A linear wake expansion function for the double‐Gaussian analytical wake model

The double‐Gaussian (DG) approach for analytical wake modeling leads to a better understanding of the wake transition mechanism within a full‐wake region behind a non‐yawed horizontal‐axis wind turbine (HAWT). To date, a key parameter of the wake expansion in the DG model still has yet to be defined explicitly instead of tuning, thus limiting its usability for practical applications. The present work aims to overcome this limitation by proposing a simple linear wake expansion function for the DG model constructed from the existing parameters based on the conservation of mass and momentum. Considering the physical and statistical approaches, the proposed function is specifically intended to approximate the wake expansion downstream of a non‐yawed HAWT under turbulence inflow. Seven case studies from wind tunnel measurements and large eddy simulations under different inflow conditions were used to examine the effectiveness of the proposed function. In general, the evaluation results in the present study show the effectiveness of the proposed expansion function for the DG wake model to predict the wake expansion and its recovery behind a non‐yawed HAWT without a prior adjustment or tuning of the wake expansion parameter.


| INTRODUCTION
Wind energy exploitation is rapidly increasing worldwide due to its tremendous potential to supply a vast amount of clean energy for large-scale power generation. The development in wind turbine technology nowadays enables project developers to manufacture and operate megawatttype horizontal-axis wind turbines (HAWTs) with hub heights of 100 m or more. 1 When those HAWTs are integrated into a cluster with a specific array arrangement known as a wind farm, they can produce a total nameplate capacity up to the Gigawatt scale. Thus, the massive deployment of wind farms in onshore and offshore terrains could significantly contribute to decarbonizing electricity generation worldwide. However, any downstream turbines inside the wind farm will experience power losses due to the upstream turbine's wakes, thus decreasing the overall power generation of the farm. 2 A HAWT wake can be defined as a plume-like region downstream of a wind turbine characterized by reduced wind speed and increased turbulence intensity. 3 Given the importance of wake aerodynamics on the overall wind farm performance, several investigations have been conducted to elucidate the wake behavior using an experimental, [4][5][6][7][8][9][10][11] numerical, [12][13][14][15][16][17][18][19][20][21] or analytical approach. [22][23][24][25][26][27][28][29][30][31][32][33] Among the other approaches, analytical modeling is known for its computationally inexpensive and less time-consuming of giving a practical prediction of the wake flow field, thus providing a relatively efficient solution for wind farm layout optimization methodology. 34,35 Due to its suitability for practical applications, such as in the wind industry sector, the present study is focused on analytical wake modeling.
The wake shape distribution from analytical wake modeling is mainly divided into two based approaches: (1) top-hat (TH); (2) single-Gaussian (SG). However, the TH approach is a less accurate assumption to represent the shape of velocity deficit behind the HAWTs since it generally forms the Gaussian distribution. 5 The SG models were designed to typically perform within the far-wake region where most downstream turbines are located. The HAWTs within a wind farm are commonly installed up to 11D for the interrow spacing. Within this interrow distance, it was observed experimentally 26 and numerically 14,27 that the single HAWT wakes expanded linearly under different turbine operating and inflow conditions. Thus, some existing wake models considered the linear expansion for the wake growth rate downstream of the turbine. 14,23,26 However, their expansion coefficients were determined empirically based on specific case studies, which may require additional tuning under more varying inflow conditions.
Since the SG-based approach is based on a practical perspective that focuses on the far-wake region where most of the utility-scale downstream HAWTs were installed, thus the existing SG models may not give a satisfying result of wake velocity prediction within the near-wake region. This region is characterized by the double-Gaussian (DG) velocity distribution due to optimum lift around the blade mid-span and very small lift around the blade's root and tip. 36 The DG distribution is formed immediately after a significant velocity deficit around the centerline due to the nacelle blockage effect recovered by mixing with the faster air surrounding the nacelle. Since this significant centerline deficit recovered rapidly over a relatively short distance, 37 thus the DGbased approach is still reasonable to represent the wake characteristic within the near-wake region.
By giving attention to this near-wake characteristic, Keane et al. laid the foundation of the DG analytical wake modeling based on the conservation of mass and momentum. 29 Later, Schreiber et al. found some issues in the original DG formulation, which resulted in a nonphysical result inconsistent with the conservation equations; thus, some of the original expressions were rederived. 30 Recently, Keane improved the original DG model, providing a viable wake velocity deficit function within the near-wake through the use of a complex solution, and introduced the effective rotor diameter as the entire extent of the DG wind velocity deficit. 31 It is worth noting that DG wake models also effectively predict the wake distribution and recovery within the fullwake region, which encompasses the near and far-wake regions. This advantage was confirmed using computational fluid dynamics (CFD) data, where the root mean square error results showed the superiority of the DG model against the SG model for lateral velocity deficit predictions within a full-wake region. 30 Most recently, the anisotropic DG model was compared with the anisotropic SG model referring to large eddy simulation (LES) and lidar data. 37 The results showed that the anisotropic DG model could perform better than the anisotropic SG model within the full-wake region, thus reconfirming the effectiveness of the DG-based wake model.
