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# Leading effect for wind turbine wake models

Ingrid Neunaber, Michael Hölling, Martín Obligado

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Ingrid Neunaber, Michael Hölling, Martín Obligado. Leading effect for wind turbine wake models. Renewable Energy, 2024, 223, pp.119935. 10.1016/j.renene.2023.119935. hal-04860971

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# Leading effect for wind turbine wake models

Ingrid Neunaber<sup>a,*</sup>, Michael Hölling<sup>b</sup>, Martín Obligado<sup>c</sup>

## ARTICLE INFO

### Keywords:

Wind turbine wake

Virtual origin

Wind tunnel

Experiments

## ABSTRACT

As wind energy expands worldwide, the demand of reliable, fast, cost-efficient wind turbine wake models is growing. This is a significant challenge as wind turbines face various inflow conditions that include turbulence, inhomogeneities/instationarities and upstream wakes. In consequence, an enormous number of engineering models, each one based on different physical concepts, has been proposed. The majority focuses on the far wake where the mean velocity recovers and turbulence decays after it built up. We argue that the most important, or the leading, parameter for wake modeling is the length scale of a virtual origin. Testing different models from the literature for data sets from laboratory wind turbines and multi-megawatt turbines obtained by LiDAR, we find that all models perform significantly better when such a virtual origin is added. Our results can therefore be used for a yet missing definition of a near wake zone.

## 1. Introduction

Wind is considered as an important renewable energy source for a future sustainable energy supply of our society [1–3]. For economic reasons, wind turbines, more technically speaking, wind energy converters, are mainly built in large farm configurations. For the energy yield, for planning, as well as for operation (e.g. [4]), the interaction of wind turbines by means of wind-reducing wakes that occur downstream of all operating wind turbines is a major concern. A profound knowledge of the wind speed reduction by wind turbines standing in front of another is essential. In consequence, in the wind energy research community, the understanding of the velocity deficit downstream of a wind turbine is one of the most attended topics, as can be seen for example by the number of competing wake models.

The wake of a wind turbine is classically divided into two major regions, namely the *near wake* and the *far wake* (e.g. [5,6]; while other authors mention up to four regions [7,8]). In the near wake, the tip and root vortices that are shed from the rotor blades, and the vortices shed from the tower and nacelle dominate the wake (Fig. 1(a)). These structures decay and the shear layers surrounding the wake expand while turbulence builds up. Once they meet, around 2–4 rotor diameters downstream of the turbine, the far wake region starts [9–11]. It is characterized by a recovery of the velocity in the wake, an approximately Gaussian velocity deficit [9], and decaying turbulence that was shown to have strong features of homogeneous, isotropic turbulence [7]. The evolution of the wake depends heavily on the

ambient conditions (e.g. [7,12–14]). It is the far wake that the vast majority of wind turbine wake models account for [9,10,15].

### 1.1. Wind turbine wake modeling

The evolution of the mean velocity deficit is the most studied quantity in the wake of a wind turbine, followed by the added turbulence intensity and the entrainment of turbulent kinetic energy (e.g. [9–11]). The aim of wind turbine wake models is a description of the evolution of the mean velocity deficit, and in some cases the turbulence intensity, in the wake based on available quantities such as the inflow averaged streamwise velocity $U_∞$, the inflow averaged turbulence intensity $TI = \sigma/U_∞$ (where $\sigma$ denotes the standard deviation of the streamwise velocity) and the thrust coefficient of the wind turbine $c_T$ that indicates the force that the wind turbine exerts in the axial direction to the incoming momentum of the flow. Particularly, in the far wake, simplifying mathematical assumptions can be made, such as self-similarity, the universality of some statistics, etc. Analytical wind turbine wake models are often derived from governing equations and basic assumptions, typically including conservation of mass and momentum.

Among these wake models, only few were adopted and are frequently used (for a more complete overview of different wind turbine wake models, the reader is referred to e.g. [9,15–17]). We remark that, while they all provide a functional form for the centerline velocity

<footnote><sup>a</sup> Department of Energy and Process Engineering, Norwegian University of Science and Technology, Kolbjørn Hejes vei 1B, Trondheim, 7034, Norway</footnote>

<footnote><sup>b</sup> Institute of Physics and ForWind, University of Oldenburg, Küpkersweg 70, Oldenburg, 26129, Germany</footnote>

<footnote><sup>c</sup> Grenoble INP, CNRS, LEGI, Université Grenoble Alpes, 1209 Rue de la Piscine, Gières, 38610, France</footnote>

<footnote>* Corresponding author.
E-mail addresses: ingrid.neunaber@ntnu.no (I. Neunaber), michael.hoelling@uol.de (M. Hölling), martín.obligado@centralelille.fr (M. Obligado).</footnote>
<figure><img src="image_5.png" /><figcaption>Fig. 1. (a) Schematic of the turbulent wake of a wind turbine, showing the near and far wakes. (b) Sketch of the experimental setup used for the wind tunnel tests in this work.</figcaption></figure>

deficit, some allow to obtain also the wake width via different relations (momentum conservation, a hypothesis on the wake profile shape, etc.). Such models will therefore provide the main relevant information for the wake's averaged quantities.

However, when applying these wind turbine wake models to wake measurements from field campaigns or wind tunnel measurements, very different results are obtained (e.g. [18–20]). Despite this discrepancy, in this work, we show by means of data analysis, that some commonly used models, in principle, still appropriately capture the decay of the velocity deficits in the recovering wakes.

Finally, we note that, while a virtual origin arises naturally in the analytical solution for the bluff body wake ([21,22]; see Section 1.2), almost no wind turbine wake model considers the existence of a virtual origin. Similar concepts include the concept of the “initial wake width”, that could also be interpreted formally as a virtual origin, though with physical limitations [23], and a “streamwise offset for the reference frame” that is added by the models proposed (or, rather, updated) in [24,25].

## 1.2. Virtual origin

Within the framework of the Townsend-George model, the analytical description for bluff body wakes, the virtual origin arises naturally from the governing equations (see [21,22] for the original description and [26] for its extension to a non-Kolmogorov energy cascade). It has recently been shown that the wake of a wind turbine fulfills the requirements necessary to apply it [27], and several works that apply this phenomenology to wind turbine wakes found a better fit compared to engineering wind turbine wake models [27–30].

The concept of a virtual origin can be related to the integration constant that arises in most engineering wind turbine wake models after integrating the streamwise momentum equation. By adding extra hypotheses, such as the applicability of actuator disk theory and the validity of the selected model at $x \le 0$, a virtual origin can be obtained and estimated (see for instance [31]).

This work proposes a simpler approach, where the virtual origin is added as an *ad hoc* quantity to selected models and is fitted to different data sets. The objective is therefore to unambiguously show that the use of such length scale will significantly improve our predictive capacity for wind turbines operated in realistic conditions. The mathematical formulation or the physics that control this value remain out of the scope of this work and should be the topic of future studies.

Further, with the present work, we want to draw attention to another important and not well investigated effect, that is, the near wake region directly in the wind shadow of the wind turbine’s rotor.

While it has been shown that the behavior of the far wake is, in principle, independent of the inflow turbulence conditions, the turbine operational state and even the wake generating object (i.e., a wind turbine and a static porous disc produce similar far wakes), in the near wake region, turbulence builds up, depending on the inflow conditions etc., until a developed turbulent state is reached from where the decay takes place. We show that it is the not well understood near wake behavior which leads to major differences in common wake models, e.g. [32], particularly since the extension of the near wake dynamically varies with the turbulence intensity, atmospheric stability and turbine thrust. To overcome this problem, we introduce the virtual origin also as a dynamic length scale for the near wake. This quantity is well known in the field of turbulence, e.g. bluff body wakes [26,33] and grid-generated turbulent flows [34–36]. Based on analyzed data we show evidence that this length parameter is the leading term to improve the wake models and, thus, the description of the wind deficit even in quite far distances behind a wind turbine.

The virtual origin can be seen therefore as the leading term to quantify the extent of the near wake. Previous works have addressed this quantity in different conditions, that include wind tunnel experiments [30,37,38] and field data [27]. Further, in a similar manner, a near wake length is defined in [39], which collapses the centerline mean velocity deficit for different cases obtained by large eddy simulations. Nevertheless, no study has been performed to highlight the relevance of such parameter for wind energy applications. To the authors best knowledge, there is neither a systematic study where different models, inflows and operating conditions are investigated simultaneously to test the relevance of such length scale.

In summary, the implementation of a virtual origin is, within the framework discussed here, a new approach relevant to the wind energy community:

* The virtual origin is not empirical like parameters of wind turbine wake models but arises from governing equations;

* The virtual origin takes into account the production and the decay of turbulence;

* The virtual origin takes into account the finite-size nature of the wake generator;

* The virtual origin takes into account the inertial conditions from a physical point of view;

* Contrary to the structure of most wind turbine wake models, the virtual origin combines inertial conditions in a single parameter.

