Investigation of the dissipation in the wake of a wind turbine array

Within the energy transition, more and more wind turbines are clustered in big wind farms, often offshore. Therefore, an optimal positioning of the wind turbines is crucial to optimize both the annual power production and the maintenance time. Good knowledge of the wind turbine wake and the turbulence within is thus important. However, although wind turbine wakes have been subject to various studies, they are still not fully understood. One possibility to improve the comprehension is to look into the modeling of bluff body wakes. These wakes have been the subject of intensive study for decades, and by means of 5 the scaling behavior of the centerline mean velocity deficit, the nature of the turbulence inside a wake can be classified. In this paper, we introduce the models for equilibrium and non-equilibrium turbulence from classical wake theory as introduced by A. Townsend and W. George, and we test whether the requirements are fulfilled in the wake of a wind turbine. Finally, we apply the theory to characterize the wind turbine wake, and we compare the results to the Jensen and the Bastankhah-Porté-Agel model. We find that the insight into the classical bluff body wake can be used to further improve the wind turbine wake models. 10 Particularly, the classical bluff body wake models perform better than the wind turbine wake models due to the presence of a virtual origin in the scalings, and we demonstrate the possibility of improving the wind turbine wake models by implementing this parameter. We also see how the dissipation changes across the wake which is important to model wakes within wind farms correctly.

scale describing the energy-containing structures. C ε = const. is therefore one of the most critical points in the theory. The scaling of the dissipation rate ε is a scaling used in many aspects of turbulence theory and in the modelling and understanding of many turbulent flows (see for example Tennekes and Lumley (1972)). As better detailed in section 2, the closure provoked by this assumption leads to the standard equilibrium streamwise scalings for ∆U and δ.
Yet, in the last years, evidence has been found that some axisymmetric turbulent wakes do not follow the standard scalings 50 of ∆U and δ. Different experimental studies and direct numerical simulations of bluff plates with both regular and irregular peripheries found that C ε is not constant, but instead it goes as C ε ∼ Re m G /Re n L (e.g. Vassilicos (2015); Dairay et al. (2015); Obligado et al. (2016); Nedic et al. (2013); Es-Sahli et al. (2020)). Re G is a Reynolds number that depends on the inlet conditions, and Re L is a local, streamwise, position-dependent one (Vassilicos (2015)). The exponents n and m have been found to be very close to unity; m = n = 1 for large values of the Taylor Reynolds number Re λ (cf. equation A4). For an axisymmetric 55 wake, Re G = √ ΛU ∞ /ν, with Λ the frontal area of the plate, U ∞ the inlet velocity and ν the kinematic viscosity of the flow.
The local Reynolds number is defined as Re L = δu /ν, with u the RMS value of the fluctuating streamwise velocity. These anomalous non-equilibrium scalings have been reported for experiments at X/ √ Λ < 50, therefore a range of interest within 1 Note that a fundamental difference between the engineering wake models and the analytical bluff body wake model is the modeling of the wake width: In wind turbine models, a growth rate is derived empirically, while in the bluff body wake analysis, an analytical model for the wake width is derived. some wind energy applications.
The presence of a different dissipation scaling of ε within wind turbine wakes would have important consequences regard-60 ing modelling of wind farms and numerical simulations: Many different numerical models implicitly assume the standard dissipation scalings (for more details, see Vassilicos (2015)). However, a different dissipation scaling also implies different streamwise scalings for ∆U and δ (see section 2). The disregard of this point shows that we have not yet fully understood the physics underlying the simplest possible configuration in wind energy: a single turbine facing a uniform, stationary, laminar flow. The approach of engineering models that are based on the conservation of mass and momentum is therefore also 65 insufficient as the wake of a wind turbine can only be fully understood if the turbulence evolution is properly characterised.
Furthermore, the fact that different dissipation scalings result in different spreading properties of the wake shows that the turbulence cascade controls how the wake is evolving. Therefore, the initial conditions, such as the wake generator and the background flow, become important as they ultimately control the evolution of the cascade. It is thus important to study how the dissipation scalings are affected by the inflow conditions.

