Towards a Simplified Dynamic Wake Model using POD Analysis

We apply the proper orthogonal decomposition (POD) to large eddy simulation data of a wind turbine wake in a turbulent atmospheric boundary layer. The turbine is modeled as an actuator disk. Our analyis mainly focuses on the question whether POD could be a useful tool to develop a simplified dynamic wake model. The extracted POD modes are used to obtain approximate descriptions of the velocity field. To assess the quality of these POD reconstructions, we define simple measures which are believed to be relevant for a sequential turbine in the wake such as the energy flux through a disk in the wake. It is shown that only a few modes are necessary to capture basic dynamical aspects of these measures even though only a small part of the turbulent kinetic energy is restored. Furthermore, we show that the importance of the individual modes depends on the measure chosen. Therefore, the optimal choice of modes for a possible model could in principle depend on the application of interest. We additionally present a possible interpretation of the POD modes relating them to specific properties of the wake. For example the first mode is related to the horizontal large scale movement. Besides yielding a deeper understanding, this also enables us to view our results in comparison to existing dynamic wake models.

The world wide wind energy capacity installed has increased rapidly in the last decade and is expected to increase much further in the near future due to environmental, political and economical reasons. More and more of this energy is produced by clusters of wind turbines containing up to hundreds of turbines. These wind farms offer many advantages over single turbines such as economical benefits stemming, e.g., from a more efficient installation and maintenance. One of the key issues of wind farms is what is known as the wake effect. A wind turbine in the wake of another turbine experiences a strongly altered inflow resulting in a reduced power production and higher loads acting on the turbine. In order to minimize these effects in farms a detailed understanding and efficient modeling of wakes is essential. This is of particular importance in the planning phase of a farm, where an optimized layout can result in a much higher energy production (see e.g. [1]) and longer lifetimes of the turbines. Furthermore, good wake models are an important tool for an efficient wind farm control [2,3], where e.g. wake deflection through yawing of the turbines can mitigate the wake effect [4][5][6]. The hydrodynamic equations describing the dynamics of a wind turbine wake and the flow through a wind farm are well-known. However, solving these equations in acceptable times is currently impossible due to the many scales involved in turbulent flows. A very successful modeling tool for turbulent flows are large eddy simulations (LES) which solve the hydrodynamic equations on the large scales of the flow but model the influence of the smaller scales by subgrid scale turbulence models (see e.g. [7]). Combined with simplified turbine models such as actuator disk and actuator line, LES have been increasingly applied in wind energy research (e.g. [2,3,[8][9][10][11]). Unfortunately, they are still too time consuming for many practical applications such as the optimization of a wind farm layout or wind farm control. Therefore, steady state wake models such as [12][13][14][15][16] are still the state of the art for many purposes. The models can for example be empirical descriptions of the velocity deficit such as [12] or mean field solutions to strongly approximated versions of the fluid dynamical equations such as [13]. More recently, mean field wakes stemming from computational fluid dynamic simulations have been used in the context of layout optimization [17] with respect to power. Instead of modeling the velocity deficit itself Frandsen [15] uses an effective turbulence intensity to model the fatigue loads in a wind park. However, all these models neglect dynamical aspects of the wake which are commonly expected to be relevant for the loads and power production of a turbine in a wake. A first step to incorporate dynamical aspects of the wake is to take the meandering movement of the velocity deficit into account. Usually, this is done under the assumption that the meandering is mainly caused by the large scale dynamics of the atmospheric boundary layer (ABL) [18][19][20][21]. While this is a promising approach [22,23], these models neglect , e.g., the form of the deficit which is also expected to be relevant, particularly for the loads on the turbine. Moreover, it is not fully clear whether the passive tracer assumption always yields a good description of the meandering, see e.g. [24]. An alternative approach to simplified modeling, used in fluid dynamics, is to apply the proper orthogonal decomposition (POD) [25][26][27][28] and develop reduced order models for specific flow problems. [25,[29][30][31]. Andersen et al. [24,32] applied the POD to LES data of a wake in an infinitely long row of turbines modeled with an actuator line approach. They found clearly structured POD modes which could be a first step to a reduced order model of wakes in a long row of turbines. Additionally, their results indicate that the low-frequency dynamics of the wake are not only caused by the large scale dynamics of the ABL, and can thus not be treated separate from the wake itself. In this work we apply the POD to LES data of a wind turbine wake. The paper is an extension of [33], putting this former work in a more general framework. Furthermore, we put a special emphasis on the comparison of different quality measures and gain a deeper understanding of our results through relating some of the POD modes to specific properties of the wake. In contrast to Andersen et al. [24,32], we analyze the wake of a single turbine modeled by an actuator disk and use a turbulent atmospheric boundary layer as inflow condition. Our main focus lies on the question whether POD can be a useful tool for developing simplified dynamic wake models. We propose to assess the quality of POD reconstructions of wakes based on the potential impact on a turbine in the wake instead of only considering the recovery of the turbulent kinetic energy. The results of this approach naturally suggest a new way of selecting modes depending on the desired application of a possible model. Furthermore, we aim to draw general conclusions on the complexity needed for useful dynamic wake models. In Section 2 of this work we describe the LES data used for our analysis. Section 3 begins with a brief description of the theory behind POD, followed by a presentation of the preprocessing (Sec. 3.2) which is applied before performing the POD analysis. Next, we illustrate the extracted POD modes (Sec. 3.3) and investigate approximations of the velocity field depending on the numbers of POD modes used for reconstruction. In Section 4 we introduce the general concept of assessing the quality of POD reconstructions through measures related to a turbine in the wake flow. We define three simple measures (e.g. the energy flux through a disk) and compare the corresponding results in Section 4.3. In Section 5 we further investigate the role the individual POD modes and try to relate them to specific properties of the wake. Finally, we summarize the main results and discuss consequences and conclusions drawn from our work (Sec. 6).