* As illustrated in Fig. 1(a), the virtual origin allows to shift the wind turbine wake model and thus takes into account the onset of the far wake, adapting the model to the surrounding conditions.
In the following, the virtual origin will therefore be added to all other models as an ad hoc assumption to quantify, at first order, the influence of the near wake and the production of turbulence (as it is already the case in the Townsend-George phenomenology and also when studying the evolution of turbulence downstream of different grids, e.g. [34,36,40]).

### 1.3. Structure of the paper

This work is therefore a systematic study that aims at showing how selected available wind turbine wake models can be significantly improved, no matter the working conditions of the wind turbine, by the addition of a virtual origin. To justify our claim, we first briefly describe the six models that will be checked in this work, and their main hypotheses and governing equations can be found in Section 2. We then compare the performance of these wake models in their original form to modified versions where a virtual origin is added. Four different data bases are used to study different wake conditions. First, experiments performed by us in a wind tunnel with small models of wind turbines in different inflow conditions (Fig. 1b) are described in Section 3.1. Additionally, Light Detection And Ranging (LiDAR) measurements of the wakes behind real multi-megawatt wind turbines from three different measurement campaigns that are available in the literature are introduced in Section 3.2. This data base is used to compare the selected wake models and discuss the impact of adding a virtual origin on the performance of the wake models in Section 4. Finally we work out that the most important or the leading parameter for wake modeling is the length scale of a virtual origin, cf. Section 6.

## 2. Wind turbine wake models

In the following, the wake models investigated here are briefly introduced. In total, six models are introduced, four commonly used wind turbine wake models and the two Townsend-George models for free shear flows. While many wind turbine wake models are available, in this study, we selected four examples that have often been used in the scientific community for comparisons (in different combinations e.g., [16,18,20,38]).

### 2.1. Townsend-George model for free shear flows

From the assumptions of self-similarity of the one-point turbulence quantities and high-Reynolds number turbulence decaying in equilibrium as postulated in [22,41,42] derive the recovery of the normalized mean velocity deficit of the far wake of a bluff body

$$ \frac{\Delta U}{U_{\infty}} = T_{-2/3} \cdot ((x - x_0))^{-2/3}. \qquad (1) $$

Here, $\Delta U = U_{\infty} - U_0$ denotes the centerline velocity deficit at downstream position x, $U_{\infty}$ is the inflow velocity, $T_{-2/3}$ is a scaling parameter, and $x_0$ is a virtual origin. We will treat both $T_{-2/3}$ and $x_0$ as fitting parameters. In addition, this model allows for the calculation of the wake width $\delta$,

$$ \delta(x) = B_{-2/3} \cdot ((x - x_0))^{1/3}, \qquad (2) $$

where $B_{-2/3}$ gives the wake growth.

However, it has also been shown by [22] that a second scaling exists,

$$ \frac{\Delta U}{U_{\infty}} = T_{-1} \cdot ((x - x_0))^{-1}, \qquad (3) $$

$$ \delta(x) = B_{-1} \cdot ((x - x_0))^{1/2}, \qquad (4) $$

where $T_{-1}$ is a scaling parameter and $B_{-1}$ is the wake growth.

### 2.2. Jensen wake model

The Jensen wake model is a pioneering wake model proposed by N.O. Jensen in 1983 [43]. It assumes a top-hat shaped velocity deficit and is derived by means of conservation of mass. The normalized velocity deficit is given by

$$ \frac{\Delta U}{U_{\infty}} = \frac{1 - \sqrt{1 - c_T}}{(1 + 2k_j x / D)^2}. \qquad (5) $$

$c_T$ is the turbine's thrust coefficient. A linear expansion of the wake is assumed, and it is described by the wake growth rate $k_j$. Onshore, $k_j = 0.075$ is suggested whereas offshore, where the wake recovery is slower due to lower turbulence levels, $k_j = 0.04$ or $k_j = 0.05$ is assumed [15,44].

Here, we will investigate the model's performance by (a) using its original form without fitting (i.e., literature values are used for $k_j$), denoted as “no fit”; (b) treating $k_j$ as a fitting parameter, denoted as original, “orig.”; (c) treating $k_j$ as a fitting parameter while also replacing $x$ with $x - x_0$ (and therefore $x_0$ is also a fitting parameter), thus adding a virtual origin, denoted “VO add.”. Also, we show in the Appendix C how the modeling is improved by adding a virtual origin as fitting parameter even when using literature values for $k_j$.

While there are several improvements to the original model, see e.g. [18] for an overview, in this study, we use the original version.

### 2.3. Porté-Agel wake model

From conservation of mass and momentum, [45] derived a model for the description of the three-dimensional far wake of a wind turbine assuming a Gaussian shape of the normalized velocity deficit,

$$ \frac{\Delta U}{U_{\infty}} = \underbrace{\left(1 - \sqrt{1 - \frac{c_T}{8 \left(k_{BP} \cdot x / D + 0.2 \sqrt{\beta}\right)^2}}\right)}_{\text{centerline velocity deficit}} \times \underbrace{\exp\left(-\frac{\left((z - z_h) / D\right)^2 + (y / D)^2}{2 \left(k_{BP} \cdot x / D + 0.2 \sqrt{\beta}\right)^2}\right)}_{\text{Gaussian velocity profile; 0 for $(z - z_h) = y = 0$}} \qquad (6) $$

with

$$ \beta = \frac{1 + \sqrt{1 - c_T}}{2\sqrt{1 - c_T}}; \qquad (7) $$

$z_h$ denotes the hub height of the turbine and $y$ and $z$ denote the span-wise and wall-normal coordinates, respectively. $k_{BP}$ is a tunable parameter that can be calculated with the equation from [9],

$$ k_{BP} = 0.38 \cdot TI + 0.004. \qquad (8) $$

Here, we will investigate the model's performance by (a) using its original form without fitting (i.e., literature values are used for $k_{BP}$), denoted as “no fit”; (b) treating $k_{BP}$ as a fitting parameter, denoted as original, “orig.”; (c) treating $k_{BP}$ as a fitting parameter while also replacing $x$ with $x - x_0$ (and therefore $x_0$ is also a fitting parameter), thus adding a virtual origin, denoted “VO add.”. Also, we show in the Appendix C how adding a virtual origin as fitting parameter, while using literature values for $k_{BP}$, improves the modeling.

The model was modified later to also describe the wake of a yawed wind turbine [49].
<figure><img src="image_6.png" /><figcaption>Fig. 2. Photographs of the wind tunnel setup. Model wind turbine (from [46]) (a), front view of the model wind turbine in the wind tunnel with hot-wire array in the background (b), and view from the back, showing the traverse with the hot-wire array (c).</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 3. Setup for LiDAR measurements at the wind farm Alpha Ventus taken from [47] (CC BY-NC-ND 4.0 [48]): Measurements were carried out in the German bay at the wind farm Alpha Ventus (a). For measurements presented by Krishnamurthy et al. [47], a LiDAR was placed at the FINO 1 platform (b), and the PPI scan pattern is shown in (c).</figcaption></figure>

## 2.4. Aitken wake model

With a simple power law approach applied to various data sets, Aitken et al. derived the following relation [9,50] for the normalized centerline velocity deficit,

$$ \frac{\Delta U}{U_{\infty}} = 0.56 \left( \frac{x}{D} \right)^{-0.57} . \qquad (9) $$

Note that this model has no tunable parameter. It will therefore be used without fitting (referred to as “no fit”); by adding a virtual origin, this model will have one fit parameter, namely $x_0$ (referred to as “VO add.”).

## 2.5. Frandsen wake model

Similar to the Jensen wake model, the Frandsen wake model assumes a top-hat shaped velocity deficit in the wake [23]. It is derived using conservation of mass and momentum, and we use the simplified version suggested by the authors in agreement with their findings for an infinite row of two-dimensional obstacles, where the normalized velocity deficit is given by

$$ \frac{\Delta U}{U_{\infty}} = \frac{1}{2} \left( 1 - \sqrt{1 - \frac{2c_T}{\beta + k_F x/D}} \right), \qquad (10) $$

where $\beta$ is given by Eq. (7). For small $c_T$, the authors state that $k_F = \Theta(10 \cdot k_j)$. Here, we will investigate the model’s performance by (a) using its original form without fitting (i.e., literature values are used for $k_F$), denoted as “no fit”; (b) treating $k_F$ as a fitting parameter, denoted as original, “orig.”; (c) treating $k_F$ as a fitting parameter while also replacing $x$ with $x - x_0$ (and therefore $x_0$ is also a fitting parameter), thus adding a virtual origin, denoted “VO add.”. Also, we show in the

Appendix C how adding a virtual origin as fitting parameter, while using literature values for $k_F$, improves the modeling.