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In contrast to the axisymmetric turbulent wake of a bluff body, no systematic work on the turbulence decay and on the energy cascade has been made yet in the turbulent wind turbine wake. However, Okulov et al. (2015) applied the standard equilibrium bluff body wake scalings derived from the Richardson-Kolmogorov phenomenology to experimental data to justify the presence of power-law decays for ∆U and δ. In addition, Stein and Kaltenbach (2019) performed a systematic study on the nature of the dissipation scaling, testing also the non-equilibrium scaling, but without measuring nor taking into account C ε in 75 the discussion. Furthermore, for the latter, the turbine studied was within a turbulent boundary layer background flow which means that the turbulence evolution was additionally influenced by the turbulent background flow.
In this paper, we present an experimental study on the axisymmetric turbulent wake generated by a wind turbine in a wind tunnel via hot-wire anemometry. We study the wake produced by a single turbine in uniform laminar inflow. Furthermore, we also analyse the behaviour of a turbine within a turbulent background: this is achieved by placing a second turbine downstream of 80 the first one. The second turbine is tested at two different radial positions, that allow us to explore the behaviour of a turbulent wake within two different, relevant configurations within a wind farm.
We perform a systematic study on the applicability of the Townsend-George theory to wind-turbine-generated wakes for the different incoming flows. For this, we explore the streamwise range in which the requirements are fulfilled. We also study the behaviour of C ε for this flow, and therefore of the scalings of ∆U and δ. By doing so, we identify the nature of the energy 85 cascade for the different wind turbine configurations. We find that all three turbine configurations are in good agreement with the Richardson Kolmogorov energy cascade. Furthermore, we verify that, to some extent, they also are within the conditions and hypotheses required to apply the Townsend-George theory.
In the following, we will introduce the Townsend-George theory in section 2. Here, the framework as well as the necessary requirements to apply the theory are introduced. Also, two important engineering wake models are introduced for a comparison.

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Then, the setup is presented in section 3.1, and the fulfillment of the requirements is investigated in section 3.2. The theory is applied in section 4 and the results are discussed in section 5. This paper ends with a conclusion in section 6.
Bluff body wakes have been the object of intensive analyses for the past 60 years, and similarly to the study of wind turbine wakes, the downstream evolution of the centerline mean velocity deficit ∆U and the wake width δ are of interest.
In this work, we will follow the revisited Townsend-George theory (Dairay et al. (2015)). The classical equilibrium predictions and the non-equilibrium predictions rely on the axisymmetry of turbulence wake statistics, the self-preservation of line velocity deficit and U c the streamwise centerline mean velocity), the turbulent kinetic energy K, the Reynolds shear stress R xr , the turbulence dissipation ε, and on a scaling law for the centerline turbulence dissipation that is determined by the behavior of C ε due to ε = C ε K 2/3 c /L. Both sets of predictions are obtained from the Reynolds-averaged streamwise momentum and turbulent kinetic energy equations leading to a closed set of equations for ∆U (X) and the wake width δ(X).
These hypotheses also allow to express the momentum conservation within the wake as (θ/δ) 2 = ∆U/U ∞ . The momentum 105 thickness θ is defined by rdr which is constant with X, and the wake's width is characterised here by the integral wake width δ defined by δ 2 (X) = 1 The equilibrium predictions for axisymmetric turbulent wakes (see Townsend, 1976;George, 1989) for the streamwise evolution (along X) of ∆U and δ are 110 where A and B are dimensionless constants, and X 0 a virtual origin (that appears naturally from the mathematical formulation and has to be the same for ∆U and δ). In contrast, the non-equilibrium predictions are The only difference between equilibrium and high-Reynolds number non-equilibrium scalings for ∆U and δ is in the scaling of the centerline value of ε: As discussed in the introduction, the non-equilibrium turbulence dissipation law is ε c = C ε K 3/2 c /δ 120 where K c is centerline turbulent kinetic energy, ε c is the centerline turbulence dissipation and C ε is a dimensionless coefficient which is constant in equilibrium turbulence but proportional to the ratio of two Reynolds numbers, Re m G /Re n L , in the case of non-equilibrium turbulence.
Another important assumption used throughout the theory is that the large scales of the turbulence are represented by the wake width δ, and therefore δ ∝ L (Cafiero et al. (2020)). This is an important, and frequently overlooked assumption within the 125 Townsend-George theory. To the authors knowledge, it has not been verified for wind turbine wakes.
To summarize, the Townsend-George theory of axisymmetric turbulent wakes relies on the following flow properties: Requirements.
1. The flow has to be decaying turbulence: we will focus on the range for which the turbulent kinetic energy is a decreasing function of the streamwise distance. 2. The flow has to be turbulent. A way to verify this requirement is to check that the velocity signal has a) a large value of Re λ (i.e. Re λ > 200), and b) a power spectral density with an inertial sub-range that decays according to the E(f ) ∝ f −5/3 power law.
3. The mean streamwise velocity and the turbulence quantities discussed above also have to be self-preserving and thus show a self-similar scaling (for a more detailed study on this point, we refer the reader to Dairay et al. (2015)). 4. The mean velocity components and the turbulence quantities have to be a axisymmetric.
5. As stated, the flow has to be in a streamwise range for which the longitudinal integral length scale is proportional to the wake width (L ∝ δ).
If all conditions are fulfilled, the theory predicts power-law decays for the velocity deficit and the wake width. Furthermore, the exponents of such power-laws can be related to the dissipation scaling of ε in the flow. In order to disentangle different 140 dissipation scalings, another requirement is to be on a streamwise range where Re λ varies.
In the following, we will refer to the equilibrium scaling as EQ scaling, and the non-equilibrium scaling as NEQ scaling.