LES Data
The LES data which will be used for the POD analysis in Section 3 has been generated by performing simulations with the parallelized LES model PALM [34] which has been widely used for studies of the atmospheric boundary layer.
In this study, a standard uniformly loaded actuator disk model is applied which has been shown to yield very similar far wake characteristics compared to the computationally much more expensive complex turbine parametrizations such as the actuator line method [11]. The simulated wind turbine has a rotor diameter of D = 100 m, a hub height of 160m. A constant thrust coefficient of C T = 0.75 was applied. A stationary and fully turbulent neutral ABL is generated in a precursor simulation without wind turbine using periodic inflow conditions. A geostrophic wind speed of u g = 10 ms −1 and a roughness length of z 0 = 0.05 m are prescribed. The Coriolis force is neglected, so that the mean flow is aligned with the x-axis in all heights. After 12 hours, a stationary inflow is achieved. The main simulation with the wind turbine is initialized with the final results of the precursor simulation. Non-periodic inflow boundary conditions are applied with a fixed inflow profile and turbulence recycling (details in [35]). The domain size of the main simulation is about 8 km x 2 km x 0.5 km with a grid size of 4 m. With this large domain we ensure that the wake is virtually unaffected by the boundaries. The turbine is placed in to center of the domain in y-direction and 2500 m downstream of the inflow. Hereafter, the origin of the coordinate system will be located at the hub of the turbine.
For our analysis, we focus on data in a y-z plane perpendicular to the main flow direction (white dashed line in Fig. 1a) 4 D downstream of the turbine. In this distance we expect only minor differences between the standard actuator disk model used here and more sophisticated actuator models [36]. Furthermore, we confine our analysis on the main velocity component yielding a spatio-temporal data field u(y, z, t). The generated data has a spatial resolution of 4 m and is written out at a rate of 2 Hz with a time series length of 11750 s excluding the first 500 s of the simulation. A snapshot of the data for t = 50 s can be seen in Fig. 1b. It shows a pronounced wake downstream of the turbine with a strong reduction of the flow velocity. After 2 to 3 D the recovery of the wake deficit sets in and at around 7 D (right border of the plot) the wake has almost fully recovered.