# 3. Methods

## 3.1. Experimental setup

The measurements presented here were carried out in the wind tunnel of the University of Oldenburg. The closed-loop wind tunnel has an inlet of 3 × 3 m² and a test section with a length of 30 m [53]. The setup is presented in Fig. 1(b), and the origin of the coordinate system is at the center of the turbine rotor. Experiments were carried out using a three-bladed horizontal axis model wind turbine of type MoWiTO 0.6 (cf. [46]) with a control system described in [54]. It has a rotor diameter of $D = 58$ cm and a thrust coefficient of $c_T = 0.70$ for optimal operation conditions at tip speed ratio $TSR \approx 6.5$. The blockage of the turbine with additional support is below 5%. When operating in the so-called low-turbulent conditions (i.e., with the rotor placed in an empty test section), the turbine is exposed to a low turbulent inflow conditions with a velocity of $U_{\infty} = 8.3$ ms⁻¹ and turbulence intensity of $TI = 0.3\%$ (and therefore a Reynolds number $Re_D = U_{\infty}D/\nu = 3.2 \times 10^5$, with $\nu$ the kinematic viscosity of air). The turbulent inflow is generated by means of a passive grid with a velocity of $U_{\infty} = 6.9$ ms⁻¹ and a turbulence intensity of $TI = 2.0\%$ ($Re_D = 2.7 \times 10^5$). To characterize the mean velocity at the centerline of the rotor $U_0$ (i.e., $Y = Z = 0$), a 1d hot-wire installed on a rack was traversed between $x/D = 1$ and $x/D = 20$ in steps of 1D, cf. Fig. 2. It was operated with a Dantec Dynamics StreamLine 9091N0102 frame with a 91C10 CTA (Constant Temperature Anemometry) Module. The hot-wire was sampled at $f_s = 20$ kHz. A hardware low-pass filter was set with a cut-off frequency of $f = 10$ kHz. The sampling time was 240 s for all cases.

A summary of the inflow conditions can be found in Table 1.
<figure><img src="image_8.png" /><figcaption>Fig. 4. Setup for LiDAR measurements taken from [51] (CC BY 3.0 [52]): Measurements were carried out in the north of France (left) at a wind farm with seven wind turbines, and a LiDAR scans the area marked in white. The wakes of turbine SMV5 and SMV6 are measured.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 5. Downstream evolution of the normalized mean velocity deficit for wind tunnel experiments in low-turbulent inflow. (a) shows the fitted curves of the six wake models (solid lines; $k$ was fitted for the wind turbine wake models) as well as the predictions of four wind turbine wake models where all parameters, i.e., $c_{f}$, $T1$ and $k$, are extracted from the literature (dotted lines). (b) A virtual origin was added as additional fit parameter in the equations for the wind turbine wake models.</figcaption></figure>

## 3.2. Field measurements

Data from three field campaigns, [47,55,56], carried out using LiDAR (e.g. [57,58]) is utilized here. This technology is capable of measuring the line of sight velocity at multiple distances nearly simultaneously, and the wind speed at a certain height can be estimated from this by sweeping the azimuth angle at a fixed elevation angle. This type of scan is called plan-position-indicator (PPI) scan. The method is thus suitable to investigate the centerline velocity downstream of a wind turbine.

All data sets and error bars were extracted from the original publications using a web plot digitizer [60]. A summary of the data sets can be found in Table 1.

The first data base, referred to as *Gallacher* in Table 1, is from a campaign carried out at the offshore wind farm Alpha Ventus in the German North Sea (see Fig. 3(a)) using two G4000 Galion LiDAR mounted on the nacelle of the wind turbine AV7, cf. Fig. 3(a) and (c). One LiDAR monitored the inflow and the other monitored the wake of the turbine. The wind turbine has a rotor diameter of 116 m and a hub height of 90 m. The PPI scans have therefore zero elevation angle and were obtained slightly above the centerline. The data was filtered according to the wind speed obtained by the LiDAR 2.24 D

upstream of the rotor ((6–8) ms⁻¹, (8–10) ms⁻¹ and (10–12) ms⁻¹) and wind direction (210°–330° for an undisturbed inflow), and the wake centerline velocity was extracted as the lowest velocity. Error bars represent the standard deviation over all samples normalized by the number of samples. The thrust coefficients and turbulence intensities that are reported in Table 1 were estimated by [59] who also use the data sets. [59] state that “the intensity from FINO1 followed the definition from IEC 61400–3 standard for offshore turbulence intensity, which is based on an approximation of the 90th percentile of the standard deviation of streamwise wind velocity at the hub height”.

The second data base, referred to as *Krishnamurthy* in Table 1, is taken from [47] and was obtained during the OBLEX-F1 (Offshore Boundary-Layer Experiment at FINO1) campaign, and the presented data was obtained in August and September 2016. The campaign was also carried out at the wind farm Alpha Ventus in the German North Sea, but with a scanning Doppler LiDAR Windcube 100S installed at the FINO1 platform 21 m above sea level. The wake of turbine AV7 (cf. Fig. 3(c)) was investigated in this study. The distance to the LiDAR is 7.4 D. The wind turbine has a rotor diameter of 116 m and a hub height of 90 m. PPI scans over a 48° sector were performed at an elevation angle of 4.62°, thus crossing the hub-height at AV7. The scan rate was 1 degree per second, giving an azimuthal resolution of 15 m
<figure><img src="image_10.png" /><figcaption>**Fig. 6.** Downstream evolution of the normalized mean velocity deficit: Data measured by LiDAR with a velocity range of 6–8 ms⁻¹, extracted from [56] with cₜ and TI values from [59]. (a) shows the fitted curves of the six wake models (solid lines; *k* was fitted for the wind turbine wake models) as well as the predictions of four wind turbine wake models where all parameters, i.e., cₜ, TI and *k*, are extracted from the literature (dotted lines). (b) A virtual origin was added as additional fit parameter in the equations for the wind turbine wake models.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Fig. 7.** Downstream evolution of the normalized velocity deficit: Data measured at the FINO 1 platform by LiDAR, extracted from [47]. Error bars indicate the standard deviation of the average over all velocity measurements. (a) shows the fitted curves of the six wake models (solid lines; *k* was fitted for the wind turbine wake models) as well as the predictions of four wind turbine wake models where all parameters, i.e., cₜ, TI and *k*, are extracted from the literature (dotted lines). (b) A virtual origin was added as additional fit parameter in the equations for the wind turbine wake models.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Fig. 8.** Downstream evolution of the normalized mean velocity deficit: Data measured by LiDAR, extracted from [16]; the wind turbine is in the wake of an upstream turbine. (a) shows the fitted curves of the six wake models (solid lines; *k* was fitted for the wind turbine wake models) as well as the predictions of four wind turbine wake models where all parameters, i.e., cₜ, TI and *k*, are extracted from the literature (dotted lines). (b) A virtual origin was added as additional fit parameter in the equations for the wind turbine wake models.</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>**Fig. 9.** Dependence of the virtual origin on (a) the turbulence intensity, with $c_T$ color-coded, and (b) $c_T$, with the turbulence intensity color coded. Data points framed in red indicate laboratory measurements, while data points framed in orange indicate a double wake scenario.</figcaption></figure>

<figure><img src="image_14.png" /></figure>

at wind turbine AV7. The range-gate size or radial resolution of the Lidar is 25 m. The data was filtered (for details, see [47]) and only data of wind directions between 330° ± 30° were taken into account. The wind direction and wind statistics of the inflow are obtained at a height of 90 m from anemometers of the met mast installed at FINO1. During the measurement period, the authors report pre-dominant winds from southwest with a bi-modal wind speed distribution with a mean velocity of 7.5 ms⁻<fnref n="1" />; the reported Weibull shape parameter is 8.11 and the scale parameter is 2.16. Also, the turbulence intensity is reported to be mostly less than 0.1. In Table 1, we therefore report $TI < 0.1$, and we use $TI = 0.1$ for estimating the wake growth rate $k_{BP}$ for the Porté-Agel wake model given by Eq. (8). The wake was identified by fitting a Gaussian distribution to measurements from an axis perpendicular to the wind turbine center-axis (that is aligned with the wind direction). Data from 252 scans was averaged without filtering for mean incoming wind speed, atmospheric stability, turbine thrust coefficient or interaction of the wake with the neighboring wind turbines. Scans were only included when the wind turbine AV7 was in operation.

The third data base, referred to as *Smarteleole* in Table 1, is from a measurement campaign carried out within the SMARTEOLE (ANR-14-CE05-0034) project in a wind farm in the north of France (cf. [16,51,55]). The geographical location and the site are shown in Fig. 4; seven SENVION MM82 wind turbines with a rotor diameter of 82 m and a hub height of 80 m are installed at the site. A pulsed scanning LiDAR of type Windcube 200S by Leosphere was positioned approximately 1320 m, or 16 D, from the investigated wind turbine SMV5 at the ground. The elevation angle of the PPI scans used here is 3.8°, leading to a variation of the measurement height below ±0.1 D in the fit region. The data was filtered according to atmospheric stability (neutral), environmental wind speed (13 ms⁻<fnref n="1" /> at 50 m above sea level, obtained from the Modern-Era Retrospective analysis for Research and Applications version 2 (MERRA-2), although the measured wind speed is around 11 ms⁻<fnref n="1" /> at hub height, according to the authors), and wind direction (207°, 233° and 246°). The data set for wind direction 207° shows the wake of turbine SMV5 exposed to the wake of the upstream turbine SMV6. For a wind direction of 233°, the wakes of the two turbines SMV5 and SMV6 partially interact, and for 246°, the wakes of these turbines do not interact. Therefore, for wind directions 233° and 246°, the wakes of both turbines are fitted separately. Based on the variations of the data points, error bars were added that indicate fluctuations of

<caption>**Table 1**</caption>

Summary of investigated data sets. Data sets in **bold** are discussed in the main document, whereas the remaining data sets are presented in the appendix. The wind speed values in parenthesis given for the Smarteleole cases are obtained from SCADA data, while the main values are obtained from MERRA-2.

