## Abstract

The dynamics of the velocity field resulting from the interaction between the atmospheric boundary layer and a wind turbine array can affect significantly the performance of a wind power plant and the durability of wind turbines. In this work, dynamics in wind turbine wakes and instabilities of helicoidal tip vortices are detected and characterized through modal decomposition techniques. The dataset under examination consists of snapshots of the velocity field obtained from large-eddy simulations (LES) of an isolated wind turbine, for which aerodynamic forcing exerted by the turbine blades on the atmospheric boundary layer is mimicked through the actuator line model. Particular attention is paid to the interaction between the downstream evolution of the helicoidal tip vortices and the alternate vortex shedding from the turbine tower. The LES dataset is interrogated through different modal decomposition techniques, such as proper orthogonal decomposition and dynamic mode decomposition. The dominant wake dynamics are selected for the formulation of a reduced order model, which consists in a linear time-marching algorithm where temporal evolution of flow dynamics is obtained from the previous temporal realization multiplied by a time-invariant operator.

This article is part of the themed issue ‘Wind energy in complex terrains’.

## 1. Introduction

The power performance and durability of a wind farm are significantly affected by the spatial and temporal variability of the wind field flowing across wind turbine rotors. Wind fluctuations can be consequent to the intrinsic turbulent nature of the atmospheric boundary layer or due to turbine wakes evolving within a wind farm. Furthermore, wind characteristics, such as turbulence intensity and wind shear, vary periodically according to the daily cycle of the atmospheric stability. During daytime convective conditions, heat flux from a warm ground surface leads to higher turbulence intensity and lower wind shear with respect to night-time stable conditions, for which turbulence is mainly mechanically produced in the proximity of the ground due to the significant wind shear. Under convective regimes of the atmospheric stability, an enhanced power production was documented for unwaked turbines located within an onshore wind farm [1]. By contrast, variability in power production as a function of the atmospheric stability becomes negligible for turbines typically invested by upstream wakes. A larger power production for convective atmospheric conditions was also estimated for the wind turbine Vestas V_{80}-2 MW by using met-tower data and a statistical model [2]. Conversely, larger power production was observed under stable atmospheric conditions for the West Coast North American wind farm, especially during spring and summer [3].

Previous works showed that fatigue loads can be significantly larger than those prescribed by the International Electrotechnical Commission (IEC) standards, for which the daily cycle of atmospheric stability is neglected [4]. Higher fatigue loads typically occur under strong unstable convective regimes [5]. However, specific loads, such as the out-of-plane blade bending moments, are typically larger under stable conditions associated with higher wind shear [6].

The scenario becomes even more complicated by considering the intra-wind-farm wind field, which is characterized by the dynamics connected with the downstream evolution of wind turbine wakes and their complex interactions. Wind turbine wakes are the result of the power capture carried out by the rotation of the turbine blades, and they are characterized by a velocity deficit and a swirling motion acting in the opposite direction to the rotor rotation. In the near wake, coherent vorticity structures are present, such as a system of helicoidal vortices shed from the tip of each blade and the hub vortex, which is a columnar vorticity structure mainly oriented in the streamwise direction. These vorticity structures evolve downstream and undergo different instability mechanisms, whose temporal and spatial characteristics are determined by the turbine loading and incoming wind conditions [7].

Early analytical studies showed that a system of helicoidal vortices is unstable to small harmonic perturbations, producing different instability mechanisms characterized by a broad range of unstable wavenumbers [8,9]. More recent studies have been focused specifically on the instability of tip vortices generated by wind turbines, showing that the dominant instability mechanism is the mutual inductance between consecutive vortices, leading to vortex pairing, merging and finally to dissipation of the vorticity structures to small-scale turbulence [10–12]. The dominant unstable modes are characterized by out-of-phase displacements of consecutive vortices, with a non-dimensional spatial growth rate of about *π*/2, and wavenumbers equal to half-integer multiples of the number of blades [11]. The hub vortex undergoes helicoidal instability [13,14], whose azimuthal wavenumber and growth rate are determined by the thrust coefficient of the turbine, the lift distribution over the blade span and the incoming wind conditions. Linear stability analysis showed that the hub vortex instability is promoted by the radial shear of the streamwise velocity, wake swirl, and it is damped by the incoming wind turbulence [7].