Wind turbine wake models
In the past, a lot of effort was put into the understanding and finding a description of the wake flow of wind turbines, especially its recovery on the centerline and the wake expansion, by carrying out experiments and simulations. Additionally, different 145 theories have been underlain. However, an analytical model that is capable of including different inflow conditions and turbine operations does still not exist. In the following, we will briefly introduce two commonly used wake models that we will compare to the EQ and NEQ wake models.
One of the first models was developed by N. O. Jensen in 1983, and it is based on the conservation of momentum (cf. Jensen (1983)). It is assumed that the normalized velocity deficit with the wake velocity U W is top-hat shaped and evolves with increasing downstream distance X according to U ∞ denotes the unperturbed inflow velocity, U W the streamwise wake velocity, c T the turbine's thrust coefficient, D the turbine's diameter and k J the wake expansion, that is assumed to be constant. While Jensen suggests k J = 0.1, k J = 0.075 is 155 suggested in Barthelmie et al. (2006).
One recent model that includes the conservation of mass in addition to the conservation of momentum is the model derived by Bastankhah and Porté-Agel (2014). Here, the recovery of the centerline velocity deficit is combined with a Gaussian velocity profile, 160 In this formula, Y and Z denote the span-wise and wall-normal coordinates, Z h denotes the hub height, and k BP is the wake growth rate that needs to be specified and is in the order of magnitude of 0.03 ± 0.02 (Bastankhah and Porté-Agel (2014), Okulov et al. (2015)). β is given by the equation In the following, we will call this wake model the BP model.

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It can be seen that the two approaches that were presented above to describe the wind turbine wake differ significantly from the classical wake models that were derived from the perspective of turbulence.

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In the following, the experimental setup used to carry out the experiments evaluated in this study is introduced.
The experiments have been performed in Oldenburg's Large Wind Tunnel (OLWiT) at the University of Oldenburg that has an inlet of (3 × 3) m 2 and a closed test section of 30 m length (cf. Kröger et al. (2018)). In the empty test section, the background turbulence intensity of the flow is 0.3%, and velocities of up to 42 ms −1 can be reached. A closed-loop control keeps the velocity constant, and to correct for changes, the flow temperature, the ambient pressure, and the humidity are constantly 175 monitored. During all experiments presented here, the uniform inlet velocity was U ∞ = (7.55 ± 0.05) ms −1 .
Two three-bladed horizontal axis model wind turbines of the same type have been used in the experiments (cf. Schottler et al. (2018)). In the following, they will be referred to as turbine 1 and turbine 2. Both turbines have a rotor diameter of D = 58 cm and a hub height of h = 72 cm (resulting in an inlet Reynolds number of Re G = 3 × 10 5 based on the square root of the rotor surface). The rotor load is controlled using a closed-loop control to ensure a performance of the turbine at the optimal tip speed 180 ratio of T SR ≈ 5.8 in the full-load range (cf. Petrović et al. (2018); Neunaber (2019)). This is achieved be measuring the rotor torque and adapting it to achieve the optimal rotational speed that can be taken from a look-up table. The thrust coefficient of As shown in figure 1, an array of six 1d hot-wire probes with sensor lengths of 1.25 mm and a sensor diameter of 5 µm was used to measure the wake downstream of three different wind turbine array configurations at the hub height with a very high downstream resolution of ∆X/D = 0.17. The data was collected using a StreamLine 9091N0102 frame with 91C10 CTA Modules. A temperature correction, as proposed by Hultmark and Smits (2010), was applied to the data. At each position, 1.2 · 10 6 data points were collected with a sampling frequency of f s = 15 kHz. A hardware low-pass filter with a cut-off 190 frequency of 10 kHz has been used.
Three wake configurations were investigated (cf. figure 1): 1. The wake of turbine 1 exposed to the uniform, laminar inflow in the range between X/D = 0.55 and X/D = 12.62.
This configuration will be referred to as turbine 1. Based on the rotor, the blockage of the wind tunnel is 3% in this configuration.

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2. The wake of turbine 2 positioned 5.17D downstream in the wake of turbine 1 between X/D = 0.55 and X/D = 8.66 with respect to turbine 2 2 (a) First, turbine 2 is exposed centrally to the wake of turbine 1. As shown in Neunaber (2019), the inflow is thus turbulent with an average inflow velocity of 3.4 ms −1 (2.1 ms −1 in the center, 5.5 ms −1 at the blade tip). This configuration will be referred to as turbine 2 mid.

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(b) Then, turbine 2 is moved 0.5D to the side so that it is exposed to an inhomogeneous half-wake inflow. As shown in Neunaber et al. (2020), the inflow is thus turbulent and intermittent. The average inflow velocity is 5.1 ms −1 with a gradient from 2.1 ms −1 at one side of the rotor to 7.4 ms −1 at the other side of the rotor. This configuration will be referred to as turbine 2 side.
In the cases of turbine 1 and turbine 2 mid, we measure one half of the wake because the wake is expected to be axisymmetric 205 in uniform inflow (Stevens and Meneveau (2017)). The horizontal position of the hot-wire array is not changed in the case of turbine 2 side so that the full wake is measured in this case.