Although several attempts have been made to improve the reliability of the DG wake model, the practical use of this model is still difficult since the wake expansion parameter has yet to be defined explicitly instead of tuning. It should be noted that this expansion parameter controls the wake recovery development and eventually determines the accuracy of the wake velocity prediction. Since the wind power varies with the cube of wind velocity, the inaccurate prediction of the wake velocity would result in significant errors in power prediction.
Given the importance of the wake expansion parameter, the present work aims to construct a simple physicallybased linear wake expansion function for the DG model incorporated from the existing parameters based on the conservation of mass and momentum. To the best of our knowledge, this is the first expansion function for the DG wake model that can be used directly without expansion parameter tuning. The validity of the proposed expansion function was examined with seven case studies of single HAWT wakes under different operating and inflow conditions. In addition, the tuning results from the existing expansion function were also evaluated for comparison. Finally, the effectiveness of the proposed function on the given case studies was evaluated using statistical measures in terms of NRMSE and R correlation coefficient.
2 | METHODOLOGY 2.1 | Linear wake expansion function for the DG analytical wake model Based on turbine micro-siting practices, the HAWTs within a wind farm are rarely installed over 11D for the interrow spacing. 37 Within this spacing distance, it was observed both experimentally 26 and numerically 14,27 that the single HAWT wakes expanded linearly under different turbine operating and inflow conditions. In analytical wake modeling, Jensen and Katic already considered linear expansion for the wake growth behind the HAWT with their top-hat wake model. 22,23 In this model, the wake radius at the HAWT's rotor plane r w0 (x = 0) was assumed equal to the rotor radius (r 0 ). However, by employing the rotor disc approach to simplify the rotor aerodynamics, the area over which the thrust T acting shall extend beyond the disc area, on the assumption that the thrust coefficient C T is fixed. 31 Hence, the previous simplification regarding r w0 = r 0 should be redefined to r w0 > r 0 for a more detailed approximation of the wake radius at the rotor plane. This consideration is crucial to locate the outset from which the wake expands linearly within the practical interrow turbine spacing. An illustration of linear wake expansion in the present study is shown in Figure 1.
In the proposed function, r w0 should satisfy the condition r w0 > r 0 . Based on the physical approach, Bastankhah and Porté-Agel 26 derived a semi-empirical formula for the standard deviation of the SG velocity deficit profile at the rotor plane σ w0 . Combining this semi-empirical formula for σ w0 with the statistical approach, thus 2.58σ w0 that covers 99% values of Gaussian distribution 28 and satisfies the condition r w0 > r 0 , is used to estimate r w0 in rotor diameter unit [D] as follows: where β is expressed as 24 Meanwhile, the standard deviation of velocity deficit at the onset of the far-wake region, σ w1 [D], can be approximated using the DG wake expansion at the stream tube outlet, 30 which here is written using the following relation: where M and N are the integral solutions for the amplitude function of velocity deficit in the DG-based approach, which relies on the conservation of mass and momentum. Their formulations will be defined later. It would not be straightforward to obtain σ w1 from Equation (3). Thus, it should be solved numerically, one of which is by employing the Matlab function vpasolve. After numerically solving σ w1 , the wake radius at the far-wake onset r w1 [D] located at x/D = x 1 can be calculated as follows: F I G U R E 1 Illustration of the proposed expansion function for the analytical DG wake model. The wake expands linearly in the lateral direction (y) at the hub height (z h ) along the downstream distance x behind the rotor.
where r min [D] denotes the radial position of Gaussian minima and its value has been determined empirically equal to r min = 0.26. 37 Meanwhile, the streamwise position of the far-wake onset, x 1 [D], is approximated using the following formula 33,37 : where the coefficients a = 0.58 and b = 0.077, 33 c = 1.2, 37 and TI x,hub is the streamwise turbulence intensity of the incoming streamwise velocity at the hub height.