The ultimate goal of our research consists in the development of a reduced order model (ROM) to predict wind velocity fluctuations within a wind farm with limited computational costs and a prescribed accuracy. Therefore, this tool might be useful for the design, control and optimization of wind power plants. In this paper, two large-eddy simulations (LES) datasets of wind turbine wakes are used as test cases: one is performed with the classical actuator line model; the other one simulates not only the blade rotation, but also the presence of the turbine nacelle and tower through the immersed boundary method [15]. The obtained three-dimensional velocity field is interrogated with two modal decomposition techniques, namely proper orthogonal decomposition (POD) [16] and dynamic mode decomposition (DMD) [17,18]. These techniques enable the detection of the most dominant coherent structures present within the wake velocity field. Furthermore, their energy content and spectral contribution are also characterized. The DMD modes of interest are then selected to formulate a ROM consisting in a linear time-marching algorithm.

The remainder of the paper is organized as follows: the LES datasets are described in §2. The principal statistics of the wake velocity field obtained with and without the simulating turbine tower and nacelle are then discussed in §3. POD and DMD algorithms are briefly summarized in §4, while their results and the formulation of the ROM are described in §5. Final remarks are reported in §6.

## 2. Large-eddy simulation dataset

The reference frame used for this work has its origin at the centre of the turbine rotor. The *x*-axis is parallel to the incoming wind direction and positive pointing downstream; the *z*-axis is in the vertical direction and positive pointing upwards; whereas the *y*-axis is in the transverse direction and oriented to produce a right-handed reference frame. Figure 1 shows a sketch of the reference frame, the turbine model, incoming wind direction and the computational domain. The velocity components are referred to as (*u*, *v*, *w*) in the (*x*, *y*, *z*) directions, respectively.

The wind turbine considered for this survey is that used for a well-known blind test based on wind tunnel experiments [19]. This three-bladed wind turbine model has a diameter, *d*, of 0.89 m. Two LES of the selected wind turbine were performed: one simulation with only an actuator line model, while for the second case a wind turbine tower and nacelle are included in the simulation via an immersed boundary method [15], which allows the flexibility of imposing complex geometries inside the domain without the need for body-fitted grids.

The streamwise velocity at the turbine location and hub height is *U*_{hub}=10 m s^{−1}, corresponding to a Reynolds number based on the rotor diameter of about 609 600. The rotational frequency of the rotor, *f*_{hub}, is 10.71 Hz, producing a tip speed ratio equal to 3 (i.e. the ratio between the tangential velocity at the blade tip and the streamwise velocity at hub height). The velocity field is sampled with a frequency of *F*_{samp}=112.36 Hz for this analysis, which corresponds to about 10.5 times the hub rotational frequency. The two LES datasets consist of 282 and 387 snapshots for the actuator line case and the case with the turbine tower and nacelle, respectively. Therefore, the former corresponds to about 27 full rotations of the turbine rotor and the latter to about 37 full rotations. A larger number of snapshots for the dataset related to the LES with the tower and nacelle was required to achieve convergence of the turbulence statistics.

The computational box has dimensions 12.5*d*×3*d*×2.1*d*, i.e. *N*_{x}=1024×*N*_{y}=512×*N*_{z}=512 grid points evenly spaced in the streamwise (*x*), spanwise (*y*) and wall-normal (*z*) directions, respectively. A uniform velocity profile is used as the inflow condition, while Sommerfeld radiation boundary conditions are imposed at the outlet as follows:
2.1
where the index *i* represents the three Cartesian directions and *C* is the phase velocity of the flow [20].