Requirements
As explained in section 2, certain requirements have to be fulfilled to justify the use of the Townsend-George theory. Therefore, in the following, we will verify that the listed criteria are accomplished.  (2019)): A hot-wire array is used to measure the wakes of 1. a single turbine (plot color: dark purple), 2. a turbine exposed to the wake of an upstream turbine (plot color: turquoise), and 3. the half-wake of an upstream turbine (plot color: green).

Turbulence intensity
In figure 2, the centerline evolution of the turbulence intensity is plotted for the three inflow scenarios. As also discussed in Neunaber et al. (2020), the turbulence intensity first decreases in the nacelle's lee, then increases when the turbulence builds up due to the expansion of the shear layers and afterwards decreases in the far wake where the turbulence decays. The first requirement that has to be fulfilled in order to apply the Townsend-George theory is fully developed, decaying turbulence 215 which is indicated by a decreasing turbulence intensity. Therefore, the data points in the near wake prior to the local maximum of the turbulence intensity will be masked in the following. In the plots, this is indicated by hatched areas.

Taylor Reynolds number
In order to apply the Townsend-George theory, we are looking for a region with fully developed turbulence where Re λ > 200 220 holds, as stated in requirement 2a). In addition, the Taylor Reynolds number is supposed to change so that the presence of equilibrium and non-equilibrium turbulence can be disentangled. In figure 3, Re λ is therefore plotted in the downstream region where the turbulence intensity decays. It can be seen that Re λ changes in the region directly downstream of the local maximum turbulence intensity that was identified as the decay region in Neunaber et al. (2020) but remains then constant in the far wake.
Therefore, the Townsend-George theory can be applied and non-equilibrium turbulence could be present in the region just after 225 the turbulence intensity peak. This region is underlain in color. We remark that Re λ is increasing in the region of interest, and while there is no specification in the Townsend-George theory how Re λ has to change, George (1989) writes that Re L has to decrease. Therefore, we also show the downstream evolution of the local Reynolds number Re L in figure 4. It can be seen, that Re L increases in the marked region where Re λ is also increasing. Afterwards, Re L is constant downstream of turbine 1, and it decreases downstream of turbine 2 mid and side. Since Re λ remains constant in the far wake where Re L is decreasing, the 230 results regarding the existence of non-equilibrium turbulence are inconclusive in that range. The results also suggest that we capture the onset of the streamwise decay of Re L and Re λ , in agreement with the theory prediction.
In summary, based on the Taylor Reynolds number and the local Reynolds number, we conclude, that this requirement to apply the Townsend-George theory is fulfilled while the requirement regarding the identification of non-equilibrium turbulence is partially fulfilled.   turbine 1 -laminar inflow; b) turbine 2 mid -turbulent; c) turbine 2 side -turbulent and intermittent. In blue, the E(f ) ∝ f −5/3 decay is marked.

Energy spectral density
Next, the energy spectral density has to be checked. It is important to have an energy spectrum with a clear inertial sub-range that decays according to a power law, as explained in requirement 2b). In figure 5, the energy spectral density is plotted for the three inflow scenarios at the respective maximum of the turbulence intensity. Clearly, an inertial sub-range that decays according to E(f ) ∝ f −5/3 is present in all three wake scenarios. In Neunaber et al. (2020), it is also shown how the energy 240 spectral density has a clearly decaying inertial sub-range for all positions farther downstream. Therefore, the next criterion is fulfilled in the chosen downstream region.

Self-similarity
As the Townsend-George theory originates in classical bluff body wake analysis, another important assumption is that of the 245 self-similarity of the wake, as stated in requirement 3. To verify the self-similarity in this study, the normalized velocity deficit U∞−U (X,Y /δ) U∞−Uc is plotted over the normalized radial component Y /δ. In the case of a self-similar behavior of the turbulence, the curves are expected to collapse. For the two cases turbine 1 and turbine 2 mid, cf. figure 6 a) and b), the curves collapse as required and thus, a self-similar behavior is confirmed. To show that the profiles collapse to a Gaussian curve as often implied when assuming a constant eddy viscosity, a fit according to was applied. In addition, the modified constant eddy viscosity (MCEV) model discussed in Cafiero et al. (2020), has been applied. With root mean square (RMS) errors of 0.013 and 0.027 for turbine 1 and 2 mid, respectively, the MCEV model performs indeed better than the Gaussian fit with RMS errors of 0.033 and 0.054, respectively. In the asymmetric, more 255 complex wake of turbine 2 side (cf. figure 6 c)), a self-similar behavior can not be found and therefore, we do not expect the Townsend-George theory to fully hold there.
In addition to self-similarity, also axisymmetry of the wake is required, as explained in requirement 4. As the measurements that we present have been carried out in one half of the wake, we are not able to directly verify the axisymmetry. However, based on the symmetric setups for turbine 1 and turbine 2 mid and other studies with similar conditions, see e.g. Stevens and 260 Meneveau (2017), we conclude that the requirement of axisymmetry can be taken as valid for these inflow conditions.