POD Theory
Subsequently, we briefly summarize the mathematical definition and important properties of the POD. Details can be found e.g. in [25]. As its name suggests the POD decomposes the field u(r, t) into a superposition of basis functions φ j : where u(r, t) is a scalar mean free field, r a spatial vector (in our case (y, z))and t a scalar representing time. In the case of the POD the basis functions φ j are called POD modes which will be defined in the following. The POD is motivated by the aim of finding the optimal basis functions with respect to the kinetic energy of the flow. In other words we are searching for best N functions φ j (r) to approximate u(r, t): Thus, we would like to minimize the mean squared error: with respect to the basis functions φ j . Using a variational approach one can show that the described minimization problem is equivalent to the eigenvalue equation of the covariance operator: This equivalency is the main reason why the POD is a simple but powerful tool. Instead of treating a complex optimization problem, we can now simply solve an eigenvalue problem of an hermitian operator. In practice Eq. (4) is often approximated by the eigenvalue problem of the discretized covariance matrix where u k is the value of u at the k-th grid point. In the POD basis the covariance operator is obviously diagonal yielding a linearly decorrelated description: Due to Parseval's relation the j th eigenvalue can be interpreted as a measure for the kinetic energy contained in the j th mode yielding for the mean squared error of a reconstruction u (N ) : In the following, we will approximate u(r, t) by N j=1 a j (t)φ j (r) using different numbers of modes N .
In our case u(r, t) = u(y, z, t) has a nonzero mean field which is removed before the POD and added again for an approximation. Thus, we define: which we will call a POD reconstruction of u(y, z, t) in the following.
For many applications the method of snapshots (see e.g. [25] ) is used instead of directly solving Eq. (4). Here, we solve the direct problem, since the number of time steps used for the analysis is of the same order as the number of grid points. Therefore, the method of snapshots is not more efficient.

Preprocessing
Since we aim to describe the wake dynamics in an ABL separately from the ABL itself, we apply two preprocessing steps to the data u(y, z, t) which has been described in Sec. 2.
As a first step, we subtract the mean field of the flow without turbine u 0 (y, z) ( Fig. 2a) from u(y, z, t). This lead to a slightly better separation from the ABL structures (Fig. 2c). Additionally, we change the sign of u which is of minor importance here. Second, we extract the deficit simply by using a (temporally local) relative threshold. This means that we set all values weaker than 40% of the current deficit maximum to zero (Fig. 2d). A similar procedure has been applied in [37]. The results presented here are relatively robust against the choice of the threshold with similar results choosing 20% − 50%. Without applying the threshold however the results change strongly, particularly the form of the POD modes. The modes then contain a lot of large scale structures of the ABL which are present in the LES also without an actuator disk present. The results presented in the following are mainly confined to the POD reconstructions of the preprocessed field after applying the threshold. However, it is important to point out that even though we choose a relatively strong threshold, we are able to reconstruct aspects of the full field without threshold application as well (see Sec. 4.5).

POD Modes
Now, we calculate the covariance matrix corresponding to the preprocessed field and numerically solve its eigenvalue problem defined by Eq.(4). The obtained eigenfunctions are the POD modes of u(y, z, t) which are shown in Fig. 3-5 and will be described in the following. The modes show clear structures with a trend from larger to smaller scales with increasing mode number. This trend is easily explained by the fact that the modes are sorted with respect to energy and that the kinetic energy in a turbulent flow typically decreases with scale. The modes resemble Fourier modes in the azimuthal direction. We observe approximately axisymmetric modes 3 and 6 and pairs showing a dipole-like structure (1, 2), quadropole structure (4, 5) and hexapole structure (7,8). These observations indicate an approximate statistical isotropy. In a statistically isotropic field, rotational invariance is carried over to the eigenspaces spanned by the POD modes. In other words, modes are either isotropic themselves or correspond to a degenerate eigenvalue with an invariant eigenspace. Here, however there is no perfect rotational symmetry, which is for example revealed by the fact the eigenmode pairs described above do not correspond to the same eigenvalue.
However, the rotational symmetry seems to be at least partially retained even though it is clearly broken by the ABL.
The extracted modes are remarkably similar to the modes found by Andersen et al. [24] in the far wake indicating that the they are a relatively robust result. We even expect that the approximate symmetry in the field will always lead to a similar set of modes. However, if the symmetry is clearly broken as in the near wake due to rotation of a turbine or if the wake interacts strongly and nonlinearly with the ground the modes will probably change.

POD Reconstruction
We now use the extracted POD modes to approximate the field u(y, z, t) according to Eq. (8) and vary the number of modes used for the reconstruction. Examining the reconstructed snapshots ( Fig. 6 and 7), we see that more modes obviously lead to a more accurate description of the wake structure. To recover the spatial small scale structures of the wake very many POD modes are necessary. However, pure visualization does not offer many clues on the number of modes necessary to obtain a useful low order description. Therefore, we need to define other quantitative measures to draw conclusions on the quality of reconstruction.  (c) Full field A first and common measure for the quality of a reconstruction is the mean squared error. Due to Eq. (7) it is given by: yielding the relative error Fig. 8a shows the relatively slow decay of this error. Since the error is directly related to the turbulent kinetic energy we can conclude that we need around a 100 modes to recover around 80% of the average turbulent kinetic energy (Fig. 8b ). This is not surprising since in a turbulent flow the energy is spread out over a wide range of scales. The POD basis does not solve this problem even though it might perform better in inhomogeneous fields than a standard Fourier basis. q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q The slow decay of ε kin also suggests that useful reduced order models need to contain a lot of modes. However, it is not clear that in the context of wake modeling, the turbulent kinetic energy really is a good indicator for the quality of a reconstruction of a wake. Therefore, we propose the use of alternative measures of quality in the next section.