<table><thead><tr><th></th><th></th><th>U<sub>∞</sub>/ms<sup>-1</sup></th><th>TI</th><th>c<sub>T</sub></th></tr></thead><tbody><tr><td>1</td><td>low-turbulent</td><td>8.3</td><td>0.0012</td><td>0.70</td></tr><tr><td>2</td><td>turbulent</td><td>6.9</td><td>0.02</td><td>0.70</td></tr><tr><td>3</td><td>Gallacher 6-8</td><td>6-8</td><td>0.082</td><td>0.82</td></tr><tr><td>4</td><td>Gallacher 8-10</td><td>8-10</td><td>0.073</td><td>0.81</td></tr><tr><td>5</td><td>Gallacher 10-12</td><td>10-12</td><td>0.068</td><td>0.79</td></tr><tr><td>6</td><td>Krishnamurthy</td><td>-</td><td>&lt;0.1</td><td>0.80</td></tr><tr><td>7</td><td>Smarteleole 207° SMV5</td><td>12.88 (9.63)</td><td>0.183</td><td>0.74</td></tr><tr><td>8</td><td>Smarteleole 233° SMV6</td><td>12.48 (11.84)</td><td>0.107</td><td>0.53</td></tr><tr><td>9</td><td>Smarteleole 233° SMV5</td><td>12.48 (11.41)</td><td>0.114</td><td>0.58</td></tr><tr><td>10</td><td>Smarteleole 246° SMV6</td><td>12.61 (11.45)</td><td>0.103</td><td>0.58</td></tr><tr><td>11</td><td>Smarteleole 246° SMV5</td><td>12.61 (11.16)</td><td>0.104</td><td>0.61</td></tr></tbody></table>

0.01 · $\frac{U_{\infty}-U_0}{U_{\infty}}$. The mean velocities obtained both from MERRA-2 and through SCADA data (reported in parenthesis), the turbulence intensity of the inflow obtained from SCADA data, and the thrust coefficient in Table 1 are taken from [16].

## 4. Results

The results of this paper are based on data collected in controlled conditions in a wind tunnel and field measurements behind multi-megawatt wind turbines obtained with LiDAR technique in three independent measurement campaigns. Indeed, to justify our claims, field data are the most important data sets, but have the drawback that the precise working mode of the turbine is not always known, nor the exact inflow condition, which is always unsteady so that mean wind velocities and the turbulence intensity may change during measuring period (cf. [61–63]). Such disadvantages can be overcome by controlled experiments with wind turbine models in a wind tunnel. The biggest disadvantage of wind tunnel experiments is the low Reynolds number but it is commonly assumed that the wake behavior resembles well that of the free field [64]. Here, we use both kinds of data: the field data from the literature mentioned above and wind tunnel data form the setup described in Section 3. We obtain consistent results for all different data sets.

<footnote n="1">¹ Note that this is the mean wind speed over all wind directions, not the mean wind speed for the wake measurements.</footnote>
## 4.1. Wind tunnel data

We start by analyzing the data obtained in lab conditions. This data set was measured in the wake of a scaled rotor for both low-turbulent and turbulent inflows; the latter can be found in the appendix.

Fig. 5 shows the results for a low-turbulent inflow, where the experimental data and the predictions of six different models can be compared. In Fig. 5(a), solid lines indicate that the wake growth rate *k* was fitted for the wind turbine wake models, whereas dotted lines indicate that the wind turbine wake models were applied as they are intended to, with *c*<sub>T</sub>, *T*I and *k* taken from measurements or the literature. The models describe the wake with different grades of accuracy, and applying the wind turbine wake models without fitting decreases their accuracy, which can be seen by means of the root mean square (RMS) errors given in Table 2. By only adding a virtual origin, Fig. 5(b), all models significantly improve their performance and decrease the RMS errors. Note that the Townsend-George models are used in their native form and thus always include the virtual origin. By just adding a virtual origin, all models except the Aitken model fit the profile equally well, almost collapsing onto a single curve with the experimental data. The fitting parameters for all fits presented here can be found in Table 3. Moreover, when changing the inflow conditions in this experiment to turbulent ones, exactly the same trend is obtained (see Appendix A, Fig. A.10).

## 4.2. Field data

We now proceed to verify if a similar improvement is observed when dealing with LiDAR field data extracted from the literature. We check several data sets of onshore and offshore wind farms that include isolated wakes (data sets 3–6, and 8–11 in Table 1), and double-wakes (i.e., wakes with an inflow that is the wake of an upstream turbine, data set 7 in Table 1). While we will discuss results for data sets 3, 6, and 7, the remaining data sets with different inflow velocities and different degrees of wind turbine wake interaction can be found in Appendix B, Figs. B.11–B.16. The fitting parameters for all figures presented here can be found in Table 3.

Figs. 6 and 7 show two different measurements of turbine AV7 in the offshore wind farm Alpha Ventus (data sets 3 and 6). Similarly to the lab data, in (a), solid lines indicate that the wake growth rate *k* was fitted for the wind turbine wake models, whereas dotted lines indicate that the wind turbine wake models were applied as they are intended to, with *c*<sub>T</sub>, *T*I and *k* taken from measurements or the literature; in (b) a virtual origin was added to the wind turbine wake models. Note that both data sets tend to a constant value from 15D (Gallacher 6–8), respectively 10D (Krishnamurthy), and therefore, fits are performed only in the region where the velocity deficit decays. First, it can be seen that the original wind turbine wake models (without an added virtual origin) perform significantly better than when applying them to the laboratory data, although the deviations of the Aitken, Jensen and Frandsen wake models are obvious around 5D. Second, from (b), it becomes obvious that an additional virtual origin improves the fit of all wake models, which is also supported by the reduced RMS errors in Table 2.

In Fig. 8, data set 7 shows the evolution of the wake of a turbine exposed to the wake of an upstream turbine. Again, in (a), solid lines indicate that the wake growth rate *k* was fitted for the wind turbine wake models, whereas dotted lines indicate that the wind turbine wake models were applied as they are intended to, with *c*<sub>T</sub>, *T*I and *k* taken from measurements or the literature; in (b) a virtual origin was added to the wind turbine wake models. For the rather small range where the fits are applied, in (a), the Townsend-George models and the Porté-Agel wake models fit the evolution quite well, whereas the remaining wake models perform poorly. This is also indicated by the RMS errors in Table 2. For all cases, the addition of a virtual origin reduces the RMS error significantly, although it should be noted that the addition of a virtual origin to the Frandsen model does not improve its performance significantly for this wake scenario.

## 5. Discussion

By means of several different data sets obtained from wind tunnel and field measurements in onshore and offshore conditions, we test the implementation of a virtual origin in four different wind turbine wake models. We also compare them to the two analytical bluff body wake models. From the data discussed in the main document and the additional data in the appendix, we conclude that, by modeling the turbulence production range with the length scale of a virtual origin, all wake models tend to give acceptable results, independently of the inflow and operation conditions for which data was collected. This is shown by the reduced RMS errors in Tables 2 and E.5 in all cases considered.

In addition, from Figs. C.17–C.20 in the Appendix C and Table E.5, we can see that by even just adding a virtual origin as a fit parameter while using values from the literature for all other parameters, all wind turbine wake models can be improved.

It therefore remains an open question if some scaling laws can also be found for the virtual origin itself. For instance, Fig. 9 shows the virtual origin plotted over (a) the turbulence intensity with a color-coded thrust coefficient, and (b) the thrust coefficient with color-coded turbulence intensity. While no dependencies can be derived based on these two variables, some trends can be discussed. First, we see that the virtual origins obtained for the different wake models are more scattered for the low-turbulent lab data sets, circled in red, than for the field data sets. A possible explanation is the later wake recovery due to the low ambient turbulence. Next, it can be seen that the virtual origin obtained for wakes in similar conditions, i.e., similar values of *c*<sub>T</sub> and *T*I, are very close. This is an encouraging result that shows the stability of our method for different independent data sets and indicates the possibility of finding scaling laws for the virtual origin from first principles. Finally, we also observe a change of the virtual origin for low thrust coefficients, i.e., when the wind turbine approaches rated power and is controlled to convert less energy, which indicates that our approach could indeed be used dynamically to account for the change in the wake evolution due to different ambient conditions. It also becomes obvious from this plot that more data, systematically spanning various *T*Is and *c*<sub>T</sub>s, is needed to identify dependencies.