Integral length scale over wake width
The last requirement that needs to be fulfilled is the behavior of L/δ, see requirement 5. As explained above, δ ∝ L is assumed in the Townsend-George theory for bluff body wakes. In figure 7, L/δ is therefore plotted over X/D for the three scenarios, and error bars are included. The errors for L/δ were calculated using error propagation. As the calculation of L is quite sensitive,  highly complex, sheared and asymmetric inflow the turbulence appears to drive the evolution of the wake towards symmetry. A deeper analysis of this will be postponed to future work. Briefly, we can conclude that the condition L/δ ≈ const. is fulfilled in the far wake case of turbine 1 and turbine 2 mid but not in the case of turbine 2 side.

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In order to fulfill the first requirement of the Townsend-George theory, we first chose the downstream regions in which the turbulence intensity is decaying. In this region, the other requirements have been tested, and by means of the Taylor Reynolds number, the energy spectral density, the self-similarity and, in the far wake, also the behavior of L/δ, we can conclude that the Townsend-George theory can be applied to the wakes downstream of turbine 1 and turbine 2 mid. In the wake of turbine 2 side, the requirements are partially fulfilled (fully developed turbulence that is indicated by the decaying turbulence intensity, 280 the high Taylor Reynolds numbers, and a decay of the energy spectral density has been verified while self-similarity and L/δ ≈ const. are not given). With the confirmation that the requirements are met met for turbine 1 and 2 mid and partially for turbine 2 side, we will apply the Townsend-George theory to the data. In addition, we can conclude for all three wake scenarios that non-equilibrium turbulence might be present in the far wake, which is indicated by a decreasing Re L , but not in the transition region where the Taylor Reynolds number is changing. To confirm or refute this, C ε has to be investigated.

Results
After verifying the requirements that need to be fulfilled in order to apply the Townsend-George theory, the next step is to check whether there is a dependence of C ε on the Taylor Reynolds number in order to differ between the equilibrium scaling for C ε scatter about ±15% around the mean value, which is normal due to the sensitivity of the calculation of C ε , no clear dependency of C ε (Re λ ) is present. Since this would be expected in the case of non-equilibrium turbulence, we can conclude that in the evaluated region in the wake of a wind turbine, we do not find evidence of non-equilibrium turbulence in any of the 295 inflow scenarios.
It can also be seen that the three different wakes have a different average C ε . This indicates inhomogeneity of the dissipation constant of the turbulence inside of a wind farm. Another consequence is that the dissipation may be instationary: When the wind direction changes, a downstream turbine can be exposed to a wake at first but after the wind direction change not anymore.
In such a scenario C ε changes with time.

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These consequences show the importance of the analysis of C ε in the wake of a wind turbine since computational fluid dynamics models normally rely on a constant dissipation coefficient.
After verifying all requirements and testing for the occurrence of non-equilibrium turbulence, finally, figure 9 shows the downstream evolution of the normalised velocity deficit ∆U/U ∞ over X/D at the centerline for all inflow conditions. An error for the velocity deficit has been calculated from error propagation where the error of U (X) comes from the calibration 305 uncertainty and the error of U ∞ is a statistical estimation of the inflow variation. The error bars are included in the plot but the error bars are very small, with the maximum relative errors being 0.0054 in the case of turbine 1, 0.0035 in the case of turbine 2 mid and 0.0043 in the case of turbine 2 side. In the following, we apply the EQ and NEQ scalings from the Townsend-∆U = A EQ · (X − X 0,EQ ) −2/3 and ∆U = A N EQ · (X − X 0,N EQ ) −1 where A and X 0 are fitted. For the Jensen and the BP models, the wake expansions k J and k BP are used as fit variables. The residual standard errors of the respective fits are given in the legends in figure 9. The fit parameters can be found in table 1.
In the case of turbine 1, the NEQ wake model and the EQ wake model predict a similar evolution of the mean centerline velocity deficit and show the best results based on the residual standard error. As the maximum error for this data set is 315 ∂∆U/U ∞ = 0.0054, we can conclude that they perform similarly. The BP wake model performs less well and it can be seen that there are deviations in the evolution of the normalized velocity deficit. The Jensen model does not capture the evolution of the mean centerline velocity deficit. It should be noted that, since the Jensen model uses a top-hat shaped wake to calculate the average wake velocity, it was never designed to accurately model the centerline velocity deficit. To give a specific example of the consequence of making an error in the estimation of the mean velocity, we will look at the BP model: In the beginning, 320 the velocity deficit is underestimated by 10% and at the end of the measurement range, the velocity deficit is overestimated by 3%. This would lead to a difference in the power estimation of 25% respectively 10% based on the centerline velocity for this specific wake scenario.
Downstream of turbine 2 mid, it can be seen that the EQ model captures the evolution of the velocity deficit best. The NEQ model performs less well and shows slight deviations from the data. In a systematic way, the model overestimates the velocity 325 up to X/D ≈ 2.5, underestimate the velocity in the range 2.5 ≤ X/D ≤ 6 and overestimate it from X/D ≈ 6. In the far wake, the velocity is overestimated by 3% which would result in an over-prediction of the power of approximately 9% for this specific wake scenario. The fits for the Jensen and the BP model show large deviations.
Downstream of turbine 2 side, the EQ and NEQ models perform best and fit the measurement points well. The Jensen model does not capture the evolution of the mean centerline velocity deficit. The BP model is not suitable for this wake condition 330 with complex inflow, and the deviations are significant: at the end of the measurement range, the velocity is overestimated by approximately 6% which leads to an overestimation of the power by approximately 17%.
Overall, it is obvious from the presented results that the EQ wake model performs well for all three wake scenarios, and in combination with the result from above that C ε ≈ const. (cf. figure 8), we conclude that the EQ model from classical wake theory is the wake that performs quite well. Note that the EQ model clearly improves the mentioned engineering models..