Alternative Measures for the Quality of Reconstruction
Our main interest obviously lies in the impact of the wake on a sequential turbine. Therefore, a measure used to draw conclusions on the quality of a reconstructed field should be related to quantities which characterize forces on a turbine or a turbine's response to the field. Such a measure could for example be given by a specific load or the power a turbine would produce if it was standing in the wake.

A general quality measure
Thus, we define a general scalar quality measure M (t) (e.g. the power) as a function of the flow u(y, z, t). In other words, M (t) results from the application of a corresponding operatorM to the flow u(y, z, t):M [u(y, z, t)] =: M (t) .
Usually, M will depend on the field u(y, z, t) confined to the virtual rotor area only (see e.g. Fig. 9) and not on the velocity in the rest of the domain. Analogue to Eq. (11), we define for a reconstructed field: We now consider a reconstructed field as "good" if M (N ) (t) is a good approximation of M (t). However, there is still some freedom in assessing this approximation. One obvious way to define a relative reconstruction error is e.g. through the 2-norm of the time series: which will be refered to as the standard error in the following. Another possible choice is motivated by the idea that for some measures the mean M t might be less important than the fluctuations M (t) − M t . It is e.g. often assumed that for the calculation of fatigue loads the influence of the mean is negligible. Hence, we define: which will be refered to as dynamical error in the following. Obviously, other error definitions are possible. We could for example be only interested in reproducing statistical quantities only, such as the mean or variance of M (t) but here, we confine ourselves to the errors defined above.

Introducing Alternative Measures
We now define three different measures as M (t) which are related to relevant quantities of a sequential turbine. To this purpose, we assume the turbine to be a disk in the wake as shown in Fig. 9. Since not only the deficit is acting on the turbine, the measures defined in the following will also add back the mean field without a turbine u 0 (y, z) defined in Sec. 3.2. To obtain a more compact notation, we will also writeũ(y, z, t) = u(y, z, t) + u 0 (y, z).
The first quantity we analyze is the effective velocity, which we define as the average velocity over the disk area: u eff (t) = u(y, z, t) + u 0 (y, z) disk = ũ(y, z, t) disk , where ... + u 0 (y, z) disk now plays the role of the operatorM in Eq. (11). In the spirit of actuator disk theory this quantity is related to the power output and the thrust force on the turbine: As a second measure, we use the energy flux through the disk given by: where A is the area of the disk. P is obviously related to the potential power output of the turbine. The last measure defined here, is related to the torque along the z-axis through the disk (Fig. 11b) and is defined by: where l(y) is the distance to the rotational axis.
In the case of a wind turbine the torque in z-direction is also called the tower top yaw-moment which is particularly important for the yaw-drive of the turbine. In contrast to u eff and P , the averaging over the disk is now weighted by l(y) yielding a dependence on the spatial distribution of the deficit over the disk.