Looking at the fitting parameters, some further trends can be identified: for the two data sets from the wind tunnel, that have a lower turbulence intensity than the data sets from field measurements, we can see that the virtual origin is diverging more from $x_0 = 0$ than it is the case for most field data sets (cf. Tables 3 and D.4). This is in agreement with observations that the wake recovers more slowly in low-turbulent inflow, as the evolution of the wake is hindered, and the near wake region is extended. Therefore, the virtual origin indeed gives an indication of the near wake length, although it has to be noted that each wake model appears to have an individual range of optimal virtual origins for different conditions. Further, it is interesting to see that the values for the virtual origin are rather small in the case of the Porté-Agel model, which agrees with the observation that its predictions are quite good even in the original form.

Another interesting result from this study is that, particularly for the field data sets, the addition of a virtual origin to the different wake models reveals the sensitivity of the fits to the different decay exponents of the models. For example, in Figs. 6(b) and 7(b), the Jensen and Porté-Agel models, that both include a quadratic function part, outperform the Townsend-George models when adding a virtual origin.

While these are interesting observations that support our claims about the importance of a virtual origin, further research is needed to identify the dependence of this parameter with the thrust, the ambient turbulence intensity, and the atmospheric stability, to allow for a dynamic correction of wind turbine wake models.

Finally, we would like to comment that field data has larger error bars and sometimes show behaviors that do not fully follow a decaying axisymmetric turbulent wake. For instance, the velocity deficits
<caption>**Table 2**</caption>

Root mean square error ×10⁻² for the six different models and three different fitting methods (fitting the original model as indicated in Section 2 (orig), adding a virtual origin to the wind turbine wake models (VO add.), and predicting the wake without applying a fit (no fit)), for the four data sets compared in the main document.

<table><caption>**Table 3**</caption><thead><tr><th rowspan="2"></th><th colspan="3">Low-turbulent</th><th colspan="3">Galacher 6-8</th><th colspan="3">Krishnamurthy</th><th colspan="3">Smarteole 207° SMV5</th></tr><tr><th>orig.</th><th>VO add.</th><th>No fit</th><th>orig.</th><th>VO add.</th><th>No fit</th><th>orig.</th><th>VO add.</th><th>No fit</th><th>orig.</th><th>VO add.</th><th>No fit</th></tr></thead><tbody><tr><td>Townsend -2/3</td><td>0.4</td><td>-</td><td>-</td><td>3.0</td><td>-</td><td>-</td><td>2.5</td><td>-</td><td>-</td><td>1.9</td><td>-</td><td>-</td></tr><tr><td>Townsend -1</td><td>0.3</td><td>-</td><td>-</td><td>1.8</td><td>-</td><td>-</td><td>1.3</td><td>-</td><td>-</td><td>1.5</td><td>-</td><td>-</td></tr><tr><td>Aitken</td><td>-</td><td>6.4</td><td>15</td><td>-</td><td>4.6</td><td>8.9</td><td>-</td><td>5.8</td><td>8.1</td><td>-</td><td>7.0</td><td>17.6</td></tr><tr><td>Jensen</td><td>4.3</td><td>0.7</td><td>22.4</td><td>6.9</td><td>1.9</td><td>10.4</td><td>7.2</td><td>1.1</td><td>9.3</td><td>9.6</td><td>1.3</td><td>24.4</td></tr><tr><td>P-A</td><td>1.0</td><td>0.6</td><td>3.5</td><td>3.2</td><td>1.8</td><td>3.3</td><td>3.2</td><td>1.0</td><td>3.5</td><td>3.6</td><td>1.5</td><td>5.1</td></tr><tr><td>Frandsen</td><td>5.3</td><td>0.7</td><td>11.2</td><td>9.7</td><td>5.5</td><td>14.0</td><td>9.5</td><td>4.7</td><td>13.7</td><td>10.0</td><td>8.5</td><td>25.6</td></tr></tbody></table>

Fit parameters for all data sets for the Townsend bluff body wake models, and the Jensen, Porté-Agel, Aitken and Frandsen wake models. For the Jensen model, the numbers marked as **bold** are values from literature (according to [9], 0.075 for onshore and 0.04–0.05 for offshore wakes), or close to literature. For the Frandsen model, it is stated that $k_F = O(10 \cdot k_j)$ [9] for small $c_T$. The column labeled “fit” indicates whether $k$ was used as a fitting parameter (without virtual origin), or both $k$ and $x_0$ were used as fitting parameters. Fits were performed in MATLAB, and iterative least squares estimation is used to estimate the coefficients.

<table><thead><tr><th rowspan="2">Data set</th><th rowspan="2">fit</th><th rowspan="2">c<sub>T</sub></th><th rowspan="2">TI</th><th colspan="2">Townsend -2/3</th><th colspan="2">Townsend -1</th><th colspan="2">Jensen</th><th colspan="2">Porté-Agel</th><th colspan="2">Aitken</th><th colspan="2">Frandsen</th></tr><tr><th>T<sub>-2/3</sub></th><th>x<sub>0</sub>/D</th><th>T<sub>-1</sub></th><th>x<sub>0</sub>/D</th><th>k<sub>j</sub></th><th>x<sub>0</sub>/D</th><th>k<sub>BP</sub></th><th>x<sub>0</sub>/D</th><th>x<sub>0</sub>/D</th><th>k<sub>f</sub></th><th>x<sub>0</sub>/D</th></tr></thead><tbody><tr><td rowspan="2">low-turbulent</td><td>k</td><td></td><td></td><td></td><td></td><td></td><td></td><td>0.011</td><td>0</td><td>0.015</td><td>0</td><td>-</td><td>0.028</td><td>0</td></tr><tr><td>k and x<sub>0</sub></td><td>0.7</td><td>0.0012<sup>a</sup></td><td>1.203</td><td>3.337</td><td>3.671</td><td>-0.655</td><td>0.024</td><td>7.109</td><td>0.014</td><td>-1.271</td><td>6.738</td><td>0.074</td><td>7.83</td></tr><tr><td rowspan="2">Gallacher 6-8</td><td>k</td><td></td><td></td><td></td><td></td><td></td><td></td><td>0.045</td><td>0</td><td>0.037</td><td>0</td><td>-</td><td>0.117</td><td>0</td></tr><tr><td>k and x<sub>0</sub></td><td>0.82<sup>b</sup></td><td>0.082<sup>b</sup></td><td>0.555</td><td>2.2</td><td>1.13</td><td>1.19</td><td>0.088</td><td>3.04</td><td>0.046</td><td>1.059</td><td>2.106</td><td>0.247</td><td>3.259</td></tr><tr><td rowspan="2">Krishnamurthy</td><td>k</td><td></td><td></td><td></td><td></td><td></td><td></td><td>0.051</td><td>0</td><td>0.045</td><td>0</td><td>-</td><td>0.128</td><td>0</td></tr><tr><td>k and x<sub>0</sub></td><td>0.8</td><td>&lt;0.1</td><td>0.479</td><td>1.91</td><td>0.897</td><td>1.093</td><td>0.11</td><td>2.639</td><td>0.059</td><td>0.984</td><td>1.435</td><td>0.314</td><td>2.876</td></tr><tr><td rowspan="2">Smarteole 207° SMV5</td><td>k</td><td></td><td></td><td></td><td></td><td></td><td></td><td>0.009</td><td>0</td><td>0.039</td><td>0</td><td>-</td><td>0.006</td><td>0</td></tr><tr><td>k and x<sub>0</sub></td><td>0.74</td><td>0.1<sup>c</sup></td><td>0.939</td><td>0.208</td><td>1.946</td><td>-1.064</td><td>0.062</td><td>2.955</td><td>0.03</td><td>-0.835</td><td>1.272</td><td>0.026</td><td>2.187</td></tr></tbody></table>

in Figs. 6 and 7 tend to a constant value instead of showing the monotonous decrease expected for an isolated wake in its far field region. This shows the difficulties in modeling realistic wind turbine-generated flows in atmospheric conditions, as the resulting flow is actually a combination of different canonical flows (atmospheric boundary layers, turbulent wakes, etc.); also, the wake properties may change during data acquisition. In consequence, models tend to perform worse than in wind tunnel conditions. Nevertheless, our main finding is that all models produce good fits by only adding a virtual origin.

## 6. Conclusion

In the present work, thanks to a few comprehensive examples, we show that the addition of a virtual origin to widely used engineering models for wind turbine wakes has the capability of improving all these models significantly. This simple but effective addition to wake models is motivated by fundamental turbulence research, and its significance extends to serving as a novel measure of the near wake region. The latter, contrary to the far wake, is not universal due to the impact of rotor properties and the operating conditions. In particular, we show how the decay law of the mean centerline velocity deficit is secondary to the implementation of a virtual origin.

The virtual origin has to be seen as a dynamic correction. The objective being that, once the underlying physics are understood, this length scale can be easily tuned to unsteady operation conditions, i.e., the inflow velocity, inflow turbulence degree and thrust coefficient of the turbine.

With the importance of precise modeling of wind turbine wakes, our findings shift the discussion of wind turbine wakes in a new direction. Therefore, the natural continuation of this work is to understand the physics of the virtual origin that are determined by the flow effects occurring in the near wake.