Here, the three primary parameters of r w0 , r w1 , and x 1 are incorporated to formulate a linear wake expansion function for the DG model. Since the wake expansion is assumed linear, the parameter α, which is the angle between r w0 and r w1 , can be calculated using the following relation: In a more general form, Equation (4) can be rewritten using the following expression: thus yielding: By employing Equation (6), The wake radius at any downstream distance, r w [D], can be expressed as follows: Using the definitions above for r w0 , r w1 , and r min , then Equation (10) Finally, a compact form of the proposed linear wake expansion function for the DG model, σ w [D], is formulated as follows: As formulated in Equation (12), the proposed expansion function can be used directly without tuning. It should be noted that this expansion function is only compatible for non-yawed HAWTs under turbulence inflow, which also corresponds to the actual inflow condition where the utility-scale turbine operates. Hence, TI x,hub becomes another input parameter for the expansion function in addition to C T . The proposed function was applied to the existing DG wake model, 37 as summarized in Table 1. where r [D] is the radial distance from the wake center defined as r y z z = +( − ) h 2 2 . The normalized wake velocity U w /U 0 within the Cartesian coordinate system can be calculated using the following relation: C σ x f r y z σ x ( ( )) ( ( , ), ( )) where U 0 [m/s] is the reference streamwise incoming velocity at the hub height. The lateral (y) and vertical (z) directions are in the rotor diameter unit [D]. It is worth noting that the rotor diameter parameter (D) in the formulation for C σ x ( ( )) from Table 1 is self-normalized, thus its quadratic value in that formula remains equal to 1. Meanwhile, the definitions for M and N in Table 1 are expressed as follows: It should be kept in mind that the wake expansion is assumed to be isotropic so that the expansion functions in the lateral direction (σ y ) and vertical direction (σ z ) are in Equation (3) are calculated using Equations (14) and (15), respectively, by substituting σ y and σ z with σ w1 for the wake expansion at the far-wake onset. In Table 1, if , the solution is solved using the absolute value or modulus of a complex number 31 :

| Validation
In this study, seven case studies of non-yawed HAWT wake resulting from LESs and wind tunnel measurements were used to evaluate the effectiveness of the proposed expansion function. Since all case studies were configured under neutral atmospheric stability, all the analytical predictions in this study are only relevant to neutrally stratified atmospheric boundary layer without any thermal effects. Further information on the turbine's operating and inflow conditions in each case is shown in Table 2. Cases 1-3 resulted from three-dimensional highfidelity LESs using a noncommercial CFD solver Front-Flow/Blue (FFB). 38 The solver uses the finite-element method (FEM) to solve the unsteady incompressible Navier-Stokes (NS) equations numerically. The reference turbine used was NREL 5 MW HAWT, developed by National Renewable Energy Laboratory (NREL) for research purposes. 39 Main components of the turbine, such as the rotor blades, tower, and nacelle, were directly modeled to provide detailed information of the flow field within the near-wake region. The computational domain of the FFB simulations consisted of about 360 million elements of structured hexahedral meshes. The simulations were running on the Japanese supercomputer Fugaku.
The data range of Cases 4-7 in Table 2 covers the fullwake region. In this study, the terminology of "full-wake region" means the wake data are selected from the near and far-wake regions, but it does not certainly mean that the analyses must include all the downstream distances and start exactly from x = 0. In Case 4, six selected downstream distances (x/D = 1.7, 2, 3, 4, 6, and 9) from wind-tunnel measurements of the wake profile behind the G1 turbine model were cited from Schreiber et al. 30 Turbulent boundary layer inflow had the hub-height reference velocity (U 0 ) ≈ 5 m/s and the hub-height turbulence intensity (TI hub ) ≈ 5%. Since no specific direction was mentioned for the inflow turbulence, the TI hub of the inflow was interpreted as the total TI hub (TI total,hub ). Thus, in the present study, the hub-height turbulence intensity in the streamwise direction (TI x,hub ) of the incoming flow was approximated using the IEC 61400-1 standard, where TI x,hub ≈ TI total /0.8, resulting in TI x,hub of about 6.25%.
Ten selected downstream distances (x/D = 1.5, 2, 3, 4, 5, 6, 7, 8, 10, and 12) in Cases 5-6 were cited from T A B L E 2 Case studies used to validate the proposed expansion function.  41 The rotor was indirectly modeled using the actuator line method. Meanwhile, the turbine's nacelle and tower were not modeled. The LES data from the last 600 s simulation time (sampling rate of 10 s) of the fully-developed wake flow field were processed for validation. The turbine was operated under turbulent boundary layer, with the reference velocity of U 0 = 7.87 m/s and the incoming TI x,hub = 5.1% in Case 5. Meanwhile, U 0 = 7.7 m/s and the incoming TI x,hub = 8.5% for Case 6. Relatively low and medium incoming TI x,hub from the selected data sets were considered realistic for offshore environments with flat terrain. Ten selected downstream distances (x/D = 1, 1.5, 2, 2.5, 3, 4, 5, 6, 7, 8) of the wake data set in Case 7 were cited from Technical University of Denmark (DTU) database for LES of NREL 5 MW wake, 42 which was obtained using the in-house incompressible finite volume code EllipSys3D. 43 The indirect method for rotor modeling was used as the actuator disk with a fixed force distribution based on a rotor-resolved detached eddy simulation. 16 The HAWT wake was simulated under turbulent boundary layer with the reference velocity of U 0 = 7.87 m/s and the incoming TI x,hub of about 16%. The data set represents an onshore situation with characteristically high TI x,hub , possibly due to high ground surface roughness and the presence of complex terrain or obstacles.

| The near-wake region
This section provides the LES-FFB results and their analytical predictions for Cases 1-3, focusing on the wake velocity distribution at the hub height in the lateral direction within the near-wake region. The wake profiles from Case 1 are shown in Figure 2.
The high-fidelity LES results captured the influence of rotor geometry on the produced wake flow field indicated by the DG shape velocity profiles formed within the near-wake region. The proposed function could give reasonable predictions of the wake recovery at the centerline (y = 0, z = z hub ), except for x/D = 0.5, where the wake velocity from LES gradually reduced towards the centerline due to the nacelle blockage effect. As the downstream distance increased, the blockage effect became insignificant, thus causing better agreement with the LES data. The LES results in Case 2 and their analytical estimations are shown in Figure 3.
Case 2 was set under the same uniformly distributed inlet velocity as the previous case but with a higher incoming TI x,hub . The wake recovery at the selected distances within the near-wake region is almost the same F I G U R E 2 Wake velocity profiles at the hub height under uniform inlet with U 0 ≈ 11.4 m/s and incoming TI x,hub ≈ 3% within the near-wake region (Case 1). as in Case 1. The maximum velocity deficit around the blade mid-span gradually reduced as the downstream distance increased. The top-hat profile from the LES result around the blade mid-span position to the wake centerline at x/D = 3 might indicate the rotor presence's attenuation on the wake flow field. At the same downstream distance, the proposed function gave a faster wake recovery than the LES result, causing an overestimation of the wake velocity prediction. However, the proposed function could still provide a reasonable representation of wake recovery within the near-wake region, as indicated by decent estimations of the centerline velocity at the selected distances.
Another LES result of the single HAWT wake under the ABL inlet with moderate TI x,hub (Case 3) is shown in Figure 4. F I G U R E 4 Wake velocity profiles at the hub height under ABL inlet with U 0 ≈ 11.6 m/s and incoming TI x,hub ≈ 7.1% within the near-wake region (Case 3).
The top-hat profile from the LES result around the blade mid-span to the wake centerline occurred earlier than in Case 2, observably from x/D = 2.5. As in Case 2, the proposed function also overestimated the wake velocity at x/D = 3, with a salient deviation within a rotor swept area. However, the resulting analytical prediction could still reasonably produce the DG velocity distribution at that distance.
In general, the proposed expansion function could give a rough yet reasonable representation of the wake profile expansion and its recovery within the near-wake region of the evaluated Cases 1-3. Salient velocity deficits were noticed around the centerline due to the nacelle blockage effect, which can be observed up to x/D = 1.0. Afterward, the bimodal distribution formed, which justifies the DG-based approach's feasibility in representing the wake velocity distribution within the near-wake region.

| The full-wake region
This section highlights the wake characteristics within the full-wake region of Cases 4-7, represented by the hub-height wake velocity distribution and its recovery under different turbine operating and inflow conditions. Figure 5 shows the hub-height streamwise wake velocity contours within the full-wake region from Cases 5-7. Meanwhile, Case 4 was excluded from contour mapping due to the limitation of measurement data points, which could affect the contour mapping accuracy. By referring to the velocity colorbar, the DG wake profile in Case 5 can be observed from the LES result and its analytical prediction until x/D ≈ 2.5. Afterward, the wake velocity gradually recovered and transformed into the SG profile at further downstream distances. It was found that the maximum velocity deficit was located around the blade mid-span position.

| Contour of streamwise wake velocity
The same wake transformation behavior was found in Case 6 but with faster recovery. Velocity deficit around the blade mid-span was observed to reduce gradually after x/D ≈ 2. Meanwhile, a prominent velocity deficit could not be found in Case 7, either from the LES result or its analytical prediction. Thus, causing the wake velocity around the blade mid-span to the centerline was less varying. This behavior might indicate that the DG profile only took place at a distance just behind the turbine before it eventually merged into the SG profile and quickly recovered at the farther downstream positions.
The discussion above gives a qualitative judgment regarding the consistency between the LES results and their analytical predictions using the DG model with the proposed expansion function for the streamwise velocity field prediction within the full-wake region. Relatively low incoming TI x,hub as in Case 5, resulted in a slow wake recovery. Meanwhile, the wake recovered faster with a relatively moderate incoming TI x,hub , as in Case 6. The high incoming TI x,hub in Case 7 caused the wake to recover faster than the previous Cases 5-6. These results reconfirms the previous numerical investigation, which showed that the HAWT wake recovers more rapidly under higher turbulence inflow. 32

| The wake width and centerline velocity deficit
In the present study, the proposed expansion function is compared against the existing expansion function for the DG wake model. 30 The existing function σ w [D] is expressed as follows: The parameter k*, which controls the wake expansion rate, is tuned by case. Here, fminsearchcon MATLAB function 44 was employed to find the k* values that give the best fit with the benchmark data sets. After applying the DG model described in Table 1, the tuned k* values that best fit the benchmark data sets are shown in Table 3.
The estimated far-wake onset values (x 1 /D) for Cases 4-7 in Table 3 were also used by the proposed expansion function. The comparison of wake expansion between the proposed and tuned expansion functions against the benchmark data is shown in Figure 6.
In general, the proposed expansion function could reasonably estimate the wake expansion with relatively small residuals against the benchmark data. The proposed and tuned function predictions for σ and r w in Cases 4-6 were nearly identical. However, noticeable differences were found in Case 7, where the wake radius resulting from both expansion functions deviated from the LES result, mainly at x/D > 6. The σ and r w from the LES result were observed to shrink, resulting in a curved expansion. This occurrence may be related to the wake contraction, which is not accounted for in the present DG wake model.
The wake expansion is related to the wake recovery. Thus, the expansion function of the DG wake model has a significant influence on the accuracy of the wake recovery prediction. Figure 7 compares the normalized velocity deficit at the centerline ΔU c /U 0 predicted by the DG analytical model using both the proposed and tuned expansion functions.
The predictions of ΔU c /U 0 from both the expansion functions are in good agreement with most of the benchmark data sets. The centerline velocity deficit ΔU c /U 0 gradually diminished as the downstream distance increased, representing the wake recovery at the selected distances. Sampled at the downstream position within the far-wake region where the flow is already stable (x/D = 8), it was found that the centerline velocity deficit ΔU c /U hub for the Cases 4-7 with the respective incoming TI x,hub of 6.25%, 5.1%, 8.5%, and 16% were about 0.3, 0.325, 0.225, and 0.15, respectively. As expected, Case 7, with the highest incoming TI x,hub had the lowest velocity deficit at the distance of interest. In contrast, Case 5, with the smallest incoming TI x,hub , gave the maximum ΔU c /U hub among the other evaluated Cases. These results confirm the strong dependency of the wake recovery rate on the incoming turbulence intensity.
From the evaluated Cases, the proposed expansion function could give comparable performance against the tuning results. For Cases 5 and 6, both the expansion functions provided almost identical results at all the selected downstream distances. The highest residuals of ΔU c /U hub between the expansion functions and the benchmark data were observed up to x/D = 2 for most of the evaluated Cases. Afterward, the analytical model's accuracy improved for all Cases 4-7.
T A B L E 3 Far-wake onset and tuned k* parameter for Cases 4-7.
Input parameters

| Streamwise wake velocity
The proposed function's performance within the fullwake region was evaluated using four benchmark data sets of streamwise wake velocity profiles at the hub height in the lateral direction resulting from the wind tunnel measurements (Case 4) and the LESs (Cases 5-7). In addition, the best-fit results from the tuned expansion function were also included for comparison. The evaluation was specifically intended to examine the performance of the proposed function, where its similarity with the fitting results could also represent its reliability. For Case 4, the measurement results at the selected downstream distances and their analytical predictions are shown in Figure 8. Salient residuals of the proposed function's prediction against the measurement data were observed at the downstream distances of x/D = 1.7 and 2, notably around the blade mid-span. Meanwhile, the tuned function better estimated the wake profiles at the same locations. At x/D = 3, the accuracy of the proposed function significantly improved, which was confirmed by its good agreement with the tuning result and the reference data. In addition, the DG distribution can still be observed clearly at x/D = 3. At further distances of x/D = 4 and 6, the wake predictions from both the expansion functions were almost identical and tended to form the top-hat shape around the blade mid-span to the centerline. Nevertheless, their accuracies were still relatively high, around 96%, referring to the measurement data. At x/ D = 9, both expansion functions fully transformed into the SG shape, and the proposed function outperformed the tuning result.
The LES results for Case 5 and their analytical predictions are shown in Figure 9. The wake expansion predictions from the proposed and tuned functions at all the selected distances were quantitatively similar.
The DG distributions were observed clearly from the LES and analytical results at the downstream distances of x/D = 1.5 and 2. However, noticeable differences were found mainly within the blade mid-span area, where the DG model predictions from both the expansion functions  F I G U R E 9 Wake velocity profiles at the hub height in the lateral direction behind an INNWIND 10 MW reference turbine within the full-wake region (Case 5). underestimated the LES results. Moreover, the LES results produced asymmetric distributions, where the wake profiles in the negative lateral direction recovered faster than in the opposite direction. These appearances could not be reproduced by the analytical model due to the axisymmetric approach being used.
Meanwhile, better wake profile prediction and recovery were obtained as the downstream distance increased. At x/D = 3, a relatively small discrepancy between the LES result and its analytical predictions were still observed within the blade mid-span area but were less significant. The LES result formed a top-hat profile within this area before eventually transforming into an SG profile at x/D = 4. In general, the accuracies of both the expansion functions for the wake profile predictions were further improved at the farther downstream distances. At x/D ≥ 6, the estimated velocity profiles have fully transformed into the SG distribution, causing accuracy improvement in velocity deficit estimations at the centerline.
The LES results and their analytical predictions in Case 6 are shown in Figure 10. At all the selected distances, the wake profiles from the proposed function almost matched perfectly with the tuning results. However, it was observed that both expansion function predictions underestimated the LES results at x/D = 1.5 and 2. The major difference was observed between the blade mid-span and the centerline position, particularly in the negative lateral direction where the LES results showed faster wake recovery, thus creating asymmetric wake profiles. At x/D = 3, the LES result tended to form a top-hat profile around the wake center, while the analytical predictions remained in the DG shape profile. At the distance of x/D = 4, less residuals were observed around the same position.
The wake profiles predicted by the DG model eventually transformed into an SG shape starting from x/D = 5, thus reducing the residuals against the LES results. Within 6 ≤ x/D ≤ 12, the wake center gradually shifted around y = 0 toward the positive lateral direction.
F I G U R E 10 Wake velocity profiles at the hub height in the lateral direction behind an INNWIND 10 MW reference turbine within the full-wake region (Case 6). This occurrence was strongly related to the wake meandering, which made random oscillations with respect to the wake trajectory. Distinct velocity fluctuations were also observed, mainly at the lateral directions of y/D ≤ −1. However, the DG model with the proposed expansion function was still relatively good in predicting the average behavior of the wake profile, especially within the far-wake region.
The LES results for Case 7 and their analytical predictions are compared in Figure 11. At the downstream distance of x/D = 1, the DG model, particularly by using the proposed function, underestimated the velocity profile produced by the LES around the blade mid-span to the centerline position. In addition, the wake fluctuation within −1.2 ≤ y/D ≤ −0.7 from the LES result increased the analytical model residuals against the LES data. At the farther distance of x/D = 1.5, the prediction accuracy from the analytical model improved, where both the expansion functions accurately predicted the wake profile around the centerline. However, the velocity fluctuation still occurred in the LES result with lower intensity.
At the downstream distance of x/D = 2, the wake profiles predicted by both expansion functions almost coincide. The profiles formed a top-hat shape around the blade mid-span to the centerline position and slightly overestimated the LES result. The same behavior was also observed at x/D = 2.5 but with lesser residuals. In addition, an obvious transition of the wake profile from top-hat to fully SG shape was observed in the LES result, which indicated a faster wake recovery, particularly compared with the other evaluated Cases. At the farther downstream distances, discrepancies between the LES results and their analytical predictions around the centerline became narrower, thus increasing the analytical model accuracy at the remaining selected distances.
The wake profiles predicted by the DG model with both expansion functions have fully transformed into SG shape at x/D ≥ 3, yielding better fitness with the LES data. By looking in more detail within 5 ≤ x/D ≤ 8, the F I G U R E 11 wake velocity profiles at the hub height in the lateral direction behind an NREL 5 MW reference turbine within the fullwake region (Case 7).
proposed function outperformed the tuning results by giving better accuracy of the wake recovery predictions within the rotor swept area. In general, the ability of the analytical model to predict the general characteristics of the wake velocity profile and its recovery for the evaluated Cases 5-7 is relatively accurate, particularly when considering the huge distinction of computational resources between computational (LES) and analytical approaches.

