the Cheng2019 models under stable and unstable atmospheric conditions. It is obvious that the wake height is consistently lower under stable conditions than under unstable conditions, due to suppressed vertical mixing in stable stratification. In contrast, unstable conditions enhance convective diffusion, resulting in a taller and more vertically dispersed wake. This trend is consistent across all models, underscoring the dominant role of atmospheric stability in governing vertical wake scale.

4. Conclusions

This study systematically validates the newly proposed analytical wake model that accounts for atmospheric stability effects. The main conclusions are summarized as follows:

(1)

By incorporating a stability-dependent turbulence expansion term including the square of a cosine function and the stability sign parameter, a wake model framework capable of dynamically responding to atmospheric stability has been established. This framework overcomes the limitation of traditional models that rely on neutral atmospheric assumptions, achieving a physically consistent description of turbulence suppression under stable conditions and convective enhancement under unstable conditions.

(2)

A newly proposed far-field decay function effectively adjusts the wake development at the near-wake region and the far-wake region, maintaining computational efficiency while significantly improving the prediction accuracy under complex atmospheric stability conditions. Validation results demonstrate that the Present model exhibits optimal overall performance in predicting wake velocity distributions on both vertical and horizontal planes.

(3)

The Present model integrates atmospheric stability parameters into its core algorithm, enabling continuous predictions across all stability conditions—neutral, stable, and unstable. Validated against field data from the SWiFT facility and the Alsvik wind farm, the model demonstrates strong adaptability. It offers a reliable tool for optimizing turbine layout and predicting power output under complex atmospheric conditions.

The model proposed in this study demonstrates good prediction accuracy for single wakes over flat, homogeneous terrain, but still exhibits several limitations. Firstly, the current validation relies exclusively on field measurements from flat and homogeneous sites (e.g., SWIFT and Alsvik). Its performance in large wind farm arrays with cumulative wake effects and over complex topography—such as areas with topographically forced shear or secondary flows—remains to be investigated and may require the introduction of terrain-adjustment parameters. Secondly, validation has been conducted only for small- and medium-sized turbines, while systematic assessment for modern large-scale turbines ($D >150 \mathrm{m}$) is lacking, where wake scale effects and atmospheric boundary layer interactions could differ substantially. Thirdly, the model depends on the Obukhov length (L) to characterize atmospheric stability via a sign parameter (S). Although effective in distinguishing stability regimes, this approach relies on quasi-steady-state assumptions and does not explicitly account for non-stationary processes like rapid stability transitions. Moreover, L may not be routinely available at all wind farm sites; linking stability inputs to more commonly measured parameters could enhance practical usability. Additionally, key model expressions—such as the far-field decay function—incorporate coefficients calibrated on existing datasets, and their universality under extreme stability or for unconventional turbine designs requires further validation. Finally, the simplification adopted for the velocity deficit profile, while beneficial for numerical stability, may limit accuracy in the very near-wake region ($x/D < 2$).

Future work will focus on four directions: first, extending the model to complex terrain and wind-farm-array scenarios through terrain parameterization and cumulative-wake modeling to improve practical applicability; second, conducting dedicated validation for large-scale turbines, supported by high-fidelity simulations such as Large Eddy Simulation (LES) to elucidate scale-dependent wake–boundary layer interactions and provide a more intuitive and in-depth understanding of vortex evolution dynamics in wind turbine wakes; third, developing dynamic stability representation methods, such as coupling with time-evolving L from mesoscale models or constructing transitional weighting functions, to better capture non-stationary atmospheric processes, while also exploring linkages to more routinely measured meteorological parameters; and fourth, refining near-wake modeling and optimizing coefficients using broader datasets to enhance robustness and universality across varying conditions. These refinements will further advance the model’s engineering practicality and physical completeness.