## CRediT authorship contribution statement

**Ingrid Neunaber:** Formal analysis, Investigation, Writing – original draft, Writing – review & editing, Visualization. **Michael Hölling:** Conceptualization, Methodology, Resources. **Martín Obligado:** Conceptualization, Formal analysis, Writing – original draft, Writing – review & editing, Project administration, Funding acquisition.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgments

Part of this work has been performed during the stay associated to fellowships from the Hanse-Wissenschaftskolleg (HWK Institute for Advanced Study, Delmenhorst, Germany) assigned to MO and IN. The authors also thank the Lille Turbulence Program (LTP) and Joachim Peinke for fruitful discussions. Ingrid Neunaber would like to thank Caroline Braud and Sandrine Aubrun and the Laboratoire de recherche en Hydrodynamique, Énergétique et Environnement Atmosphérique (LHEEA) of the Centre national de la recherche scientifique (CNRS) at Centrale Nantes.

<footnote><sup>a</sup> $k_j$ used for the Porté-Agel model since the equation for estimating $k_{BP}$ is not valid for very low TIs.</footnote>

<footnote><sup>b</sup> Values taken from [59].</footnote>

<footnote><sup>c</sup> For the calculation of $k$ according to the Porté-Agel model, $TI = 0.1$ has been used here instead of the value obtained from SCADA data, $TI = 0.18$, to reach convergence of the fit.</footnote>
## Appendix A. Additional wind tunnel data

<figure><img src="image_15.png" /><figcaption>**Fig. A.10.** Downstream evolution of the normalized mean velocity deficit. Data set 2: Wake of model wind turbine in turbulent inflow generated by a regular grid (cf. Section 3.1). (a) shows the fitted curves of the six wake models (solid lines; *k* was fitted for the wind turbine wake models) as well as the predictions of four wind turbine wake models where all parameters, i.e., *c*<sub>T</sub>, *T*I and *k*, are extracted from the literature (dotted lines). (b) A virtual origin was added as additional fit parameter in the equations for the wind turbine wake models.</figcaption></figure>

## Appendix B. Additional field data

<figure><img src="image_16.png" /><figcaption>**Fig. B.11.** Downstream evolution of the normalized velocity deficit: Data measured at FINO 1 by a nacelle-mounted LiDAR with a velocity range of 8–10 ms<sup>-1</sup>, extracted from [56] with *c*<sub>T</sub> and *T*I values from [59]. Error bars indicate the standard deviation of the average over all velocity measurements, normalized by the square root of the number of measurements. (a) shows the fitted curves of the six wake models (solid lines; *k* was fitted for the wind turbine wake models) as well as the predictions of four wind turbine wake models where all parameters, i.e., *c*<sub>T</sub>, *T*I and *k*, are extracted from the literature (dotted lines). (b) A virtual origin was added as additional fit parameter in the equations for the wind turbine wake models.</figcaption></figure>
<figure><img src="image_17.png" /><figcaption>**Fig. B.12.** Downstream evolution of the normalized velocity deficit: Data measured at FINO 1 by a nacelle-mounted LiDAR with a velocity range of 10–12ms⁻¹, extracted from [56] with *c*<sub>T</sub> and *TI* values from [59]. Error bars indicate the standard deviation of the average over all velocity measurements, normalized by the square root of the number of measurements. (a) shows the fitted curves of the six wake models (solid lines; *k* was fitted for the wind turbine wake models) as well as the predictions of four wind turbine wake models where all parameters, i.e., *c*<sub>T</sub>, *TI* and *k*, are extracted from the literature (dotted lines). (b) A virtual origin was added as additional fit parameter in the equations for the wind turbine wake models.</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>**Fig. B.13.** Downstream evolution of the normalized velocity deficit: Data measured onshore by LiDAR within the SMARTEOLE project, extracted from [16]; the wake slightly interacts with the wake of a downstream turbine. Error bars are indicating variations of 0.01 of the normalized velocity deficit. (a) shows the fitted curves of the six wake models (solid lines; *k* was fitted for the wind turbine wake models) as well as the predictions of four wind turbine wake models where all parameters, i.e., *c*<sub>T</sub>, *TI* and *k*, are extracted from the literature (dotted lines). (b) A virtual origin was added as additional fit parameter in the equations for the wind turbine wake models.</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>**Fig. B.14.** Downstream evolution of the normalized velocity deficit: Data measured onshore by LiDAR within the SMARTEOLE project, extracted from [16]; the turbine slightly interacts with the wake of an upstream turbine. Error bars are indicating variations of 0.01 of the normalized velocity deficit. (a) shows the fitted curves of the six wake models (solid lines; *k* was fitted for the wind turbine wake models) as well as the predictions of four wind turbine wake models where all parameters, i.e., *c*<sub>T</sub>, *TI* and *k*, are extracted from the literature (dotted lines). (b) A virtual origin was added as additional fit parameter in the equations for the wind turbine wake models.</figcaption></figure>
<figure><img src="image_20.png" /><figcaption>**Fig. B.15.** Downstream evolution of the normalized velocity deficit: Data measured onshore by LiDAR within the SMARTEOLE project, extracted from [16]. Error bars are indicating variations of 0.01 of the normalized velocity deficit. (a) shows the fitted curves of the six wake models (solid lines; *k* was fitted for the wind turbine wake models) as well as the predictions of four wind turbine wake models where all parameters, i.e., *c*<sub>T</sub>, *T*I and *k*, are extracted from the literature (dotted lines). (b) A virtual origin was added as additional fit parameter in the equations for the wind turbine wake models.</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>**Fig. B.16.** Downstream evolution of the normalized velocity deficit: Data measured onshore by LiDAR within the SMARTEOLE project, extracted from [16]. Error bars are indicating variations of 0.01 of the normalized velocity deficit. (a) shows the fitted curves of the six wake models (solid lines; *k* was fitted for the wind turbine wake models) as well as the predictions of four wind turbine wake models where all parameters, i.e., *c*<sub>T</sub>, *T*I and *k*, are extracted from the literature (dotted lines). (b) A virtual origin was added as additional fit parameter in the equations for the wind turbine wake models.</figcaption></figure>

## Appendix C. Fitting only a virtual origin

In this section, in Figs. C.17-C.20, we present the results from adding a virtual origin as fitting parameter to the engineering models while using theoretical values for the wake expansion rates *k*. The values used for *k* and obtained for *x*<sub>0</sub> can be found in Table D.4. Note that, while we aimed at keeping *k* close to the theoretical values, we adapted them to give reasonable results for the fits.
<figure><img src="image_22.png" /><figcaption>**Fig. C.17.** Downstream evolution of the normalized velocity deficit. Only a virtual origin is fitted for the engineering wake models. (a) low-turbulent inflow; (b) turbulent inflow generated by a regular grid (cf. Section 3.1).</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>**Fig. C.18.** Downstream evolution of the normalized velocity deficit: Data measured at FINO 1 by a nacelle-mounted LiDAR, extracted from [56] with $c_T$ values from [59]. Only a virtual origin is fitted for the engineering wake models. Velocity range of (a) 6–8 ms⁻¹; (b) 8–10 ms⁻¹; (c) 10–12 ms⁻¹.</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>**Fig. C.19.** Downstream evolution of the normalized velocity deficit: Data measured at the FINO 1 platform by LiDAR, extracted from [47]. Only a virtual origin is fitted for the engineering wake models.</figcaption></figure>
<figure><img src="image_25.png" /><figcaption>Fig. C.20. Downstream evolution of the normalized velocity deficit: Data measured onshore by LiDAR within the SMARTEOLE project, extracted from [16]. Error bars are indicating variations of 0.01 of the normalized velocity deficit. Only a virtual origin is fitted for the engineering wake models. (a) wake of a turbine exposed to the wake of an upstream turbine; (b) the wake interacts slightly with the wake of a downstream turbine; (c) wake of a turbine partially exposed to the wake of an upstream turbine; (d) wake of a wind turbine; (e) wake of a wind turbine.</figcaption></figure>

## Appendix D. Fit parameters

<caption>Table D.4</caption>

Fit parameters for all data sets for the Townsend bluff body wake models, and the Jensen, Porté-Agel, Aitken and Frandsen wake models. For the Jensen model, the numbers marked as **bold** are values from literature (according to [9], 0.075 for onshore and 0.04–0.05 for offshore wakes), or close to literature. For the Frandsen model, it is stated that $k_F = O(10 \cdot k_j)$ [9] for small $c_T$. The column labeled “fit” indicates whether $k$ was used as a fitting parameter (without virtual origin), both $k$ and $x_0$ were used as fitting parameters, or just $x_0$ was added to the original model and treated as fitting parameter. Fits were performed in MATLAB, and iterative least squares estimation is used to estimate the coefficients.