LES of the wind turbine wake were carried out by using Smagorinsky sub-grid scale model with a coefficient *C*_{s}=0.18. Van Driest damping was used near the top, bottom and lateral walls of the numerical domain, and an immersed boundary method [15] was used to simulate accurately the presence of the solid walls of the wind tunnel related to the reference experimental dataset [19]. The numerical code uses a second-order finite-difference scheme in space with a fractional step, and third-order Runge–Kutta for the time advancement [21,22]. Flow around the wind turbine blades is modelled using the actuator line model [23]. Lift and drag forces are obtained from look-up tables of the turbine model used [19] and by computing the local angle of attack *α* and velocity amplitude at each blade section. The forcing produced by the blade rotation is distributed radially along three lines representing the wind turbine blades. To reduce numerical oscillations and ensure numerical convergence, the body forces representing the blade aerodynamic loads are spread through a Gaussian function with a reference length equal to four times the local grid size [24].

## 3. Statistics of the wake velocity field

In this section, the effects of including the turbine nacelle and tower in the LES of wind turbine wakes are investigated. The LES data have been validated against the experimental results of blind test 1 performed at the Norwegian University of Science and Technology, Trondheim, Norway [19,25]. Regarding the time-averaged velocity field, figure 2*a* shows that the wake velocity produced for the LES without the tower and nacelle is practically axisymmetric, while a swirl velocity persisting far downstream is present. An unrealistic higher velocity is observed at the wake centre, which is due to the lack of a turbine hub for the actuator line model used.

By including the tower and nacelle in the LES, the time-averaged velocity field is significantly modified, as shown in figure 2*b*. The wake produced by the tower is evident, while the rotor wake is characterized by lower velocity values than the previous case without the presence of a jet at the centre of the wake. The wake swirl is enhanced in the near wake, which then decreases moving downstream. The swirl velocity is slightly larger in the lower part of the rotor wake, which is an effect of the interaction with the tower wake. Further downstream, the wake is slightly skewed towards negative *y*-positions, which is consistent with the swirling velocity within the wake. The maximum deflection of the wake is observed in the proximity of the bottom blade-tip where the rotor wake crosses the tower wake. The wake produced by the tower undergoes a significant transverse expansion, which is not observed for the rotor wake.

Useful information about the wake turbulence, wake recovery mechanisms and interaction between the rotor and tower wakes is obtained from the analysis of the Reynolds shear stresses in figure 3. For the LES data without the tower and nacelle, wake turbulence is clearly connected with the presence of the helicoidal tip vortices at a radial distance roughly equal to the rotor radius, and to the wake swirl, hub and root vortices in correspondence to the wake core. Also the wake turbulence reveals the roughly axisymmetric morphology of the wake.

By including the tower and nacelle in the LES, all the stresses generally increase, which is a consequence of the enhanced velocity deficit in correspondence to the rotor and tower wakes. Interaction of the rotor and tower wakes produces enhanced shear stresses in the lower part of the rotor wake, and just after the intersection between the tip vortices and the vortices shed from the tower, i.e. the area for negative *y*-coordinates and heights between the hub and blade bottom tip. This feature suggests that by including the turbine tower the mechanism of wake recovery is substantially different from that for the previous case of a roughly axisymmetric wake. For the latter, wake recovery is mainly governed by the downstream evolution of the tip vortices, hence their instabilities and decay to small-scale turbulence. Breakdown of the tip vortices then promotes flow entrainment from the surrounding, hence wake recovery. By including the tower, a new physical mechanism is responsible for a faster wake recovery. Indeed, the rotor–tower–wake interaction produces a region which is in the proximity of the blade bottom tip and after the interaction zone between the rotor and tower wakes, where the wake vorticity structures are suddenly disorganized and less coherent, leading to an enhanced flow entrainment from the surrounding flow. This mechanism is already present in the very near wake; thus, it affects significantly the overall wake recovery.

## 4. Modal decomposition techniques

The dynamics of the wake velocity field are investigated through two different modal decomposition techniques, namely POD [16,26–28] and DMD [17,18]. Modal decomposition was performed by considering the three velocity components over the domain −0.5≤*x*/*d*≤6.5, −0.75≤*y*/*d*≤0.75 and −1≤*z*/*d*≤0.7.