| Statistical evaluations
In this section, statistical measures are used to quantify the effectiveness of the proposed expansion function for the evaluated Cases. The normalized mean square error (NRMSE) was used to measure the residuals from the evaluated expansion functions against the benchmark data sets. Meanwhile, the R linear correlation coefficient was used to measure a linear dependence between the benchmark data sets and their analytical predictions. It must be stated that these statistical measures only provide a general correlation between the model and the benchmark data from the sampled points. Those statistical measures are formulated as follows: where E denotes the analytical estimation either using the proposed or tuned function, B denotes the benchmark data, subscript i denotes the ith point from the total N points, and the overbar ( -) means an average from all N points. Meanwhile, σ B and σ E refer to the standard deviation from the analytical and benchmark data sets, respectively. The NRMSE and R, as formulated in Equations (18) and (19), were measured at each selected downstream distance. There were 22 points from −0.82≤ y/D ≤ 0.82 for Case 4, 201 points from −2.12 ≤ y/D ≤ 2.12 for Cases 5-6, and 121 points from −2 ≤ y/D ≤ 2 for Case 7, measured at each selected distance for NRMSE and R analyses. A perfect model would result in NRMSE and R values of 0 and 1, respectively. It is worth noting that the total number of points (N) used in each Case for these statistical evaluations varied according to the data availability. The comparison of NRMSE and R between the proposed and tuned expansion functions for the evaluated Cases 4-7 is shown in Figure 12.
For Case 4, NRMSEs of the DG model with the proposed function were higher than the tuning results at x/D = 1.7 and 2. These errors occurred due to a significant underestimation of the wake velocity, particularly around the blade mid-span area. In contrast, the R coefficients for the proposed function were highly correlated and remained high at the evaluated distances. This condition happened due to the high linearity between the analytical and measurement results. Meanwhile, within the evaluated distances of 3 ≤ x/D ≤ 9, the accuracy of the proposed function significantly improved where the average NRMSE ≈ 0.03. The same trend applied for the tuned function, except at the distance of x/D = 9 with an NRMSE of about 0.06 higher than the proposed function's NRMSE.
In Case 5, the results of NRMSE and R obtained from both expansion functions are in close agreement for all distances considered. Prominent NRMSEs from both functions were observed within the distances of x/D ≤ 2 and significantly reduced to less than 0.03 at the remaining selected distances. Meanwhile, the R coefficients of wake shape distribution between expansion function predictions and the LES results show high linearity at all selected distances.
In Case 6, no significant discrepancies between the expansion functions were observed in NRMSEs and R results. The highest NRMSEs from both the functions against the benchmark data sets were approximately 0.085 at the distance of x/D = 1.5, with relatively high R coefficients of about 0.98. At the distances of x/D = 2 and 3, the NRMSEs from both functions were ≈0.055 and reduced further to about 0.04 at the remaining selected distances. Besides, the R correlations from both the expansion functions within 2 ≤ x/D ≤ 5 were approximately 0.98 and substantially decreased to about 0.82 at x/D = 12. This steep reduction was caused by the fluctuation of the wake profiles and shifting of the wake center in the LES results.
In Case 7, the highest residual from both expansion functions was observed at x/D = 1, where the proposed function's NRMSE was about 0.035 higher than the tuning result. Their NRMSEs at the remaining selected distances almost coincide within the range 0.045-0.055. Meanwhile, the R coefficients resulting from both expansion functions were relatively high at all the selected distances. The amplitude of R coefficient fluctuation was found to increase with the downstream distance. It should be noted that the R coefficients obtained from the proposed function were higher than the tuning results at x/D ≥ 5. This was consistent with the NRMSEs within the same distances where the proposed function predictions for the wake profile were more accurate than the tuned function.

| Comparison of wind power potential
In a wind farm, energy potential from the wake velocity behind the upstream turbines is extracted by the respective downstream turbines for power generation. By referring to the wind energy theory, wind power varies as the cube of the wind velocity. Therefore, the validity of the power prediction of the downstream turbine is strongly determined by the prediction accuracy of the upstream turbine wake. The wind power potential within the wake region P w is calculated using the following formula: where ρ is the air density and A 0 denotes the rotor swept area. In this study, the streamwise wake velocity at the hub height spanned in the lateral direction within the rotor swept area (−0.5 ≤ y/D ≤ 0.5) was averaged to represent the potential wake velocity U p,w, thus the wind power potential P w at each evaluated downstream distance can estimated using Equation (20). The same method was applied to calculate the reference wind power potential P ref,w resulting from the reference potential wake velocity U ref,p,w within the rotor swept area A 0 . The practical region, 38 located within the downstream distances of 2.4 ≤ x/D ≤ 11, covers the range of actual HAWTs micro-siting in both onshore and offshore wind farms. Thus, it was considered in this study as the region of interest to be evaluated. The normalized wind power potential P w /P ref,w from the benchmark data sets and the DG model using both expansion functions within the downstream distances of 2.4 ≤ x/D ≤ 12 were calculated and compared in Figure 13.
In Case 4, four downstream distances ranging from 3 ≤ x/D ≤ 9 were selected. Compared to the tuned function, the proposed function provided a better prediction of wind power potentials at x/D = 3 and 9 with P w /P ref,w of about 0.96 and 0.97, respectively.
F I G U R E 12 Comparison of the normalized root mean square error NRMSE and the linear correlation coefficient R between the proposed and the tuned expansion functions.