<table><thead><tr><th rowspan="2">Data set</th><th rowspan="2">fit</th><th rowspan="2">c<sub>F</sub></th><th rowspan="2">T<sub>I</sub></th><th colspan="2">Townsend -2/3</th><th colspan="2">Townsend -1</th><th colspan="2">Jensen</th><th colspan="2">Porté-Agel</th><th colspan="2">Aitken</th><th colspan="2">Frandsen</th></tr><tr><th>T<sub>-2/3</sub></th><th>x<sub>0</sub>/D</th><th>T<sub>-1</sub></th><th>x<sub>0</sub>/D</th><th>k<sub>j</sub></th><th>x<sub>0</sub>/D</th><th>k<sub>BP</sub></th><th>x<sub>0</sub>/D</th><th>x<sub>0</sub>/D</th><th>k<sub>f</sub></th><th>x<sub>0</sub>/D</th></tr></thead><tbody><tr><td rowspan="3">low-turbulent</td><td>k</td><td rowspan="3">0.70</td><td rowspan="3">0.0012<sup>a</sup></td><td rowspan="3">1.203</td><td rowspan="3">3.337</td><td rowspan="3">3.671</td><td rowspan="3">-0.655</td><td>0.011</td><td>0</td><td>0.015</td><td>0</td><td>-</td><td>0.028</td><td>0</td></tr><tr><td>k, x<sub>0</sub></td><td>0.024</td><td>7.109</td><td>0.014</td><td>-1.271</td><td>-</td><td>0.074</td><td>7.830</td></tr><tr><td>x<sub>0</sub></td><td><strong>0.027</strong></td><td>7.500</td><td>k<sub>j</sub></td><td>4.108</td><td>6.738</td><td>3 · k<sub>j</sub> = 0.081</td><td>8.250</td></tr><tr><td rowspan="3">turbulent</td><td>k</td><td rowspan="3">0.70</td><td rowspan="3">0.02</td><td rowspan="3">1.426</td><td rowspan="3">2.377</td><td rowspan="3">4.552</td><td rowspan="3">-2.239</td><td>0.009</td><td>0</td><td>0.014</td><td>0</td><td>-</td><td>0.020</td><td>0</td></tr><tr><td>k, x<sub>0</sub></td><td>0.021</td><td>7.574</td><td>0.011</td><td>-2.578</td><td>-</td><td>0.057</td><td>8.176</td></tr><tr><td>x<sub>0</sub></td><td><strong>0.024</strong></td><td>8.109</td><td>0.012<sup>b</sup></td><td>-2.195</td><td>6.894</td><td>2.5 · k<sub>j</sub> = 0.060</td><td>8.256</td></tr><tr><td rowspan="3">Gallacher 6–8</td><td>k, x<sub>0</sub></td><td rowspan="3">0.82<sup>c</sup></td><td rowspan="3">0.082<sup>c</sup></td><td rowspan="3">0.555</td><td rowspan="3">2.200</td><td rowspan="3">1.130</td><td rowspan="3">1.190</td><td>0.045</td><td>0</td><td>0.037</td><td>0</td><td>-</td><td>0.117<sup>j</sup></td><td>0</td></tr><tr><td>x<sub>0</sub></td><td>0.088</td><td>3.040</td><td>0.046</td><td>1.059</td><td>-</td><td>0.247</td><td>3.259</td></tr><tr><td>k</td><td><strong>0.075</strong></td><td>2.803</td><td>0.035<sup>b</sup></td><td>0.753</td><td>2.106</td><td>5 · k<sub>j</sub> = 0.375</td><td>3.508</td></tr><tr><td rowspan="3">Gallacher 8–10</td><td>k, x<sub>0</sub></td><td rowspan="3">0.81<sup>c</sup></td><td rowspan="3">0.072<sup>c</sup></td><td rowspan="3">0.609</td><td rowspan="3">2.108</td><td rowspan="3">1.265</td><td rowspan="3">1.026</td><td>0.040</td><td>0</td><td>0.034</td><td>0</td><td>-</td><td>0.104</td><td>0</td></tr><tr><td>x<sub>0</sub></td><td>0.081</td><td>3.231</td><td>0.042</td><td>0.941</td><td>-</td><td>0.229</td><td>3.477</td></tr><tr><td>k</td><td><strong>0.075</strong></td><td>3.145</td><td>0.032<sup>b</sup></td><td>0.932</td><td>2.198</td><td>5 · k<sub>j</sub> = 0.375</td><td>3.749</td></tr><tr><td rowspan="3">Gallacher 10–12</td><td>k, x<sub>0</sub></td><td rowspan="3">0.79<sup>c</sup></td><td rowspan="3">0.068<sup>c</sup></td><td rowspan="3">0.625</td><td rowspan="3">1.910</td><td rowspan="3">1.347</td><td rowspan="3">0.683</td><td>0.039</td><td>0</td><td>0.035</td><td>0</td><td>-</td><td>0.103</td><td>0</td></tr><tr><td>x<sub>0</sub></td><td>0.073</td><td>3.088</td><td>0.039</td><td>0.508</td><td>-</td><td>0.221</td><td>3.476</td></tr><tr><td>k</td><td><strong>0.075</strong></td><td>3.126</td><td>0.030<sup>b</sup></td><td>0.779</td><td>2.049</td><td>5 · k<sub>j</sub> = 0.375</td><td>3.745</td></tr><tr><td rowspan="3">Krishnamurthy</td><td>k, x<sub>0</sub></td><td rowspan="3">0.80</td><td rowspan="3">&lt;0.10</td><td rowspan="3">0.479</td><td rowspan="3">1.910</td><td rowspan="3">0.897</td><td rowspan="3">1.093</td><td>0.051</td><td>0</td><td>0.045</td><td>0</td><td>-</td><td>0.128</td><td>0</td></tr><tr><td>x<sub>0</sub></td><td>0.110</td><td>2.639</td><td>0.059</td><td>0.984</td><td>-</td><td>0.314</td><td>2.876</td></tr><tr><td>k</td><td><strong>0.075</strong></td><td>2.012</td><td>0.042<sup>b</sup></td><td>-0.070</td><td>1.435</td><td>5 · k<sub>j</sub> = 0.375</td><td>3.876</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>0.009</td><td>0</td><td>0.039</td><td>0</td><td>-</td><td>0.006</td><td>0</td></tr></tbody></table>

(continued on next page)
<table><caption>Table D.4 (continued).</caption><thead><tr><th rowspan="2">Data set</th><th rowspan="2">fit</th><th rowspan="2">c<sub>T</sub></th><th rowspan="2">TI</th><th colspan="2">Townsend -2/3</th><th colspan="2">Townsend -1</th><th colspan="2">Jensen</th><th colspan="2">Porté-Agel</th><th colspan="2">Aitken</th><th colspan="2">Frandsen</th></tr><tr><th>T<sub>-2/3</sub></th><th>x<sub>0</sub>/D</th><th>T<sub>-1</sub></th><th>x<sub>0</sub>/D</th><th>k<sub>j</sub></th><th>x<sub>0</sub>/D</th><th>k<sub>BP</sub></th><th>x<sub>0</sub>/D</th><th>x<sub>0</sub>/D</th><th>k<sub>j</sub></th><th>x<sub>0</sub>/D</th></tr></thead><tbody><tr><td>Smarteole 207° SMV5</td><td>k, x<sub>0</sub><br>x<sub>0</sub><br>k</td><td>0.74</td><td>0.18<sup>d</sup></td><td>0.939</td><td>0.208</td><td>1.946</td><td>-1.064</td><td>0.062<br><b>0.075</b></td><td>2.955<br>2.916</td><td>0.030<br>0.042<sup>b</sup></td><td>-0.835<br>0.136</td><td>-</td><td>0.026<br>k<sub>j</sub> = 0.075</td><td>2.187<br>2.322</td></tr><tr><td>Smarteole 233° SMV6</td><td>k, x<sub>0</sub><br>x<sub>0</sub><br>k</td><td>0.53</td><td>0.11</td><td>0.488</td><td>0.777</td><td>1.002</td><td>-0.429</td><td>0.022<br><b>0.075</b></td><td>0<br>2.792</td><td>0.046<br>0.041</td><td>0<br>-0.403</td><td>1.272<br>-</td><td>0.070<br>0.182</td><td>0<br>2.328</td></tr><tr><td>Smarteole 233° SMV5</td><td>k, x<sub>0</sub><br>x<sub>0</sub><br>k</td><td>0.58</td><td>0.11</td><td>0.891</td><td>-1.262</td><td>2.138</td><td>-3.280</td><td>0.005<br><b>0.075</b></td><td>0<br>2.822</td><td>0.046<sup>b</sup><br>0.046<sup>b</sup></td><td>-0.069<br>-0.142</td><td>-0.270<br>0.362</td><td>3 · k<sub>j</sub> = 0.225<br>k<sub>j</sub> = 0.075</td><td>2.446<br>2.184</td></tr><tr><td>Smarteole 246° SMV6</td><td>k, x<sub>0</sub><br>x<sub>0</sub><br>k</td><td>0.58</td><td>0.10</td><td>0.493</td><td>-0.094</td><td>0.952</td><td>-1.038</td><td>0.040<br><b>0.075</b></td><td>0<br>1.751</td><td>0.073<br>0.042<sup>b</sup></td><td>0<br>-0.969<br>-1.331</td><td>-</td><td>0.126<br>-0.712</td><td>0<br>1.312</td></tr><tr><td>Smarteole 246° SMV5</td><td>k, x<sub>0</sub><br>x<sub>0</sub><br>k</td><td>0.61</td><td>0.10</td><td>0.568</td><td>0.148</td><td>1.161</td><td>-1.099</td><td>0.024<br><b>0.075</b></td><td>0<br>1.973</td><td>0.058<br>0.042<sup>b</sup></td><td>0<br>-0.994</td><td>-0.184<br>-</td><td>0.061<br>3 · k<sub>j</sub> = 0.225</td><td>0<br>2.018</td></tr></tbody></table>

a k<sub>j</sub> used for the Porté-Agel model since the equation for estimating k<sub>BP</sub> is not valid for very low TIs.

b Calculated with the equation from [9]: 0.38 · TI + 0.004.

c Values taken from [59].

d For the calculation of k according to the Porté-Agel model, TI = 0.1 has been used here to reach convergence of the fit.