POD allows an orthonormal basis to be achieved for the turbulent velocity field obtained from LES. Velocity fluctuations with zero time average *u*_{i}(*x*,*y*,*z*,*t*) (*t* is time) are represented through a linear combination of deterministic functions, the POD modes, *ϕ*_{j}:
4.1
where *N* is the total number of snapshots. The eigenfunctions of the covariance matrix of the velocity snapshots, *ϕ*_{j}, represent the typical realizations in a statistical sense. The parameters *a*_{j}(*t*) are uncorrelated coefficients denoted as principal components, which represent the amplitude of each mode as a function of time. POD provides a modal decomposition that is completely *a posteriori* data dependent and does not neglect the nonlinearities of the original dynamical system, even being a linear procedure. Furthermore, the POD basis is orthogonal and optimal, i.e. among all linear decomposition techniques, it provides the most efficient detection, in a certain least-squares optimal sense, of the dominant components. POD is computed with the method of snapshots [29], and the POD modes, *ϕ*_{j}, are expressed as a linear combination of the snapshots with the respective eigenvectors of the covariance matrix.

The other modal decomposition technique used for this work is DMD, which is based on the assumption that the temporal evolution of a dynamic system is generated through a linear, time-invariant operator applied to the previous snapshot. It is worth pointing out that the assumption of a linear system for the DMD time-marching algorithm is not equivalent to performing a linearization of the dynamic system, as performed for instance with linear stability analysis. For DMD, all the nonlinearities in the evolution of the system dynamics are retained, and only the advancement in time is linearized. Snapshots of the velocity fluctuations are arranged as columns and two data matrices are generated:
4.2
The time-invariant linear operator defined in DMD for the time advancement of the velocity dynamics is *A*, which leads to
4.3
If *r* is the dimension of the intended ROM, with *r*<*N*, a matrix *F* can be defined through DMD in order to be the optimal representation of the matrix *A* in the basis spanned by the POD modes of *Ψ*_{0},
4.4
*U** denotes the complex conjugate transpose of the POD modes of the snapshot matrix *Ψ*_{0}, obtained with an economic-size singular value decomposition,
4.5
where *Σ* is the *r*×*r* diagonal matrix of the singular values. The matrix *F* can be determined by minimizing the Frobenius norm of the difference between *Ψ*_{1} and *AΨ*_{0} using the optimal DMD algorithm [30,31],
4.6
The different dynamics of a wind turbine wake flow are then properly characterized through the DMD modes, whose frequency is equal to , and the growth rate is equal to , where *μ*_{i} are the eigenvalues of the matrix *F* and *F*_{samp} is the sampling frequency of the available LES dataset.

## 5. Results of the modal decomposition and reduced order model

The results obtained through the modal decomposition techniques are presented in this section. For the LES case performed by neglecting the presence of the turbine nacelle and tower, an effective way to investigate the physical contribution of the various POD modes to the wake dynamics is achieved by plotting the projection of the instantaneous transverse vorticity, *ω*_{y}, over the wake vertical symmetry plane (*x*, *y*=0, *z*). In figure 4*a*, a snapshot of *ω*_{y} projected onto the POD modes 1 and 2, which are coupled (i.e. they are orthogonal POD modes representing roughly the same contribution in energy and frequency), is reported. The first two POD modes enable 50.56% of the total turbulent kinetic energy (TKE) connected with the turbine wake to be extracted. A very complex system of tip, root and trailing vortices is captured through POD modes 1 and 2, which also unveil a reach scenario of vortex dynamics within the turbine wake. In the very near wake, tip and root vortices shed by the turbine blades are characterized by vorticity of opposite sign. The core size and vorticity magnitude of the tip vortices are noticeably larger than those for the root vortices. Within the area included between consecutive vortices shed from the blade tip and root, vorticity cores with opposite sign are formed. These secondary vortices are mainly produced from the velocity induced by the primary vortices and the relatively small pitch of the helicoidal vortices, which is a consequence of the relatively low tip speed ratio of the turbine. In the near wake, the streamwise wavelength connected with POD modes 1 and 2 is roughly equal to 0.276*d* and 0.274*d* for the tip and root vortices, respectively.

Moving downstream, slightly outwards with respect to the radial position of the root vortices, nuclei with vorticity of same sign as the adjacent root vortices are formed. These vortices are the result of the roll-up of the trailing vorticity shed by the blades, which is induced by strain of the root vortices with vorticity of the same sign. Further downstream, a small amount of vorticity is observed at the wake core.

While the tip vortices are advected downstream, their vorticity cores seem to be deformed, indicating that an elliptic short-wave instability might occur. For the outer part of the tip vortices, namely towards larger radial distances, shear layers departing from the cores of the tip vortices become gradually more evident by proceeding downstream, which might be the footprint of a centrifugal instability. Moving downstream, the trailing vorticity becomes more intense, and for *x*/*d*>5 trailing vorticity and tip vortices clearly interact splitting each tip vortex into two smaller vortices. In this case, this instability of the tip vortices might be ascribed to a shear-driven instability mechanism. In the far wake, root vortices, trailing vorticity and a double system of helicoidal tip vortices with vorticity of the same sign and located at slightly different radial distances around the blade tip location are observed.

In order to estimate the spectral contribution connected with each POD mode, the Welch spectra of the temporal coefficients, *a*_{j}(*t*), of the various POD modes are reported in figure 5*a*. It is evident that the dominant frequency for POD modes 1 and 2 corresponds to *f*/*f*_{hub}≈3, which is related to the shedding of vorticity structures from the three-bladed turbine.

POD modes 3 and 4 capture 11.3% of the total TKE of the analysed data and show dynamics affecting mainly the tip vortices, for which the typical streamwise wavelength is about half that of POD modes 1 and 2, i.e. 0.136*d* (figure 4*b*). The frequency associated with these POD modes is about *f*/*f*_{hub}≈4.5, as shown in figure 5*a*. Similarly to POD modes 1 and 2, the evolution of an elliptic instability of the tip vortices is also detected with POD modes 3 and 4, which then leads to splitting of the vorticity cores, and finally to double concentric systems of helicoidal tip vortices. POD modes 5 and 6 capture 1.73 % of the total TKE and represent a long-wave modulation for both tip and root vortices (figure 4*c*), with frequency about *f*/*f*_{hub}≈1.5 (figure 5*a*).

This modal–spectral analysis suggests that POD modes 1 and 2 represent the carrier with frequency *f*/*f*_{hub}≈3 of the shedding of tip and root vortices from the three-bladed turbine, while POD modes 3, 4, 5 and 6 represent a modulation with frequency *f*/*f*_{hub}≈1.5. The reconstruction of the transverse vorticity, *ω*_{y}, by using all the considered six POD modes in figure 4*d* shows highly complex wake dynamics. Focusing on the top-tip area (for the low-tip region holds a similar description but with vorticity of opposite sign), in the near wake the mutual induction between consecutive positive–negative tip vortices leads to the generation of a secondary core with positive vorticity (red) at *x*/*d*≈2. Moving downstream, the secondary vortex becomes stronger and moves slightly inboard. The mutual induction between two consecutive vortices with vorticity of the same sign induces a rotation of the secondary vortex around the primary until it achieves an equilibrium position for *x*/*d*≈6. At this downstream location, two coaxial systems of helicoidal tip vortices with vorticity of the same sign are present.

The scenario of the vortex dynamics within a wind turbine wake becomes even more complex by including in the LES the wind turbine tower and nacelle. The main difference with respect to the previous dataset consists in the interaction between the rotor wake and the alternate vortex shedding from the turbine tower. The vertical vorticity, *ω*_{z}, of the most energetic POD modes (1–4) is reported in figure 6*a*, showing that the dominant POD modes capture the alternate vortex shedding from the turbine tower. The vorticity magnitude, *ω*, in figure 6*e*, highlights that these vortices are roughly vertical and significantly bent upstream in the region crossing the rotor area, which is due to the velocity deficit present in the rotor wake. These vortices become significantly weaker at a downstream distance of *x*/*d*≈2, but still noticeable further downstream. In figure 5*b*, the Welch spectra of the temporal coefficients of the considered POD modes show a dominant frequency of *f*/*f*_{hub}≈1.7.

In figure 6*b*, it is shown that POD modes 5 and 6 are related to the shedding of vorticity structures from the turbine blades, which occurs with a frequency of *f*/*f*_{hub}≈3 (figure 5*b*). The main effect produced by the interaction of the rotor wake and the alternate vortex shedding from the turbine tower consists in a disorganization of the wake vorticity structures in the lower half of the rotor wake. Tip vortices are only observed at the top-tip region of the wake, while root vortices are not observed. This flow feature indicates that the presence of the turbine nacelle inhibits the formation of coherent vorticity structures at the wake core. The vorticity cores of the tip vortices are deformed while proceeding downstream, which can be connected to a short-wave elliptic instability, while the radial location of the tip vortices seems to be affected by a long-wake fluctuation (instability). In contrast with the previous LES dataset obtained without the presence of the turbine tower and nacelle, no splitting of the tip vortices is observed in the far wake. The vorticity magnitude reported in figure 6*f* shows that, with the first six POD modes, the dominant wake vorticity structures are captured, which represent 15.56% of the total TKE.

Various POD modes with different energy content and representing instabilities of the tip vortices have been detected, which are reported in figure 6*c*. By combining these POD modes with the above-mentioned modes characterized by higher energy content, a pairing mechanism of the tip vortices is unveiled, as shown in figure 6*g*. For this LES dataset, tip vortices are not evenly separated in the streamwise direction, but mutual inductance of consecutive vortices leads to an increase/decrease in the distance between consecutive vortices. Therefore, including the turbine tower and nacelle for simulating wind turbine wakes affects not only the mean and turbulent kinetic energy, but also fluctuations in coherent vorticity structures and their instability mechanisms.

Finally, specific POD modes representing the interaction between the rotor wake and the vortex shedding from the turbine tower are detected, such as those reported in figure 6*d*. These POD modes represent the deformation of the wake vorticity structures due to their interactions, and they are mainly located at the blade bottom tip.

The vorticity structures detected through the DMD analysis are very similar to the POD modes presented in this section. They are not presented here for the sake of brevity. DMD has the advantage of directly providing the frequency associated with each DMD mode, without performing the spectra of the temporal coefficient as for the POD modes. In figure 5*c*, the DMD spectra show that similar spectral contributions to those of the POD modes are obtained for both LES datasets.

As described in §4, DMD allows a time-marching algorithm to be defined for prediction of flow dynamics through a linear operator (equation (4.3)). By leveraging equation (4.4), a ROM can be formulated by including in the linear operator only the DMD modes of interest, similarly to the POD modes presented above (for more details, see [32]). As a general procedure, we recommend developing ROMs with an increasing number of DMD modes in order to avoid inaccurate conclusions in case complex multi-component dynamics have been only partially singled out with a limited number of DMD modes.

In figure 7*a*, the DMD spectrum obtained from the LES dataset without the tower and nacelle, and using 200 snapshots, is presented. The highest number of DMD modes achievable is 199. The ROM was formulated by gradually removing an increasing number of DMD modes. As shown in figure 7*a*, for one case modes with *f*/*f*_{hub}>4.7 were removed, for another only low-frequency modes ( *f*/*f*_{hub}<1.4), then both high- and low-frequency modes, finally keeping only DMD modes in the proximity of the dominant contributions with *f*/*f*_{hub}≈1.5, 3 and 4.5.

Accuracy of the ROM is estimated for the streamwise velocity over the wake vertical symmetry plane at *y*=0. Error is estimated as the percentage difference between the RMS value of the instantaneous velocity field estimated through a ROM with respect to the original LES data. The RMS error should not be interpreted as a bad performance of the ROM, rather as a level of approximation consequent to the number of DMD modes selected for the ROM. In figure 7*b*, it is shown that when all the DMD modes are selected for the ROM, then the RMS error is identically null for each time step. Conversely, if the number of selected DMD modes is reduced, the error increases and it is not constant as a function of time. Selecting 60 DMD modes, the error can be as large as 40% of the original LES data. The performance of the ROM consisting of a different number of DMD modes has been estimated only with respect to the available LES dataset. However, it is worth highlighting that, according to equation (4.6), the DMD modes have been obtained as the optimal basis only for the provided LES dataset. Thus, it is not possible to ensure levels of accuracy similar to these reported in figure 7 for future time periods. A possible procedure to extend the applications of ROMs to forecasting is that proposed by [32]. In that work, a ROM obtained through DMD applied to an available data set is combined with a Kalman filter in case new observations are available during the forecast in order to perform an update of the flow predictions.