Meanwhile, the worst performance was found at x/D = 4, where both expansion functions resulting almost the same P w /P ref,w of about 1.09, which was approximately 0.09 higher than the reference value.
In Case 5, results from the proposed function were similar to the tuning results within all selected distances. At x/D = 3, residuals of about 0.11 lower than the reference value were obtained from both expansion functions. As the downstream distance increased, the performance of both expansion functions improved with the residuals below 0.04 against the reference values. Meanwhile, the highest accuracies from both expansion functions were found at x/D = 12, where their predictions were almost identical to the reference data.
For Case 6, predictions of P w /P ref,w by the proposed function were also in good agreement with the tuning results. The highest residuals from both expansion functions were about 0.19 higher than the reference value at x/D = 3 and significantly reduced to approximately 0.07 at x/D = 4. It was also observed that the predictive ability improved at the distances of interest within the far-wake region. The accuracy of both expansion functions at the distances of x/D ≥ 5 was almost constant. The average P w /P ref,w was approximately 0.94, except at

| The performance comparison of the evaluated Cases 4-7
The result of mean values of NRMSE, R coefficient, and ΔP w /P ref,w within the full-wake and practical regions are summarized in Tables 4 and 5, respectively. For all the selected distances in Cases 4-7, the tuned function resulted in a NRMSE of 0.046, which was better than the proposed function's NRMSE of 0.053. It was mainly due to higher residuals from the proposed F I G U R E 13 Comparison of the normalized wind power potential P w /P ref,w between the analytical predictions and their benchmark data sets.
function's prediction at the downstream distances near the turbine. However, within the practical region, the proposed function could give a NRMSE of 0.036, slightly better than the tuned function's NRMSE of 0.037. Meanwhile, R coefficients of the proposed function were better when compared to the tuning results in both the full-wake and practical regions.
Within the full-wake region in Cases 4-7, both the expansion functions yielded relatively high residuals for the mean normalized power difference P P Δ /

| CONCLUSION
A linear wake expansion function for the DG wake model is proposed to estimate the wake expansion and recovery behind a non-yawed HAWT without the expansion parameter tuning. Seven case studies from the LESs and wind tunnel measurements of non-yawed HAWT's wake under a variety of turbulent inflow conditions were used to examine the effectiveness of the proposed function. For the near-wake analyses in Cases 1-3, the proposed function could give rough but reasonable estimations of the DG wake profile expansion and its recovery from the high-fidelity LES results. For the full-wake analyses in Cases 4-7, the existing DG expansion function with a tunable parameter for the wake expansion was also added for comparison. In general, the full-wake predictions from the proposed and tuned functions for standard deviation σ and the centerline velocity deficit ΔU c /U 0 were in good agreement with those of the benchmark data sets. As a result, both expansion functions could reasonably predict the wake expansions and their recoveries at most of the selected distances. It was observed that both expansion functions underestimated the wake velocity near the turbine, mainly due to the strong influence of the near-wake turbulence, which is not considered in the current analytical DG model. The performances of both expansion functions for the evaluated Cases 4-7 were evaluated and compared using the NRMSE and R coefficient statistical measures. For all selected downstream distances in the full-wake region, the tuned function resulted in a mean NRMSE of 0.046, T A B L E 5 The mean values of NRMSE, R coefficient, and ΔP w /P ref,w within the practical region. which is lower than the proposed function's mean NRMSE of 0.053. It was mainly due to higher residuals from the proposed function's prediction at the downstream distances near the turbine. However, within the practical region, the proposed function could give a mean NRMSE of 0.036, slightly better than the tuned function's mean NRMSE of 0.037. Meanwhile, the mean R coefficients of the proposed function were better when compared to the tuning results in both the full-wake and practical regions. Those conducted statistical analyses could represent the proposed function's effectiveness in providing a reasonable estimation of the wake expansion and its recovery under a variety of inflow conditions in Cases 4-7. Moreover, both the expansion functions could provide the accuracies of mean P w /P ref,w predictions above 92%. Specifically, the proposed function yielded a mean ΔP w /P ref,w of 0.066, which was more accurate than the tuning function prediction with a mean ΔP w /P ref,w of 0.077. These results become evidence regarding the feasibility of the proposed expansion function to give a reasonable prediction of P w /P ref,w .

Case
For future works, the DG model with the proposed function could be combined with the available wakemerging methods to predict the power of clustered HAWTs in a wind farm. Meanwhile, the proposed function could be extended to include the anisotropic nature of the wake expansion, thus allowing its usability under different atmospheric stabilities. In addition, more data from high-fidelity LESs of HAWT wakes under varying inflow conditions and atmospheric stabilities would be necessary for further evaluations of analytical wake modeling.