Discussion
In this work, we have introduced a different approach to model the wakes of wind turbines that originates in the theory of bluff body wakes. For this, we verified first that the requirements necessary to apply these theories are fulfilled (see requirements 1-5). Then, we searched for turbulence with non-equilibrium scaling by investigating the evolution of C ε . While we did not find evidence for non-equilibrium turbulence, we could show that both the EQ and the NEQ bluff body wake theories fit the 340 wind turbine wake remarkably well. While it is hard to disentangle the difference in the power-law exponents from the EQ  and NEQ models (in the end, they are both similar in the limited streamwise range studied), we show that they both predict a functional form that properly fits our data: a power law with a no-zero virtual origin.
To further expand the discussion of the turbulence present in the wake of a wind turbine, we did calculate C ε for all measurement positions in the three wake scenarios in the regions where the energy spectral density shows an inertial sub-range with a 345 power law decay. The results are presented as interpolated contour plots in figure 10 for the three wake scenarios. In addition, the positions of the blade tips are marked as horizontal red dashed lines, and the position of turbine 2 is marked as a vertical red dashed line in the wake of turbine 1. For all wakes, C ε does not show strong variations inside the wake. At the centerline downstream of turbine 1 and turbine 2 mid, C ε is larger than around the blade tip position. Downstream of turbine 2 side, the variations of C ε inside the wake are even less pronounced and the value is smaller as above-mentioned. The variation of C ε 350 due to the inflow turbulence means that different dissipation scalings are present inside a wind farm as already discussed in section 4, and during a change of the wind direction, even a temporal fluctuation of the dissipation scalings can occur. Another very interesting effect is visible at some points in these plots: at the wake edges at the turbulent-non-turbulent interface, C ε is significantly higher than inside the wake. This can be interpreted as a ring of large C ε surrounding the wake 3 , similar to the ring of high intermittency that was found to surround the wake in Schottler et al. (2018) and that was shown to be traceable 355 along the whole measured range in Neunaber et al. (2020). This shows that beside the wake core that is characterized by equilibrium turbulence, there are distinctive regions with non-trivial turbulence that can be found in the wake of a wind turbine.
One consequence of the variation of C ε in the wake is that the dissipation changes radially. Even with the complex inflow of turbine 2 side that is also exposed to this C ε ring, the wake center shows equilibrium turbulence. This confirms again the finding that the turbine creates an own, dominant type of turbulence (Neunaber et al. (2020)).

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With the knowledge that the wake core is characterized by equilibrium turbulence, we also applied the different wake models to the evolution of the velocity deficit at another radial position in the wake core, Y /D = −0.21. For the BP model, we only use the centerline velocity deficit part of equation 7. The results are presented in figure 11 for the three wake scenarios, and the coefficients are given in table 2. The plots include error bars that are obtained similarly to the errors in figure 9, and the maximum errors are 0.0062 for turbine 1, 0.0077 for turbine 2 mid, 0.0066 for turbine 2 side and, in addition, 0.0055 for turbine 365 2 side at Y /D = 0.31. It can be seen that all wake models except the Jensen model perform very well in the case of turbine 1.
With respect to the errors, the NEQ and BP models perform similarly and the EQ model shows a slightly worse fit. The Jensen model does not capture the evolution of the centerline velocity deficit, which is expected as it was developed to estimate wake losses for the whole wake. In the case of turbine 2 mid, the EQ model performs best, and the NEQ shows slight systematic differences, as discussed above. Both the BP and the Jensen model do not capture the evolution centerline velocity deficit, but 370 the Jensen model performs reasonably well in the far wake.
In the case of turbine 2 side, two radial positions, Y /D = −0.21 and Y /D = 0.31, have been used to test the wake models.
It can be seen that in this complex flow, the models have greater difficulties to perform well, and the evolution of the mean velocity deficit is not captured very accurately in most cases. At Y /D = −0.21, the bluff body wake theories perform best.
For Y /D = 0.31, the error is the smallest for the EQ and NEQ models and the velocity is predicted well at the end of the  turbine 2 mid -turbulent; c) turbine 2 side -turbulent and intermittent. Different models are fitted to the recovery region, and the residual standard errors are presented. Table 2. Fit parameters and RMS errors for the different models predicting the evolution of ∆U/U∞ for the three scenarios at the radial position Y /D = −0.21. Note that for turbine 2 side, in addition, the parameters for the radial position Y /D = 0.31 are included.
turbine 1 turbine 2 mid turbine 2 side turbine 2 side is not captured very well which is also indicated by the RMS errors. Overall, this test illustrates that even though most of the models have been designed to fit the centerline velocity deficit, they also perform quite well in the wake core at inner radial positions. This is an interesting result because it is not easy to measure along the centerline in field measurements, and with the concept of a wake core as introduced in Neunaber et al. (2020), the application of these models is expanded.