## 6. Final remarks

In this paper, LES data of the wake flow generated by a wind turbine have been investigated through two different modal decomposition techniques, namely POD and DMD, in order to detect the dominant wake dynamics and develop a ROM for prediction of wake dynamics.

One LES case was performed with the classical actuator line model, while a second case was performed for the same turbine and under the same operating conditions of the previous one, but including the turbine tower and nacelle. Statistics of the velocity field show that, for the LES case performed with the classical actuator line, an unrealistic speed-up is observed at the wake centre. Conversely, for the LES case with the tower and nacelle, an enhanced velocity deficit is observed, which is also extended to the tower region. The analysis of the Reynolds shear stresses allows different mechanisms of the wake recovery to be highlighted for the two LES datasets. For the case without the tower and nacelle, the wake is practically axisymmetric and the wake recovery is significantly affected by the instability of the tip vortices and their diffusion. Conversely, for the case with the tower and nacelle, already in the near wake a region with enhanced Reynolds stresses is observed at the blade bottom-tip region where the helicoidal tip vortices interact with the vortices shed from the turbine tower. This region is considered to be a source for an increased flow entrainment from the surrounding flow and for a faster wake recovery.

The modal decomposition techniques have been very effective in detecting and extracting the main flow dynamics occurring within the turbine wake. For the LES case without the tower and nacelle, the main dynamics of the wake vorticity structures are well captured by selecting only the first six POD modes. The tip vortices are characterized by a split instability mechanism, leading in the far wake to double helicoidal vortex systems, which are concentric and located at a distance approximately equal to the blade radius. The source of this instability mechanism is the mutual inductance between consecutive vortices combined with the axial shear present within the turbine wake. From a spectral standpoint, the helicoidal tip vortices are characterized by a carrier with frequency equal to three times the rotor frequency and a modulation of 1.5 times the rotor frequency.

Also the root vortices are characterized by the generation of secondary co-rotating vortices, which are generated by the roll-up of the trailing vorticity. Moving downstream, the main root vortices are diffused relatively quickly, while the secondary vortex system is still present further downstream.

By adding the wind turbine tower and nacelle, the system of wake vorticity structures is significantly altered with respect to the previous case. The tip vortices are clearly detected only in the upper half of the rotor wake, while they are completely disorganized in the lower half, due to the interaction with the alternate vortices shed by the turbine tower. In this case, tip vortices do not exhibit any split instability mechanism. Conversely, the main instability is pairing characterized by alternating longer and shorter streamwise spacing between consecutive vortices. Furthermore, the radial location of the tip vortices is affected by a long-wave instability. For both LES datasets, the cores of the tip vortices are gradually deformed by proceeding downstream, which might be a symptom of a short-wave elliptic instability.

For the LES case with tower and nacelle, root vortices are practically not observed. Alternate vortex shedding occurs in the tower region, and the vortices are bent upstream from the vertical approaching the rotor wake, which is due to the local velocity deficit. Moreover, specific modes allow the interaction between helicoidal tip vortices and alternate vortex shedding to be characterized.

DMD analysis has produced similar results to the POD one, with the advantage that DMD modes of interest can be selected to formulate a time-marching algorithm to predict wake dynamics. It has been shown that different levels of accuracy can be achieved by selecting different modes and a different rank of the reduced order model.

## Authors' contributions

M.D. and G.V.I. contributed to the conception, design and data analysis of this study. C.S. and S.L. provided the LES datasets. G.V.I. coordinated the study and drafted the manuscript, which was then revised and approved for publication by all the authors.

## Competing interests

The authors declare that there are no competing interests.

## Funding

We received no funding for this study.

## Acknowledgements

S.L. and C.S. were supported by National Science Foundation grant no. 1243482 (the WINDINSPIRE project). TACC is acknowledged for providing computational time.

## Footnotes

One contribution of 11 to a theme issue ‘Wind energy in complex terrains’.

- Accepted December 14, 2016.

- © 2017 The Author(s)

Published by the Royal Society. All rights reserved.