Results and Discussion
We now evaluate these measures for our data and the POD reconstructions. While the dynamics of u eff and P cannot be captured using 2 or less modes, three modes already grasp the basic dynamic features. (Fig. 10). Consequently, the dynamical error shows a sudden decline from above 80% to below 20% for both measures (Fig. 12a). For τ z , grasping the basic dynamics is even possible including only the first mode (Fig. 11a) with a dynamical error of below 20% (Fig. 12a). Including 6 modes yields a dynamical error of below 10% for all three measures followed by a slower convergence to zero.
These results clearly indicate that models based on only very few POD modes might be possible. A model based on the first three modes for example could already describe the basic dynamics of all measures defined here. It is also important to note that the relatively good description with only 1-6 modes occurs despite the fact that a much smaller part of the kinetic energy has been reproduced. For example, less than 50% of the energy is reproduced with 6 modes when the dynamical error is already below 10%. Our results also show that the dependence on the numbers of modes used is different for different measures. Therefore, models of different complexity might be needed when aiming for the correct description of different measures. The improvement of dyn when adding a mode to the reconstruction might be seen as an indicator of its importance with respect to the considered measure. This indicates e.g. that the third mode plays a special role for the description of u eff and P while the first one is of major importance for τ z . Therefore, again depending on the relevant measure, possible models based on POD could also differ in the individual modes chosen and not only in the number of modes. This is further discussed in Section 4.4.
The standard error shows some major differences to the dynamic one as can be seen in Fig. 12. For u eff and P the standard error is already relatively low when only using the mean field of the flow, with around 10% for u eff and 20% for P . P shows a higher standard error than u eff for all N . For τ z , the standard error is much higher than the dynamical error with, e.g., around ε std ≈ 40% and ε dyn ≈ 20% when including the first mode only. Furthermore, it also strongly exceeds the standard error for u eff and P .
The strongly different behavior of the errors shows that assessing the quality of a reconstruction has to important steps. Besides choosing the time-dependent measures such as P, u eff relevant to a specific application we also need to carefully choose an error for the reconstruction of these quantities. This error should be chosen with respect to the properties of e.g. P (t) which we need to preserve in the reconstruction.
The reason for the different behavior of dyn and std obviously lies in the fact that dyn neglects the mean values of the measures in contrast to std . For u eff using only the steady mean field already yields a perfect reconstruction of u eff t : Since the fluctuations of u eff are much smaller then the mean value, the correct description of the mean obviously leads to a low standard error (see Eq. (13)) due to the normalization. For the dynamical error (Eq. (14)) the recovery of the mean obviously does not play any role. For P , P t cannot be fully recovered using just u(y, z, t) t due to: Consequently, the standard error is always higher for P than for u eff . Similarly, we have for τ z : For τ z , the fluctuations are much higher than for P . This yields a large difference between u 2 t and u 2 t resulting in the large standard error compared to P and u eff . The short discussion above does not only explain the different behavior of ε dyn and ε std , but also nicely illustrates a major challenge for steady mean field models. Since the conversion from wind to relevant quantities such as power or loads can be a nonlinear process, the mean velocity field does not necessarily offer a good description of other mean quantities.

Reconstruction using single modes only
In Section 4.3 we already mentioned that our results suggest that different modes are relevant for different measures. However, we included our modes ordered from high to low energy content as usual for POD reconstructions. For an optimal reconstruction with respect to the kinetic energy this is obviously the best choice. However, for the measures we introduced this might not be the optimal order. For example, using only the third mode actually yields a better reconstruction of u eff than only the first one (Fig. 13). It is not even theoretically guaranteed that we have a monotonous improvement meaning that an additional mode always improves our result. Generally, we could now ask which N modes to take for an N mode reconstruction optimized with respect to the measure of interest. Detailed and systematic answers to these questions however, are complex and beyond the scope of this paper. However, our results suggest that for a POD based model, the modes should not necessarily be chosen in an energetic order. Thus, not only the number of modes but also the individual modes used for a possible model can in principle be selected dependent on the measures of interested which are defined by the application of the model.

Reconstruction of the field before threshold application
The results presented above correspond to POD reconstruction of a preprocessed field including an application of simple threshold (Sec. 3.2). As already mentioned in Sec. 3.2, it is however also possible to use the POD modes extracted through threshold application for a reconstruction of the field before application of the threshold. For this purpose, we only need to use the field before applying the threshold as u(y, z, t) in Eq. (8). Fig. 14a show that the results stay essentially the same illustrating that the modes do not only work after a rather artificial threshold application. This shows that the extracted modes are not only able to approximate properties of an "artificially" preprocessed field but do contain information of the full wake. From this point of view the threshold could be seen as "trick" to extract the modes corresponding to the wake only. This reconstruction as well as the one in the former sections do obviously not perform well outside the wake region where additional ABL models have to be used when aiming for a more complete description.

Interpretation of the Modes
For a better understanding of the results in the last section, we now relate the POD modes to dynamical properties of the wake deficit. Particularly, we show evidence why the first three modes can be enough to capture basic dynamical behavior.
The first dynamical property we analyze is the meandering, which corresponds to the large scale movement of the velocity deficit. For this purpose, we determine a center of the deficit as which we evaluate for every time step. This definition can be interpreted as the center of energy. Other approaches are possible and discussed cf. [20,37,38]. To investigate the relation to the POD modes, we compare the trajectories of the center to the amplitudes a i (t) of the POD modes. The horizontal trajectory is strongly correlated to the amplitude of the first mode (Fig. 15a) yielding a linear correlation coefficient of ρ(y c , a 1 ) = .78 (Fig. 15b). The correlations to other amplitudes are almost negligible. Analogously, we find ρ(z c , a 2 ) = .75 for the vertical movement and the second amplitude (Fig. 16). We can now also interpret the deficit position e.g. y c (t) as a quality measure in the sense of M (t) in Section 4.1. It turns out that the first mode really is enough to capture the basic dynamics of the horizontal movement (Fig. 17). However, large amplitudes can not be recovered. The connection of the first mode to the horizontal meandering also yields a deeper understanding of the behavior of τ z . The torque in z-direction and thus the measure τ z are obviously strongly influenced by the horizontal position of the deficit. A large movement to the right for example yields a strongly asymmetric force on the disk inducing a positive torque in z-direction. Since one mode is enough to capture basic dynamics of the horizontal movement, it is also sufficient for an approximate description of τ z .
The results in Sec. 4.3 show that for the effective velocity the third mode also plays an important role. It turns out that we find a relatively strong (anti-)correlation of a 3 (t) to the average amplitude of the deficit (Fig. 18), which we define just as the velocity spatially averaged over the complete deficit (extracted through the threshold application). It is not really surprising that the effective velocity is strongly influenced by such a quantity since both are defined through averaging processes over different areas. However, it is still noteworthy that the amplitude fluctuations of the deficit do play an important role for the approximation of the effective velocity and thus probably also for other quantities relevant to a turbine in the wake. Therefore, including a fluctuating amplitude could possibly improve existing simplified dynamic wake models which have meandering as their only dynamic feature.

Conclusions
We applied a POD analysis to the large eddy simulation of an actuator disk in a turbulent atmospheric boundary layer. Our results strongly support the idea of building simplified dynamic wake models based on POD. Such models could be a useful alternative to dynamic models mainly based on the meandering process (see e.g. [19]).
We believe that the extracted modes are strongly influenced by an approximate statistical rotational symmetry in the flow and thus suspect them to be a rather robust result. This is also supported by similar modes obtained by Andersen et al. [24] in a slightly different setting. Models based on similar modes would thus have a relatively wide range of applicability.
We propose that the quality of POD reconstructions should not only be assessed by considering the recovery of turbulent kinetic energy but should be based on measures relevant to a sequential turbine in the wake. The basic dynamics of such relevant quantities (e.g. the energy flux through a disk) of the wake could be captured with only a few modes (1-6) even though only a small part of the turbulent kinetic energy was recovered. This indicates that the POD could in principle yield useful models of very low order. Furthermore, our results support the idea of simplified dynamic wake models in general since we show that relevant information is contained in only a few degrees of freedom.
Additionally, we showed that the modes used for a model could possibly be selected dependent on the application. This is based on the result that the approximation errors of the different measures showed a qualitatively different dependence on the number of modes used. Furthermore, it was for example shown that different modes were important for different measures and that including modes in the usual energetic order is not generally the best choice.
By relating the POD modes to specific properties of the wake a deeper understanding of our results could be gained. The first and second mode were found to be responsible for the horizontal and vertical meandering movement of the wake while the third one seems to be related to the average amplitude of the wake. From this perspective we can, e.g. for the measure related to the tower top yaw moment, connect the importance of mode 1 to the special role of the horizontal movement of the wake. Furthermore, this interpretation enables us to view our results in comparison to other dynamic wake models. For example, a description with the two first POD modes is related to a dynamic wake model which includes meandering as its only dynamic feature. The importance of mode 3 and thus the average amplitude of the wake indicates that meandering alone might not be sufficient to capture the dynamics of e.g. the thrust on the turbine. Thus, a fluctuating amplitude might be a useful extension to a meandering model of the wake.
To show the robustness of our results, this study could obviously benefit from the analysis of additional data sets, such as more detailed simulations or high speed PIV data from wind tunnel experiments. The quality measures introduced, even though related to the performance of a turbine in the wake, are very simple. Therefore, aeroelastic code could be used, as a next step, to obtain better measures of quality. So far we have mainly investigated the quality of reduced descriptions of the wake dynamics. Building a reduced order model based on the extracted POD modes is a very challenging task. Since we need to model the temporal dynamics of the expansion coefficients of the modes (a j in Eq. (8)). To do this, we are going to consider the expansion coefficients as stochastic processes and estimate the corresponding model equations in the spirit of e.g. [39][40][41].