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To further test the applicability of the models, in the next step, we will calculate the velocity deficit at the radial position Y /D = −0.21 (and additionally Y /D = 0.31 in the case of turbine 2 side) from a radial correction: The BP model already has an implemented radial correction. The Jensen model is a top-hat shaped model and we therefore forego a radial correction. The EQ and NEQ models can be expanded by including the radial profile information from equation 10 to calculate the normalized and ∆U (Y /D) from the fit parameters obtained for the centerline velocity deficit and the wake profile. For this, the evolution of the wake width has to be known, and it is given by equations 2 and 4. Here, we apply the equations to the evolution of δ(X/D), which 390 is presented in figure 12 for the three scenarios, and we apply in addition a linear fit of the form δ(X/D) = α + β · (X/D).
It can be seen that all three models perform very similarly in the measurement range, and also, that δ(X/D) increases almost linearly. From the fit parameters given in table 3, we see that the fit coefficients B EQ and B N EQ of turbine 1 and turbine 2 mid are similar. As expected, the results are different for turbine 2 side where the prediction of the wake width is not very accurate due to the complex inflow. In addition, it can be seen that the virtual origins are different from the ones obtained for the fits of 395 the normalized velocity deficits. Therefore, while the Townsend George theory overall holds for a wind turbine wake, there are certain details that deviate and that will need further investigation in the future.
In figure 13, the normalized mean centerline velocity deficit at the radial position Y /D = −0.21 is plotted for the three scenarios, and the predicted evolutions are added for the different models. In the case of turbine 1, the EQ and NEQ models capture the evolution very well from X/D ≈ 7. The best fit is achieved with the NEQ model. The BP model captures the evolution 400 of ∆U/U ∞ in the far wake but shows deviations closer to the rotor. As expected, the Jensen model does not perform well.
In the case of turbine 2 mid, the EQ and NEQ models perform similarly well and capture the evolution of ∆U/U ∞ in the measurement range, but the curves start to deviate far downstream. The BP model shows a big deviation because already the centerline velocity deficit (cf. figure 9) was not captured well. The Jensen model also shows strong deviations. In the case of turbine 2 side, Y /D = −0.21, the EQ and NEQ models do underpredict the velocity deficit in the far wake but the result is 405 close to the measured data. The Jensen model is again strongly deviating. The BP model fits the evolution of ∆U/U ∞ best in this situation. In the case of Y /D = 0.31, none of the models perform well because all rely on a decreasing velocity deficit towards the free flow, but in the sheared inflow situation, the velocity deficit actually increases.
As we have seen, the EQ and NEQ bluff body wake models are significantly better in predicting the normalized mean velocity deficit at Y /D = −0.21 than the engineering wind turbine wake models. One reason is that more parameters need to 410 be obtained from fits of the wake shape and the centerline velocity deficit. Another reason is the implied virtual origin that is common in free shear flow analysis. It gives these models a high flexibility to adapt to different wakes. Therefore, we checked whether the Jensen and BP models can be improved by including a virtual origin, thus substituting X by X − X 0 . The results are shown in figure 14, and the fit parameters are given in table 4. For both models, the implementation improves the accuracy  of the fits significantly so that they are now in the order of magnitude of the errors obtained for the EQ and NEQ models in 415 some cases. Nevertheless, a similar systematic deviation can be identified in the wakes of turbine 2 mid and side where the centerline mean velocity deficit is overpredicted in one part of the wake and underpredicted in an other part of the wake. A clear difference in X 0 can be seen for laminar and turbulent inflow. This shows the potential of the concept of a virtual origin to include the background turbulence conditions.