## Appendix E. Root mean square errors

<caption>Table E.5</caption>

RMS errors ×10⁻² for the different fit configurations for all six wake models and all eleven data sets.

<table><thead><tr><th></th><th>Data set</th><th>fit type</th><th>Townsend -2/3</th><th>Townsend -1</th><th>Aitken</th><th>Jensen</th><th>BP</th><th>Frandsen</th></tr></thead><tbody><tr><td rowspan="4">1</td><td rowspan="4">Low-turbulent</td><td>Orig.</td><td>0.4</td><td>0.3</td><td>-</td><td>4.3</td><td>1.0</td><td>5.3</td></tr><tr><td>VO added</td><td>-</td><td>-</td><td>-</td><td>0.7</td><td>0.6</td><td>0.7</td></tr><tr><td>fit only VO</td><td>-</td><td>-</td><td>6.4</td><td>1.0</td><td>6.1</td><td>0.3</td></tr><tr><td>No fit</td><td>-</td><td>-</td><td>15.0</td><td>22.4</td><td>3.5</td><td>11.2</td></tr><tr><td rowspan="4">2</td><td rowspan="4">Grid</td><td>Orig.</td><td>0.3</td><td>0.2</td><td>-</td><td>4.4</td><td>1.7</td><td>5.3</td></tr><tr><td>VO added</td><td>-</td><td>-</td><td>-</td><td>0.6</td><td>0.5</td><td>0.7</td></tr><tr><td>fit only VO</td><td>-</td><td>-</td><td>8.9</td><td>1.4</td><td>0.5</td><td>0.7</td></tr><tr><td>No fit</td><td>-</td><td>-</td><td>17.6</td><td>25.0</td><td>4.5</td><td>11.1</td></tr><tr><td rowspan="4">3</td><td rowspan="4">Gallacher 6-8</td><td>Orig.</td><td>3.0</td><td>1.8</td><td>-</td><td>6.9</td><td>3.2</td><td>9.7</td></tr><tr><td>VO added</td><td>-</td><td>-</td><td>-</td><td>1.9</td><td>1.8</td><td>5.5</td></tr><tr><td>fit only VO</td><td>-</td><td>-</td><td>4.6</td><td>2.5</td><td>3.3</td><td>6.2</td></tr><tr><td>No fit</td><td>-</td><td>-</td><td>8.9</td><td>10.4</td><td>3.3</td><td>14.0</td></tr><tr><td rowspan="4">4</td><td rowspan="4">Gallacher 8-10</td><td>Orig.</td><td>4.0</td><td>2.5</td><td>-</td><td>7.7</td><td>3.1</td><td>10.6</td></tr><tr><td>VO added</td><td>-</td><td>-</td><td>-</td><td>1.2</td><td>1.7</td><td>6.2</td></tr><tr><td>fit only VO</td><td>-</td><td>-</td><td>4.9</td><td>1.5</td><td>3.6</td><td>7.3</td></tr><tr><td>No fit</td><td>-</td><td>-</td><td>10.1</td><td>12.0</td><td>3.6</td><td>15.5</td></tr><tr><td rowspan="4">5</td><td rowspan="4">Gallacher 10-12</td><td>Orig.</td><td>3.5</td><td>2.4</td><td>-</td><td>6.6</td><td>2.2</td><td>9.1</td></tr><tr><td>VO added</td><td>-</td><td>-</td><td>-</td><td>1.6</td><td>1.8</td><td>4.7</td></tr><tr><td>fit only VO</td><td>-</td><td>-</td><td>4.3</td><td>1.6</td><td>3.2</td><td>6.2</td></tr><tr><td>No fit</td><td>-</td><td>-</td><td>8.8</td><td>11.6</td><td>4.7</td><td>14.7</td></tr><tr><td rowspan="4">6</td><td rowspan="4">Krishnamurthy</td><td>Orig.</td><td>2.5</td><td>1.3</td><td>-</td><td>7.2</td><td>3.2</td><td>9.5</td></tr><tr><td>VO added</td><td>-</td><td>-</td><td>-</td><td>1.1</td><td>1.0</td><td>4.7</td></tr><tr><td>fit only VO</td><td>-</td><td>-</td><td>5.8</td><td>3.9</td><td>3.5</td><td>4.9</td></tr><tr><td>No fit</td><td>-</td><td>-</td><td>8.1</td><td>9.3</td><td>3.5</td><td>13.7</td></tr><tr><td rowspan="4">7</td><td rowspan="4">Smarteole 207° SMV5</td><td>Orig.</td><td>1.9</td><td>1.5</td><td>-</td><td>9.6</td><td>3.6</td><td>10.0</td></tr><tr><td>VO added</td><td>-</td><td>-</td><td>-</td><td>1.3</td><td>1.5</td><td>8.5</td></tr><tr><td>fit only VO</td><td>-</td><td>-</td><td>7.0</td><td>2.4</td><td>4.5</td><td>9.3</td></tr><tr><td>No fit</td><td>-</td><td>-</td><td>17.6</td><td>24.4</td><td>5.1</td><td>25.6</td></tr><tr><td rowspan="4">8</td><td rowspan="4">Smarteole 233° SMV6</td><td>Orig.</td><td>0.5</td><td>0.5</td><td>-</td><td>4.3</td><td>4.8</td><td>4.3</td></tr><tr><td>VO added</td><td>-</td><td>-</td><td>-</td><td>0.5</td><td>0.5</td><td>3.5</td></tr><tr><td>fit only VO</td><td>-</td><td>-</td><td>2.3</td><td>3.8</td><td>4.5</td><td>3.6</td></tr><tr><td>No fit</td><td>-</td><td>-</td><td>3.8</td><td>17.4</td><td>4.7</td><td>16.0</td></tr><tr><td rowspan="4">9</td><td rowspan="4">Smarteole 233° SMV5</td><td>Orig.</td><td>1.0</td><td>0.8</td><td>-</td><td>5.2</td><td>1.2</td><td>5.5</td></tr><tr><td>VO added</td><td>-</td><td>-</td><td>-</td><td>1.0</td><td>0.9</td><td>3.0</td></tr><tr><td>fit only VO</td><td>-</td><td>-</td><td>3.2</td><td>1.1</td><td>1.1</td><td>3.1</td></tr><tr><td>No fit</td><td>-</td><td>-</td><td>3.4</td><td>11.8</td><td>1.2</td><td>10.0</td></tr></tbody></table>

(continued on next page)
<table><caption>Table E.5 (continued).</caption><thead><tr><th></th><th>Data set</th><th>fit type</th><th>Townsend -2/3</th><th>Townsend -1</th><th>Aitken</th><th>Jensen</th><th>BP</th><th>Frandsen</th></tr></thead><tbody><tr><td rowspan="4">10</td><td rowspan="4">Smarteole 246° SMV6</td><td>Orig.</td><td>0.6</td><td>0.4</td><td>-</td><td>3.2</td><td>1.9</td><td>3.5</td></tr><tr><td>VO added</td><td>-</td><td>-</td><td>-</td><td>0.4</td><td>0.4</td><td>0.6</td></tr><tr><td>fit only VO</td><td>-</td><td>-</td><td>1.3</td><td>0.4</td><td>0.4</td><td>0.7</td></tr><tr><td>No fit</td><td>-</td><td>-</td><td>1.7</td><td>11.4</td><td>10.5</td><td>10.4</td></tr><tr><td rowspan="4">11</td><td rowspan="4">Smarteole 246° SMV5</td><td>Orig.</td><td>3.6</td><td>2.8</td><td>-</td><td>6.9</td><td>5.6</td><td>7.5</td></tr><tr><td>VO added</td><td>-</td><td>-</td><td>-</td><td>2.0</td><td>2.2</td><td>5.4</td></tr><tr><td>fit only VO</td><td>-</td><td>-</td><td>5.6</td><td>2.2</td><td>2.5</td><td>5.3</td></tr><tr><td>No fit</td><td>-</td><td>-</td><td>7.8</td><td>9.3</td><td>14.5</td><td>8.9</td></tr></tbody></table>

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