420
In this paper we have introduced a new approach to look at the evolution of the wake of a wind turbine, the Townsend-George theory that has originally been derived from bluff body wakes to predict the evolution of the mean centerline velocity deficit in a laminar incoming flow. For this, we performed wind tunnel measurements of the wake downstream of a single turbine exposed to laminar inflow, and a turbine exposed to the wake of an upstream turbine and the half-wake of an upstream turbine.
First, we have presented a systematic study on the applicability of the Townsend-George theory to wind-turbine-generated 425 wakes for the different incoming flows. For this, we explored the streamwise range in which all requirements are fulfilled. We found that all five requirements are satisfied in the cases of turbine 1 and turbine 2 mid in the far wake and partially in the transition region. The requirements are partially fulfilled in the case of turbine 2 side. Next, we studied the behaviour of C ε for this flow to draw conclusions regarding the centerline evolution of ∆U and δ (i.e. equilibrium or non-equilibrium scaling). By means of C ε , we found evidence for equilibrium turbulence. In the following step, we applied the Townsend-George theory 430 of ∆U to estimate the velocity deficit in the three wake scenarios using the equilibrium and the non-equilibrium predictions.
In addition, two engineering wind turbine wake models have been applied to the data. It was found that the Townsend-George theory does describe the wake of a wind turbine and even a second-row turbine very well. In particular, the equilibrium scaling gives good results for all three wake scenarios. Thus, the nature of the energy cascade is closer to the equilibrium scaling than to the non-equilibrium scaling for the different wind turbine wake flows. Especially in the more complex situation of a wake 435 inflow, the equilibrium bluff body wake model outperforms the wind turbine wake models despite the fact that the former has been derived for the wake evolving downstream of a static bluff body exposed to uniform laminar inflow. Due to the cubic dependence of the power on the wind velocity, even rather small deviations in the velocity estimation can lead to large deviations in the power prediction.
To further investigate the applicability of the wake models, we also applied them to a different radial position, Y /D = −0.21. 440 We saw that the equilibrium bluff body wake model again outperforms the common wind turbine wake models. As the centerline is not always captured perfectly in field measurements, this result is very interesting for the application. Also, we could show that a radial correction can be used to predict the wake at Y /D = −0.21 using the fit coefficients obtained for the centerline. In comparison with the BP model that has a built-in radial correction, the Townsend-George theory shows again better results.

445
The virtual origin used in the Townsend-George theory was identified as one main advantage of the bluff body wake models.
Therefore, we also discuss how the implementation of a virtual origin may improve the Jensen and the BP models, especially as we see that it can be used to include the inflow turbulence in the wake models.
A final result of this analysis is that C ε does change in the wake both with respect to the radial position and with respect to the inflow condition. A consequence is that the turbulence dissipation inside a wind farm is inhomogeneous and instationary 450 which needs to be considered in simulations to further improve wind farm modeling.
Overall, this work shows in a detailed manner the consistent applicability of the equilibrium turbulence bluff body wake model from turbulence theory to wind turbine wakes, and thus the importance of correctly taking into account C ε . While the Jensen model is derived from conservation of momentum, and the BP model includes conservation of mass and momentum and considers the shape of the wake, we show that the additional inclusion of the turbulent nature is crucial. Nevertheless, we also see 455 that the models fail in multi-wake scenarios where the inflow is not symmetric and where thus superposition approaches would as well fail. This and the additional dependence of the wind turbine wake on the operational state of the turbine are difficulties occurring inside find farms that need further investigation.
For the future, a more detailed investigation of C ε inside a wind farm will be interesting in order to identify possible regions with non-equilibrium turbulence and their dependence on the inflow turbulence and the operational state of the turbine (e.g. the 460 thrust coefficient). In addition, the possibility of expanding engineering wind turbine wake models with a virtual origin should be further explored, especially with respect to the inflow turbulence. Finally, with the extremely asymmetric example of turbine 2 side, we could show that the application of the equilibrium wake model works to a certain degree because the turbulence tends to evolve towards a symmetric wake as far as possible. This gives a good indication for new theories to model half-wake and multi-wake scenarios with asymmetric inflows where C ε can be included. Competing interests. The authors declare that they have no conflict of interest.
Data availability. The data are available upon request.

Appendix A: Estimation of turbulence quantities
To estimate the integral length scale L and the Taylor length λ, we use the one-dimensional energy spectra as proposed by Hinze (1975). As the upper boundary of the inertial sub-range. the integral length scale can be estimated calculating the limit of the energy spectrum in the frequency domain for f approaching 0, u is the RMS value of the fluctuating streamwise velocity U (t).
The Taylor length is defined as where U = U (t) − U are the fluctuations of the velocity. · denotes the ensemble average. The energy spectrum in the space domain can be used to calculate (∂U /∂X) 2 , with the wave number k.
By means of the Taylor length λ, the RMS value of the fluctuating streamwise velocity u and the kinematic viscosity ν, the Taylor Reynolds number is calculated, The energy dissipation rate ε can be estimated by ε = 15ν (∂U /∂X) 2 (cf. Pope (2000)) under the assumptions of isotropy in small scales and the validity of Taylor's hypothesis of frozen turbulence. The derivative is calculated using formula A3.
Using the RMS value of the fluctuating streamwise velocity as estimate of the turbulent kinetic energy, C ε is calculated: